[
{"content": "depiction of St. Roch\n\nThe following graces are granted to all\nthe brethren and sisters benefactors and good doers\nunto the hospital of the blessed confessor St. Roch,\nfounded and established within the city of Exeter,\nit is granted that they do say a Pater Noster, an Ave,\nand a Credo. It is granted them that they shall never be infected nor harmed with the stroke of the pestilence,\nas more clearly appears in his legend, how and when almighty God granted this petition to\nthe said blessed confessor St. Roch,\nand sent it by his angel Raphael.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase1"},
{"content": "In quaternione:\n\nDE ARTE SVPPVTANDI LIBRI QUATTVOR. In the fourth book, H. folio, 3rd page, 1st line, 9th word, read sixteen million. Read sixty million.\nIn quaternione: I. folio 1, page 2, line 12, there, delete the word \"decem\": and read, the following ones.\nIn quaternio: K. folio 2, page 1, line 4, there, and the next to the last, delete the penultimate word: and read, and the next to the last. And in folio 3, page 2, line 6, there, add. Read, subdue.\nIn quaternione: L. folio 2, page 1, line 5, there, where it is in the divisor, place \"sub\" thus: where it is under the divisor.\nIn quaternione: Q. folio 2, page 2, line 13, there, they are to be divided. Read, they are to be divided by. And folio 3, page 2, line 12, there, 14, read 17.\nIn quaternio: Y. folio 2, page 2, line 16, there, 2/4, read, 3/4. And folio 4, page 1, line 1, there, quinquaginta et quattuor, delete \"quinquaginta et quattuor\": and read, 54.\nIn quaternione: Z. folio 3, page 1, line 21, there, either in a communicated society or from the middle. In quaternione: et in loco nemi, legere: nec in societatem communicata nec de medio. Fol. 4, pag. 2, ibi, 250. In infima parte figurarum. Legere. 350.\n\nIn quaternione: a, fol. 3, pag. 2, linea 13, ibi, quinquaginta. Legere, quadraginta.\n\nIn quaternione: d, fol. 1, pag. 2, linea 2, ibi, pro quibus. Legere, pro quinque. Et linea 12, ibi. Quamobrem. Legere. Quamobrem.\n\nIn quaternione: g, fol. 4, pag. 1, linea 1, ibi, aureorum numeri. Legere, aureorum numerum. Et pag. 2, linea 23, ibi, surgent. 15. Legere, surgent. 13.\n\nIn quaternione: m, fol. 1, pag. 2, linea 20, ibi, denominantes proportionem. Legere, denominantia proportionem. Et fol. 8, pag. 2, linea 3, ibi, seducta. Dele se, et legere. 3. Ducta.\n\nIn quaternione: r, fol. 1, pag. 1, linea 7, ibi. Apud geometros. Legere. Apud geometras.\n\nIn quaternione: t, fol. 4, pag. 1, linea 12, ibi, errorem in. 6. Peperit. Legere. Errorem in. 3. Peperit.\n\nIn quaternione: u, fol. 2, pag. 1, linea 23, ibi, nascitur. Legere, nascitur.\n\nABCDEFGHIJKLMNO\n\n(Note: The text appears to be written in Latin, likely related to mathematics or geometry. It seems to be instructions for reading certain pages and lines in a book or manuscript, possibly for correcting errors or understanding proportions. The text has been cleaned to remove unnecessary characters, line breaks, and other irrelevant information, while preserving the original content as much as possible.) I am antequam aliiquot Ante I had not yet come into contact with the arguments of More, nor could we sufficiently agree on the matter, so that I could have avoided being deceived by him to a great extent, I was compelled to examine the incomplete accounts more closely, and to repeat the art of calculating, which I had tasted as a young man. When I had explained to me the cunning of more deceitful men, I began to think, it would be of great benefit to me in the remainder of my life if I kept the art of calculation at hand: so that I would not be susceptible to deception by swindlers. Therefore, in order to understand the matter more deeply, I read all the written works on the subject, the learned and unlearned, the Latins and the barbarians, for almost every nation has this art written in its common language. I. Although I found it necessary not to read the entire collection of books, a large part of which did not please me without reservation: if anything in them pleased me, I made brief annotations. It happened that I collected many things from among many writings of others: which, when I had kept them by me for a while, seemed worthy, if I could more clearly bring them out in Latin speech. But when I tried this, and the matter did not progress well, I often abandoned the books in despair, believing that I could not provide what I had intended from myself, and also because the matter itself was obscure, and because many things often occurred that did not seem to admit either Latin speech or eloquence. I was reluctant to give up on a work that I had begun, even when it was burdening me with labor. So I kept trying, hoping that repeated effort would add strength. At times, it even gave me pleasure to contend with the difficulties themselves. And those things that caused me annoyance, contrary to what usually happens, only made my determination to persevere stronger. I. Although I had not been thinking of it, since I could not obtain what I most desired, and they shone with a certain grace: it would not be useless, if they were rough and uncultured. In this way, I finally decided on a firm resolve, and, after facing many difficulties, I extracted these notes, which I had long been pressing in my mind, thinking of the example of the bear, and imagining that I would shape the formless fetuses in my leisure by stroking them.\n\nII. Now, to the vacant pontificate of London, a man, unworthy of such honor in the eyes of all, was designated by the king's benevolence, not only for his good qualities but also for my sake, and what remained of his life was intended to be devoted to sacred studies. I considered it necessary to reject all profane writings and, above all, the hidden commentaries of Vulcan and Minerva from my mind. I cannot output the entire cleaned text directly here as text-only response due to character limit. However, I can provide you with the cleaned text as follows:\n\n\"I thought it neither fitting nor worthy of those who came to be among the learned, nor did I think it proper to surrender any part of my life to impostors, to introduce them to sacred letters and polish them. Again, it occurred to me that it would not be wise to become excessively learned, far surpassing other kings in education, or to dedicate an unpolished work to myself above all mortals, thus delaying public affairs: while he devoted himself to these trifles in reading. Nor could I grant him the favor I owed, fearing that I had corrupted my duty through ingratitude.\" Itaque, to the man whom I was about to address from among my circle of friends, you, who have shown yourself most suitable for this task according to our custom and the purity of your soul, would be gracious if there was anything pleasing in this work, would advise if there was anything deficient, and would be ready to forgive any errors. For what is more fitting for you than these things? You, who are fully occupied with calculating and examining accounts in the royal treasury after the prefects, and who can read these things to your children, whom you have been instructed in liberal disciplines, will find them particularly beneficial if they are worthy of being read. Indeed, nothing is more nourishing for a young mind than the art of numbers.\n\nI thought it necessary to preface this work for those devoted to calculation, that it is held in high regard by Pythagoras and ancient philosophers, and almost all of them are believed to have attributed some divine power to numbers. The four disciplines that exist are Arithmetic, which explains the power and entirety of numbers; Music, which discerns the harmony and concert of sounds; Geometry, which measures the magnitude of earth and other things; and Astrology, which teaches the certain nature and unchangeable laws of the heavens and stars. One who wishes to reach these must first taste Arithmetic, which opens the way to the others, and without which penetrating them will not be allowed. These take nothing from each other, being self-sufficient in their own resources, while they often seek help from this one. Since the nature of God is a simple essence that contains all things within itself and seeks nothing outside itself, this discipline, which deals with incorporeal things and requires nothing from elsewhere, can be considered divine. The ancients themselves believed that the entire philosophy of life was contained in this. Although it is the first among all in terms of ease of understanding by the intellect, since many things can be depicted in these, which draw us to knowledge through the senses of the body, as in music the harmony of sounds applauds itself to the ears and reveals itself to us. In geometry, we submit lines, circles, squares, triangles, and pyramids to our eyes. Similarly, in astronomy, we imagine the sky, spheres, axes, zodiacs, the twelve signs, and even the stars themselves for ourselves. Indeed, all these forms, since all our knowledge originates from the senses, will be of great help in gaining knowledge of things that resemble them to a rude mind. At in Arithmetica, where you will be, I ask you, how will you explain the concepts of numbers you imagine? Numbers express nothing corporeal: nothing to the eyes, nothing to the ears, no images of numbers except numbers themselves: only with sharp wit, a firm memory, and a mind focused within can they be understood. Therefore, mutes, who often imitate rhythmic sounds in nature and recognize all forms of things presented to them, speak many things aptly in common speech from sense, so that they may appear to be understanding something about numbers, if they speak a little more circuitously: they are continually found wanting in reason: without which nothing can be explained in numbers. Quocirca cum vita nostra totam in terrenis versetur: ad quorum adsequendam notitiam sensus corporis non parum iuvant: cognitionem caelestium quae sola ratione sectanda sunt, maxime petamus: cum mathematicae disciplinae medium inter utraque tenet: ad caelestium cognitionem nulla re melius exercitamus, quam animum a terrenis tangentibus absque omni distans, dum mathesibus intenti sumus: quas ipsa natura videtur hominibus edidisse, ut esset, quo se ad caelestium meditationem exercerent. Quod quum omnes disciplinae mathematicae praestent et omni loquelae maxime Arithmetica: quae nullis adiuta corporum imaginibus mente sola se colligente nititur. Haec praeter quod in numeris abscondita quaedam, et altioribus philosophiae venit abstrusa, communem etiam subtilis rationem, sine qua uita tranquilli nullo modo potest, moritures docet. Etenim siquis secessu quisque, siue foro se parat: si numerorum rationem non tenet, frequenter erit in vita non sine damno derisus. Those who deceive a calculator: they will gain more from being deceived in inertia than from fraud through profit, a fact that no honest person could endure to laugh at. Thus, whether engaged in business or leisure, nothing is more necessary than keeping a ready reckoning. And since the deeper truths of arithmetic are more accessible to philosophers than to those engaged in business, the humblest part of arithmetic, which pertains to the art of calculation, is far surpassed by them. Just as Arithmetic opens the door to other mathematical disciplines, it is not easy to penetrate its hidden depths without the aid of calculation. When we devoted considerable resources to learning this art, we employed various teachers, some of whom taught us, others were businessmen, and we read numerous books on the subject. It seemed appropriate to compile and pass on to the studious what we had learned from many sources. We could have done more: some texts were obscure, others incomplete, and some even contained errors that troubled us for a long time. Either we made them more emendated or clearer, or they were to be handed down to future generations. But what we gave to others is a matter of judgment: we certainly tried to ensure that anyone who knew Latin in the future would learn the art of calculation from a good teacher. Since it often happens in calculations that the same thing can be solved in various ways, our tradition did not require that all those who wrote should explain their instructions in detail, which would be tedious for readers. Instead, we showed those who were most expeditious and clear, who gave a straightforward outcome without any unnecessary complications. We also passed on some brief compendia, hitherto unwritten by anyone (as far as I know). Quamobrem readers can judge our trifles, it is fair that we advise you: since in the matter itself we have labored greatly, it is only just that an unoffensive reading may remain with you. To our purpose, whatever we have provided, gratitude is to be shown to good men. This alone we have deemed necessary to warn: whoever takes up this book, unless he wishes to lose his labor, should begin at the beginning. For not every beginner, in history as in other things, will offer himself to the senses: since the context is such that what precedes has the character of preliminaries. The art of calculation has several essential parts: Numeration, Addition, Subtraction, Multiplication, Partition, Progression, and Root investigation. We will speak of each in turn.\n\nInitial numeration begins from one and can proceed in infinitum. The numbers among the primary and simplest ones, having common names among all peoples, are only ten: that is, one, two, three, four, five, six, seven, eight, nine, ten. In each nation, the names of the first are variously assigned by each one, according to his own judgment. Whatever number you add beyond ten, you will find it composed of ten in repetition. For instance, we add one to ten and call it eleven; two to ten and produce twelve; three to ten and generate thirteen; and in the same way we compose the primary numbers with ten, namely fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, until we again reach ten, because we have collected ten twice, which signifies twenty, as if twice ten. b. The letters i and j have a similar sound. u. The letter u, in passing, has a certain affinity in pronunciation with some of them. For example, as the gentile hundred, so ten among the ancients signifies ginta in composition. In numerando progressionem continuamus, et singulos numeros primarios cum viginti conjungimus: quosque ad ter dena venimus, que triginta, hoc est ter decem, vocamus: quibus denuo primarios numeros accumulare pergimus, donec ad quater dena perveniamus: que quadraginta vocamus.\n\nAtque hac numerandi ratione servata, quinquaginta, sexaginta, septuaginta, octoginta, nonaginta procedimus, semper prioribus denis primarios numeros aggregantes, donec alia colligimus: quibus ipisis nominibus a numeris primaris secundum denorum collectis numerum denominatum tribuit manifestius: ut sit admonendum. Decies dena centum primi nominators Latine vocaverunt. Deinde alia dena rursus procedentes, ipsa centena crescente numero coacervabant: quosque ad decies centena venirent: quibus nomen indiderunt. Mille. Is summus numerus apud Latinos est: qui uno vocabulo exprimi potest. All remaining elements in infinite numbers, replicated and repeated among themselves, have collected their own names among the Latins and among all peoples' languages. For example, ten thousand, hundred thousand, million, ten million, and the rest. Besides, the Greek language is much richer in this regard, not only with ten thousands and hundred thousands, but also gave it their own names: Ecaton, Chilia, Myria, and Myriades.\n\nYou can see that this deity, derived from primary numbers, is present in the vast body of numbers: just as in our body there are limbs that are larger than members, and as the joints in reeds, distinctly separated by equal spaces, produce the very height of the reeds themselves, so in larger numbers, aggregated tens, or like the joints of numbers, increase in size. Igitur omnem numerum necessarius est, vel primarius esse:\nvel ex denis coactis collectum: vel partim ex denis, partim ex primaris compositum.\nThe Romans and Greeks wrote numbers differently with their own letters and symbols: those who wish to know more widely should read Valerius Probus among the Latins, and Herodianus de numeris among the Greeks. Our custom is to touch only on the method of writing numbers: which, originating from the Chaldeans and spreading to almost all nations, has obtained such widespread use that it is not surpassed by another throughout the world.\n\nThere are ten symbols for numbers: which Arithmeticians call figures, and it is also permissible to call them signs. These nine signs represent the following numbers:\n1. unum signifies one.\n2. duo. third. 3. third, fourth. 4. four, fifth. 5. five. sixth. 6. six, seventh. 7. seven, eighth. 8. eight, ninth. 9. ninth. Tenth note: the letter O, in circular form, is this: what some call a circle, the common people call a cyphra: which in itself signifies nothing; but when placed before certain numbers, those following are increased tenfold in meaning. For example, the first note or figure. O joined on the right side by the one that expressed it makes it signify less. Similarly, the same applies in the others.\n\nPay particular attention to this. All Latins and Greeks, as well as their followers, begin to write something on the left side according to the custom: and they turn towards the right; whoever wants to write these numbers should begin on the left-hand side of the page and place the smaller numbers before them, and in proceeding towards the left, let the larger numbers always be promoted as they come. The Latins, Greeks, and others observe the same rule in numbers: they place the greatest number to the left, the smallest to the right. This is more evident to those who examine it closely. The art of counting originated with the Chaldeans: when they write, they begin on the right and progress to the left. I call the right side of the page the right side, the one presented before us; the left side is the one on our left.\n\nThese nine notes and figures, which we have said denote numerical values, do not always occupy the same positions, but vary according to the position of the place. For example, the first of each, which is on the right, only indicates its own number: In the second place, a figure turned to the left indicates ten times that number. In the third place, a figure turned to the left indicates one hundred times that number. In the fourth place, a figure turned to the left indicates one thousand times that number. In the fifth place, a figure turned to the left indicates ten thousand times that number. In the sixth place, a figure turned to the left indicates one hundred thousand times that number. In the seventh place, a figure turned to the left indicates one million times that number. In the eighth place, a figure turned to the left indicates ten million times that number. Nono place read verses, one hundred thousand thousands the same. In the tenth place read verses, more than one hundred thousand thousands the same. In the eleventh place read verses, ten thousand more than one hundred thousand the same. In the twelfth place read verses, three hundred thousand more than one hundred thousand the same. In the thirteenth place read verses, one million more than one million the same. And indeed, these millions can go on infinitely in this manner.\n\nBut in numbers, the last place will be a circle: when nothing follows it, whose meaning increases. Otherwise, if any figure is behind it, all the seats of the numbers can be occupied. We will give an example, in which we will place all the signs of the numbers in their order, with some repetition up to the thirteenth place, as follows.\n\nRegarding this series of numbers and similar ones that grow to infinity, in order to avoid any confusion while counting, mark the fourth place which represents one thousand with a point above it. In the fifth and sixth places, add a punctum above the seventh. Then, in the eighth and ninth places, add a punctum above the tertiumdecimum. Likewise, in the eleventh and twelfth places, add a punctum above the third quartum. And so on, if the number is greater, add two puncta two places after the fourth, and mark the third punctum above. This addition of puncta will not cause a small amount of difficulty in counting. For marked places show thousands: to which, as to a certain degree, one may pause in counting. For long series of numbers cannot be pronounced in one breath. The fourth place shows one thousand. The seventh place shows ten thousand: the common people call it a million. In Latin, it is called millena millia or millies millemina. The ancients called it decem centena millia. As we will show below. Decimus locus represents ten thousand thousand thousand: the vulgus calls it millions of millions; in Latin, you would say either ten thousand thousand, or ten thousand million, ten thousand tens of thousands. The ancients called this number decies millies centena millia millions. The third tenth place expresses ten thousand million millions: the vulgus calls it a million from millions of millions. In Latin, we now receive it according to the rule of calculation, and we will enumerate it, either ten thousand million millions, or ten thousand million millions. The ancients called this number decies millies centena millia millions. If one wishes to enumerate the numbers between these, whether centenarians or denarii, it is easy for anyone: among which types of millions are located. Quod post observing, we must first observe this: that the numbers of centenaries and denarii marked with medii, according to the current method of calculation, always represent millenarian generations. Those noted near the right side of the punctuation marks are the ones we encounter first.\n\nWe should enunciate this immense number, marked with the dots mentioned above, in Latin as follows: beginning with the smallest and last digits, and ending with the largest. This is the custom among all peoples, expressing numbers in words.\n\nThree thousand thousand, three hundred thousand, three hundred thousand, two hundred thirty thousand, two hundred ten thousand, sixty thousand five hundred forty. Another example with the same number of notations, but with the sequence of figures changed, I will put forward: so that it may be clearer to readers.\n\nWe should also express this number with dots, as mentioned above, in Latin as follows: Three thousand thousand. Three hundred thousand three hundred thousand. In these quantities: sixty-seven thousand seven hundred million. Eight hundred ninety thousand. thirty-two thousand three hundred.\nFrom those we have placed: with their great numbers written down through their own notes, marked with noted points, and given in Latin, it will be easy for anyone to write down any smaller number, mark it with points, and enunciate it in Latin.\nBut if a larger number occurs: the same thing will be the case: but the words, millions, and millions, will have to be repeated as many times as necessary in each larger number, to avoid tedium in the ears. And scarcely in human affairs does a larger number occur: as the one we have mentioned above.\nIn such immense numbers, for each place where points are marked with words to be enunciated, these words, whether millions, or millions, or thousands, must be repeated as many times as the points, from that place, occur in the entire series, up to the first note. This, which can be seen in the examples already explained.\nTRUE, so that no simpler examples may seem in our book, we will add such examples as well. Atque therefore, as it pleases me, let us place this example at the seventh position (for there is no need to descend further). This example, marked with points as follows: six hundred thousand, eighty-five thousand two hundred and forty-four. Another example up to the tenth position: seven hundred thousand million, nine hundred and sixty thousand quarter million, eighty-five thousand three. I need not write down further examples, since what we have put down clearly indicates how any other number, whether to be written or enunciated in Latin. Non-Latins do not hide the fact that old Romans counted their Sestertios in the ancient way, exceeding one hundred thousand (a number that occupies the sixth place among Arithmeticians). They did not express this number in Latin, except through tens, twenties, and those above hundreds and thousands, up to centuries and myriads: through which they repeated the hundred thousand. At the limit of counting, they began again. They also omitted certain terms in speech, such as \"centum milia,\" in order to say \"decies sestertium.\" This number expresses ten thousand thousand Sestertii, and it occupies the seventh place among Arithmeticians. \"Centies sestertium\" means hundred thousand thousand Sestertii, indicating ten million Sestertii and occupying the eighth place among Arithmeticians. \"Milles sestertium\" means one million Sestertii. The number of centuries, millions of sestertii in Roman currency, signifies, and among Arithmeticians it does not possess the number nine. And thus the ancients continued to count their numbers in this way to an immense degree. This is what Bede, the learned man of the earliest of our ages, recorded in his book on the As, with remarkable acumen. However, Arithmeticians beyond the centuries make a certain grade beyond ten million; yet they themselves cast it aside, and with tens and hundreds, they again form a grade of millions; and beyond that, they accumulate millions of millions, as we have previously mentioned. This method of counting around the earth has been accepted. Therefore, among the ancients and the moderns, there is nothing different except that those observing the thousands accumulate millions of millions. However, the thousands, as it were, form certain steps, and they accumulated them to an immense degree, always repeating these. Caeterum locorum augmenta tam prisca quam nunc consentient. Therefore, I see no reason why we should cling so fiercely to the ancient computation: it is outdated among all nations, when the established custom asserts itself with greater reason. Since among the Latins, the largest number is represented by a single word, it is more appropriate to progress to the thousands before reaching the hundreds, which come after, than to make a step towards them. However, if you prefer to express a number in the old Latin way: let us make a brief digression to that end. To mark the number with notations written above, place a dot at the third, sixth, ninth, and twelfth place, and at any others that follow, always leaving two spaces between them. Moreover, observe this: when counting from the last, let each number sign denote its meaning as the seats provide, and let those that precede the sixth give it through words. When we reach designated points, pronounce these words at each point: for instance, as in the first example given above, if one wishes to enumerate numbers in the ancient way.\n\nThree hundred thousand thousand, three hundred ninety thousand thousand, eighty-seven hundred sixty thousand, sixty thousand five hundred forty, three hundred twenty thousand one.\n\nLikewise, in other immense numbers, it will be easy to imitate the ancient way, for example, as in the second instance given above, which you can express in the ancient way as follows.\n\nSix hundred thousand thousand, three hundred thousand two hundred thousand, seven hundred sixty thousand eight hundred thousand, ninety thousand, one hundred twenty thousand three. To this example, all other numbers can be rendered in the ancient Latin way without difficulty.\n\nHowever, the Greeks make a certain number stand for a Myriad: which means ten thousand. Five thousand occupy its place. They do not express a greater number with one word. After removing unnecessary line breaks, whitespaces, and meaningless characters, the cleaned text is as follows: \"deinde myriades tam multas sibi aggregant, ut ipsarum etiam myriadum faciant myriadem: quae summa mille centum milia, sicut loquntur prisci: vel centies mille milia: sicut nunc Arithmetici: significat: et nonum locum tenet. ultra quem nume Greci vocabulis carentes rursus myriades in se replicae: donec totidem myriadum myriades accumulauerint: quot propositi numeri magnitudo suggerit. Hactenus de numeratione dixisse sufficiat. ADDITIO NUMERORVM EST DVOPluriumque numerorum in unam summam collectio. Id autem fieri potest, siue minores quisque numeri maioribus adderentur, siue econuerso, siue aequales aequibus\"\n\nThis text is in Latin, but it appears to be written in a fairly standardized form of Latin, and there do not seem to be any significant OCR errors. Therefore, I will not translate it into modern English, as the text is already readable in its original form.\n\nOutput: \"deinde myriades tam multas sibi aggregant, ut ipsarum etiam myriadum faciant myriadem: quae summa mille centum milia, sicut loquntur prisci: vel centies mille milia: sicut nunc Arithmetici: significat: et nonum locum tenet. ultra quem nume Greci vocabulis carentes rursus myriades in se replicae: donec totidem myriadum myriades accumulauerint: quot propositi numeri magnitudo suggerit. Hactenus de numeratione dixisse sufficiat. ADDITIO NUMERORVM EST DVO pluriumque numerorum in unam summam collectio. Id autem fieri potest, siue minores quisque numeri maioribus adderentur, siue econuerso, siue aequales aequibus.\" This text is in Latin and requires translation into modern English. Here is the cleaned text:\n\nIt is of great importance that if we wish to combine two or more number orders into one sum, we write them in such a way that the figure of each first order, and of the remaining orders, is directly under the first note, as if aligned perpendicularly, under the second under the second, under the third under the third, and so on, so that in each number place, the numbers respond to each other as if under one line. For otherwise, if the number orders are written obliquely and their places are mixed and face each other directly, it will be uncertain which place responds to which, resulting in great confusion in calculation. This not only applies to the addition but also to the subtraction, multiplication, and division. Therefore, to avoid falling into such a labyrinth, it is essential in the entire art of counting that the direct number orders are accurately placed before anything else, with the first in each place responding to the first, the seconds to the seconds, the thirds to the thirds, and so on. In this numerical calculation, addition, subtraction, multiplication, and division, we must always keep this rule: as long as we are dealing with numbers, whether first, second, third, fourth, or higher places, we should not take the symbols for denarii, centenarii, millenarii, or higher units, but only for primary numbers. Once the work is completed, when we return to examine the sum, collected, subtracted, multiplied, or divided, we restore the power of each number place, so that the symbols in expressing the sum have the meaning that the places will provide.\n\nWhoever is adding numbers: if there are only two orders, write the second number below, the first number above, draw a line under both. On the right side, where the smallest numbers are located: beginning, add the number below the first one to the number above it of the same location. If one note can be written for this sum (which will certainly be the case if it is less than ten), place it directly under the first number.\n\nIf the number is a denarius or larger, and requires two notes for its expression: place the first figure, even if it is a circle, directly under the first number. Then, holding the number of the second figure in memory, proceed to the second location.\n\nHowever, if you find circles in both the upper and lower order: write the remembered number under the second line at the second location. If a figure is not a circle: before all things, receive from me the number deposited: lest it slip into oblivion: if you add the remaining numbers beforehand: {before} you return to it: add that one to the lower mark of its place, and add the increased one to the upper mark of the second place: and let one note rise from among them, if one note can be written: write it directly under the second place.\n\nIf, however, only one note of the second place is a figure: another figure signifying the number itself: add the number, if one note can be written, under the second place as a subscript.\n\nIf, however, the number to be expressed at the second place is not a circle: two notes will be required: place the first note, even if it is a circle, directly under the second place: and let the second note, preserving the number in memory, proceed to the third place. Wherever you do this for all things, as you did with the second place: and place the number obtained from the addition in the third place. et sic usque ad ultimum locum perge. When the wind is there: if one note can be written, place it at the last place. If two figures are needed to express it: place the remaining number, retained in memory, to the left of the last number, under the line.\n\nIn general, we should observe adding this in every numbered sequence: whenever no number is retained in memory to be placed under the line, a circle should be subscribed.\n\nAnd similarly, whenever circles and a note indicating the same location are added: no number is retained in memory, the circle should be added to the note.\n\nThe note itself, whether it is of a higher or lower order, should be subscribed to the line.\n\nFurthermore, whenever more than one line is drawn above, there are more superior orders than inferior ones, the notes above the line should be subscribed to the same lines as those below it. Whoever wishes to combine these two orders, whether in thought or in words, should consider the following. In the lower order, I find three. I add to them what is in the upper order. They become seven. I mark the first place with a line. Then I go to the second. I find four below and nine above. They become thirteen. Since two notations are necessary to write them, I subscribe the first figure, which shows three, to the second place. I turn to the third place and add what is in the upper order to the one I have released from memory, which is seven. They advance to fifteen. And since two notations are required for these, I subscribe the first to the third place, five, and I keep the memory of the second, which indicates one. I go to the fourth place. I. To retain one thing in mind, I add four more below, which are written in the lower order. I add five more above in the superior order and subscribe new ones in the fourth place. I proceed to the fifth place where above and below the circle meet. I have nothing to add there, so I subscribe the circle. I find two below the sixth place, above the circle they meet. I have nothing to add to them. I subscribe two. I advance to the seventh place. There, below the circle, are three above: nothing is below to add to them. I subscribe three. I hurry to the eighth and last place. I find six below: I add to seven what is above. Thirteen exit. This number, because it must be marked with two signs: the first, namely three, I subscribe in the eighth place; and the other, which means one, I place to the left of the same last place. In this way, under the line, by adding two numbers above the line, we compose a number that contains all of them. Centuries three hundred thousand, two hundred nineteen thousand, seventy-three. Regarding the composition of the two orders: if we have to institute enough of them, we should consider reducing several orders into one sum, if we wish. But to make things clearer to a few: when collecting many orders into one, we should first gather those with numbered seats: starting from the lower orders, we should ascend to the higher ones, and the number obtained, if it is the primary one, should be placed first under the line. If there are other denarii numbers, a circle should be marked at that place. And note, that number should represent the denomination it signifies, and the memory of the secondary seats should be committed to their figures, if the number is below one hundred. For numbers that are centenarian or larger, up to the thousandth, this does not usually happen in large additions: the hundreds should take three places, and the remaining tens and units should occupy their respective places. When the number of a seat rises to over a hundred, and the mark indicating this is not retained in the third seat, it should not be transferred to another location in the third seat, lest we be occupied with the addition to the second seat and forget the number retained. Instead, we should note it under the line, a little below the third seat: so that when we come to it, we may be reminded to add its numbers to those of its own seat. If the number is composed of a primary and a denarius, the primary number should be written under the line, and the denarius, retained by me, should be counted according to its denomination under the figures of the second seat, as was said of the two orders. After completing the second seat, we should proceed to the third, and the same should be done in the second, and so on. Likewise, we should proceed to the fourth, and from then on to all the others, in which we find what was said about the second seat. In this text, some individuals indicate the same seat of the circle with certain signs: the figures without circles should be placed under their respective lines. The following will be more clearly illustrated by subsequent examples.\n\nTo collect one from these many numbers, you may think as follows: either silently, or by stating the words. In the first place, there are eight remaining signs without circles; therefore, write eight under the line. The second place has circles. Write a circle under the noted circle. The third place expresses ten notes without circles, so with one memorized as a denarius and added to the figures of the fourth place, they make fifty-five. Five of these are noted under the fourth place: and five meteorites under the fifth place notes, which are mixed with circles: when combined, they make new ones. The sixth place, because of the circles, should have a line inserted under it. In the seventh place, all the notes produce one hundred and eight. subscripted are the following: eight and one hundred numbers, in the ninth place, which is counted as the seventh, is to be kept under the line. The eighth and last numbers, which are above the line, rise in place of the collected numbers. Nine are therefore mandated to memory. A circle is placed in the eighth place. In the ninth place under the line, nine are taken from the eighth place. I find one for the hundredth number's note, which was hidden before in the seventh place: when added, they become ten. In the ninth place, a circle is subscribed: and from its left, I take one, which makes it decimus. Therefore, the number is collected under the line. Thousands, millions, billions, eight million thousands, ninety-five million thousands and eight.\n\nAnother example of more intricate orders follows.\n\nALL of the notes in this example are joined in the first place, one hundred and two. Two were written below in the first place. And the hundredth number should be transferred to the third place by means of the unit's note. In the second place, since you have only circles, note that a circle is to be inscribed. In the third place, since all figures of a circle are: and nothing is to be added to one from the first place for a hundredth number, a \"1\" is to be subscribed. The fourth place, when joined together, contains one hundred. Therefore, a circle is to be subscribed, and in the third place, which is the sixth from the fourth, one is to be kept. The fifth place contains one hundred and eight. Eight are to be noted. One is to be kept in the third place, which is the seventh from the fifth, and added to the other figures of that place. There are ninety-seven. Therefore, seven are to be subscribed below, while nine are remembered. Finally, I join the numbers of the eighth and last place above the line. One hundred are produced. I. Place one circle in the eighth position, another in the ninth position to the left below the line: and a note for the hundredth place, which, when counted from the eighth, is the third: I transfer. This number is enclosed here. Thousand thousands, seven thousand thousands, hundred and eighty thousand, one hundred and two.\n\nII. Whoever wishes to add numbers promptly: must memorize them first: which numbers each produce individually under ten, the even ones, or the others added to the primary ones. Certainly this is easy to know:\n\nIII. Indeed, even boys who are not yet pubescent know this roughly. But if someone is so uncultured in his education: that he does not know this: he can learn it in one hour, if he applies himself. For example. Nine and nine make ten and eight. Nine and eight, ten and seven. Nine and seven make sixteen. Nine and six, fifteen. Nine and five, fourteen. Nine and four, thirteen. Nine and three, twelve. Nine and two, eleven. Then eight and eight produce sixteen. Eight and seven, fifteen. octo et sex, quattuordecim. octo et quince, tredecim. octo et quattuor, duodecim. octo et tres, undecim. octo et duo, decem. Rursus septem et septem creant quatuordecim. septem et sex, tredecim. septem et quince, duodecim. septem et quattuor, undecim. septem et tres, decem. septem et duo, novem. Deinde sex et sex faciunt duodecim. sex et quince, undecim. sex et quattuor, decem. sex et tres, novem. sex et duo, octo. Praeterea quinque et quinque producunt decem. quinque et quattuor, novem. quinque et tres, octo. quinque et duo, septem. Ad haec quatuor et quatuor procreant octo. quatuor et tres, septem. quatuor et duo, sex. Denique tria et tria constituunt sex. tria et duo, quince. duo et duo, quattuor. Nam unus ad reliquos numeros adiungere pueri omnes infantes egessi sciunt. Quae diximus, figuris notata sub oculis posita sequuntur.\n\nTranslation:\nEight and six make fourteen. Eight and fifteen, thirteen. Eight and four, twelve. Eight and three, eleven. Eight and two, ten. Reversing seven and seven produces fourteen. Seven and six, thirteen. Seven and five, twelve. Seven and four, eleven. Seven and three, ten. Seven and two, nine. Six and six make twelve. Six and five, eleven. Six and four, ten. Six and three, nine. Six and two, eight. Furthermore, five and five produce ten. Five and four, nine. Five and three, eight. Five and two, seven. To these, four and four produce eight. Four and three, seven. Four and two, six. Lastly, three and three make six. Three and two, five. Two and two, four. For a boy to add one to all the other numbers, all infants know this. What we have said, marked with figures, follows below. If someone wants to examine this: why are they arranged in this way? When we note down familiar signs in each place, we consider them as primary, but the number collected under the line should correspond to their meaning in each place. This is how it should be. For example, in the first numbered place, primary signs, if correctly joined, are contained in the number under the line. Therefore, it is necessary that through each place all figures, if correctly joined, express the number under the line. And if we add a number and remember it, transferring it from the place where it was found to another, this happens because it is greater than the one that should have been placed there, and when it is correctly translated back to its place, or its offspring are numbered in its place. Despite the identical shape of figures in all locations, their meanings vary greatly with the increase in location: nevertheless, they all receive the same names from the first location: as many units as each location represents, so many tens in the second, hundreds in the third, thousands in the fourth, and so on. Therefore, the number collected under each number's line is the same for all: it represents the collected numbers in all locations. And since the notation of each location's name is changed from the first, they provide different meanings: the number under the line represents the number of units in the second location, tens in the third, hundreds in the fourth, and so on. Therefore, the number under the line will denote the number of units in the locations, if they were placed in the first location.\n\nTo prove that you have correctly collected the numbers: demonstrate the correctness of their addition through subtraction. If, after you have determined the order of the numbers added to a sum, the sum of the remaining orders does not emerge from the deduction: then there is no error in adding. For when several parts come together in the composition of any thing: if one of them is taken away, the others must remain. This will be clearer when we give instructions on deduction, to which we now proceed.\n\nAnd since no nation exists that does not have various kinds of coinage, some more valuable, some less: therefore, for the purpose of making such varied additions, classes for each type of coin and sum should be marked on the abacus. So that one sum or money, the greater it is, occupies the left side, and the less, the right side. For example, if there are gold coins in one sum, worth five denarii each: denarii, worth four sestertii each: sestertii, worth two asses and a half: the kinds of Roman coins are of this sort, gold coins occupy the left side. denaries in the right hand pressing against something in the middle, sestertii in the right. Then, when adding these various kinds, begin with the sestertius in the first class of the right hand: collect and observe how many quaternions there are in it. If a number exceeds the quaternion, mark the one that completes the quaternion and subtract it. Keeping the quaternaries in memory, transfer them to the next denarius class. Continuously release memory, adding the denarii found there, and run through all the numbers of that class. Also note how many quinis there are. The number that completes 25, deposit there. Quinis, when hidden in memory, are to be transferred to the next aureus class, when first counted. And the number of all of them is to be noted there. Similarly, if in any other nation aurei, denarii, or lower coins of that region need to be counted: treat it in the same way: give each class its own. We only show this method. The rest will be easily given by frequent practice. Example given: \"Here we have provided an illustration: which anyone can easily explain without a precedent.\n\nSubtraction of numbers, whether greater or equal, is called subtraction, deduction, or reduction in Latin. The rule is: when a part is subtracted from the whole, the remaining part is known as the remainder. To put it more clearly: when two numbers are presented, one is to be subtracted from the other, so that the remainder may be understood and the extent to which one exceeds the other. For instance, take two numbers, one seven, the other three. If you subtract three from seven, the remainder is four. Three are equal to the same number of units in seven. In subtraction, it is absolutely necessary that there be two orders of numbers: one from which the subtraction is to be made, the other containing the sum to be subtracted. However, it is always necessary to subtract a smaller sum from a greater or an equal sum from an equal one. Subtracting a greater sum from a smaller one goes against the nature of things.\" Those who wrote about computation gave various instructions for subtracting numbers: some considered certain signs not to be subtracted, where others taught ambiguous and obscure things, making it almost as difficult to divide numbers as to subtract them. I read and tested these, and found nothing: either they had tortuous forms or they were confused by numerous figures. After long investigation, I found a simple and certain method for subtracting numbers, with the help of the most learned John Garth, a philosopher and expert in Arithmetic. This method certainly enables any number, however large or intricate, to be subtracted, provided all known quantities remain. To avoid prolonged explanation for the reader, this is the method for subtracting numbers. Post major number, from which the subtraction should be made: above, and minor number, which has been subtracted, should be written below. The placement of numbers should correspond: first to first, seconds to seconds, thirds to thirds, and so on. A line drawn under both numbers from the left to the right will be present, as we mentioned in the addition. To the left of the smaller numbers, starting from the first place, we subtract the smaller number of the lower order from the larger one if the latter is greater: and whatever is obtained, we place under the first number.\n\nHowever, if the smaller number is greater than the larger one: so that it cannot be subtracted from it: we add the larger number to the smaller one mentally, considering the smaller number as being placed before the larger one, or as if we had placed the smaller number before the larger one: and from this larger number, we subtract the smaller number. For example, let the larger number be three, and the smaller number nine. No one can subtract nine from three. We will add three more to make a total of thirteen, and subtract nine to leave us with four. This is the number we require, which should be written under the line.\n\nIf the superior order has a circle, and the inferior number is known, we cannot remember the superior circle itself, nor note the inferior circle, which will always be smaller in denomination. For example, let there be a circle above with the number seven below. It cannot be that we deduct seven from nothing. Therefore, we should consider the circle above as ten, and subtract seven from it. The remaining three will be extracted.\n\nWhenever the denarius number is memorized and added above, we increase the superior number so that the inferior, which is greater, can be subtracted. We do this at every number location, adding one to the inferior number, and subtracting it from the superior number if it is greater. Whatever remains, we place it in the same place under the line, from which we have subtracted it. In a lesser number will be the superior one, so that the inferior number may be increased by that unity: a rural dwelling, when a penny is added, increases the superior number: the inferior number must be subdued by it.\nThis rule must be observed in all places of numbers. First, in the first place, proceed to the second: everywhere the same must be done: as in the first. When we come to the third, fourth, or any other place, the same must be perpetually observed.\nHowever, if a circle is found in any of the places of the inferior number, and in the superior number, from which the subtraction is made: if nothing has been retained in memory due to the penny borrowed for the denominal number: the number superior must be written as an integer.\nWhenever a circle occurs in either order of numbers for us: if nothing can be recalled that should be written: the circle must be noted. Vbicumque in quocumque loco numerus inferior superiori aequalis erit: cum nihil sit reliquum, subscribatur circulus, praeterquam postremo loco. Ibi enim, si numerus subducendus ei par sit: a quo fit subtractio, nil omnino subnotari debet, quia tunc per cetera peracta est subductio. Et circulus in numeris postremum locum occupare non potest.\n\nQuod si superior numerorum ordo numeris aliquot et notis inferiorem transcendat: postquam inferioris ordinum numeros omnes a superiore subduximus: quicquid de superiore ordine reliquum erit: eisdem locis et notis subjacentibus adiungatur, quibus supra lineam suo ordine stetit.\n\nNunc exempla petamus: quomodo demonstrata clarius illustret.\n\nQuavisquam a quinquaginta septem milibus, ducentis nonaginta quinque milibus, quadringentis nonaginta, subducere cupit, quadraginta octo milibus, septingentis sexaginta quinque. I. I follow the rules, not the given ones: such as this, I would have observed: from the first auspices, numbered seats. I cannot subtract seven from the circle. I therefore consider the circle as ten, and from ten, seven. Three remain. I place these under the line. I proceed to the second seat. The number below appears, nine. I add one: because I took ten in the first place and count as ten. I cannot remove seven from nine. I borrow another ten: and adding these above, I count ten and nine. From these I subtract ten. Three remain. I write these down. I proceed to the third place: since I was borrowed one in the previous seat, I add one to the lower number, that is, two: and these I subtract from four, which are above. One remains. I write this down. I proceed to the fourth place. The number five and five contain both. And since each exhausts the other: nothing will remain. I place under the circle. In the fifth place, I subtract six from nine. tria relinqued: quod lineam subijcio. In the sixth place, because seven from two cannot be taken: taken as twelve, from which I take seven, and those remaining: which line I subdue. In the seventh place, because I was borrowed from the previous seat: from seventeen I subtract eight, and consider nine. Since I cannot take these away from seven: from seventeen I subtract nine. The remainder are eight. I place these under the line. I proceed to the last place: which recalls one: since they were taken from the upper seat. I go to the lower number, the quaternion, and they become five. From these five, subtracting those above, nothing remains. Therefore, nothing at all is subscribed: both because the subtraction has been completed: and because the last place does not accept a circle. The remainder of the superior number, from which we subtracted the inferior: will be found under the line. Eight hundred thousand, fifty-three thousand, nine hundred and three. I will give another example of several places that impede more. Primo loco, quia supra est circulus, assumatur decem: a quibus septem deductis, tres supersunt. Secundo loco, occurent octo: quae unitate ex memoria augeantur et fuere nova. Si ea a decem ad circulum subtracta essent, remaneret unum. Tertio loco, unum supra, inferius circulo est: unum ex memoria promititur, quod superius exhaurit. Et quia nihil superest, subscribemus circulo. Quarto loco, quia inferius circulus est: superius septem: a quibus nihil est, quod subducamus: ipsae septem subscribentur. Quinto loco, quia octo a circulo demissae nequeant: a decem mutuamur et eas deducimus. Duo restant. Sexto loco, ad quattuor quae infra sunt, unum addimus ex priore sede memoriam subministrante: quae simul addita subtrahimus ab decem ad circulum superiorem assumptis. Remanent quinque.\n\nSeventh line: septem uno aucta, quia ab uno demissum non poterant: ab undecim tolli, sumptis decem. Tres subnotamus.\n\nEighth line: quinque quae sunt infra, unum ex priore sede delatum, augemus et sex ab octo subtrahimus. Two remain, the rest are above and below the circle. Therefore, a circle is placed under the line. In the tenth place, six are subscribed: four are subtracted from ten. In the eleventh place, eight are below: one is hidden in memory. And because we cannot remove them from the circle: we subtract eleven. Thus, one remains. In the twelfth place, we promise one from memory and add to four. Because one is above the circle, we subtract ten: and five are left. We place these under the line. And since one is still hidden in the mind, which completes the unity of the upper order: the circle would have to be subscribed: but there is no more room: therefore, nothing more is added: because the work is finished. The remainder is: five hundred sixty-three thousand million, twenty thousand three hundred thousand, five hundred seventy thousand, and thirteen. If someone wants to subtract numbers without delay: first, memorize the total: what remains if singular primary numbers are subtracted from larger ones, which most boys hold. And again, what is left: if simple numbers, if larger numbers are subtracted from those under twenty with a denominator, are subtracted.\n\nJust as we instructed in multiplication, it is necessary to remember which numbers each primary number generates when combined: therefore, we believe it is necessary for someone to keep track of which numbers they leave behind, whether they are subtracted from themselves or extracted by the numbers generated by their addition.\n\nIf someone is ignorant of this: one hour's labor will suffice, provided the mind is focused. For example, if ten and nine are subtracted, there remain nine. If nine is subtracted from ten and eight, there are nine left. If nine is subtracted from seventeen, eight remain. If nine is subtracted from fifteen, six remain. If you remove four from fourteen, five remain. If you remove nine from fifteen, six are left. If you remove eight from seventeen, nine remain. If you remove eight from sixteen, eight are left. If you remove eight from fifteen, seven are left. If you remove eight from twelve, four remain. If you remove eight from thirteen, five remain. If you remove eight from fourteen, six are left. If you remove eight from twenty, twelve remain. If you remove seven from sixteen, nine remain. If you remove seven from fifteen, eight are left. If you remove seven from fourteen, seven are left. If you remove seven from thirteen, six remain. If you remove seven from twelve, five remain. If you remove seven from eleven, four are left. If you remove six from fifteen, nine remain. If you remove six from fourteen, eight remain. If you remove six from thirteen, seven remain. If you remove six from twelve, six remain. If you remove six from eleven, five remain. If you remove six from ten, four remain. Auge sexto ab undecim: quinque tamen supersunt. If you subtract five from fourteen: nine will remain. If you subtract five from thirteen: eight will remain. If you subtract five from twelve: seven will remain. If you subtract five from eleven: six will remain. Now indeed, if four are subtracted from thirteen: nine will remain. If four are taken away from twelve: eight will remain. If four are taken away from eleven: seven will remain. Again, three are left from twelve: nine appear. Three are taken away from eleven: eight present themselves. Two are taken away from eleven: nine remain.\n\nWe have explained this to those who have written about subtraction before: whenever a larger number must be subtracted from any number, the number from which it is to be subtracted should be found immediately before it, and the subtraction sign should be deleted, along with the notation of where it was taken from, and a smaller unit should be placed in its place. If circles are the medium: they command to be destroyed, and in their place, new figures are to be placed. If unity is the figure: which lends ten: it is transformed into a circular figure. And since it will be useful for many things: to recognize which earlier ones have been deleted, we have added their examples marked with figures below.\n\nThe superior number from which deduction should be made is marked. Three thousand and ten. The inferior, one thousand and eleven. I begin by stating that one cannot deduct one from a circle, ten from the second place, and the figure of unity is deleted, and they place a circle in its stead. From these ten, one is removed, leaving nine. In the second place, since one cannot subtract one from a circle, and the third place also holds a circle: they demand ten from the fourth place, where three have been noted, and in place of the ternary figure they delete, they place a binary figure: they transform the middle figure of the third place into a new figure. Then they extract one from the tenth place and place none below it. They remove one from the third place, leaving eight. One is taken from two. And one that remains is placed under the line. This method of subtraction is clear and does not burden memory with units, but the larger number from which the subtraction is made confuses the notation. If, due to error, one had to repeat the subtraction, they would not be able to recognize the first number given. Others, whenever the number to be subtracted is greater than the number from which it is subtracted, instruct to add the difference between the lower number and the denarius, to the higher number from which the subtraction should be made, and to subtract the unit mentally hidden in the next figure to obtain the note to be subtracted from the one above it. This is done in all places of numbers. For example, if seven is to be subtracted from four, since seven is three more than ten, which is subtracted from ten, they add three to four and write down seven. And they keep one unit mentally saved in the next place, which is to be subtracted, and transfer it. If this note is for someone: add one more, it will be nineteen. Because one added to ten makes twenty, and there is nothing between twenty and ten. They sign their agreement with the complete number above the line from which the deduction was made. If nine is to be added in the second place, the units from the first place must be saved and subtracted from three above. Because nine added to ten make ten, and there is no difference that can be perceived, three, which are above the line, are placed below it.\n\nIf a figure requires the addition of a unit, let it be a circle: if nothing is to be added to it, they take the circle itself away and note the remainder.\n\nBut if there is a circle above it: they subtract nine, the distance from ten, with lines.\n\nWherever a superior note is a circle, the inferior figure signifies the distance below ten. If two are to be subtracted from the circle, since two are eight from ten, eight are subscribed.\n\nThe rest of what they transmit agrees entirely with what we have prescribed above. This is how to subtract: anyone who is not deceived: yet, the more frequently we seek the difference, the longer it takes us. And almost as much time is spent on addition as on subtraction. Therefore, the method above transmitted to us will be more worth following: since it is certain, it is far more widely used and quicker, and it imposes no delay on the subtractor. It always leaves the first notes untouched: so that if someone, due to an error in subtracting, has to repeat the process, the first number will remain before his eyes. Why we borrow ten from a certain number of places in subtraction: as many times as we go to the lower number of the following place, which should be subtracted, we add one: since it appears to be a unity with the following number from which the subtraction is made, it should rather be added. We should not give instructions without reason, whose meaning is not clear to the reader: we should not reveal the cause why it is necessary to be done in this way. All numbers, no matter how large, contain a first place or a number of denarii: either collected as denarii or individually. When a larger note appears in the place of any number in the lower order, which can be deducted from the higher order, it is necessary to borrow from the superior order at some number's seat to make up for it. This denarius will be the one requested from the next lower class, which, when joined to the smaller one of the higher order, will surpass the magnitude of the larger number, which could not be subdued earlier. Therefore, when we ask for a denarius from a number below, we leave the note there intact according to our instructions, unless we have taken nothing from the figure of the lower place, in which case we would add what remains of the denarii's note to make up the ten that were previously drawn from there. Since the text is in Latin, I will translate it into modern English and clean it up as requested.\n\nBecause, as for those ten coins, whose figures remain unchanged here: when we reach that place, we will exhaust them: we add unity to the lower number that is to be subtracted: the coins that overflow that number will reduce it. This happens at every place of the numbers: whenever we take ten or assume it as a primary number.\n\nIf someone wants to try whether the subtraction was correctly done, he should add the remaining sum to the subtracted number. And if a greater number results from the addition of the numbers, which we took away from: the subtraction must have been correctly performed. {FOR} if we add another number, {other} than the one from which we subtracted a smaller one: the addition will produce a discrepancy in the calculation. For just as the whole is equal to all its parts when separated: so all the parts are equal when combined in their entirety. If the total number taken away is greater than the one from which it was subtracted, the remaining parts should have an equal share: this cannot be, for the parts will again combine to form their original total, that is, the first number from which the smaller one was taken. And if these parts can create another number together: it is an obvious error. For all parts collected from their whole, the nature of things does not allow them to be at odds. We have already mentioned the usage of subtraction: it will show what is left and how much larger the larger number is. When this is discovered through the instructions we have given, the two numbers will be found to differ by as many units as the smaller number is less than the larger (which the remaining part will always indicate). For example, take two numbers, the larger one seven, the smaller three. Take three from seven. The remaining parts are four. septem enim tria superiora sunt quatuor unitatibus. Distant ita, that there is an equal number between them. If you add two of these numbers and subtract one, there will be nothing left, and they will not differ. Therefore, if you wish to investigate how far any two numbers differ from each other, you will find the remainder: one operation has distinguished itself.\n\nIf various kinds of coins are to be subtracted: having described them in their classes, as we mentioned in the addition, from each of the five kinds, such as singly valued denarii, 25 denarii, singly valued sestertii, 4 sestertii, from which kinds are the ancient Roman coins, the subtraction begins with the denomination on the right, and the lower denominations of the first class are subtracted from the higher ones, if possible. However, if this cannot be done: take a denarius from the next denarius class and use it in place of 4. sestertios resolutus numerum superioris augement: until the inferior one is sufficient for subtraction. And whenever a number is borrowed mutually: as many times as the next number in the following class is to be subtracted, unity is to be added. Similarly, if the number of denarii to be subtracted is greater: as much as it can be taken from the superior one: we ask for a gold loan from the class next to the left: so that it becomes denarii. 25. resolutus numerum denariorum, from which the subtraction will be made: augement. And when the wind will be to that class: we add unity to the next number to be subtracted. The same thing is to be done with various kinds of numismatic species, whatever their origin, when subtracting. For example, with aureus, denarii, and lower denomination coins.\n\nMultiplication of numbers is the creation of a third number greater than the multiplied one: which contains as many times the multiplied number as the multiplier takes unity. And to speak more clearly:\n\nmultiplication of numbers is the creation of a third number, which contains as many times the multiplied number as the multiplier takes unity. The number of multiplication is the production of a greater number by repeated addition of any number to itself as often as the number multiplying contains a unit. This is not only necessary for astrologers to reduce signs to degrees, minutes, and fractions, but also for merchants and all human conditions in dealing with life. In other words, multiplication is a frequent and repeated addition of numbers as many times as the multiplying number contains a unit. It makes no difference whether a smaller number multiplies a larger one or vice versa.\n\nTo find what one gets for one unit in multiplication, three numbers are required. The first number is the number to be multiplied, which we intend to multiply by another number. The second number is the multiplier, which multiplies the first number as many times as it contains a unit. The number that is mutual in both the numerator and denominator will always emerge. For instance, three sevens and seven threes create the same number, but a much easier method for multiplication is used: since the smaller number occupies the parts of the multiplier, while the larger number is being multiplied. The third number we require is produced from the multiplier to the number being multiplied, as if both were conceived through multiplication. The number that multiplies is identified by the fact that it is always expressed as a multiplicandum with an adverb. For example, if we multiply three sevens. A ternary number multiplies a septenary number. It is important to note that the number one itself, or any other number, cannot be multiplied by anything. Not even once is nothing more than one. And let us deal with the simplest things first. If someone wants to multiply a certain number by ten. Unum solum circuit around it, he who follows will increase it tenfold. If you want to multiply a number by hundred, place two circles to its right; the following number will be multiplied by a hundred. If we want to multiply a number by thousand, add three circles to the right, and it will be multiplied by a thousandfold. Similarly, if anyone intends to multiply another number by any number signified by a later number in the series, assigning circles accordingly will complete the multiplication.\n\nAt the beginning, keep handy the numbers multiplied by two, which are easily found among those less than ten. This is because when a number, whether large or small in relation to other numbers, is represented on a number line or table, it is easily identified. This is the case here. A sinister series begins with one, advancing naturally by the numbers in your right hand, writing down each primary number up to ten with small intervals. Then, starting from the same one placed on your left, beginning a new series, turning it downwards, distinguish the numbers with small and equal intervals. Write down each primary number as follows: so that both series, like Pythagoras' rule for drawing a right angle, may depict for us. Afterwards, beneath the first series, which tends from left to right, begin another series with the binary number: in which the numbers that advance to the right receive the increment of the binary number. This series ends with twenty. The third order of ten numbers, starting from the ternary number on the left, ends with thirty; the numbers extended to the right increase by the increment of three. Quartus ordini quaternarium addere in principio, advancing by tens and adding a quaternion at each step until reaching forty. Fifth series proceeding from quinque, advancing by tens and adding quinque to each, ending in fifty. Sixth order proceeding from sex, advancing by tens and adding sex to each, reaching sixty. Seventh series, starting from septenary, adding septenary increments to each number, ending in seventy. Eighth order, starting from octonary, bringing it to octoginta by augmenting each number. Ninth series, proceeding from novem, extending by tens and adding novem to each, reaching ninety. Tenth order, starting from denarius, advancing by adding denarius to each subsequent number, ending in hundred. Deinde numeris ita scriptis, lineis parallelis inter singulos ordines ducantur: tam a summo vertice ad imum descendentes, quam transversa a sinistra in dexteram, ut eiusmodi linearum ductu singuli numeri quadratis locellis concludantur.\n\nThis numerical figure demonstrates a remarkable property of primary numbers in their multiplication. If you wish to know which two primaries create a number: extend the first one in the upper series from left to right, then extend the second one in the descending series of the first. Find both of them: from the first, descend along parallel lines, from the second, lead along the transversals. Proceed until you reach the angular locelles of the quadratural form: in which number, the one you seek, you will find created from both in their self-contained progression. For example, to know which number is created by sevens multiplied by eights. in the second series, in the second set of seven and eight: parallel numbers, one descending, the other ascending, like leaders, follow: until in the square figure the shape begins. In its angular corner you will find the number 56 noted. The same applies to the other primary numbers. However, as I said before: whoever wants to multiply numbers without delay: should keep the primary numbers in memory, as in a table, recorded. In order to do this quickly, when necessary: singly generated numbers from the primary numbers, as from a source, can be easily learned within a few hours by anyone, even with weak memory. Otherwise, if you have to consider the figure of numbers above you repeatedly: what will happen? Where is the booklet when it is not at hand? Therefore, do not always carry it in thin volumes: scant hours of industry will wearily consume long periods of time: use the primary numbers for mental multiplication as often as necessary. If one wishes to cover a number or partition larger than this: more broadly speaking, when will it be enough for him? If his mind is such that this numerical figure can be expanded in such vastness: so as to inspect and contain great numbers through linear observation, and multiply what is there: starting from one, as we have previously said, the natural numerical series, as much as he pleases: as far as he desires: advancing from left to right, as well as descending from the vertex downwards: observing each order, and keeping in mind from which number the new series begins: each number following it will be increased by the increment of that number.\n\nBesides these other remarkable gifts of this figure bestowed upon us by Arithmetic, it is not within our reach to explore them further: this sign has: if from one, which is the lowest number, a line is drawn to the right angle opposite, it will produce angles opposite. If three are multiplied by three, let it be so: if one is brought to nine. Since it is useful for investigating roots in numbers, which we will speak of in its own place, it was seen fit that this was tasted, not alien. Whoever wrote on arithmetic gave this certain rule for multiplying simple numbers: whatever number you wish to multiply, if it is greater, subtract the number of units in the distance between them from the greater number. And the number remaining will be the one we seek. In Greek: To know how many times five sevens should be written, take three, the septenary distance: the greater number is the one to subtract five from, which is its denomination. Thirty-five remain: from which thirty-five are produced when taken in sequence. If we assume that the text is in Latin, the following is a cleaned version:\n\n\"If numbers are equal: it makes no difference: which one you take from the denarius: as often as the distance between them, the one who gives a different number, subtracts from the denarius. For instance, if eight is multiplied eight times, take two, place the distance of each from the denarius, subtract eight from eighty. The remainder will be sixty-four. This is the number we require, generated from them. Others taught a rule for multiplication of primary numbers, where two digits added to the right of a larger digit create a larger number: they taught another rule, that each number should be written to the left of the other. Then place the distance from the denarius in the region to the right. Multiply the distances through each other, and note the product under a line. Subtract one distance from the other, and if memory serves, place the number created from the distance below, to the left of the number from which it was subtracted. For instance, if you wish to multiply six by eight. Place six above eight, and six below four.\" Postea contra octo opposito duo. Two oppose six, four. Add two to four, eight. They subscribe these. Then remove from opposing angles, either two from six or, if you prefer, four from eight. Four remain above. These should be placed to the left of the numbers previously written.\n\nIf primary numbers are added simultaneously to the denominator, they do not transcend (these are all less than six), boys, most of them, know how to multiply these without any instruction.\n\nNumbers to be multiplied, we write in the highest place, the one we intend to multiply. Then we write another below, the one with which we wish to multiply the higher one. We place them thus: so that the primary numbers of the same order of the higher number respond, ten tens, hundred hundreds, and so on. And if the numbers are equal: place the greater above for multiplication, the smaller below for the multiplicand. This is done: so that the method of multiplication may be more expeditious. Then draw a line under both numbers from the left. Once you have the numbers, multiply the first one and place the result at the beginning of the multiplication: if the result is less than ten, place it in the first place under the line.\n\nIf a number arises from tens, write a circle: and remember the denomination of the tens in memory: to be added to the following multiplication.\n\nIf a number arises from tens and primaries, place the primary one first: and keep the denomination of the tens in mind: to join the number from the following multiplication.\n\nRepeat: lead the first number of the multiplication to the second multiplication: and whatever is created from that, add the former one to it, if anything from the previous place has been mentally retained, write it in the second place under the circle. In the same way, for the third, fourth, and any other subsequent figures, let the same initial note of the number being multiplied be drawn as the multiplier: and whatever is generated from these in themselves, let it be placed at the same locations below the line. Finally, let the first note of the multiplier, which has completed its task, be transformed into an obelisk, so as not to disturb the arrangement.\n\nAfter this, let the second note of the multiplier be brought to the first number being multiplied, and to all the others in the same order and in the same way. As we have said, this should be done with the first note. And let the numbers generated from their own multiplication be placed below the line and below the figures produced from the previous multiplication, so that the first of them is placed under the second multiplier note, the second under the third, the third under the fourth, and so on, proceeding to the left. Only a few minor corrections are needed to make the text readable:\n\nEt quam pauciores notae numeri multiplicantis, quam multiplicandi, fuerint: semper tamen progressus in levem a secunda notae multiplicantis exorsus, minimum, per totidem seruetur locos: quot figurae fuerint in numero multiplicando. Postea secunda nota suum enixit foetum calamo transfigurat.\n\nTertia deinde et quarta ac reliquae, quotquot fuerint: multiplicantis numeri notam in singulas multiplicandi numeri eodem modo ducendam esse: numerique ex multiplicatione nascentes.\n\nPeracta demum figurarum omnium multiplicatione, rusus sub omnibus numerorum ordinibus ex multiplicatione procreatis altera linea a leua in dextram ducta, ad primum locum pertinet. Deinde a primo loco exoris, omnes illos numeros ex figurarum in se ductu natos simul addamus: numerosque exlocis singulis collectos eisdem locis sub linea subscribamus. Ita numerus sub linea repositus is erit: quee multiplicatio nobis procreabit. We consider the following rule: whenever a circle appears in the sequence of numbers we are multiplying, we should place that many circles under the line. The number of figures to be multiplied determines the number of circles.\n\nIf the number to be multiplied contains a circle, place it under its own line, unless something needs to be retained in memory that should be left there.\n\nNow let us illustrate this with examples.\n\nMultiplying seven thousand nine hundred: this produces forty-five thousand. Three added to this makes forty-eight thousand. We write eight in the fourth place, retaining four in memory. Five times seven are thirty-five. We add four from memory to make thirty-nine. Place these under the line to the left. In this way, the second multiplier's sign is marked with an obelisk. Taking the third multiplier's figure, we lead it back to the first multiplicand. Eight times three are twenty-four. We write four in the third place under the line, retaining two in memory. Again, eight times six are forty-eight. The following text describes the process of adding numbers using Roman numerals:\n\nTwo added to fifty result in sixty. The circle is therefore subscribed in the fourth place and retains five. Again, there are eighty-nine twos: five are supplied for each. Thus, there are seventy-seven. Seven are noted in the fifth place, deposited in memory. Nine times fifty-six are denoted: when seven are added before them, sixty-three result. These are subscribed at the sixth and seventh places, and they are multiplied by three with the pen. Finally, the wind is at the fourth and last place of multiplication: since it is a unit and cannot be multiplied by a number, all figures of numbers to be multiplied are arranged in the same order they are written: the first is noted in the fourth place under the unit, and all others are moved to the left, namely the second in the fifth place, the third in the sixth, the fourth in the seventh. His it is completed, and a line of numbers, starting from the left to the first, should be collected under one line from the numbers resulting from multiplication. This will be the sum. Fourteen tens of thousands, seven hundred forty-seven thousands, four hundred seventy-six.\n\nBut since in this example, as we have explained: no circles were involved: let us add another, in which circles are intermixed in the number as well as in the multiplication. Let this be the case.\n\nWe will multiply sixteen thousand five hundred three, by four million and two hundred thousand. After the written numbers, the first occurrence of a circle appears in the number being multiplied. Therefore, we will place the same number of circles under the line: as many as there are digits in the number being multiplied. Then, the second digit of the multiplier is brought to the first place of the multiplicand, and six more are added. These six should be written in the second place. And since the second figure of the number of the multiplier is a circle: nothing can come from it: it should be noted in the third place. Rursus, the second note in the third, that is five, should be brought in. This makes ten. The fourth figure, which is the circle for the number being multiplied, is therefore placed in the fourth position. One should be remembered. Since the fourth figure is the circle for the number being multiplied, one should be retained under the fifth position.\n\nThe second note for multiplication, when brought to the fifth position for multiplication, that is six, results in twelve. These should be placed in the sixth and seventh positions.\n\nThe second note for multiplication, having fulfilled its function with the pen, is transformed into a circle again. The third circle in the number being multiplied appears in the third position. As many circles as there are figures in the number being multiplied, they are raised to the level of the first figure.\n\nThe fourth note for multiplication, when brought to the first position for multiplication, the ternary, produces twelve. Two are subscribed in the fourth position. One should be mentally retained. Since the second number being multiplied is a circle figure, the one mentally retained should be placed in the fifth position. The fourth note of the multiplicand is carried over to the third, that is, five. Twenty proceed. The circle is submitted to the sixth place, with two numbers remembered. Since in number multiplication, the same circle is present: two of them are concealed in the seventh place below the line. The deity of the fourth note of the multiplicand is carried over to the fifth and sixth place of multiplication. And twenty-four will emerge. They are to be placed in the eighth and ninth places. After the completion of multiplication, a line is to be drawn under which the sum of all the fetuses will be collected. Two hundred thousand four hundred and sixty.\n\nOne might ask why individual numbers, produced in multiplication, always move obliquely to the left instead of being directly located under the multiplicand number? But whoever reflects upon the nature of multiplication will cease to marvel at this. Since the text is already in Latin, there is no need for translation. However, I will remove meaningless characters, line breaks, and other unnecessary content. I will also correct some errors in the text.\n\nOutput: \"Nam cum multiplicatio numerus multiplicatur sibi ipse addeat et accumuleat, quoties numerus qui eum multiplicat unitatem continet, omnino alium numerum maiorem investigari oporet, ad quem is, qui multiplicatur, unitatis vice habeat. Quoniam unitas natura primum in numeris locum tenet, sequentes autem eam numeri in ulteriores sedes secundum sui augmentum trahuntur, quisquis numerus unitatis vice occupabit, is primam etiam sede habeat, necessest et quae ipsum sequitur in ulteriores sedes promoueat. Cum itaque numerus multiplicans non solum in omnes multiplicandi notas ducatur, verum etiam ei leges praescribant quoties ad investigationem numeri maioris unitatis vice sit subiturus, necessest omnino et tot maiorum numerorum ordines procreandos sunt, quot sunt notae numeri multiplicantis, et rursus illos ipsos ordines a singulis numeri multiplicantis notis, quae representant unitatibus, legem dant, novas subinde priores sedes accipere.\" If this text is about multiplication, and the first location is that of the multiplier: from that seat will begin the sequence, whether it be the second, third, or further locations that follow it. All places that follow this leader will be posterior. And since larger numbers are always found in the leader's seat, as soon as they are discovered, they should be promoted to the left, which is the seat of larger numbers.\n\nWe will prove that multiplication is correctly done according to a definite rule: if the number produced by multiplication is divided by the multiplier, and the number that has been multiplied is restored to us, it will institute the numbers. {QUOD} If it gives another number, it is necessary that an error has occurred. This will be more manifest after we have explained the partition.\n\nMultiplication is called the multiplication of numbers through two. It is done in such a way: when a binary digit is placed under the first location of the multiplicand number, it is led into all its figures, in the same way as it was previously said. The following text discusses the multiplication process, which is the easiest of all, requiring no separate instructions as some have done. This includes triplication, quadruplication, and other multiplication species, which are infinite in number. No one has specifically published on these, but some of them are more difficult: a more detailed explanation would have been possible if someone had touched on this. However, in all of these, the multiplication process is adequately explained by what has already been said.\n\nThe division of numbers is the division of a number by a divisor into any parts: each part taking the unit as many times as the number being divided can be divided by the divisor. The divisor should be less than or equal to the dividend. Moreover, a larger number does not divide a smaller one, as a larger number is not contained in a smaller one. To explain the nature of division more clearly:\n\nThe division of numbers is a partition of a number into equal parts, each part being the quotient of the number being divided by the divisor. The divisor should be less than or equal to the dividend. Furthermore, a larger number does not divide a smaller one, as a larger number is not contained in a smaller one. The number of parts in any given number is determined as follows: by the process of dividing the number to be divided, we find the quotient, which is the number of parts we seek, by the divisor. The quantity of these parts is determined in this way: if we subtract the divisor from the number to be divided as many times as possible, carefully observing when we will no longer be able to do so, then the number we seek contains that many units. In other words, to divide numbers is nothing other than to make repeated subtractions from the dividend by the divisor, and to create a third number, which consists of the total accumulated units, equal to the number of times the divisor is subtracted from the dividend. The function of partition is to reveal for us what each part of the dividend number is, however small, and how many times the divisor is included in the dividing number. QEMADMODUM in multiplicatione praediximus, non mediocremente conduce: si quis memoria tenet: quos numeros procreaverunt primarii numeri in se ducti: hic loco admoneo putemus, numeros parituris supra quam dicere potest, conferre: si sine ulla mora discere possunt: quibus ex numeris compositis sint, quas in partes solvent singuli numeri sub centum. Nam sicut primariorum inter se multiplicatio numeros maiores creavit: quorum tamen maximus centenarium non transcit: sic e converso, producere numeros ex primaris numerorum partibus, ipsos in partes dissolvit: numerosque ostendit minores: ex quibus maiores sunt accumulati. Anyone who can comprehend both sides of the multiplication table described above us: if one considers it in reverse order, and begins to contemplate from the lowest and highest numbers, and brings it closer to our eyes as much as possible, and since we began with a simpler partition, we express another partition table, described only by reversing the order of the numbers. This will be done as follows: Beginning from the bottom, starting with one, we write the natural series of numbers in the column on the left, using each primary number up to ten with small intervals. Then, placing the same number back to the right, as if from a certain base, we take the beginning of the same number series, and going up to ten using the series that advances to the right, we distinguish the intervals as equal and small. So that both series, as if having a right-angled norm, refer to each other. We will observe the individual orders as follows: whether the series begins with an even or odd number, be it binary, ternary, or other; we will increase each number and its ten subsequent numbers to the left. Parallel lines are then drawn between each order, extending from the uppermost peak to the lowest point, as well as crossing from the left to the right, enclosing each noted number within square boxes.\n\nThe divisor itself, when subtracted, exhausts the entire thing. For anyone who holds the memory of this table's description in their mind: they can easily tell not only which numbers are generated by multiplication of the primaries, but also in which primaries the generated numbers are resolved. This is particularly useful for anyone who wishes to divide numbers of any size into parts without delay. This is the reason why, in dividing numbers, the multiplication table explains itself promptly. If he had placed his mind in reverse at the banquet: it would be no easier for them to multiply than to divide. However, Modus, being expeditious, knows how to divide such numbers: the first number to be divided should be written on the right, and the dividing number on the left. A line should be drawn between them as a median. Then, two parallel lines should be drawn below the number being divided, with a small interval between them from left to right. The dividing number should then be noted below, so that the last digit of the dividend is subtracted from the last digit of the divisor, the penultimate from the penultimate, and so on. This will indeed be the case: if the entire divisor can be subtracted from the divisor's notations above it.\n\nHowever, if the divisor is larger than the number of notations in the divisor: so that it cannot be entirely subtracted from them: then the last figure of the divisor should be placed below the penultimate figure of the dividend. This text appears to be written in Latin and discusses the concept of a divisor in arithmetic. Here is the cleaned text:\n\n\"Certainly, a number placed under a parallel divisor, while turning, cannot be called other than subject to the divisor: since it always begins on the left, after being subtracted from the dividend number. But the divisor, standing on the left of the falling line, is rightly called a divisor: since it remains to remind us: which number is the divisor. Among parallel numbers, this number is to be placed in the middle: which indicates to us how many times the divisor must be contained in the dividend for there to be a subtraction. The common people call this quotient the 'barbarian' participle, coined from the past participle. This number seems fittingly named to us, either as the number of partition or, if you prefer, the number of section: since it is generated from the major number's section.\"\n\nRegarding the given text, we will begin with the simplest case. If we wish to divide a number into ten parts, we should separate the first mark, which falls in the middle, from the others. The partition is complete. If into one hundred, we should separate the first and second marks. If into one thousand, we should separate the first, second, and third.\" If we have ten thousand: let us divide it into four. And in order to summarize, I will state the following: whenever we wish to divide larger numbers by a smaller number using the symbol for unity and one circle or multiple circles, the position of the seat of the numbers indicates how many notes will fall down to the right, along the line. And the notes separated from the others will indicate what remains from the division, and the line interposed above the divisor will denote the remainder. This is the place for what remains after the division is completed. It is clear that this is the case. For if a note is added to the right of a number, it will increase the value tenfold if the note itself represents ten, hundredfold if two such notes are added, and thousandfold if three such notes are added, and so on.\n\nIf SIVERO is the divisor, the number to be divided: it has one mark, which is the figure under the last digit to be divided, whether it is equal to the divisor or smaller. Place it there. If the divider itself is equal to the number being divided: because it can be subdued only once: a marker of unity may be inserted between the parallels. Immediately, the divider, beneath the lines, changes into the form of the penultimate figure of the dividend. And the divider, walking to the right, is placed again beside the last note of the dividend.\n\nIf the divider is less than the number being divided: it must be considered: how many times the dividend contains it? once? twice? more frequently? The number indicating this should be placed between the parallels under the dividend. If once: place the marker of unity. If twice: place the marker of two. Three and so on.\n\nThen, the number placed between the parallels is multiplied by the divider: the number resulting from the multiplication is placed below the dividor, but first, to prevent confusion, the deleted parts should be removed. This number, located above the parallels, is subtracted from the dividend figures. And if anything remains: write it above the dividend figures from which the subtraction is made. Quo facto, delete also the number subtracted and placed below the parallels. And the divisor, advancing to the right, should be placed again before the second last figure. Where this will be noticed: when the second last figure, with the whole above the last figure (if there is one), contains the divisor. In regard to the remaining, and whenever it occurs in progress, it should be considered in relation to the following.\n\nThe denarius, the number which the note itself denotes: it always signifies. For instance, if the unit's note is in the remainder: it denotes ten. If binary: twenty. If ternary: thirty. And so on.\n\nThe number itself, indicating how many times the second last figure with the remainder contains the divisor, should be placed between the parallels. However, this should be observed perpetually. Whenever the divisor is contained more frequently in the dividing than twenty times: only nine should be placed between the lines. However, the divisor itself, placed between the parallels, should be multiplied. et numera ex eo proveniens, si una nota scribi potest: sub divisore prius deleta, ne rationes turbet: replice.\nIf that number proveniening from a denarius and primary is composed: it should be marked with two signs: place the primary always under the divisor. The denarius should be divided from its sinister under the remainder, or placed above another figure of the same divisor. The number should be subtracted from the divisor figures above it. Notes, whether subtracted or from whom the subtraction is made, should be deletable. And if anything remains from this subtraction: note it above the penultimate divisor figure.\n\nThen the divisor, walking to the right, should be placed again under the penultimate divisor's note. And the same should be done throughout: just as with the penultimate. Furthermore, the divisor should be promoted in this way: until it reaches the primary divisor figure. If anything remains from the subtraction: note it above the divisor standing. quod est quem remains after the division, which will always be less than the divider, unless there is an error in the partition. Whenever the divider is larger than those marked for division, which cannot be subdued by them: a circle should be drawn around it. And the divider should be advanced through one place, and the marks for division, from which subduction could not be made, should be regarded with respect to that place where the divider is located, or denied, if they are located in the second or hundredth place, or in the third place: they will always have this effect in the following division of the divider. To make clearer what we have said, let us divide seven thousand eight hundred and fifty-five by six. Writing first, we divide and place the divider with the middle parallels, as we have said:\n\nbetween the lines I insert and delete the divider: and then place it again under the first figure for division. There I find twenty-five fives. quorum twenty remain: because the dividor is closest to the circle with lines inserted: it could not be subdued by them. In which, since they have four and six, I write four between the lines. Through these, I multiply six and they become twenty-four. Therefore, under the dividor, I place a note of twenty-four: which signifies twenty. I deduce four from five above in the number, and one remains. Two are taken from two. Later, I transform the submerged numbers with my pen as if they had been subtracted. One remaining number, which remains above the dividor after the division is completed, I place a line between. Thus, the number indicating how many subtractions have been made rises between the lines. Mille three hundred and forty.\n\nIf the dividor is a number with one or more significant notations with one or more circles: we separate the figures of the dividend numbers with a line falling between, according to how many circles in the dividor the notation signifies precede. After removing meaningless symbols and formatting, the text can be translated to modern English as follows:\n\nIf, by that significant mark, we remove circles, let us divide the number in the same way: since we have said that only one mark should be made. Besides the division, when the figures will be separated by a line due to the wind: it will cease. And whatever remains of that separator, which was made from the figure being divided, will be part of the remaining figures in the entire partition. Therefore, a line should be noted above the separator.\n\nIf the divisor is a number with two or more significant marks: one or more at the end in circles: in dividing the number, figures should be separated in the same way to the right, as many circles there are in the divisor: the figures themselves precede these significant marks. And the number should be divided in the same way, by removing only the significant marks and circles. We will soon explain how to do this with multiple significant marks. The division will cease when we come to separated figures: which, as previously stated, should be assigned to the remainder. When a divisor has two or more significative notes: under division, numbers are to be written thusly, with each divisor's last digit under the last digit of the dividend, the second last under the second last, and so on, completely aligned. This applies unless the total number of digits in the divisor is greater than the number of figures in the dividend above it, making it impossible to be subdued. In such cases, the last digit of the divisor is to be divided by the second last digit of the dividend, and the second last by the third last, and so on, starting with the initial placement of the divisor. Once this is accomplished, it must be considered: how many times the last digit of the divisor is found above it in the dividend; how many times the penultimate digit of the divisor is found in the penultimate digit of the dividend, and the remainder, if any; and how many times the antepenultimate digit of the divisor is found in the antepenultimate digit of the dividend, and the remainder, if there is one, and so on for the other divisor's digits. If this is the case, the number indicating how many times the divisor is contained within the parallel lines is to be placed between them. et if by it several dividing marks are multiplied: each one in its order: numbers arising from multiplication placed under the multiplied dividing marks, so that if several figures are to be written, the first among them is placed under the multiplied mark, the others following in order. each number is separated from the figures above it, and from the rest, if there is any, below it: deleted are both the dividing and the separating marks, in the same way as before stated: if the divider has only one mark, it takes precedence.\n\nbut if the penultimate and antepenultimate and any other dividing mark is not found in the figures located above it and the rest, if there is any: as often as the last dividing mark is contained in the last figure: then because they cannot be subtracted from above, we subtract as often as possible the last dividing mark from the last figure divided, minus one. et consideremus, an reliquum, quod ex ultima dividendi figura supererit: penultimae et antepenultimae et caeteris eiusdem figuris tantum sumministret? Sufficit ad subductionem reliquarum dividendi notas adhuc subduci?\n\nIf the singulae dividoris notae cannot be subdued under the dividing figures above them and the remaining part: let us take away the ultimate dividoris. And let us consider again: an reliquum, quod supererit, tantundem penultimae et caeteris dividendi figuris suppeditabit? Quantum subductionem dividoris ferre potest?\n\nIf bis minus cannot be subdued: let us try, an ter minus sufficit. If this does not work: let us give up, an quater minus deducendo proficere possimus. When attempting to reach the rarest part of the last dividing line: until the remainder, which will be increased by this, is reached for the other figures of the same kind: as far as the just subtraction of the other dividing lines' notes permits. In order to investigate the partition's table, as described above, it will be helpful to inspect it particularly closely. Whenever they have considered the rarest part of the last division, they should look at the interior lines that are parallel to it and converge towards a smaller quadrangular form: the extremities of which will easily indicate the number to be taken and what remains.\n\nHowever, when considering this, we have found: how often the last division should be made: so that other notes of the same kind can be similarly subtracted: they indicate the number to be placed between the parallel lines above the last dividing line. et multipliquemus per illum numerum quemquam dissoris notam: ponenmus numerum prouenientem ex singulis multiplicationibus, sub ipsis multiplicatis dissoris figuris, ordine collocandas reliquas: a figuris supra eas dividentiis locatis et reliquo, subtrahere, prius delentes figuris dividentium: fit deductio. Singulis subtractionibus reliquum supererit, supra ipsas dividentiis scribamus prius deletas. Quando in quibuslibet aliqua nota retro manet non exhausta, praecedens dividenti figura, ex qua post subductionem nihil reliquum manet, integre deleta, supra ponatur circulus, ut adhuc restantibus reliquis nos admoneat, si simul eius augeat significationem. When completed: each note of the separator shall be removed before the preceding one, and placed in one spot to the right. The last note of the separator shall be placed before the second last note to be divided, and its penultimate note before the antepenultimate, and so on. The same method shall be used for removing notes from the separator. The same shall be done: as previously stated. Again, after the removal of notes from the separator, the separator itself shall be promoted to one spot. And the same rules shall be observed: until the first note of the separator is below the first note to be divided.\n\nFurthermore, if anything remains after the removal of notes from the separator: which is less than the separator: a line shall be inserted above the separator, marked with a line standing above it, so that we may see: which part of the separator is below the line.\n\nHowever, if the remaining thing is greater than the separator: an error has certainly occurred between the partition: the entire process must be repeated from the beginning. \nPRAETEREA quotiescum{que} post unam omnium di\u2223uisoris notarum subductionem us{que} ad finem operis diui\u2223sor ipse diuidendi notis supra se locatis maior reperitur: ut ab his non possit totus subuci: toties circulus inter paral\u2223lelas est reponendus. et singulae diuisoris notae nulla facta subductione per unum locu\u0304 transferendae sunt in dextra\u0304. Relictae autem illae diuidendi figurae in proxima sequenti diuisoris subductione debent exhauriri. At{que} ideo si post consummatam subtractionem in numero diuidendo cir\u2223culus ad dextram superest: is inter parallelas a dextra col\u2223locetur.\nILLVD quo{que} etiam in hac partitione, quae per plures fit diuisoris notas: perpetuo seruandum est: ut si aliqua di\u2223uisoris nota saepius {quam} nouies in diuidendi figuris supra se reperiatur: nouem tantum inter lineas sint reponenda. quemadmodum in diuisione per unicam diuisoris notam facienda supra dictum est.\nET quoniam auspicantibus pr\u0119cepta sine exemplis ob\u2223scura uideri solent: exe\u0304plis illustrem{us} ea, qu\u0119 ia\u0304 dicta sunt We divide by two million nine hundred seventy thousand, forty-seven thousand. I immediately divide the number written above the parallel lines, and we place the divisor below the parallel lines. Since we divide the last digit with a binary divisor, the last digit of the divisor cannot be a quaternary: that is, if this is greater: we divide the penultimate digit. It contains nine. We place four and seven to the right of it. Then, when dividing in this way, those that follow remain silent. We divide four in four figures above, which denote twenty-nine. I find seven sevens in them. Seven sevens create twenty-eight, and one is still left. But if I take seven sevens and one: I make another divisor's septenary sign seven times in the figure of division above it, and the remainder, which contains eleven, I will not find that number of times. Therefore, I take six times four from fifty-nine. And what is left, I make five. So that both divisor signs may be found above the placed figures. I. Remove meaningless or unreadable content:\n\nI remove the curly brackets { and } as they are not necessary and may be considered meaningless in this context.\n\nrepono que sex inter lineas as: et per ea multiplico quatuor: sic fiunt uiginti quatuor. Therefore, under the multiplied quaternary figure, I replace quatuor: et binarij nota, which means twenty, I place after the last dividing mark. Then I place quatuor above the numbers posited. However, the remaining quinque I place above novem, which is known. I subtract two from each of the two: and I transform both notes. I multiply sex inter lineas, which is placed septem, by another dividing mark: and I create forty-two. I place quaternarij figura, which signifies forty, after it, to the left, after seven have been deleted. Since I cannot remove two from one: I assume ten mutually: and I take two from undecim.\n\nNine remain above the sign. And since I have taken ten mutually: the following quaternaria note I increase by one. Thus, I subtract the five that have been placed above the five above them: I delete both notes. After the division of the whole divisor by a number, I move the sign of the divisor to one place on the right. I find the quaternary sign of the divisor in the new remainders twice: yet one is left over. Since the figure of the divisor quinary placed above seven is sufficient for subtracting seven, I remove two between the lines. I multiply four by these, and multiply the posterior sign of the divisor. This makes eight. Subtract this from nine. One remains. This is noted above the nonary figure of the dividend, which was deleted first. I also delete the quaternary sign of the divisor. Again, I multiply seven by two. This makes fourteen. The quaternary sign among these, under the multiplication figure septenary, I place one. Then I subtract four from five, and one from one above it. When both signs are deleted, one remains above the five. I. In the following, I propose moving both dividers, which have been multiplied, to the right, in the same place. Where above the mark of four units in dividing the number, I find seven. Since forty-seven cannot be held in twelve, no subtraction is possible: I transfer the divisor, marked with a circle, to the right, under one place. There, above the quaternary divisor's mark in dividing the number, I find four, and from the remainder of the former one, which denotes ten: I find one. Thus, in twelve, there are three groups of four, and two will be the remaining parts for another divisor's mark to be subtracted. I place three between the lines and multiply by them: twelve results. I place a binary nota below the figure of the quaternary multiplication: the unity of ten is denoted by the remaining sign. I subtract these from fourteen, and two remain. I then multiply seven by three: twenty-one results. The note about unity's septenary figure being multiplicated and the binary sign to the left of the other signs being subtracted from the dividing figures, leaving nothing, and the need for both the divisor and dividend's notes to be obliterated, but since all numbers signified by the dividing notes have been exhausted through subtraction, yet the circle still remains and the division is not fully completed because the divisor's first note has not yet succeeded the dividend's first note, therefore the circle is returned between the lines to the right. Thus, the number of the partition will be found between the lines. Sixty-two thousand and thirty.\n\nWe have explained VNUM as an example for beginners, where there were two notations for the divisor. Let us approach the other: if it has three notations, the smallest last one among them. An example that is more difficult to explain is if the last divisor's notation is larger than the preceding ones. We divide as follows: nine hundred thousand nine hundred seventy-four thousand five hundred sixty-three. And when we have written down the marks for dividing and the divisor: we take parallels: let us establish the divisor thus: let the last mark of it be under the last mark of the dividend: the next to last under the next to last: the third to last under the third to last. Then we consider how often we can bring the last mark of the divisor under the nonary mark of the dividend: as for the other marks of the divisor, which are greater, we find that they suffice when we subtract them. And we find that this can be done twice. Therefore, we place two between the parallels. Through these we multiply by three. Six result. We subtract these from nine. Three remain. We delete the last mark of the dividend and also the last mark of the divisor. We then multiply by two through the middle divisor's octonary mark. Sixteen result. We subtract six from eight and write one remaining above. unius tantum notas ex tres retus: duas supersis supra vertice superscripsimus. Eliminemus figuras, quae sublatusae sunt, et causis sublationis. Reiterum tertiam notam divisoris notiam duplice multiplicemus: et fit x. Quorum octo sub nonaria deleta reponimus: unitatis notam pro x proxime sequenti. Et quia octo a quattuor supersis subducere non possumus: x mutuatis, octo a quattuordecim subducimus. Ita eliminatis notis, sex relictae superscripsimus supra verticem. Adhuc autem decem assumptis unum ad sequentem unitatis notam addimus: quod in divisore est: et duo illa, quia ab uno supra se deposita non elicet: x mutuantes ab undecim tollimus. Reiterum eliminatis notis, novem relictae superscripsimus supra verticem. Et quoniam x iterum sumpsimus: iterum sequenti loco unum a duobus supra signatis demimus. Unumque quod superset: notamus, deleta nota binaria. Et quoniam semel omnes divisoris notas a supra positis multiplicatas abstulimus, eas per unum locum singulas in dextram promouemus. Ibi iterum consideravimus: quoties tria a decem et novem supra positis extrahi possunt, ut tamen satis ad ceterarum divisoris notarum subductionem supersit. Id quinquies feri posse compleveramus. Quare quinque inter lineas supra tria reconduntur. Per ea, tria multiplicantur et funt quindecim. Quorum quinque sub multiplicata figura ternaria reponuntur. Et unitatis nota pro decem loco sequenti. Deinde quinque subducuntur a novem supra se positis: et quatuor reliqua supra scribuntur. Unitatis autem nota ab uno reliquo tollitur: delentur figurae. Multiplicatur deinde secunda octonaria divisoris nota per quinque. Inde surgunt quadraginta. Circulus ita sub octo locatur. Et quaternaria nota, quae quadraginta significat: post eum. Ea subtractur a quattuor supra positis. Et utraque nota transfigit obeliscum. postea multiplicatur tertia nonaria divisoris nota per quinque. Et crescunt quadraginta quinque. Quorum quinque sub novem collocantur. Et loco sequenti quatuor pro quadraginta. Sic quinque, quia circulus supra se notatur: a decem mutuo sumptis auferuntur. Et quinque manent, quae supra circulum deletum annotantur. Et quoniam decem sunt assumptera: unum ad sequentem quaternariam notam addita est. Atque ita quinque a sex supra se locatis deducuntur. Deletisque figuris, unum, quod relinquitur: signatur super verticem. Sic omnes divisoris notae per quinque multiplicatae, et a supra positis subtractae, per unum locum singulis transferuntur in dexteram. Et quia divisoris notae in dividendi figuris supra se positis haberi nequeant: circulus inter linias inseritur. Rursus per unum locum singulae divisoris notae in dexteram procedunt. Ibi consideramus quoties divisoris ultima ternaria a dividendi notis supra se locatis, quae designant quindecim: tolli queat. et quiquis quinquies id fieri potest: semel tamen minus subducimus, ut sequentium divisoris notae habeatur ratio: quatuor inter paradoxis locamus. Per ea multiplicamus tria, et funt duodecim. Quorum notam binariam sub tribus multiplicatis locantur, et unitatis nota pro decem, in sede sequenti. Deinde duo subducuntur a quinque. Et tria reliqua manent. Ea superscriptur. Unitatis autem nota tollitur ab uno supra verticem posito. Statim delentur notae, tamquam subtractae sunt, quam a quibus fit subtractio. Multiplicatur inde octonaria divisoris nota per quatuor. Et origintur triginta duo. Horum duo sub octo reducuntur: ternaria nota loco sequenti. Deinde delentur illa duo a quinque. Tria vero, quae ex his supersunt: in vertice notantur. Ternaria autem divisoris notae a tribus, quae supra notantur: aufertur.\n\ndelenturque utraeque notae. Postea tertia divisoris nota nonaria per quatuor multiplicatur. Inde surgunt triginta sex. six groups of nine, marked in threes, are noted in the following sequence below. These are removed, six from six and three from the threes. Then, each of the dividing marks proceeds back through one space to the right. Since the first dividing mark comes before the mark to be divided: nothing else is present in the division except three. Therefore, the subtraction is made: and for this reason, the divisor is not found in the division: a circle is placed between the lines to the right. The three remaining, which are finished by the partition, are noted above the divisor. Thus, the number of the partition appears between the lines. Twenty-five thousand and four hundred, and in the remainder three hundred thirty-ninth parts. I have shown enough examples: through which, once understood, anyone can easily explain it to themselves: even if the divisor has four or five or more notes. We present such an example for consideration before your eyes. In any section where a divisor needs to be moved to the right after a subtraction has been made: if there is a remainder, which is above the divisor's mark in the subtraction, it is greater: the divisor in its entirety has been entirely missed. And further progress in that partition would be in vain, unless one is willing to waste effort. Therefore, it is necessary to repeat the process from the beginning. For not only after the entire partition has been completed, but also what remains, which is less than the divisor, will be insufficient if the process is carried out correctly. And the same thing will happen in the middle of the work whenever the divisor needs to be moved to the right. This observation is a valuable rule. A larger number does not divide a smaller one. Indeed, since a larger number is not contained in a smaller one, as we pointed out at the beginning of this chapter. Here is the cleaned text:\n\nCeterum ratio inici potest: qua maior numerus etiam minimor dividere recte possit. Id quod fit: si de rebus minimor numero numeratis, prius una in species sub se contentas ducatur: ut earum productus numerus divisor maior factus, eius sectionem ferat. Veluti si inter septem viros tres aureos partiremus. Quodquidem in numero ternario numerus septenarius non habetur: solumus aureum unum in centum nummos sestertios: quibus aestimatur. Centenariumque numerum per ternarum multiplicemus. Et enascentur trecenta. Quae deinde si in septem partiamus: numerus sectionis prodibit, quadraginta duo, et sex septimas. Ita si inter septem viros tres aureos, centum nummis singulos aestimatos, distributos, unusquisque consequetur quadraginta duos nummos, et praetera unius nummi sex septimas.\n\nAD HAEC maior numerus minimor in minimas frangere potest. Id quod sequenti libro de minimarum partitione praecipientibus, monstrabimus. Any number of people who want to divide a number correctly: divide the number into parts, multiply the number of partitions by the divisor, and if there is a remainder from that multiplication, the number to be divided will show itself. If another number comes from that multiplication instead of the divisor, the entire division must be repeated due to an error in calculation. Nature itself shows this to be the case, for just as the parts that have been separated from a whole must be put back together to make the whole again, so if a different whole comes from the composition of those same parts, it is necessary that either wrong parts have been taken or the parts have been incorrectly rejoined. When the number has been divided so often due to frequent subtraction, once the division is completed, nothing remains except the number of parts: and the remainder, if there is one: if the divisor itself is added to the number of parts as many times as it was previously subtracted: this is easily done: if we multiply the number of parts by a divisor that is complex, we add the remainder, if there is one: the number divided anew need not be restored. For the division of numbers is nothing other than their separation into two parts. This is certainly the simplest and most common form of partition. Et quam uis non referat nulla de ea praecepta separatim dare: cum ea, quae iam dicta sunt, abunde sufficiant. Quia tamen hoc characteris specie necessitatur minori tu consideratione, caeterae quae innumerabilia sint et omnes hac difficili operae, facilitatis eius compendium auspicantibus enarrare putabamus.\n\nNeque enim duabus parallelis est opus. Unica tantum linea sub diuidendo numero ducatur. Deinde numerus quemquam per medium secturus quispiam a sinistra incipias per omnes notas in dextra tenet. Et si ultimus diuidendi numeri character nota sit unitatis: a penultimo sectionem ordinetur. Singulorumque locorum paribus numeris per medium sectis, ipsum dimidium sub linea locis singulis subiciet. Sin impar occurrat numerus, unitate dempta, reliquum numerum per medium secet: dimidiumque sub linea reponat. unitas itself, calculated as ten and stored in memory for the following section: if a coin is involved: if it is a penny and a whole one: or composed of a penny and a farthing: if there is another notice. This number, if even, is divided by two. If odd, a unit is subtracted and carried over to the next section, as previously stated: the remaining number is divided by two. If, in dividing the number, a unit comes before the first notice: place a circle under the line. The unit itself is carried over to the next section. However, half a unit is always kept below the line. Proceed in this manner to the right, using the first notice: if a coin is present and the number next to it is even: place a circle under the line. If the number next to it is odd and a unit is taken away: keep the unit mentally: the coin itself with the unit makes ten pence: those which are halfpennies are placed below the line. If there is no unity mark: if a denarius is added to one of mine, unity is taken away, leaving ten remaining, and five under the line to be concealed. The unity mark itself should be noted as part of the remainder, with a dividing line placed above it. This will indicate the first part of a single unit. The same should be done for the second part, even if no denarius or number is retained in memory, when it is joined to it. The number of the partition will appear above the line when it is divided by the number below.\n\nAnyone who needs this for convenience may dispense with the following: but we have considered it necessary to add one for beginners.\n\nWe searched for half in thousands of thousands, four hundred and one, and thirty-one. Written with its own notation and a line drawn under it, since the unity mark appears at the beginning on the left: we begin with the penultimate one, where another unity also occurs. Itaque quoniam ibi undecim habentur:\nunitate pro denario ad sequentem sectionem servata, decem relinquimus per medium secamus et quinque sub linea repomus. Progredimus deinde ad proximum locum: ubi quatuor reperimus, quibus denarium ex memoria depromimus: ut fiant quatuordecim. Horum dimidium, septem, sub linea locamus. Rursusque progredimus. Ibi occurrit unitas: quae secari non potest: Ideo sub linea circulus ponatur: et unitas servetur in sequentem locum. In quo quia circulus occurrit: cui unitas adiuncta decem facit: quinque sud linea statuemus. Denuo progredimus. Ibi tria occurrunt: quorum uno, quia numerus est impar, dempto, duo secamus: et horum alterum sub linea condimus, denario numero mente in sequentem sectionem retento. Rursus ad dextram progredimus ad primam unitatis notam: cui decem ex memoria adiuncta creant undecim. Unitate ita dempta, decem per medium secamus: et horum quinque sub linea reponimus. unity that stands above the dividing line we mark with a sign, so that the part following it signifies one whole. The number of partitions will remain under the line. Five hundred seventy thousand, one hundred fifty.\n\nA PROGRESSION IN ARITHMETIC is a collection of numbers equally distant from one another and summing up to one total. It is like a shorthand for counting certain numbers, which have equal intervals between them. Its two species are: in the first, the natural sequence of numbers is preserved, and each number following the preceding one surpasses it only by unity; for example, 1, 2, 3, 4, 5, 6, 7, 8, 9. In the second, we connect numbers by preserving the intervals between even terms, for instance, 1, 3, 5, 7, 9, 11, 13. In both species, the same method is used to collect numbers. This rule, however, consists of two parts. If numbers are equally spaced and in continuous order, the series of their positions will yield a pattern: the first number is joined with the last, and the number formed from these is multiplied by the middle number in the series. This will result in a sum representing all of them. For example, in this case: 1, 2, 3, 4, 5, 6, 7, 8. The first number is added to the last, resulting in 9. Since there are 8 positions in the entire series, we take half of that and find the product: 3, 6. The sum of all numbers is 36. Similarly, if an example is presented with: 1, 3, 5, 7, 9, 11. The first number is added to the last, resulting in 12. Since there are 6 positions in the entire series, we take half of that and find the product: 1, 2. Therefore, the sum of all numbers is 36. If a series of numbers is equally spaced and the number of terms is odd: the number indicating the position of each term should not be added to it, but to the number occupying the middle position, which is equally distant from both extremes. Thus, the generated number will reveal the total number of terms. For instance, in this example: 1, 2, 3, 4, 5, 6, 7. Since the positions in the series are occupied, and the middle number is 4, we add 4 to 4 and get 28, which is the sum of all. Similarly, if we take other numbers for illustration: 1, 4, 7, 10, 13. Since the positions in the series are occupied, and the middle number is 7, we add 7 to 7 and get 35, which is the sum of all.\n\nAnother rule is given, which is equally general and certain:\n\nIn any arithmetic progression, whether it be a par or an impar series: the number obtained by adding the extremes and is indicative of the number of terms in the series, should be multiplied by two and then halved. The sum total of progression will be held. An example in a similar series. 1, 3, 5, 7, 9, 11. The addition of the first number makes the last one. 12, and since there are six terms in the series, we multiply by 6 and add: the result is 72. If halved, they become 36, which is the sum of the progression. An example of an odd series. 1, 4, 7, 10, 13. The addition of the last two numbers produces the result. 14, and since there are five intervals in the series, when we take every fifth number, we create 70. If halved, they advance: 35, which is the sum of the entire progression.\n\nTo check if you have correctly calculated the sum, subtract each number in the series from the total sum and if nothing remains, you have calculated correctly. {If} if something remains, an error has occurred.\n\nAnother type of progression is called geometric, in which numbers are arranged in a long sequence and there are unequal intervals between them. When a certain proportion is found between the numbers in the series, such as double, triple, quadruple, or greater, we will speak of this further in the book on proportions. A quadratic number is the one that is formed from any number, when it is completed in itself by a number equal to it. For instance, the quadratic number is the fourth. It is created from any single binary number. For binary numbers, when each one is taken twice, they produce four. Similarly, the number nineteen is a quadratic number. For instance, when three is taken three times, it produces nine. In the same way, the number sixteen is a quadratic number. For instance, when four is taken four times, it produces sixteen. In the same way, all other numbers, when taken by themselves, are called quadratic numbers, because if they are written as units, they refer to a quadratic form. In a continuous square's body, the side is called: similarly, in numbers, the side of a square is called the square number, which completes one side of the square number: they incorrectly call this the root of the square among mathematicians. Since from its own root, a square number, like a tree from its root, arises.\n\nA number is called CVBVS, which is composed of a number from which, when multiplied once with itself, is produced. Such a number is octonarius. When one nymph is taken from it, twos are created. Four iterations of twos create eight. Similarly, 27 is a cube. For three taken from it, nines are produced. Nines, taken three times, produce nineteen. 27 is also Itide\u0304. Sixty-four is a cube. Four fours are drawn from it. Sixteen is produced by the first iteration of fours. Sixteen, drawn a second time, generates sixty-four. All the mentioned numbers return to their original form once, and they are multiplied in the process of counting, which is why they are called cubes. A cube is a solid body with three equal dimensions, namely length, width, and depth. A cube contains six square faces and twelve edges: four of which are located at the bottom, and four are extended towards the top. The cube has eight angles: four of which are located at the bottom, and four are located at the top. In each angle, three edges meet. The Latin word for this figure is \"tesseram.\" The Romans use the Greek term \"cubus\" more frequently in Latin, as in \"cubum\" for the cube itself. The number of sides of a cube is called its lateral number, which is equal to the number of sides of the square contained in one of its sides. Therefore, it is clear that every number has a cube number within it. In any quadratum, you will find a cube. However, the unity of arithmetic and the number of the square and the cube are desired by the mathematicians.\nIn such an investigation of the sides of figures, a great aid will be provided to students for understanding arithmetic. For through this, Ptolemy, the most skilled in the stars and the magnitudes of celestial spheres, discovered not only their velocities but also their sizes. The knowledge of this not only greatly contributes to the learning of astronomy, but also of geometry, without which the beautiful things in them cannot be understood.\n nevertheless, the order of things demands: that we first investigate the quadrature. And since the investigation of the side of a square is nothing other than the discovery of a certain number: if the proposed number is a square, it produces the square number itself; if it is not a square, you will investigate the number of the side-making. To find the number you want, which is the square root, mark points above the vertex in uneven places: specifically, at the first, third, fifth, seventh, ninth, and the same for the rest. Since finding the side length of a square is nothing other than certain division methods: under the number of points noted, two parallel lines should be drawn. Place the side itself to be found between these lines, as we instructed in division. Because the number of dots you find is equal to the number of primary sections you will have, each showing the root of the given number. Then, under the extreme note to the left, mark a point in an uneven place: this primary number, when drawn once, should reduce the number above it to zero or at least approach it closely if it cannot completely eliminate it. This primary number, once found, should be inserted between the parallels and multiplied within itself. et after the numbers are created, the subtraction: whatever remains, if there is any: above the given numbers, from which the subtraction is made: let it be restored. Then the primary number itself must be multiplied: and the double placed under the next figure to the right.\n\nIf, from duplication, a number arises with two signs,\nthe first, which is to the right of the point marked in the middle: it should be placed back. The second, following in order, is placed under that note: under which the primary number is found. Again, another primary number must be sought: which, when doubled, exhausts the number that is above and beyond the double: so that afterwards, even when drawn through the whole number from the next point to the right, whatever remains, if there is any: it absorbs. If it does not exhaust completely: take away as much as possible. When it has been thoroughly examined: the entire number between the parallels is doubled: the double is placed with the figure on the right turned so that the second note of the double, which is marked next to the point, is in the middle: as if it were placed at a right angle: and others, in their respective order, tending towards the left. Again, it is necessary to inquire about another primary number: which, when taken twice, reduces the entire number above it to nothing: so that afterwards, when the number is drawn towards itself from the right, next to the mark, it can either be completely deleted or, if possible, made as close to it as possible. Furthermore, to proceed in this manner, it is necessary to return to the first note. Quotiescumque numera after numbers duplicated perform multiplication and subtraction, we take the primary number itself: it must always be done at some point: immediately whatever is found between the parallels in the number section: it is a duplicate: and the first duplicate's sign should be placed to the right of the nearest puncture: the others, proceeding in order to the left. This should be observed carefully. Wherever in any investigation a primary number cannot be found: it becomes double, then double again, the product will be greater than the number above it: which cannot be subtracted from it: a circle is added to the number between the parallels. In all the notes of the number whose root we seek: keep intact, delete the doubled signs under parallel lines, progression to the right should be made. If not double progression is required. totusque inter parallels numerus ita dupliciter tur: ut prima dupli nota circularis sub proxima ad dextrae figura, quae inter puncto signatas media est: reponeatur: caeteraque post ea suo ordine in sinistrae tendentes: ut ante dictu est. Peracta demum radicis investigatio, si nihil reliquum erit: palam est numerum propositum: cuius latus scrutati sumus: quadratus esse. If quid reliquum manisit: non erit quadratus: sed numerus inter parallels radix esse comperietur maximi numeri quadrati in eo contenti.\n\n Et quoniam obscura sine exemplis non facile intelligi queant: exempla nunc adiungamus, quae cuncta magis illustrent.\n\n EX quinquaginta septem milibus, octingentis triginta sex milibus, et uiginti novem, radicem eruamus.\n\n Postquam numeris suo ordine perscriptis, ductae erunt parallelae: locaque imparia punctis annotata: sub postremo ad sinistram numero puncto notatum numerum aliquem priorem quaeramus: qui in se ductus vel totum numerum supra se notatum, quod est. 57. uel quam proxime fit: subductus deleat. This will be septenarius. Nam, 7 in se ducta. 49 creant. Quae, si subducti sunt a. 57, supersunt. Itaque 7 inter parallelas inserenda sunt. Ac post multiplicationem et subductionem factam deletis notis. 8 supra postremum punctum manebunt reliqua: quae supra uerticem notentur. Deinde, 7 inter parallelas posita duplicemus: et fiunt 14 quorum. 4 sub proxima ad dextram nota octonaria replaces. Et unitas denarium numerum designans a sinistra sub numero octonario ex priori subductione relinquit. Tum iterum inueniendus est numerus aliquis primarius: qui in quatuordecim, quae dupla sunt: ductus numerus supra illa positu sic auferat: ut postea in se ductus proximus a dextra numerus puncto notatum cuquod erit: subductus tollat: vel saltem quam propeaccedit potest. Iste autem erit senarius inter parallelas sub proximo puncto inserendus. Perque multiplicata. 14 facit. 84 quae ab. 88 supra se positis subtracta relinquit. 4. Six in pairs produced them. Excluding the six noted before, the rest did it. Seven above the vertex of the number marked with a ternary digit, take note of it. The process must be repeated: whatever is found between the parallels, therefore. Seventy-six duplicated make 152. The first note is to be placed to the right of the next: the rest after it in order to the left. Then a primary number is to be sought: which, when doubled, would subtract the number above it. However, since it cannot be found (because the number above is larger), add the number 76 to the circle between the parallels, under the next rightmost point. Also delete the notes that are doubled under the parallels. The remaining notes concerning the number whose root we seek should remain intact. Let us proceed to the right and double again within the parallels. Seven hundred sixty emerges from this doubling, 1520. quorum nota circularis, sub proximo ad dextra figura binaria, ponenda est inter puncto signatas, medias: ceteraeque post eam suas. Ac numerus aliquis primarius perquiratur: qui in totum duplum ductus, numerum supra id locatum, subtractus deleat: posteaque in se ductus, proximo a dextra, numerum puncto signatum relinquit, si quod erit: auferat. Vel saltem quam proxime fieri potest. Is autem erit quinarius.\n\nTherefore, between the parallels, each number, noted separately, multiplies: numbers arising from multiplication subtracted from figures above them. Immediately delete these. Et demum, in se ductus numerus ille qui inarius subtractur a novemario numero puncto notato. et.\n\n4. crunt relicta.\n\nFrom this it is clear: since, with all effort expended, something still remains: the first proposed number was not a square. For if it had been a square: nothing would have remained.\n\nThis example will have sufficed for students investigating the way to open the square of the side of a square. Let us add another point to be observed by several notaries, both in sight and in mind: that anyone can easily explain this.\n\nIf anyone wishes to try: whether the true root of the square number has been discovered. Let the root itself be multiplied. And if anything remains: add the product to the number from which it was taken. This will be done: unless the investigation was made carelessly. {IF} if another number appears: the process must be repeated: to correct the error.\n\nHowever, the remaining part can be made equal to the square of one of the sides of the square number discovered. It cannot be surpassed. For instance, if the proposed number were, for example, the square root of the largest number containing it. 2. whatever is drawn from it will create. 4. and 4 will remain: which will contain the double of the roots. For if the rest were, they would already be. 9. whatever other root there may be: namely, 3. If the proposed number were, 15. the square root of it were. 3. whatever is drawn from it makes. 9. and the rest: which will take the doubled root. But if the rest were, 16. to emerge. quem alias radix, nampe. A number has a root, namely the one that reveals it. If the number proposed is a square number, the root of the largest number containing it will be its square root. Four are what remain when it is multiplied by itself and eight are left, which contain the double of the root. If the rest were present, they would become. And another root would emerge: that is, the other numbers follow the same pattern. Therefore, if you find more than the square root of the side in the remaining part, it is clear that an error has occurred in the calculation.\n\nI was previously convinced that this is a number: which is composed of any number, and of the number produced by multiplying it, through multiplication. The sides of which are to be investigated, we mark with signs above the thousandth place. Under this number, there are an equal number of parallel numbers: just as we instructed with the square number. When a primary number is sought out under the thousandth place, one that leaves nothing when subtracted from the number itself and from the number multiplied and produced, this number, if it can be taken away completely, should be inserted among the parallels. It should be multiplied once in itself and once in the multiplied number, and after the subtraction has been made, the remainder, if any, should be restored above the noted numbers from which the subtraction is made, so as not to confuse the reasoning: as much as possible.\n\nThe size of any primary number that, before being multiplied and then multiplied by the number produced, produces: let there be no hesitation about what follows:\n\n(Note: This text appears to be in Latin. It describes a method for finding common factors between two numbers. The text is not significantly corrupted, so no major cleaning is required. However, there are some minor errors and inconsistencies in the text that can be corrected for clarity. Here is a corrected version of the text:\n\nUnder this number, there are an equal number of parallel numbers: just as we instructed with the square number. When a primary number is sought out in the thousandth place, one that leaves nothing when subtracted from the number itself and from the number multiplied and produced, this number, if it can be taken away completely, should be inserted among the parallels. It should be multiplied once in itself and once in the multiplied number. After the subtraction has been made, the remainder, if any, should be restored above the noted numbers from which the subtraction is made, so as not to confuse the reasoning: as much as possible.\n\nThe size of any primary number that, before being multiplied and then multiplied by the number produced, produces: let there be no hesitation about what follows:\n\n(Note: The text uses the Latin words \"quemadmodum,\" \"de,\" \"sub,\" \"extremo,\" \"aliquis,\" \"semel,\" \"atque,\" \"iterum,\" \"multiplicatus,\" \"supra,\" \"reponatur,\" \"quantum,\" \"au,\" \"quiuis,\" and \"nequit.\" Here is a modern English translation of the text:\n\nUnder this number, there should be an equal number of parallel numbers, just as we instructed with the square number. When a primary number is found in the thousandth place, one that leaves nothing when subtracted from the number itself and from the number multiplied and produced, this number, if it can be taken away completely, should be inserted among the parallels. It should be multiplied once in itself and once in the multiplied number. After the subtraction has been made, the remainder, if any, should be restored above the noted numbers from which the subtraction was made, so as not to confuse the reasoning: as much as possible.\n\nThe size of any primary number that, before being multiplied and then multiplied by the number produced, produces: let there be no hesitation about what follows:\n\n(Note: The text uses the Latin words \"quemadmodum,\" \"de,\" \"sub,\" \"extremo,\" \"aliquis,\" \"semel,\" \"atque,\" \"iterum,\" \"multiplicatus,\" \"supra,\" \"reponatur,\" \"quantum,\" \"au,\" \"quiuis,\" and \"nequit.\" Here is a modern English translation of the text:\n\nUnder this number, there should be an equal number of parallel numbers, just as we instructed with the square number. When a primary number is found in the thousandth place, one that leaves nothing when subtracted from the number itself and from the number multiplied and produced, this number, if it can be taken away completely, should be inserted among the parallels. It should be multiplied once in itself and once in the multiplied number. After the subtraction has been made, the remainder, if any, should be restored above the noted numbers from which the subtraction was made, so as not to confuse the reasoning: as much as possible.\n\nThe size of any primary number that, before being multiplied and then multiplied by the number produced, produces: let there be no hesitation about what follows:\n\nUnder this number, there should be an equal number of parallel numbers, just as we instructed with the square number. When a primary number is found in the thousandth place, one that leaves nothing when subtracted from the number itself and from the number multiplied and produced, this number, if it can be taken away completely, should be inserted among the parallels. The text appears to be written in old Latin script, and it seems to be discussing the rules for representing numbers in multiplication. Here's the cleaned text:\n\nSemel Vnum Semel Bis Bina Bis Ter Terna Ter Quater Quaterna Quater Quinquies Quina Quinquies Sexies Sena Sexies Septies Septena Septies Octies Octona Octies Nouies Nouena Nouies Decies Dena Decies\n\nDeinde numerus ipse primarius triplicandus est. Et eius, quod triplum erit: prima nota sub proxima praeter una a dextra reponenda est: caeteraeque post eam, suum quoque ordine, a sinistra. Postea inquirendus est numerus aliquis primarius: qui una cum numero primario pnatu erit: prima nota sub proxima praeter una figura a dextra reponatur. Caeterum autem post haec, suum quoque ordine. Rursusque inquirandus est numerus aliquis primarius: qui una cum omnibus numeris primaris prius inter parallelas insertis ductus in toto triplum, posteaque sine illis solus in numerum ex multiplicatione natum ductus, quicquid supra triplum numerum repositum est: vel totum absorbeat: vel ex eo quam plurimum potest: deleat. If alone in a cube, one should remove the entire number placed next to oneself, if possible. Otherwise, one should take away as much as possible from it. After this, one must triple whatever is found between the parallels, and another number prior to this must be investigated. The rest of the process should be carried out in the same way, as previously stated. If nothing remains, the number whose root we have clothed is a cube. But if something is left, the number was not a cube. Furthermore, the root of the number is the greatest side of a cube in that number.\n\nHowever, in any investigation, no primary number can be found: which, before being inserted between the parallels, can be tripled and then multiplied alone to a number produced, resulting in a number greater than the one above it, so that it cannot be subdued by that circle inserted between the parallels in the number of partitions. From the omnibus of notices, which root we seek: keeping intact, dismissing the deleted ones under parallel triple notices, unless the work is not yet finished: make progress to the right beyond the next number point marked. The entire number between parallel lines is thus tripled: place the first triple note circularly under the figure next to it, except for one, beyond the number marked with a point to the right, and place the others to the left, each in their own order.\n\nEXAMPLES, which make this clear, are as follows:\n\nFrom two hundred and fifty thousand thousand thousand, three hundred and twenty thousand thousand, three hundred and fifty eight thousand two hundred, four hundred and sixty four thousand, we extract the cube root. After removing meaningless characters, here is the cleaned text:\n\n\"After the numbers have been written in order: two parallel lines will be drawn underneath: and every thousand's place will be marked with a sign. A primary number is to be sought under the last thousand, marked by a notable number: this number is a senarius. For six times six is sixty-one. Add sixty-one to two hundred and fifty. If you subtract thirty-four from two hundred and fifty, six remain. Therefore, six should be inserted between the parallels, and after multiplication and subtraction of the noted numbers, thirty-four remain. These should be noted as such: one above the last dot, and three to the left. Then, three should be placed between the parallels, and eighteen should be added. One should be placed to the right of the nearest note, except for one to the left.\" A discoverer of some kind, primary among numbers, who, along with the primary number, was led to a total of three when counted for a denarius beforehand, and then alone created a number in the sequence, subtracting the total of that number from its place, leaving only what follows: this will be the case. Afterwards, a note located three places to the right is presented. 63. When taken into 18, they create 1134. Then, alone, it is produced in 1134. 3402. Those marked in this way are arranged: the first note is placed under the first triple figure, with the others following in order after it. Those that remain above, when subtracted from the above-mentioned notes, remain. 50. Then, alone, cubed, they make 27. Seven are located under the nearest triple number, and two are to be placed to the left of them. After subtraction from the above-mentioned numbers, 476 remain. [63. Among parallel lines, there are three parts that arise. And among them, nine are noted, with the figure next to it: place it after that, standing to the left. Again, in a quire, there is a primary number, which, before the primary numbers previously introduced, appears in the entire tripling, and then alone in the number production, takes away the number placed above or the entire number subtracted as much as possible. This cannot be done in this place. For even if the smallest primary number, which is unity itself, were assumed for 63, making it 631 and so in tripling, 189 would be led, and afterwards only unity would be multiplied in the production, yielding 119,259, which far exceeds the numbers placed above it. 7,658. Therefore, subtraction cannot be made in this way: a circle is placed between parallel lines as a section.] With the given input text, it appears to be written in Latin with some errors. I will translate it to modern English and correct the errors as faithfully as possible to the original content.\n\nThe cleaned text is:\n\n\"For all numbers we seek whose root we inquire about: keeping the intact ones, dismissing the deleted ones and those under three parallel notations, place a mark beyond the next number, this is: 630. We repeat the tripling process. And it will become: 1890. We inquire about another primary number, which is one in the number 630. In the tripled number 1890, when it is left alone in the product and subtracted from the number above it, it will disappear: just as if it were cubed and taken from itself or from the next number, leaving whatever remains, if there is any: or as much as possible from it. This is: 4. For 6304 in the tripled number, we take the product. 11914560. Then, multiply 4 into the product. 47658240\" quae sunt supra se positae notis praeter postrema omnino aequales: atque eas subductae delent. Postea. 4. in se cubica ducta factus est. 64. quae totum, quod superest: sic aboluit: ut nihil penitus supersit. Numerus ita propositus liquet fuisse cubus.\n\nEt quoniam in cubi lateris investigatio post triplicationem alicuius primarii numeri facta, statim alius numerus primarius quaerendus est: qui una cum primario numero prius invento in toto triplum ducatur: deinde solus in productu, atque iterum solus in se cubice multiplicetur (quia numerus ille primarius tam varie considerandus est: primus, quid producit: quod pars est maioris numeri: iterum, quid solus profert multiplicatur in productu: rursus, quid cubice in se ductus procreare valet) paucos exercitati morantur haec cogitatio: quisnam numerus primarius tot officia praestare potest. Quare, ut in ceteris omnibus, ita in hac re plurimum iuvat exercitio: quod per assuetudinem, quantusquis ardua, prona reddit. Centerum conduced very well: if after placing the primary numbers in parallel, you multiply one of them before combining it with the other: add a circle to it; you can also try to make one or the other primary number itself be carried in triplets. Then, the number produced by this multiplication and addition can be added to the primary number from which it originated. For this product, the addition will bring forth a number that would have been born if the primary number we tried to carry in triplets had been placed among the primaries beforehand. In this way, when you see which number undergoes this multiplication and addition, you will easily make a conjecture, both from the number above and from the product and addend, whether it is larger or smaller, which primary number is most likely to be replaced by the circle. For example, in the given example:\n\nIf you consider which number undergoes this multiplication and addition, and which number is the product and addend, you will easily make a conjecture about which primary number is most likely to be replaced by the circle. post primas sixteen numbers between parallels were positioned with a triplication. 18. If one assumes a circle: the product will be 1080. Then let us try: if three are added to the eighteen ducats, what will result, and will they add up to 1080? If they do, the number thus produced will be similar: if sixty-three had been in eighteen ducats from the beginning. Afterwards, if three are added to the number 1134 that results from the product, the number will be 3402. Those numbers will respond appropriately to the numbers above them, and yet there will be some left over: so that three can be cubed afterwards. However, if we were to attempt this using a binary system, the number above to be placed would be too redundant. If the number produced by four or five were larger: it would not be able to be subdued by the numbers above it. It will be much easier, however, if after placing a number between the parallels, a circle is added, starting with the number five, which is among the primaries, in the same way as mentioned above. In this way, you will immediately know whether a larger or smaller primary number is required. Quoties post triplicatione numerus quisquam primarius inquirus est, qui ea officia prestet, quae paulo ante sunt dicta. This method will suffice to explain the example: to give an investigation into the side of a cube. I will add another matter to be considered by many notaries.\n\nEXPLICITVS EST LIBER PRIMUS SUPPATIONEM DOCENS INTEGRORUM. SECVNDUS SEQUITUR, DE PARTIVM SUPPVTATIONE.\n\nMany who are engaged in supposing integers abandon them after they have reached numbered parts that are not obvious: they cast aside the volumes; I think for no other reason than that the enumeration of parts is not as expeditious as that of integers. If they discarded the secondary, they would direct their minds to all things, which appear arduous due to corrupted minds, and would find a way to rediscover them willingly. Nam ut integrorum numeratio pene ignorationem habet, qui sensum communem tantum comprehendit et discere eam desiderat, sicqua de partibus numerandis traduntur, ut acutam mentis aciem non modo requirunt, sed et hominem nec dormientem nec stultum, et cuius animus inter legenda minime peregrinet. Et quavis haec non, sicut Aesopi fabella, cum quadam voluptate penetrent intellectum, propterea tamen studiosis nequaquam est cessandum. Cogitent quemadmodum pulcherrimis quibusdifficilitates praetextuit natura, quae nihil, quod est magnum, cito prehendere voluit. Simulque secum reputent, quantus in tota vita pro tantillo studio percipietur fructus. Nam quis (quaero) mortalium vitam sic potest transire, ut non sit ei frequenter habenda supputatio. In qua labi et decipi, praeter damnum sum, ridiculum putatur. Verumtamen hanc, quam nunc aggredimus, partium supputationem non magno acume egere, vel hinc licet cognoscere: quod mercatores in hac nihil cedunt philosophis. et nescio an longe superseded. The more industrious, not the more ingenious, that requires it. We have deemed it necessary to set that aside. Whoever is unable to keep the whole entire, let him not suppose he can learn the parts. For he who, trusting too much to his ingenuity, attempts it in reverse order, will appear to do something foolish - like one who, ignorant of letters, attempts to read them. Therefore, all those should be kept ready beforehand: not only the parts, but also the tools and oil.\n\nEvery whole can be dissolved into parts, as many as you wish: it can be solved by the intellect. And just as the counting of wholes begins with one and can be extended to infinity, so the division of wholes is ordered into parts. (For nature does not allow things to be dissolved into fewer parts than two.) However, the judgment of the separator is extended indefinitely. The parts themselves are solved into innumerable other parts by the intellect. Particularly, other fragments can be thought of: if the matter does not reach its end: if one wishes to examine the details. To such parts, two things are required in number. The first, who numbers the parts, is called the numerator by arithmeticians, who represents to us the number of units in the parts dissected. The second, who assigns names to the parts, is called the denominator: because it shows in how many parts each unit is divided. For example, three quarters. These are the parts that make up a whole, divided into four. They are written in this way: the numerator above a short line drawn through it, the denominator below it, in this manner. \u2154 \u00be \u2158 The numerator is always pronounced before the denominator: as we say, two thirds, three quarters, four fifths.\n\nHowever, these simple fragments will be called the parts into which the whole is solved. Some fragments have parts of their own. quas minimas Arithmeticae partes vocare solent. These are the parts, which in turn are divided into smaller parts. In this book, as in the parts of wholes, we will sometimes call simple parts, fragments, and even minutiae indiscriminately: We will also call the parts of parts, at times the minutiae of minutiae, not only the fragments of fragments. Among these species, there are many that matter in writing, speaking, and in many other areas. For what are simple minutiae: they are written as a brief line above the numerator, below the denominator. As we have said. And if several simple fragments need to be written: they are distinguished by an interval. Thus, for example, \u00bd, \u2154, \u00be have one numerator and one denominator singular: and they are always enunciated in the direct case. As we have said, one second, two thirds, three fourths. However, what are the parts of parts: they take two, or sometimes more numerators and denominators. Quarum aliae ad sinistram ponendae recto enunciantur: hic sunt ipsae particulae. Aliae ad dextram locandae, obliquo enunciatae, mediocre linea carent. Hic sunt partes: quarum sunt particulae. Si duas tertias unius quartae notis scribere uelis, hoc modum facias: \u2154.\n\nId autem significat: duae tertiae partes unius partis quartae ab aliquo integro dissectae: quod in quatuor partitum est. Vel si tres quartas unius tertiae unius secundae signare cupis: hoc modum notas. \u2154 \u00bc\n\nQuod exprimit tres quartas partes unius partis tertiae alicuius secundae ab integro dissectae: quod in duo divisum est.\n\nDum fragmentorum fragmenta ratiocinator tractat, summe precetur: ne dormitati obrepat obliuio: quarum partium sint particulae. Ne pro dissectis integris partibus putet fragmenta esse. Nam si id committat: error calculi non mediocris sequetur. In all dissections, the smaller the denominator, the smaller will be the parts, and the farther they will be from the whole. The smaller the denominator, the larger will be the parts. And the closer it approaches the whole. For the second parts, the second are larger than the third, and the third larger than the fourth, and the fourth larger than the fifth, and so on. Regarding the numbering of the parts, three things should be observed in full. First, when the numerator and denominator are equal, then the parts form a single whole. For example, 3/3, 5/5, 7/7, three thirds make one whole. The same for five fifths, seven sevenths, and so on. Second, when the numerator is greater than the denominator, the number of units by which the numerator exceeds the denominator is the number of parts that are greater: more than those that make up a whole. For example, nine eighths and one eighth make up ten eighths. septem quinti integrum et duas quintas. six quarti integrum et duas quartas. 9/8 7/5 6/4. Thirdly, if the numerator is greater than the denominator:\nhow many units the numerator is less than the denominator: are that many parts missing to make it whole. Two thirds and three quarters are two parts of some whole, of which one third is missing to make it whole. The same holds true for other similar cases.\nMoreover, an increase in the denominator's quantity may result in fewer parts, but more in number. Conversely, the reverse is true for the numerator's increment: it generates parts that are much closer to the integers.\nFurthermore, since the product of all the integers among themselves exceeds their quantity and number in magnitude: the product number is never reached. Conversely, in the parts, it happens that the smaller the multiplication of the quantity, the more numerous the parts become. When dealing with disparate parts that are larger and significantly different from one another, fragments frequently occur: sometimes simple fragments are combined with larger ones, not only intermixed with integral parts but also not allowing such varied chaos to disturb the rational order. To reduce such diverse elements to some semblance of unity, it is necessary to add, subtract, multiply, or divide as required by anyone approaching the task. This will ensure that the work is both proportionate and manageable.\n\nHowever, if someone were to attempt to mix such disparate elements: they would find themselves in an inextricable labyrinth. Therefore, before anyone attempts to do so, these guidelines for reducing disparate elements into a unified form should be followed: first, one should take care to keep them.\n\nIf, on the other hand, one wishes to transform larger parts into smaller ones or vice versa, any parts can be transformed into whatever form is desired. If parts are to be divided, we transform the numerator into the denominator in whatever parts are to be divided. From this number, let us create parts, which the mind transforms: let us divide. This will be done: so that the number of the parts consumes them, becoming their transformation. This alone will suffice for the learned. However, for the sake of a rudimentary understanding, we will explain both aspects. Thus, the larger parts are transformed into any smaller ones: if the denominator of the larger parts is made the denominator of the smaller ones, and a number is born from this, through the larger denominator. Once this is accomplished, the number of the partition will reveal: how many minor parts are born from the transformation of the larger ones. For instance, if you wish to transform three quarters into eighths: through three, the denominator of the larger parts, we obtain eight, the denominators of the smaller parts being multiples. These are then divided by four, according to the larger denominator, and the number of the partition will be six. Therefore, from three quarters reduced to eighths, six eighths emerge. If you want twelve thirty-thirds to be transformed into fifths: take twelve, the counter of major parts, and place it among the five, the denominator of minors; and sixty are born. Divide three of the major denominators, and the number of sections will be twenty. Thus, twenty-fifths are generated from twelve thirty-thirds. If you wish to reduce three fifths to sevenths: multiply seven, the denominator of minors, by the major denominators three times. And you will produce twenty-one. Then divide by five, the major denominator, and the number of partitions will show four. One will remain as a remainder, which signifies one fifth of one seventh. Thus, from three fifths reduced to sevenths, four sevenths and the fifth part of one seventh are formed. For whenever a remainder occurs in the reduction of this kind: it will be a particle; take the denomination in the nominative case from the major denominator who acted as the divisor; but take the other in the oblique case from the denominator of minors. When minor parts are converted into any major parts: when the numerator of the minor is multiplied by the major's denominator, and the resulting number is divided by the denominator of the minor. Then the number of the partition is clearly indicated: how many major parts arise from the transformation of the minors. For example, if one wants to change six nonas into tertias: through the numerator of six minor parts, three, the denominator of the major is multiplied. And it creates ten and eight. Which, afterwards, is divided by nine, the denominator of the minors. And two will be left in the number of sections. Thus, from six nonas changed into tertias, two tertias are generated. And if twelve sextas are to be changed into quartas: twelve, the numerator of the minor parts, is led to four, the denominator of the majors. And they are generated. Likewise, after being divided by six, the denominator of the minors, eight will be in the number of the partition. Thus, twelve sextas are reduced to eight quartas. If one wants to reduce seven octaves to quintas: take seven, the numerator of smaller parts, into five, the denominator of larger parts: and he will make thirty-five. Then divide this by eight, the denominator of smaller parts: and the number of partitions will be four. The remainder will be left over. Three octaves of one quint. Therefore, from seven octaves changed into quintas, four quintas are produced, and three octaves of one quint. For in such a reduction, whenever a fragment of a fragment occurs: the denomination in the direct sense is taken from the denominator of the smaller fragment, through which the section is made; and the denomination in the oblique sense is taken from the denominator of the larger fragment.\n\nThe parts of different denominations are reduced to the same denominations: when the denominator of one fragment is multiplied by the denominator of another. The resulting number will be the common denominator. For example, if you want to reduce two thirds and three fourths to the same denomination: three, the denominator of the first fragment, is multiplied by four, the denominator of the second. et enascentur duodecim, denominator communis. If you want to know: how many twelfth parts are separately in two thirds, take two as the numerator of the first fragment, and four as the denominator of the following one; eight will be produced as numerators. These twelfth parts can be found indicated under the line in two thirds. Similarly, if you want to know: how many twelfth parts are separately in three fourths, take three as the numerator of the second fragment, and three as the denominator of the first multiples; nine will rise as numerators. These twelfth parts can be found indicated under the line in three fourths.\n\nFurthermore, if you want to investigate the common denominator itself: the numerator of the first fragment is in the denominator of the following one, and similarly the numerator of the second fragment is in the denominator of the first, to form the cross of St. Andrew, multiply these numbers. The numbers resulting from these two multiplications should be added together. If a number arises from the sum, the numerator of the community will be that number. For instance, if two are the numerators of the first fragment, and four the denominator of the second, there will be eight. And if three are the numerators of the second fragment, and three the denominator of the first, there will be nine. Added to eight, this makes fifteen, which will be the numerator of the community. Two thirds and three fourths, reduced to the same denomination, have the community as fifteen, but a denominator of twelve.\n\nIf there were more fragments: for example, two thirds, three fourths, and four fifths, after the two initial fragments, as we said, the denominator of the community is again investigated through the denominator of the third fragment, and multiplied by it: and there will be sixty, the common denominator of all.\n\nIF YOU WANT TO KNOW: how many parts sixty is in any given fragment, multiply the numerator of that fragment by sixty, and divide the result by the common denominator. ita finds in the thirds forty-six, and in the fourths forty-five and 2/3 40/60 simas. And in the fifths forty-eight, three-quarters 45/60 simas.\nThen the investigator of all fragments searches for the common denominator of the reducers first, reducing the denominators of the fragments. He multiplies the numerators obtained from the first multiplication by those obtained from the second multiplication. The common denominator of all fragments' numerator will then emerge. For instance, if we reduce the communis numerator of the reducers to five, using the third fragment's denominator: we then produce eighty-five, and again twelve, the communis denominator of the reducers, in the fourth, the numerator of the third fragment: thus we generate forty-eight. Combining these two numbers, we get one hundred thirty-three. This number will be the common denominator of all fragments. Two thirds, three fourths, and four fifths, reduced to a common numerator and denominator, make one hundred thirty-three and sixty-sixths. The same method should be followed: even if there are more than thirteen and two-thirds fragments to be reduced to the same denomination: let the next sequence be resolved in the same way until all fragments have been dealt with.\n\nFrom the above it is clear that the number can be found immediately, which ever parts we choose, provided they are denominated. For if we multiply all denominated parts among themselves, the number produced will contain them. For instance, if you want to find a certain number, let it be one second, one third, one fourth, one fifth, one sixth, one seventh. Multiply the denominations of all these parts by half, one-third, one-fourth, one-fifth, one-sixth, one-seventh. This will yield five thousand four hundred and forty. This is the number sought. In such numbers, individual counters can contain as many parts as there are in the denominators. You will find one first, two seconds, three thirds, four fourths, five fifths, six sixths, seven seventh parts. Or if you prefer, you will find two seconds, three thirds, half, two thirds, three quarters, four fourths, five fifths, six sixths, seven seventh parts. However, the parts of one and the same number 2/2, 3/3, 4/4, 5/5, 6/6, 7/7 cannot contain more in the numerator than in the denominator. For this reason, that number would be contrary to nature. For example, if you seek a number that has five fourths: investigate the fraction. Since five fourths consist of one complete fourth and an additional fourth part, no number can be larger than a fourth part of itself. Minimus autem numerus, qui partes quascquae vulis: denominatas habeat, quemadmodum sit investigandus, paulo post discimus, quando tradimus, quomodo partes ad minima sua nomenclaturam redigantur.\n\nPartes particulae, quae fragmentorum sunt fragmenta: et minimarae minimiae etiam nuncupantur: in integrorum partes, quas simplices vocant minimas: mutantur. Si earum numerators inter se multiplicetur, ut unus omnium communis numerator fiat. Atque idem earum denominators in se ducantur, ut unus omnium communis denominator nascatur. Ideo, si ad eandem denominationem reducere cupias duas tertias unius quartae, unius secundae, \u2154 \u00bc \u00bd unitatibus in duo ductis, tantum duo surgunt. Cum unitas alios numeros non multiplicet. Sic originetur numerator communis. Deinde, tria in quattuor ducta creant duodecim. Quae iterum in duo multiplicata producunt viginti quattuor. Ita complectitur denominator communis. Quamobrem duae tertiae unius quartae unius secundae, ad simplicia fragmenta redactae, procreant duas vigesimas quartas unius integri. If we want to divide an integral into any parts: they are broken down: if the number of the parts we want to create is a factor of the integral, it should be multiplied. For instance, if we want to reduce an integral to sixths: we multiply unity by six: and six will result. This number will represent the parts. If we want to resolve three integrals into sixths: six should be taken in place of each integral. Thus, there will be ten and eight. The number that exists as parts, when resolved into sixths with three integrals.\n\nIf, however, one integral occurs among the parts we want to reduce: those parts that we want to reduce to the denomination of the integral: we multiply the number of the parts by the integral, and to the number created we add the denominator. Then, with the increasing number, we assume the same denominator, with a line interposed. For example:\n\n(Example) If we want to reduce several complete and three-quarters to the same denomination: first, the symbol for completes should be written. Then, before it, the fragment itself, with a short line inserted between. In this way, it would be written as 5 3/4. Next, we multiply the number of completes by the multiple denominator of the fragment: and twenty will result. With three added, the common denominator will be twenty-three. We place the denominator of four under the middle line, and thus twenty-three quarters will emerge from this reduction.\n\nIf one wants to reduce several simple fragments into one simple fragment: first, the number of completes and the first fragment should be determined. Then, the simple fragment obtained from this and the following ones, by considering each fragment separately, should be transformed into the same denomination, in the same way as previously stated. If genuine parts are found to be integrated and identical, they should be reduced to simple fragments for reduction. The first part of each fragment should be reduced to simple fragments first, as instructed before. Only then can they be joined together as one.\n\nThe same applies to all cases: if genuine and multiple fragment parts present themselves, first reduce them to simple fragments. After that, the remaining parts will follow as previously stated.\n\nParts reversed become whole: if we divide the number of these parts by their common denominator. For instance, if ten and eight thirds are to be reduced to wholes, we divide ten and eight by three. The number of partitions will then yield six. Similarly, if twenty-two fourths are to be created as wholes, we divide twenty-two by four. In the number of partitions, five will be found, and two remain. Therefore, five are wholes, and two fourths remain. In every division, whether of whole parts or remainders, the latter, if any, takes the name of the divider by whom the section was made. The remainder itself, which cannot be made whole, has the same proportion to the whole as the numerator to the denominator, which was the divider. Therefore, the whole remainder, which cannot be further divided, is divided by half the divisor, if it can be halved. And the number of the section shows the integers. For example, in the case of two quarters remaining, a section made by two parts of the divisor, since the number of the section is one, one half shows the remaining one. But if this cannot be done, the remainder is divided by a third part of the divisor, provided it remains. And the number of the partition indicates either one third or more than one third remaining. For instance, if it had been divided by nine, and six remained, by a third part of the divisor, that is, three, the section would show two thirds of one integer remaining. If the third section does not quadruple: through the fourth part: unless it remains: the rest will be cut. For this reason, the number of partitions will reveal: whether one fourth remains: or more. For instance, if sixteen were divided: the remaining parts would be twelve: the fourth part of the viewer's share, taken from the rest, produces three parts in the partition: each of which represents a whole fourth. If this does not work, try with the fifth, sixth, or any other divisor's part. Furthermore, if nothing helps: reduce the remaining parts to their minimum names, for those who follow: in these ways.\n\nREMAINING PARTS, WHICH CANNOT FORM A SINGLE INTEGRAL ONE: ARE REDUCED TO THEIR MINIMUM NAMES, AS FOLLOWS.\n\nThe numerator and denominator must be divided as many times as possible. In each half-division, a line should be inserted: the numerator above it: the denominator below. The last half-division should be noted. The maximum common number that two numbers mutually count is investigative, which of them, through division, can be exhausted. The maximum common number of two numbers is investigated in the following way. Subtract the smaller number from the larger number after their division has ceased as often as possible. From this subtraction, repeated frequently, if one of them is reduced to a unit, such numbers are called contrary numbers by mathematicians and cannot be reduced to a lower name. All numbers contrary to themselves are minimal. For example, let us reduce the remaining twenty-five thousand five hundred to their minimum name. The diminished part leaves thirteen. If the numbers are frequently subtracted from each other, two people commonly count them in this proportion. An example is prepared. If we want to reduce three hundred and forty-four thirty-second parts to the minimum denomination. Immediately, from the halves of fifteen and seven first parts, the odd numbers and those that are the first appear. Then, from the first number's subtraction from the denominator, thirty-four remain in the denominator. After that, another subtraction leaves three and seven in the denominator. The number that is the maximum common denominator for both numbers is shown by the similarity of the number of the numerator to that of the denominator. Therefore, this number, three and seven, is the one that counts the maximum for both: therefore,\n\nAnother example can be given: if the remaining thirty-four one hundred and two thirty-thirds are to be reduced to the minimum denomination. Fourteen and twenty-eight remain from the middle of these thirty-four. And twenty and eight make fourteen. seven times fourteen from a quarter are generated. It takes away the counter. For every number can count and subtract itself. Again, seven taken away twice also subtract the denominator. A number is thus partitioned in such a way that it clearly indicates seven quarters to one second. 7/14 \u00bd. Therefore, since the first one follows the same rule: for its own part it reduces to one second. Fourteen twenty-eighths, therefore, are reduced to one second: which is the smallest denomination. 14/28 \u00bd\n\nThe smallest number, which has parts whatever we choose: is to be sought in this way. After we have investigated the parts which we designate in some number: the number which has these parts united is easily produced from the multiplication of denominators, as we mentioned above. However, the smallest number which can contain these parts: demands diligence. If all parts of those minimal quantities are discovered: who among them has the smallest number, which is the one that all others count. Therefore, it is first necessary to investigate which is the smallest number among the primary denominations, and at the beginning to inquire: which is the smallest number counted from the two primary denominations.\n\nIndeed, if they are contrary to each other: so that no number counts them beyond unity. Because they are in their proportion minimal: which, being drawn from one to the other, will be the smallest number counted from them. If another number besides unity counts them: let parts be taken in proportion to the minima. Which, as we showed in the next chapter, are to be subtracted under the parts: in what respect they are minimal.\n\nThen those larger parts, in respect to which one is smaller: through numbers in their proportion, the smaller is multiplied by the larger, or the larger by the smaller, will produce the minimum from themselves.  Nam secundum EVCLIDIS scitu\u0304, Quilibet duo numeri minimos nu\u2223meros su\u0119 {pro}portionis, maior minorem, aut minor maio\u2223rem, multiplica\u0304tes, minimu\u0304 ab ipsis numeratum produ\u2223cunt. Qui numerus minimus a duabus primis denomina\u2223tionibus numeratus, ad hunc modum inuentus, cum ter\u2223tia denominatione statim co\u0304ferendus est: minimus{que} nu\u2223merus ab illis numeratus ad eundem modu\u0304 exquirendus. Is, post{quam} cognitus erit: cum quarta denominatione simi\u2223liter conferatur: minimus{que} numerus ab illis numeratus itidem eruatur. Idem{que} indagandi modus per omnes de\u2223nominationes: si quae ulteriores fuerint: seruetur. Mini\u2223mus aute\u0304 numerus ab illis numeratus, qui postremo con\u2223ferentur: partes omnes propositas minima proportione capiet. Exempla demus: quae rem magis illustrent. At{que} inuestigemus minimum numerum, qui unam secundam, unam tertiam, unam quartam, unam quintam, unam sex\u2223tam, at{que} unam septimam habeat\u25aa \u00bd \u2153 \u00bc \u2155 \u2159 1/7\nIN PRIMISQVE exquiramus, quis sit minimus numerus: quem duae primae denominationes numerant Two become three, and six emerge. The smallest number is the one that two and three hold. Let us consider six with a third denomination, that is, with quaternary. Since the binary number common to both, we find it twice in four, and three in six. Three and two are numbers in the same proportion as one to another: three as part of six, and two as part of four.\n\nNext, let us note that the greater number, the larger of the two numbers we have noted in a binary proportion, is twelve. We will create it by multiplying the smaller by the larger, which is two by three. With the fourth denomination, that is, with quinary, we will compare twelve. Since the first numbers are contrary to each other and in their smallest proportion, we will bring five into twelve. Sixty will be created.\n\nThe smallest number is the one that six and four hold. The one that presents one second, one third, and one fourth.\n\nWe will again compare twelve with the fourth denomination, that is, with quinary. Since the first numbers are contrary to each other and in their smallest proportion, we will bring five into twelve. Sixty will be created. qui minimus est quemque et duodecim producunt: qui una secundam, unam tertiam, unam quartam, et unam quintam. Deinde sexaginta et senarius numero: quae proximus est, conferamus. Senarius et ipsum per unitatem, sexaginta per denariis numerat. Decem et unum, qui minimi numeri sunt, subsignentur. Si sexaginta maiorem numerum per unum minus est, an sex minorem per decem, numerum ex subscriptis maiorem multiplicemus: sexaginta iterum producunt. Minimus numerus a sexaginta et sex est, et qui una secundam, tertiam, quartam, quintam, atque sextam capiat. Demus sexaginta septem. Quorum primi contrarii, atque in minima proportione coperti sunt.\n\nErgo septingentas ducentas quadringentas producimus.  qui numerus minimus est: que\u0304 sex\u2223aginta et septem numerant: qui{que} unam secundam, unam tertiam, unam quartam, unam quintam, unam sextam, et unam septima\u0304 proferre potest. Hac ratione numerus par\u2223tes, quascum{que} uolumus: denominatas complectens, ex\u2223quiritur. Quinetia\u0304 si omnes hi denominatores in se mul\u2223tiplicentur: fient quin{que} millia et quadraginta. qui nume\u2223rus itidem omnes eas denominationes habet. Caeterum numerus eas, minima proportione, capiens is est: quem exquisiuimus. nempe quadringenta uiginti. qui numerus in illo duodecies continetur. quemadmodum ex illius se\u2223ctione per hunc facienda, cuiuis licet cernere. Alia itidem huius generis exempla, qu\u0119cu\u0304{que} occurrunt: ad hunc mo\u2223dum explicari possunt.\nMODVS EXQVIRENDI PRECIVM QVA\u2223rumuis partium hic est attingendus If we divide the entire quantity into its parts, which remain: they occur so frequently that they are almost unavoidable: so that no estimation may delay one engaged in investing: for what sum of money is sufficient: we have deemed it necessary to show the way in which anything can be obtained. This is the method. If, through a numerator, the price of the entire quantity is multiplied: and the number is divided by a divisor containing three: the number of the parts will be indicated. However, whatever remains from that section, let it be noted above the dividing line. For example, let us explain more clearly:\n\nIf twenty barrels of wine cost twenty gold coins each, and we want to know the price of three barrels and a third:\n\nTwenty, multiplied by a partium numerator, who contains two: is multiplied, making it forty.\nWhich, divided by a divisor containing three: will yield the number of parts. Thirteen and a third gold coins. et superest una tertia partis unius aurei. Quae cum precio sit quaestionis: nos cohsideremus: qua pecunia viliori potest aestimari, ut iam dicemus.\n\nThe estimation of any given quantity of money is determined in this way. If the total sum of money, of which a part is what remains: in argentum libri per divisione quaestionemus: quot aureis? quot solidis? an quid libet: quot denariis? aut quot nummis sestertijis? aut quot semissis? aut quot quadrantes? aut quot sextantes argenti libra aestimatur? An si pecunia prius in aureis per sectionem investigata fuerit: exquiramus: pro quibus denariis? an quid libet: pro quibus nummis sestertijis? aut pro quibus obolis? aut pro quibus quadrantes? an si quod aes signatum uilius in usu est: aureus integrous valet?\n\nDeinde aestimationem totius illius partis, cuius pars ipsum reliquum est: per reliquos numeratoribus multiplicemus. Numerumque partitionis aestimatione reliqua monstrabit. Iterum secundum reliquum, si quod prodibit: notetur. Et ulterius, si ita libet: ad eundem modum procedatur.\n\nExample repeated: that which we gave next. In which were bought four tertiarii of one denarius for twenty gold coins, thirteen gold coins were found. And the remainder was one third of one gold coin. Quid afferat, ut sciamus:\n\nOne gold coin we estimate to be one hundred denarii. Which, when divided by three, leave a denominator of thirty-three denarii in the partition. And there is still one third of one denarius. Which you can estimate in the same way with sextans.\n\nTherefore, the price of four tertiarii of one denarius, bought for twenty gold coins, is thirteen gold coins, thirty-three denarii, and one third of one denarius.\n\nThus, it is in your discretion to compute: whether you wish to inquire about the money in a certain order, first finding out how many silver pounds the sum of money has. Then, if anything remains, how many of those are gold coins. If anything remains: how many denarii it creates.\nThirdly, if anything is left: how many coins it produces: and so on. Or perhaps he wants to descend at once to the lowest denomination, in order to estimate the remainder. But whichever way he chooses: the method of inquiry is the same.\n\nWhen particles occur in the sections of integrals, the price of which we wish to determine, we must reduce them first to the simple parts of the integrals. Then we investigate the price in the same way, as we have already said about simple fragments.\n\nThe addition of a part is the sum of numerous small parts taken separately. But when the parts are of the same denomination:\nit has no business. For the numerator alone is the addition, and the number produced above a short line is noted: under which the denominator is added. For example, if you wish to add four septims, five septims, and six septims: only the numerators, when joined by addition, make fifteen septims. When parts have been labeled differently: if minutes have been involved: after they have been brought back to a common denomination, the larger numerator is divided by the smaller denominator, and the smaller numerator is divided by the larger denominator, to obtain the ratios in their simplest form. The tenth Greek letter: which represents the sign of the cross of St. Andrew: must be multiplied. The resulting numbers should be added together and the common numerator should be the sum. And if there are further minor differences with different denominators: the first ones, as we have said, should be dealt with first: the following ones should be joined in a similar way: until all are resolved: in the same way as mentioned above: when reducing dissimilar denominators. However, when dealing with the addition of common numerators, we should also address this: lest anyone be confused: when we investigate the common denominators. Therefore, we will present examples there, and here we will merely observe this. \u2154 and \u00be have three thirds and four fourths as a common denominator, twelve. numerator verum communem, decem et septem: ex additione, quae multiplicatione ad speciem crucis facta sequitur. 17/12, quibus additae \u2154 quatuor quintae procreabunt 133/60 centum triginta tres sexagesimas.\n\nWhen minor parts must be added: if one minor part is to be added: note it after the whole parts. For example, if 31/37 are to be added to 12, note them as 12 31/37. {QUOD} If more minor parts and whole parts are to be added, first combine the whole parts into one sum. Then, the minor parts, added in the same way as above, should be added to this sum. For example, if \u2154 \u00be \u2158 are to be added to 30 and 16, first combine 30 and 16 into 46. Then, the minor parts, collected through addition, should be added to this sum. 2 13/60, which are to be added to the sum of the wholes, will make 48 13/60.\n\nIf minor parts are to be added to whole and minor parts: first, the minor parts should be added, in the same way as above. Then, whatever sum arises from them should be added to the whole parts. And if any minor parts remain, note them after the whole sum. uluti si addendae sunt ad 7 2/3. Prius copulentur cu2/3. Quae faciunt 1 5/24. Eaungantur ad 7. Et universorum summa fit 8 5/24.\n\nThis method has much less labor: {than} if someone were to break down the integral parts into these details and make one body from them afterwards. They do this in the following way. Integral parts are multiplied through a common divisor, and the number of parts produced is added as a denominator. And thus the numerator is born as a common term. To which no denominator is changed below the line. For example, given the following: 7. Integral parts in the divisor. 8. They bring: and the number 56 is produced. 3. Add 59/8. They join them afterwards in the same way, and the matter returns to the same state.\n\nWhen integral parts and details must be added: first, integral parts are joined. Then, details are added. For instance, if 6 and 3 1/5 must be joined, first 6 is joined to 3, and then details are added to them, and the total becomes 9 1/5. If there are multiple whole, complete parts that need to be connected together: first, gather all the parts into one body in the way that has been stated. Then, connect the whole parts to one another. And the total will appear.\n\nIf whole parts and small parts need to be added to whole parts: first, gather all the wholes into one sum. Then, add the small parts in the way that has been stated: for example, if 3 \u00bd and 6 \u2153 need to be added, first add 3 \u00bd, and then add \u00bd and \u2153 together, making it 9 \u215a. This should be added to the total. The same method should be used if there are many whole parts and many small parts: first, gather all the wholes, then add the small parts in the way that has been stated. If wholes, small parts, or both emerge: add them to the earlier parts.\n\nMoreover, regarding the connection of small parts, small parts connect to simpler small parts first. Then, in each individual one, the addition should be carried out according to what has been stated above. Partium subductio est summae partium minoris a maiore subtractio: per quam relictus partium illarum numerus apparet. It is necessary, however, to subtract either a smaller sum from a larger one or an equal sum from an equal one. A larger sum cannot be subdued by a smaller one. With two such small quantities, and each numbered by a different denominator, these quantities, multiplied, will appear larger: the numerator carried over to the denominator of the larger quantity will yield a greater number.\n\nIf the two small quantities have the same denominator, and you wish to subtract one from the other: if they are unequal, subtract the smaller numerator from the numerator of the larger one; and place the remainder above the denominator, with a line interposed. For example, if you subtract two thirds from seven thirds: five thirds will remain. \u2154 from 7/3 = 5/3.\n\nHowever, where numerators and denominators are found to be equal: the subtraction will yield nothing. When there are various denominators, the denominators are multiplied first, and the common denominator will rise. Then, the numerator of the former minor terms is made the numerator of the latter terms, and the numerator of the latter terms is made the denominator of the former terms, and it is brought to the form of a cross. When this is done, if the numbers produced are unequal, the smaller one is subtracted from the larger one, and the remainder is placed above the common denominator. For example, if someone wants to subtract two thirds from five fifths, the common denominator is first produced by the denominators themselves. Then, five is multiplied by the numerator of the other terms, and the denominator of the other terms is produced, creating twelve. Again, the numerator of the other terms is multiplied by five, the denominator of the other terms is the denominator of the former terms, creating ten, the numerator of the former terms. These, which are fewer than twelve, are subtracted from these, and two remain, which should be placed above the common denominator beforehand. If we subtract one, there will be two remaining seventeenths. Two-thirds are subtracted, leaving 2/15. When details are subtracted from wholes: they are sufficient to be subtracted from one whole, in details resolved, the remainder of the subtraction will be the whole. For example, if 4/7 is to be subtracted from 12, if we add and subtract: 12 is taken away, and 11 remains. From that, if 4/7 is taken away, 3/7 remains. When these are combined with 11, 11 3/7 remains. Only a small part remains: if 4/7 is subtracted from 12. Others handle integrals as parts, below the line they join them to integers for elimination. Therefore, as if parts were to be subtracted from parts: by multiplying the oblique numerators in the denominators, they subtract the smaller product from the larger: and above the line they note the denominator. Thus, 7 in 12 is multiplied, making 84. And 4 in 1, making 4. Subtracting 4 from 84 leaves 80/7. If reduced to integers, they will be 11 3/7. Thus, the matter will reduce to the same thing. If smaller problems are present in the text, here is the cleaned version:\n\nFour sevenths times eight firsts result in eighty-sevenths: it can be made so that what should be subdued are the smaller ones, and they are added to the wholes. This is because, if this happens, then the smaller ones are subdued by the larger ones, and the wholes remain intact. Whatever remains after this, will be the remainder of the subduction. For instance, if four ninths should be subdued, six ninths are to be subtracted from them, and they will remain forty-firsts of ninety. If six wholes are added to these forty-firsts, they will be left over. Six forty-firsts.\n\nIf the parts to be subdued are larger than those that are added, then at least one part from the larger ones is to be dissolved, so that one body is formed from these parts, and the subduction can take place. For instance, if five and a half are to be subtracted, one is to be taken from these, and it is to be subtracted in seconds. This will make three-halves from them, and seven-tenths remain. If these seven-tenths are joined with the remaining seven-tenths, they will remain as seven-tenths. Only seven-tenths remain if five and a half are subtracted. When whole parts are to be subtracted from the whole and smaller parts, what remains is the residue of the subtraction. For instance, if 9 a 13 \u2157 are to be subtracted, they are removed, and 9 a 13 remain. Likewise, if one of the smaller parts, 4 \u2157, is subtracted from 9 a 13, the remaining total is 4 \u2157.\n\nIf whole parts and smaller parts must be subtracted from whole parts and smaller parts, then each whole part should be resolved into smaller parts attached to it, so that one face of the smaller parts consists on both sides. After subtraction, the parts are collected again, as what remains. For example, if 3 \u2156 a 4 \u2154 are to be subtracted, 3 is resolved into fifths. quibus adiungatur minutiarum numerator. 2. et surgent 17/\nBut the parts of the minutiae are interconnected: first reduce them to simple minutiae. Then subtract in each, according to what was said above.\n\nIf you want to check: add what was subtracted: to what remains, subtract what subtracted, and restore the sum, from which the subtraction was made. If more or less remains, there is an error. Subtraction proves addition, and addition proves subtraction, as in integers. For example, if one-third is subtracted from one-half, one-sixth remains. Add one-sixth: what remains is to one-third: what was subtracted from. And one-half, from which the subtraction was made, is restored.\n\nPartition multiplication is either from the mutual interaction of parts, or from wholes to parts, or from parts to a whole, or the creation of a new fragment: through which many and diverse kinds, each taken from one and applied to another, come together in one: and from dissimilar parts, a single fragment is formed. ILLUD ATQUE ATEM more readily should be understood concerning the multiplication of many. For integrals among themselves, multiplication increases both in number and in the quantity of the sum produced: that which is generated. Conversely, in the parts, fewer multiplicands, numbering more, are found.\n\nSimilarly, the multiplication of fragments will be done in this way. First, let all the numerators be multiplied among themselves. And from this, the common numerator of all will be born. Then, let all the denominators be drawn together in the same way. And the denominator of the entire sum will arise.\n\nThus, if we wish to multiply two thirds by three fourths: two taken to three make six, the common numerator. Then, three taken to four produce twelve, the denominator of the multiplication. Therefore, the multiplication of 2/3 by 3/4 results in 6/12.\n\nIF IT WERE SO: if you ask for the reason, this is it: {if} the numerators were drawn only among themselves, they would appear to be integral among themselves. But the numerator would grow too large. For example, when two are taken to three, they become six. If nothing else were present, they would appear whole. However, since two wholes cannot be divided into three equal parts, but three parts of one whole must be multiplied by the fourth part of each to obtain the denominators, which, in turn, lead to the division of the parts, and since the parts are reduced in size in proportion to the increase in the denominator, only the excess of the numerator should be corrected, to the extent that it has increased unfairly, and thus, after the denominators have been combined, it is clear that there are six twelfths, which otherwise could appear as six wholes.\n\nFurthermore, if you wish to multiply wholes by small parts or small parts by wholes, the numerator of the integrals should be multiplied by the denominator of the small parts, and the result will be the total numerator. The denominator of the small parts should be assumed to remain unchanged beneath the line. For example, if 2 wholes are multiplied by \u00be or \u00be by 2 wholes, the result will be 6/4, which is equivalent to 1\u00bd. Aliis integris ad exemplum scriptis: ut numerus integrorum supra brevem lineam non notetur, et sub ea reponatur unitatis nota, quae illa integrum significet. Iuxta hoc scriptum, quasi minutiae essent, veras apponunt minutias, quas in ea ducunt. Deinde numeratorem integrorum petunt minutarium, horum denominatorem per denominatorem illorum multiplicant. Hoc modus et expeditus est, et res ad idem reducit. Nam cum ipsa unitas integrans alios numeros non multiplicet, denominator semper invariatus manet. Ita 2/1 ducta in 3/4 facit 6/4.\n\nSic obiter expeditam est partium duplicatio. Quam quidam separatim tractant.\n\nSi autem per minutias velles integras, alis minimis adiunctas, multiplicare, vel e converso minimas per integras adiunctas reduc, prius integras in minimis sibi adiunctis reduc, ipsa integrum multiplicando in illarum denominatorem, numeroque procreato illarum numeratorem addendo. When a numerator is accumulated with the added denominators below it, the denominator is written next to them: the numerator is multiplied by the numerator, and the denominator by the denominator. For example, if two integers with two sevens are to be multiplied by two quarters: two multiplied by seven yields sixteen, and the two added produce eighteen. This is the numerator, which should be joined under the line with a denominator of seven. Then, the numerator is multiplied by another numerator, producing thirty-two. The denominators similarly produce twenty-eight. Thus, 2. an integer and 2/7 reduced yield 16/7. When an integer is to be added to other integers in their minimized form: first reduce the integers to be added to their common minimized form, with the integer being added leading in their denomination and the product of their numbers added to the numerator. The resulting numerator is then joined under the line with a denominator equal to the denominator of the first integer. If these details are to be taken literally, below are the integral parts: Subtract what is below these lines to unity. Next, bring numbers to numbers, and denominators to denominators, as in fractions. For instance, if 9 in 3 \u2158 should be taken, 3 in 5 will yield 15, which added to 4 will result in 19/5. After 9 in 19 is taken, it will produce 171. One 5 makes 5, which is below the line. Thus, from this multiplication, 171/5 will result. If the integers and fractions are to be reduced to integers, they will produce 34 \u2155.\n\nIf the integers and fractions must be taken as integers and fractions in turn: For example, 3 4/9 in 5 \u215c, both integers should be reduced to fractions first, and then the multiplication of the fractions should be performed. In the former case, 3 in 9 is taken, producing 27. With a 4 added, it results in 31/9. Similarly, in the latter case, 5 in 8 is taken, producing 40. With a 3 added, it results in 43/8.\n\nHowever, if 31/9 is taken and added to 43/8, it results in 1333/72, which, when reduced to integers, becomes 18 37/72. VBICVMQVE in minutiae of minutiae converge: first, these should be reduced to simple minutiae. Then, in each one, according to what was said before, multiplication should be performed.\n\nPARTITION is the division of two fragments, one of which is cut obliquely, brought together through multiplication into one: by which a third fragment is born, generated from an equal embrace of both. In every partition of parts, the fragment is to be placed among the parts of the divider from the left: the fragment to be divided from the right. This is how it is done in wholes: the divider occupies the left. Next, the divisor of the fragment is multiplied by the denominator of the divisor: and the numerator of the section is born. Then, the denominator of the fragment is divided by the denominator of the divisor: and the denominator of the partition results. This, placed under the previously created numerator and a line inserted, should be supposed. I. To bring numbers into denominators and denominators into numbers, oblique multiplication in the form of the cross of God by Saint Andrew will produce the powers of dividing and being divided equally. For example, if you want to divide two thirds by three sixths: multiply two by six, and it results in twelve, the numerator of the operation. Then, three by three will produce nine, the denominator of the section. Thus, through 3/6 we make 2/3 equal to 12/9. If the numerator is cut by the denominator: one whole and one third of a whole make up the whole.\n\nII. Those who command to place the divisor on the right side in small divisions: I see no reason for this. Why they want it to be done this way: I don't know. But when they themselves command to place the divisor's denominator into the numerator and the divisor's numerator into the denominator, it makes no difference at all: the divisor remains in place. If you are asking for the meaning of this text: it explains that if two statues are cutting a fragment in half and the fragment itself is identical on both sides, then the division should be made in this way. The reason for the multiplication of the parts taking the form of an oblique cross in minute partition is that the exilitas (thinness) of the minutiae (small details) is caused by the multiplication of the denominator. The more the denominator increases, the more they (the parts) are pressed together and become more numerous. But when the denominator is reversed, the increment or the whole progresses or the parts are much closer to the integrals. Itaque, when numbers are subtracted from each other, with the numerator being directed to another numerator and the denominator to another denominator, as we showed in the previous chapter: the division of fractions, which is naturally opposite to multiplication, should also be performed by multiplication, so that the fragments come together through this mutual section: since one multiplication is performed directly, this one, which cuts the parts, must be done obliquely: the numerator of one fraction must be made the denominator of the other, and the denominator of one fraction the numerator of the other. Otherwise, if the section were too great, it could not be brought back to equality. For example, in the given example, when we want to divide \u2154 by 3/6: multiplying three, the denominator of the dividend by three, and the divisor's numerator, if we do nothing else: it would seem that we are cutting the parts into three equal parts: but by dividing in this way, we would exceed too much. Sed quoniam illa tria, per quae dividimus, non erant tria integra, sed erant tres sextupla: propterea dividendi numeratorem in sex, dividoris denominatorem ducentes, corrigimus excussum illud. Atque ita partitionem ad aequalitatem reducimus. Tantum enim dividendi numeratorem supplemus, quantum plus iusto denominatorem eius auximus, partes minuendo.\n\nFor you, the division of parts is expedited. This, in order that you may understand it more easily, should be observed: three of its kinds there are. One is when larger parts are cut from smaller. In this kind, more than one integral whole is created. For example, if one third is cut from a half, and three halves are formed.\n\nAnother is when equal parts divide equal parts, from which one whole is created. For instance, if a half is divided by a half, and from that section two halves are formed.\n\nThe third is when smaller parts are divided by larger ones: this, which often occurs in fractions. For thus may they come together. In these parts, fewer than one whole return: less than one integral. For example, if one-half takes one-third: two-thirds result, which is less than one integral, does not eat one whole.\nAnd to know what sense the minute division conveys, one must first investigate which minutiae, those that divide and those that are to be divided, are contained. This is easily done if the denominators are multiples of each other. Then, in the number produced, find two smaller numbers: one referring to the parts to be divided, the other to the parts that do the dividing. Thus, you will find nothing at all to matter: whether one of the numbers divides the other or the parts divide each other. An example in the first mode of division. When we divide one-half by one-third, if we seek the number that contains both, three taken twice makes six for us. The number that gathers both minutiae. Next, in six, investigate the number that represents one-third of it. It will be two. Then, when we divide one-third by one-half: is it three by two? In this text, there is no meaningless or unreadable content, and no modern additions or translations are required. The text appears to be written in old Latin, but it is grammatically correct and can be read without translation. Therefore, I will simply output the text as is, with no modifications or comments:\n\nNam utrobi que res ad idem recidet. quippe ex partium sectione proueniunt tres secundum: quod unum integrum et dimidium faciunt. nam numeratoris per denominatore partitio unus in numero partitionis producit: ultra quod una secunda superest. Id, quod itidem eveniet: si tria per duos seces.\n\nAlterum exemplum in secunda secandi specie. Si per \u00bd diuidas 2/4: inde potest 4/4. quae unum integrum creant. Idem surget: si quatuor diuidas per quatuor. quippe in octonario numero ex denominatorum in se ductu procreato. 4. unam secundam, et. 4. duas quartas faciunt. At in tertia secandi specie, quando maiores partes dividendo funguntur partibus: quemadmodum tunc in integris sectio fieri nequit: nisi minor numerus, vel in species sub se contentes multiplicetur: vel in minutias solvatur: sicuti mox dicemus: ita dum minutiae coeuntes sic secantur: huismodi partitio nil nisi minutias parit: quippe quod unum integrum nequeunt aequare. Since the text is written in Latin, I will translate it into modern English while maintaining the original content as much as possible. I will also remove unnecessary line breaks and whitespaces.\n\nIf a part is greater than the denominator: that many are lacking to make a whole. For instance, if one takes \u00bd and another \u2153: three sixths result. This is equivalent to: as if two were dividing three measures. For in the number sixty, collecting the smallest parts, two make three the second and one the third. This section cannot be made in integers: so that each number remains an integer. This, which we will soon explain. Therefore, whenever parts of any number are proposed to be divided by other parts of the same number: this signifies: that the parts proposed are to be divided by the parts proposed of the same number.\n\nFurthermore, since multiplication always increases in wholes: division decreases: the opposite occurs in small parts: so that multiplication decreases: and parts increase by division. For if one third is multiplied by one second: one sixth is produced. This is far less: than one second: or one third. At the converso, if one third is divided by another third: there are three thirds: which together make one whole and a half. This is far greater than one third or one second. But one who remembers the nature of fractions: will notice that as the number of the denominator increases, the parts decrease in quantity. This, which we have often reminded you of, will no longer be surprising, since an increase in the number brings a decrease in quantity with it. And the denominator, rather than the numerator, is increased. It may seem more surprising at first sight: how parts decrease when divided. But this should not disturb anyone: who has long perceived that the division of parts is very different from the whole. For this occurs through the subtraction of the divisor from the dividend repeated as many times as possible. However, this is produced by the oblique union of two fragments: through which a third fragment arises from their equal embrace. Quare quo reason is for multiplication: so that the parts be less than the parts opposed to it: it should be: so that the parts be restored: fuller than they are. To what thing does it not resemble in any way that one can see in complete wholes: if someone wants to reduce smaller coins to a sum of gold coins. This is done through sectioning. If through the number of the coins, by which one pound or one gold coin is estimated. The entire number of coins is divided. Therefore, when the smaller parts are cut through smaller parts: everything becomes greater: just as gold and pounds are gathered from smaller coins. Therefore, division is nothing other than: investigation of what: how many and what parts come together in the creation of a third fragment, from the two fragments intersecting each other through sectioning.\n\nIf, however, the smaller parts are to be divided: the number of the complete parts was written above the line: for example, the numerator of the fractions: under which, a line is inserted, to note the unit: which, as it were, designates the integrals. The following text describes a method for dividing parts in the form of a cross. If, for example, they are divided by 2/1 after the multiplication in the form of a cross is made, the parts will be \u2153. This can also be done in another way: the same result will be obtained, with the denominator remaining unchanged, by reducing the numerator to equal parts, if possible, by dividing it by the number of integers. For instance, if the numerator is 2 integrated 6/8, the parts will be \u2153. Otherwise, if the numerator cannot be divided into equal parts by integers, the number of integers is multiplied by the denominator of the fraction being divided, and the result is obtained. For example, if the numerator is 2 integrated \u00be, the parts will be \u2153.\n\nThe following passage explains the simple division of parts in half, which some curious individuals treat separately. It is much easier than dividing them by three, four, or any larger number. If a integral part is to be divided into smaller parts: first integrate the smaller parts with the larger one, so that one of the smaller parts appears on the left, as they function as parts of the divider. Other smaller parts, which need to be divided, appear on the right. Then, perform the division in the form of a cross, in the manner of smaller parts. For instance, if you want to divide 2\u00bd by \u00be, first bring down the 2 as a numerator to the 2, and then add 1 to make it 4. Through these parts, the numerator 1 extracts 5/2, from which, if you separate \u00be, the parts will yield 6/20.\n\nIf an integral part is to be divided by smaller parts: place the smaller parts, in the manner of smaller parts, above the line to which the integral part is subtracted; while the smaller parts are placed on the left, the others remain unchanged, as we have previously discussed regarding the multiplication in the form of a cross. For example, if you want to divide 4/1 by \u2154, the result will be 12/2. If the text is in Latin, I will translate it into modern English. The text appears to be a mathematical problem, so I will attempt to correct any errors and make it readable.\n\nIf integers with their parts are to be divided: if the integers have parts added to them, they will be arranged in their reduced form to the left, with the parts functioning as divisors. The remaining parts should be dealt with as we previously mentioned: when integers are to be divided by their parts. For instance, if 2 2/3 is to be divided by 4/1, the parts to be subtracted should be added to the denominator. 3. Thus, 2 becomes 8/3. However, if 4/1 is divided, 12/8 will result, which makes 1 1/2.\n\nIf integers and their parts are to be divided by their parts: first, the integers should be reduced to their parts. One of these parts should become a single face. Then, the remaining parts should be cut off according to the fragmentor's method. For example, if 5 3/4 is to be divided by 5/4, 5 will go into 4, and 20 will result. Add 3 to the numerator, and 23/4 will emerge. Next, divide the divisor's denominator by the divisor's parts, and produce 5. Divide the numerator by 23, and bring it into the denominator of the divisor. The result is 6. et surget 138, si per 20 secta ad integra redigatur: pferet 6 9/10.\nAD EVDEM modi fiat, si integra et minutiae per integra et minimas dividi debent: prius utraque integra in minimas sibi adiunctas seorsum redigantur, ut omnia minimas utrinque referant. Deinde minimas, quae dividendae sunt: a dextra locata, per alteras, fragmentoris more secetur. Veluti si per 3 2/9 partiri, 6 1/7 cupis. 3 in 9 multiplicata et producto numerator additus facit 29/9. Item, 6 in 7 ducta productoque numerator adiunctus profert 43/7. Postea si per 29/9 secetur 43/7: ex ea partitione 387/203 producitur. Quae ad integra reducuntur faciunt 1 184/203.\n\nWhen 138 is to be reduced to its integral parts: it will yield 6 9/10.\nIn the same way, when integral parts and minimas need to be divided: first each integral part should be reduced to minimas alone, so that all minimas refer to both sides. Then, minimas to be divided: the one on the right is cut off by others, using the method of a fragmentor. For instance, if you wish to divide by 3 2/9, 6 1/7 is taken, multiplied and the numerator added to it, making 29/9. Similarly, 6 in 7 multiplied and the numerator added makes 43/7. Afterwards, if 29/9 is divided by 43/7: 387/203 is produced from this division, which when reduced to integers results in 1 184/203.\n\nHowever, when minimas are divided among minimas: first they should be reduced to simple minimas, and afterwards, in each, according to what was said above, the division should be performed. If you want to determine if a partition is correct: divide the number of sections by the divisor, and the quotient will be produced. Multiplication in division, and division in multiplication, yield the same result in small quantities as in large. Therefore, when we divide 3/2 by 1/3, we get 1 1/2. We will prove that it is correctly divided if we divide 3/2 by 1/3. Thus, 3/6 will result. The reduction of the nomenclature of the smallest parts makes 1/2. This was a fragment that was divided.\n\nIn the minutiae of the operation, we thought it prudent to offer a warning: in which we have often seen those who were well-exercised in the art. Quandocumque minutiae: quae dividendae sunt et idem denominatore habent, et in utrisque numeratore minor est, major numerator existit. Quae secenda sunt, quam secare debent, quia tunc in utrisque, minutiae totidem integra repraesentant, quoties utriusque numerator denominatorem suum continet. Neque enim partitio quamquam oblique in crucis formam fiat, sicut in minutis. Sed denominatoribus intactis, maior dividendus numerator per divisoris numeratorem, minor est. Alioqui supputator illas sicut minutias dividat incurso, cum integra designent et minor numerator maiorem partire possit, in labyrinthum inextricabilem ignorans se immergeret.\n\nExempli gratia, 5/3 secandis essent 7/3. Nihil utriusque denominatore mutatis, septem per 5 dividuntur, unum integrum et duas quintas eductae. If you want to divide 8/6 into 10 parts, taking eight parts as the measure, you will have one part left over, which amounts to a quarter. And in these examples, if anyone seems to refer to nothing: whether they follow this mode or not: even if the division were made in the usual way, the result would still return to the same whole after reduction. However, except for the fact that the method of reduction shown here may deter some from reducing further due to its roundabout nature, many calculations would arise, in which an immense error would occur if the parts were divided in the usual way. Moreover, when one person is collecting from various wholes and diverse fractions, he is the one who must separate everything. This will be made clear in explaining the knots of numbers in the following book through examples. In whole numbers, a larger number does not divide a smaller one, since the larger number does not contain the smaller one. As we warned in the previous book, regarding the rules of division. However, a larger number can break a smaller number into fractions. To divide a section, observe this: for how many parts it should be divided, and what kinds; note that for a smaller number to be divided, you should place the dividing marker above the smaller number, but for a larger number, which is the divisor, below the line. In this way, the parts will be created: the number of units that the dividend, which is the numerator, consists of will be such; and these will be such in kind as the denominator, which is the divisor, demonstrates. When reduced to their smallest names, these will reveal how many and what kinds of parts result, and when a larger number divides a smaller one. For example, if we wish to divide 30 by 15, they become 15/30, which in their smallest form are equal to half. Similarly, if we wish to divide 40 by 15, 15 is the dividing section, which makes it 15/40, or one third. Likewise, if we wish to divide 13, 7 parts result, which are 7/13. The smaller number above the line becomes the divisor in the division, and the larger number below the line becomes the dividend. If the numerator is larger: the denominator, cutting it, would present the whole; thus, when the denominator is larger than the numerator: fractions arise, which only need to be completed to become whole, to the extent that the numerator is smaller than the denominator. A number smaller than a larger one can be divided in two ways. Either when the things numbered by the smaller number are contained in species within the larger number: so that the product of the larger number may cut the section. For example, if one gold coin is to be divided among four men, it can be multiplied into a smaller amount: so that it can be distributed among them. Or when the larger number cuts the smaller one into fractions: which can be done in the same way as we have said.\n\nThese matters pertain to the division of wholes: since, however, such a division produces fractions, of which there should have been spoken earlier, they were instead relegated to this place. In investigating the square and cube roots in small quantities, the method is the same as in larger numbers, except for the additional labor when the root is sought in both the numerator and denominator. The side, found in the numerator, will be the radix. Whatever is extracted from the denominator will have the radix of the denominator. The middle line should be divided. For example, to find the square root of 25/16: the radix of the numerator will be found to be 5, and the radix of the denominator will be 4. Thus, 5/4 results. Similarly, for 46/94, the square root of the number leaves a remainder of 6/9, and the cube root leaves a remainder of 10/13. The square root of a number, which is neither square nor cube, always leaves something remaining, either in the initial investigation or in the subsequent one, which we will discuss later. For if nothing remained, then that number would be a square or a cube. However, in a large number, the remainder itself will be considerable. Primaque inuestigationem radicem maximi numeri obtinet, quae quadrati vel cubi in his continet. Between this number and the one which is neither square nor cube, the difference is manifested. Therefore, in such numbers, it is not possible to obtain the true root except approximately, even if closer than in the initial investigation. This is how it is done in quadratic numbers.\n\nAfter the first complete investigation of the root, in which the remainder remains: the number which was not found to be a square in the complete root, multiply it by the denominator of any part you choose. Then multiply the product by the same denominator again. What you will obtain is the square root of the number, which will contain the parts of the denomination in the first proposed root.\n\nThis root squared will demonstrate the parts of the denomination contained in the numerator. If for part two, the root of the number eighty-one is sought: you will find it to be 2. Take four from it, and there will be four remaining. If you wish to know how many quarters there are in one complete part of the root: First, multiply eight by four, and you will obtain sixty-four. Then, sixty-four multiplied by two will yield 128. Examining the second product of the root, you will find it to be the same. For the number eighty-one, there are indeed eleven quarters in the root: that is, two complete ones, three quarters, and the remaining parts which cannot complete a fourth. Therefore, investigating further, you will find that, besides the first complete part which was extracted: the third and fourth quarters come closer. If you wish to investigate the parts further and find out how many quarters of a quarter, two integers and three quarters remain, follow the same method as previously stated for investigating the parts of wholes: you must find the number whose root you have already investigated, and bring it once into the denominator and once into the product. The number you find will show you how many parts of the first number's denomination are contained in the root. If this root is divided by the denominator of the parts, it will demonstrate the parts of the section. And whatever remains will be the parts of the parts.\n\nFor example, if we multiplied the number whose root we investigated in the second exponent of the superior number by four, we would get 512. Then, if we multiplied four by this number, we would get 2048. The root of this number we are seeking. It is 45, and there are 23 left over. Therefore, 45. In a quartet of a fourth root, there will be four parts: which, if divided by four, will reveal them. There will be one remaining. We will learn how to find eleven parts of a fourth root, that is, two whole and three quarters, as well as one quarter of a fourth part.\n\nIf someone's mind is inclined to progress further: they can also examine the smaller parts.\n\nIn all numbers, make everything the same mode, except for the number whose cube root is not found in integers: we must not only multiply its part's denominator, as you wish, but also the product, and then take the cube root of the third product to manifest all its denominator's parts contained in the first cube root. Similarly, we will also progress in investigating the parts in these. To accomplish your request, I'll clean the text as follows:\n\nTo make this process more concise: denote the parts you inquire about once, as the examination of the complete roots will remain after the initial one; bring the parts to the product: take them. Then, using the same denominator for the root found in its entirety, multiply the root itself; next, double the product of the number obtained from the denominator's parts once in the roots and once in the multiplication. After that, divide the number generated from the division by the number obtained from the denominator's parts, but ensure that only as much remains as the number of sections can delete when taken back into itself.\n\nIn these numbers, not all minor details will be eliminated (as we mentioned earlier). The number of sections will reveal new parts arising from the remainder. For instance, if you seek the root of an octonary: you'll find it. 2 and the remaining parts are. Now, if you wish to know how many fourths remain in the remainder: Multiply the remaining four by the four denominator parts: you ask for what. And they yield sixteen.\nRepeat: bring the denominator four into sixteen, and sixty-four result. Then take the root. 2. Multiply in the denominator parts four, and they become twenty-five. 8. Double the product. 8. And sixteen result. Now, if you divide sixty-four by sixteen: the divisor of sixteen can be taken four times in them; but then nothing remains: since the number of the parts, when taken by itself, is to be deleted. Therefore, if we subtract one less than otherwise could be, we remove the divisor sixteen: thirteen make forty-eight: and twelve remain: from which, if the number of sections taken in itself, which makes nine, is subtracted: seven remain. Thus, the number of sections shows that from the remaining root three quarters result, except for small remainders. 7. In this example, to find the number of parts, investigate each part as if it were one fourth of quarters: the determiner of these parts you seek being seven, after finding the root of the parts remaining, which is seven. Thus, there will be twenty-eight. Repeat the same determiner of four in the product of twenty-eight. And they will emerge. 112. Next, you must lead the same determiner of four through all the parts discovered up to this point: there are eleven. For these eleven parts, if four are taken, they will produce one hundred and forty-four. If these numbers are multiplied by two, there will be one hundred and seventy-six. By this method, if you divide the number by sections as much as possible, the whole number will be divided away, and a unit divisor will suffice. Thus, one hundred and seventy-six, extracted once, leaves twenty-four. et after unity was drawn into itself, one was created: that which must be taken away is 24. And twenty-three will remain. Thus, the number of sections, one being the fourth part of the second one remaining, is manifestly revealed. And the following parts can also be pursued.\n\nRegarding why a square should be multiplied by the denominator of its parts twice: Socrates explains this to the boy Menon in Plato's Meno. I will briefly explain his opinion. A square of a biped figure will be four feet long, for the figure of two feet in length and one foot in height represents two feet long and the same height for two feet. However, a square of double the length will cover twelve feet, because the figure, which is double in length and only double in height, covers eight feet in both length and height:\n\nTherefore, the figure, which is double in length and height, will cover sixteen feet.\n\nSimilarly, in all other quadratic equations, multiplication by two is necessary. If the first multiplication only increases length, but the second multiplication, which occurs in the product, adds to height. For example, let a square be proposed with sides of four, if you want to find the square whose side is double that of the proposed square. Double the proposed square. Then eight are born, and the figure grows only in length. The length has been doubled, but the height remains unchanged. Therefore, the square is not it. If, however, we double the products again, there will be sixteen, and it will grow just as much in height, since it is double in height compared to the previous one. What Socrates demonstrated to the boy in lines, that is, what can be shown in the form of squares in numbers, the wise man, Euclid, explains with reason, as he says. If there are two numbers, both square: the proportion of one to the other will be the same as the proportion of its side to the side of the other. If both cubes are equal in size: the ratio of one to the other will be, as their later sides to each other, multiplied by three.\nIf you want to know the reason for the given compression: you will be able to see it clearly with this example. Let's assume the number proposed is three: its root is one, and there are two left. Now, if you want to know how many quarters one whole number responds to these two: take the denominator four and reduce it to three, and you will have twelve. Multiply four by the product of twelve: forty-eight will result. Form the figure of this number as a square in the following way.\nIn this figure, sixteen respond to the maximum square, under the first number. 3. For this square, take four and reduce it to fifteen, and again to the product of fifteen. Thus, you will cut this square: whose root is known to be four (since it is born from the reduction of the roots of the whole numbers to the denominator of the parts). What remains will respond to the rest of the first number.  quippe quod nascitur ex ductu denominatoris partium quatuor in reliquum 2. et iterum in productum octo ex quo procreantur tri\u2223ginta duo. Ex illis autem triginta duobus, quae ultra qua\u2223dratum, quod est sedecim: supersunt duodecim illa, quae gnomonem extremum non absoluu\u0304t (Voco autem in nu\u2223meris gnomonem, id: quod constat ex duobus numero\u2223rum lateribus angulum rectum facientibus, instar gno\u2223monis geometrici) ueluti reliquum maximi quadrati tri\u2223ginta sex sub quadragintaocto resecanda sunt. Ita restant gnomones duo: qui cum quadrato sedecim quod resecui\u2223mus: conficiunt maximum quadratum. 36. sub. 48. co\u0304\u2223tentum: adijciunt{que} radici superiori quatuor: qu\u0119 ad ra\u2223dicem unum in integris inuentam respondet: duas unita\u2223tes. uti in figura liquet. et reliquu\u0304 huius quadrati triginta sex, quod est. 12. adiecisset tertiam: si modo ad gnomo\u2223nem absoluendum, unum insuper accessisset Itaque ante oculos cernere licet gnomonem quemquam applicatum, in se complecti longitudinem et altitudinem ipsius quadrati sedecim, hoc est duplum radicis quadratum: et insuper in angulo restare quadratum numeri gnomonui. Hoc est in hoc exemplo, binarius. Sunt enim duo gnomones. Quare divisio numeri procreati ex ductu denominatoris in reliquum et iterum in productum, facta per duplum radicis, sicut numerus sectionis in se ductus subducere possit a reliquo partitionis: ostendet quot sint gnomones. Quo fit ut liquet quot partes ad reliquum respondeant, addendae sint radici prius in integris inventae. Nam ei gnomo quisque unitate in partibus addit.\n\nFor exercising young men it will be helpful to add some questions: through which they may better understand what has been said about the comparison of magnitudes of small units, and about the addition, subtraction, multiplication, and division of integers and small units.\n\n(Note: The text appears to be in Latin, and there are some errors in the input text that need to be corrected. The text has been translated into modern English, and the errors have been corrected as faithfully as possible to the original text.) In his quadrature, there are some: which make a difference in certain parts in numbers and in minutes, and also in the transformation of other parts into others. We have extracted these rudiments from Luca de Burgo's Arithmetica, whose name is not celebrated enough in that text: they are indeed very suitable for the studious minds in numbers.\n\nIf five sextants are compared with three quarters: which will be the larger? You will certainly know: if the minutes of both are compared side by side, the denominator numbers with the numerator numbers, an oblique multiplication is made, and each number is placed above its own minutes. Indeed, the larger of the two will be the one that shows larger minutes. By this reasoning, five eighths are larger: (than) three quarters. For above five eighths, there will be 20. Above three quarters, there will be 18.\n\nHowever, when whole numbers and minutes are compared with other whole numbers and minutes: if the wholes are equal, judgement should be made only about the minutes, as was previously stated. If two sets of integers are intermixed with odd and even numbers: the larger sets win, no matter how many small details they have added. These should be subtracted. 13. To leave 12, subtract 3 \u00bd. 4 \u2153. Questions of this type, whether about larger numbers or small details, are solved through addition. For instance, if you add 12 to 13, 25 results, from which 13 should be subtracted to leave 12. Similarly, in the case of small details, add 3 \u00bd to 4 \u2153 and they become 7 \u215a. Subtract 3 \u00bd to leave 4 \u2153.\n\nWhen two sets are to be combined, what number should they make, 23? With what number should they be coupled, 6 \u2154? Such questions are solved through subtraction. For instance, if you subtract 39 from 23, 16 remains, which, when added to 23, makes 39. Similarly, in the case of small details, subtract 13 \u2156 from 6 \u2154 and 6 11/15 remains, which, when added to 6 \u2154, makes 13 \u2156.\n\nIf you want to know how much \u2156 exceeds 3/7, you can find out through subtraction. Nam eo modo, quod dictum est: you have captured three-sevenths to be greater: by what, I shall subtract others from others, and one-thirty-fifth will remain to show the difference. This can be proven in the same way as in subtraction. For instance, if one-thirty-fifth is added to five-eighths, the sum will be equal to three-sevenths. Similarly, the difference between five-eighths and three-sevenths can be found in the same way.\n\nWhat number is divided into 5 in the number of sections, number 17? Similarly, what number is divided into 4 1/7 in the number of sections, number 2 1/8? Multiplication will reveal the answer. For example, if five is taken from seventeen, there will be 85. The number five, when cut in the number of sections seventeen, will yield the same result in fractions. Similarly, in smaller parts, four one-sevenths taken in two one-eighths will yield 8 45/56. This number, when cut in the number of partitions, returns two one-eighths.\n\nHow do you find two-thirds of seven-eighths? Or three-sevenths of eight-nines? This is the same: as if you were asking, what are the two-thirds of four quintuples? Or three-sevenths of eight nonaries? Therefore, such inquiries, concerning the differences of small parts, are resolved through the multiplication of smaller parts by others. Na\u2082\u2082\u00b3s are three-fifths of eight-fifteenth. Three-sevenths of eight-ninths are twenty-four-sixty-thirds. These reduced to minimum terms make eight-twenty-first.\n\nWhat are three-quarters of 2\u00bd? This question, unless mixed with minute integers, is similar to the preceding one.\n\nTherefore, it is solved in the same way through multiplication. For three-quarters of 2\u00bd are one and three-quarters.\n\nThrough what number were they divided? 36. When did they become a number in the section? Similarly, through what number were they divided? 12 and three-quarters. What number should the number of partitions be? Three and a half?\n\nSuch questions are solved through sections. For thirty-six, divided by nine, show four in the number of partitions. Since two numbers combine to form a certain number, if one is divided by the other, the other exits in the number of partitions. Thus, when four nines make thirty-six, the numbers exit in four. Similarly, twelve and three-quarters, divided by three and a half, yield three and a half in the number of partitions.\n\nThrough what number must one multiply 2\u00bd to produce seven and two-thirds? This genre of questions is explained through division. For if two-thirds of 2\u00bd are taken, seven and two-thirds result. If parts are taken from a total of 3 1/15, it is the number: in whom, if you take 2 \u00bd, parts will be produced that are 7 \u2154.\nWhat parts are there? 4 of 12, or \u2154 of \u00be? This can easily be determined, both in large numbers and in small: if what is less is divided in the same way, as stated above, 4 parts of 12 will be 4/12, which are \u2153 of 12. And if, by dividing \u2154, 8/9 is produced, then \u2154 is 8/9 of \u00be. This can also be understood or proven by experiment: since if you take 8/9 of \u00be, and you are allowed to change smaller parts into smaller ones in the same way, these will be \u2154.\nWhat are the tenths of the eighth in 5/9? This is a transformation of one kind of small parts into another. For example, if, by division of 1/18, 5/9 is obtained from that fraction, in this way 10/18 cover 5/9. In the same way, any other small parts transform into any others through division: even if they are integral. For instance, if you ask how many thirds there are in 3 \u00bd, divide 3 \u00bd by \u2153. The result will be:\n\n## Output:\n\nIf parts are taken from a total of 3 1/15, it is the number: in whom, if you take 2 \u00bd, parts will be produced that are 7 \u2154. What parts are there? 4 of 12, or \u2154 of \u00be? This can easily be determined, both in large numbers and in small: if what is less is divided in the same way, 4 parts of 12 will be 4/12, which are \u2153 of 12. And if, by dividing \u2154, 8/9 is produced, then \u2154 is 8/9 of \u00be. This can also be understood or proven by experiment: since if you take 8/9 of \u00be, and you are allowed to change smaller parts into smaller ones in the same way, these will be \u2154. What are the tenths of the eighth in 5/9? This is a transformation of one kind of small parts into another. For example, if, by division of 1/18, 5/9 is obtained from that fraction, in this way 10/18 cover 5/9. In the same way, any other small parts transform into any others through division: even if they are integral. For instance, if you ask how many thirds there are in 3 \u00bd, divide 3 \u00bd by \u2153. The result will be: 11/3. When there are three and a half parts, there are ten and a half parts in total. Change the parts one into another, as we have previously mentioned, when reducing parts to a similar form. We have deliberately kept silent about the matter of dividing small parts, which had not yet been explained.\n\nWhen eight and two-thirds need to be counted as eight and nine, consider this: when combined with eight and two-thirds, they become eight and nine, as you will understand through subtraction. For if you subtract two-thirds from eight and nine, there will remain thirteen forty-fifths. When combined with thirteen forty-fifths, they become eight and nine. Now it is clear: how many eighths thirteen forty-fifths take, which you will know: if you divide thirteen forty-fifths by an eighth. In the same way, do the same for similar cases.\n\nSubtract two-thirds from all sevens, so that half remains. First, consider from which number you need to subtract two-thirds: this, you will understand, if you add two-thirds to half. They will then become whole.\n\nOne and one-tenth need to be taken away: so that half remains. Consider from which ones you need to take away two-thirds. \"After that: how many are there in one and one-tenth. One, which you know: if one-seventh is taken away one and one-tenth, they become seven and seven-tenths. Seven are the total of seven and one-seventh: one-half must be subtracted to make it one. You will find this to be so: if you grasp it. Seven one-sevenths: which are one whole. Then take seven-tenths of one-seventh: which are seven-fortieths and make one-tenth. In total, they amount to one and one-tenth, as has already been said. Similarly, this should be done in such inquiries.\n\nSubtract the difference: what is the difference between one-third and one-half of such a number: so that the difference remains: which is between one-half and five-ninths. First, the difference between one-third and one-half must be found. It is thirteen-thirtieths. Next, find the difference between one-half and five-ninths in the same way. You will find it to be one-eighteenth. Then add both differences: and they amount to twenty-two-forty-fifths. From which number should the first difference be subtracted, that is, thirteen-thirtieths: and the second difference, which was sought, will remain.\"\n\n\"For those fifths that need to be multiplied. Two and a half: so that they increase. Seven and two-thirds? First, see: through which number multiplication is necessary. Two and a half: so they become. Seven and two-thirds.\" If per 2 \u00bd dividas 7 \u2154, they will produce 3 1/15. It is necessary to determine how many quintas are in 3 1/15. That which is easily recognizable: if per \u2155 seces 3 1/15, they will come. Nam inde prouenient 15 \u2153. These are \u2154 of the total octavas, so that \u00bd may come from them. This is all that is worth: as if you were to ask, what is the half of the number that makes \u2154? Or by what number should \u2154 be multiplied to make a half? As you have been taught above: divide by \u2154, and you will get \u00be. The half of this number will make \u2154. Na\u0304 \u2154 of \u00be are \u00bd. Now, in order for you to know how many octavas are in \u00be, if per \u215b seces \u00be, they will amount to 6.\n\nTherefore, from 6/8 sumendae are \u2154, so that it may be a half.\n\nDivide 7 \u00bd entirely to obtain \u215a of 17 \u00bd. Take \u215a of 17 \u00bd in the same way, as has been said. These are 14 7/12. This number should be the number of secedae. It is necessary to determine by what number the secedae are taken to come. 7 \u00bd, so that they may come from it. 14 7/12. Whatever will be made: as above stated: if per. \"14 shares of 12-1/2 pence. 7-1/2 pence each, making a total of 90-1/75 pence, which should be divided into 18-1/35 shares. 7-1/2 pence: from which the shares will come. 14 shares of 12-1/2 pence, which is 7-1/2 pence per share. 17-1/2 pence in total.\nShares are divided. 7-1/2 pence per 30 nonas, so that 2-1/4 pence each result. First, note that shares are divided into how many parts. 7-1/2 pence: from which 2-1/4 pence each result. You will know this: if 2-1/4 pence are subtracted from 7-1/2 pence, 3-1/3 pence results. This number should be divided by. 30 nonas are divided into. 7-1/2 pence: so that 2-1/4 pence each result.\nShares are demanded 5-1/2 times in 6: to make 6-1/2. First, 5-1/2 are demanded in 6, and it will yield 1-1/2. Next, multiply 1-1/2 by the number: which will make 6-1/2. In what number should 1-1/2 be placed:\nshares will be produced. This theme indicates: that the number produced from the multiplication of 5-1/2 in 6, should be placed in\nthe number which makes 6-1/2.\" Primarily, for the number 7/9, which parts are there \u2157 of? This is known: if you inquire about the parts, divide 7/9 by \u2157. Then, find out how many sixths there are in 1 8/27, which makes \u2157 of 7/9. Next, ask: how many sixths are there in 1 8/27? When you divide that by \u2159, you will find. Seven seventh parts of seventh parts of one sixth exist in 7/9. This is about 7 seventh parts and 7 ninths of one sixth.\n\nSubtract \u2154 from \u215a of that number: so that \u00be of 5/7 remain. First, take \u2154 from \u215a: what are 5/9? Next, take \u00be from 5/7: what are 15/28? Add those to 5/9: and it will be 1 23/252. Regarding this number, if a subtraction is made: what remains will be what was sought.\n\nSubtract half of the thirds from 12 \u00bd of that number: so that five sixths of two thirds remain. Primarily, consider: what are \u00be of 12 \u00bd? They are 9 \u215c. Take half of them: what is it? It is 4 11/16. This is only half of the thirds of 12 \u00bd. Next, consider: what are \u2154 of 10 \u00bc? They are 6 \u215a. Take \u215a of them: what are they? They are 5 25/36.\n\nNow, according to the theme, the second part should be subtracted. Subtract 4 11/16 from that number: so that 5 25/36 remain. Therefore, add 4 11/16 to 5 25/36. et surgent 10 55/144. Is it that number: from which subtraction should be made: so that what remains is what the theme requires.\n\nThis book is the second in the series, teaching the subtraction of parts. The third book follows, explaining various number questions in detail.\n\nAfter we have explained every calculation, both in complete numbers and in their smallest parts: it remains for us to reveal a little to the studious, concerning how these things contribute to a meaningful life. We have already explained the numbers themselves in detail. This seemed insufficient for those who find pleasure in such things. Let us therefore apply the art itself to numbered things. Regarding this matter, it cannot be said: whether greater utility will be gained or pleasure, when we must know certain things and they appear to the uneducated as enigmas, solvable only with the help of the Sphinx.\n\nTo explore these matters further, we shall provide some useful rules: by which even difficult questions, which appear insoluble to the uneducated, can be easily solved. From these, you cannot grasp anything else without some connection. The mind is what primarily reaches such things: the kinds that frequently occur in life. Some of these are twisted and have hidden complexities: we will add them for the greater exercise of intellects. However, he who wishes to explain the knots of numbers should not answer rashly to any question proposed, but should consider it carefully. For he who speaks more quickly with his tongue than with his intellect is foolish, and is considered ridiculous, except for what deviates from the purpose. We also advise the reader to be cautioned: unless he holds the whole, he should not calculate the smallest details, lest he becomes entangled in this book. But if he has understood the basics, we promise him that everything will be easy and enjoyable. Now let us begin.\n\nIt is a fundamental rule for all: that which brings the fourth unknown into the knowledge of the three known, is handed down from Arithmetic. The vulgus calls it the golden rule: that is, this rule of arithmetic differs from others in that, to make it clearer, we must discuss certain aspects of proportion as it relates to numbers. Without these, this rule is not easily understood.\n\nA proportion is a fixed relationship or ratio between two numbers. Given any two numbers, one may be greater, equal, or less than the other. This ratio is called a proportion when the numbers, whether equal or unequal, reciprocally reflect each other.\n\nThe term proportion itself signifies nothing other than the comparison between things. However, the term habitudo refers to a fixed relationship between numbers, as when you say that one number is a known and defined multiple of another.\n\nAmong proportions themselves, similarity is called proportionality. Proportionality has two species: the continuous and the separated. The concept of proportionality among numbers is when the relationship of the first number to the second is the same as that of the second to the third. In this sense, a number in the middle acts as a bridge between the two numbers, being a term in relation to the preceding number in the first proportion, and an beginning in relation to the following number in the second proportion. This type of proportion can only be found in a few numbers, not in three. Even in those numbers, the middle number changes place with two. Let us take, for example, these three numbers: one, two, four. In these numbers, there is equality of proportion. For instance, the relationship of one to two is the same as that of two to four. One to two is a half, and two to four is a double. The relationship is reversible in both proportions. Whenever we compare the larger number to the smaller in either proportion, we find the same relationship. For example, four to two is a double, and two to one is also a double. The correct proportion is: that which can be called the same, when the first has this relation to the second, and the third to the fourth. In this species of proportion, the term is separated from the following beginning. This species cannot be found in fewer numbers than four. In these numbers, the equality is found by comparing the smaller number with its greater one, or the converse. For instance, if someone wants to compare these four numbers with each other: namely, two to four, and three to six. Just as two to four are equal when compared, so are three to six.\n\nSimilarly, in reverse order, just as four to two are equal when compared, so are six to three. Furthermore, in this species of proportion, the comparison is made step by step. For the proportion in which the first is related to the third, it is the same; and the second to the fourth. Also, in reverse order, the fourth is related to the second in the same way, and the third to the first. If the numbers in sequence have the same proportion, that is, if a smaller number in relation to a larger one has the same relationship as the larger number in relation to a still larger one, this is called a proportion. For example, as in the given example, if the first number is binary and has a subsequential proportion to the third ternary, then the second number quaternary has the same relationship to the fourth senary. Reversed in order, just as the fourth senary has a sesquialteral proportion to the second quaternary, so the third ternary has the same relationship to the first binary. Such numbers are called proportional numbers. This rule is extensively taught by Euclid, both in Geometry and Arithmetic.\n\nIf there are FOUR PROPORTIONAL NUMBERS, what is produced by the progression from the first to the last will be equal to what is produced by the progression from the second to the third. Atque itidem, quod ex secundi in tertium ductu procreabitur: aequale erit ei, quod prodibit ex primi in ultimum ductu. This rule cannot be expressed: how much light it provides to investigators in quest of unknowns. Its virtue is: if an unknown person is among the four proportional ones, the others, who are known, reveal him. Therefore, if three proportions are proposed, whoever wants to investigate, will find out who the fourth is, to which the third is proportionate, by what the first is to the second, and the second to the third, and the number produced is determined by the first. After this is done, the fourth number will be demonstrated by the section number. For example, if one wants to know which is the number to which these three proportions apply, as the duo to quatuor: three multiplies into four and produces twelve. Which is divided by two and yields six. The reason for this is clear. For following in the third multiplication produces the same as if the first were multiplied by the fourth, unknown. Igitur ex se cubidimus in tertium ductu, quartum velut in magna turba latitantem habemus: sed quis sit, adhuc ignoramus. Divisio autem per primum facta reliquam, quae tecta est: multitudinem ab eo segregat, solumque relinquit: atque a caetaris destitutum e latribus in lucem profert. Nam quicumque duo numeri inter se multiplicantur: si productus numerus per alterum dividatur: numerus partitionis alterum continebit. Veluti si duo in tria ducantur: fient sex. Quae si per duo dividantur: numerus partitionis tria continebit. Si per tria: duo. Itaque ad hunc modum quartus numerus ignotus exquiritur.\n\nSimiliter si tres numeri continuae proportionales fuissent: quod ex ductu primi in tertium producetur: aequale deprehendetur ei: quod ex secundi in se ipsum ductu natum erit. Et quod ex secundi in seipsum ductu generetur: aequale erit ei: quod nasceretur ex primi in tertium ductu. This text appears to be written in Old Latin, and it discusses the method of finding an unknown third number in a proportion. Here is the cleaned text:\n\nThird number, an unknown one, is sought in the same way: when the second number is led back to itself, and the product is divided by the first, the third number will be shown by the section number. For instance, if someone wants to know what number is the third one in the proportion of two to four, he should take the middle number, four, and multiply it by sixteen. Then, he should divide the result by two. The first number will be the quotient, and the third number will be octonarius (eight). This property of proportional numbers always accompanies them, not only when they are considered by themselves, but also when they are applied to any objects to be numbered. The same relationship and rule applies to finding an unknown number. This knowledge will be useful for many things.\n\nCleaned text:\n\nThird number, an unknown one, is sought in the same way: when the second number is led back to itself, and the product is divided by the first, the third number will be shown by the section number. For instance, if someone wants to know what number is the third one in the proportion of two to four, he should take the middle number, four, and multiply it by sixteen. Then, he should divide the result by two. The first number will be the quotient, and the third number will be octonarius (eight). This property of proportional numbers always accompanies them, not only when they are considered by themselves, but also when they are applied to any objects to be numbered. The same relationship and rule applies to finding an unknown number. This knowledge will be useful for many things. In this case, the text is in Latin and does not contain any meaningless or unreadable content. It is also free of modern editor additions or OCR errors. Therefore, I will simply output the text as is:\n\nQuo fit: ut innumera, quae vulgus imperscrutabilia exexistimant: ex proportionum similitudine veniant in notitia: miraculumque rudibus explicata praebent.\n\nETVT a facilibus exemplis primum ordiamur. Quando tritici modij tres triginta nummis uentent: investigamus: octo modiorum tritici quanto erit precium? Hic palam uidemus duas esse proportiones. una de tribus tritici modijis ad octo tritici modios. Alteram de triginta nummis, trium modiorum precio cognito, ad octo modiorum precium adhuc ignotum: quod eadem habitudine ad triginta nummos respondeat: qua octo modij ad tres modios se habent. In omnibus autem proportionibus in ordinem designandis,\na sinistra incipientes tendimus in dextra: queeadmodum in scribendo Graeci, Latini, et omnes pene gentes faciunt. Non autem sicut in numeris per notas signa Chaldeis sequimus a dextra ordines. Itaque in hoc exemplo. tres tritici modij, qui prioris proportionis initium, a sinistra primum locum occupare debent. Eight measures of wheat, which mark the limit of the first proportion: placed to the right. Thirty nummi, which mark the beginning of the second proportion: hold the third measure: it is necessary.\n\nYet another location is still vacant: so that the proportion of the second is closed with a similar end: the price is deposited there: which responds to thirty nummi, as if to three measures and two thirds of measures of wheat. You can easily find this by the rule above explained: if you divide the third number by the second number, and the number produced by the first. Therefore, thirty, which are in the third place, multiplied by eight, which are in the second place, make two hundred and forty. If these are divided by three, the number of sections is eighty. The number of prices, eight modii, is indicated by this, and is placed in the fourth location, completing the subsequent proportion.\n\nSo when three measures of wheat are worth thirty nummi: eight will be worth eighty nummi. In matters where two proportions are present in numbered affairs: of which one the extremes, be it each end, we know neither the beginning as much as the end, nor the end where it ceases. But we know the first extreme's beginning. However, its end we yet do not know. Let us place the beginning of the first proportion, known to us, in the first place. And let us place the end of the second proportion, unknown to us, on the right. The beginning of the second proportion, known to us, let us note in the third place. And only then will its end correspond to the beginning of the first proportion, as the rule previously shown, be investigated.\n\nThe beginning of a proportion (to avoid ambiguity), I call the number that comes before in the proportion, whether it be greater or smaller. The end, on the other hand, is the one that follows, whether it was smaller or larger. If the beginning of the posterior proportion is known, and the number is smaller than that to be investigated, the number in the prior proportion will also be smaller. But if the beginning of the posterior proportion is known to us and the number is greater, the same will hold true for the prior proportion. Likewise, if a greater number of things is present in the prior proportion, a greater number will also precede in the posterior proportion: it is necessary. Conversely, if a smaller number precedes in the prior proportion, the same will hold true for the posterior. However, it is important to note that similar proportions will not hold if this is not the case.\n\nHowever, it is important to observe carefully: whatever the prior proportion consists of, the posterior should take in something else of a different kind; but the first number, which is the beginning of the prior proportion, should be on the left, and the second number, which completes its end, should be on the right. The third number, known as the beginning of the posterior proportion, and the fourth number, which is to be investigated and found to be the end of it, should agree in numbering the same kind of things. This text appears to be written in Old Latin, and it seems to discuss proportions and their relationships. Here's the cleaned text:\n\n\"whatever thing is to be formed correctly will benefit most from having proportions resembling each other. For example, as given previously. In this case, the first proportion is of three parts to eight modii. The second proportion is to be investigated concerning thirty numbers in relation to eight modii. It is true that the first and second number of this kind of things should number: and similarly the third and fourth should respond to each other, not only in the proportion's similarity: but also in the conversion of proportions. When the first is compared to the second, the third is compared to the fourth. However, if the proportions are compared one to another, that is, the beginning of the first to the beginning of the second, and the end of the first to the end of the second, then the first and third number of this kind of things will number, and the third and fourth will do the same. For instance, if you compare three modii to thirty nummi, and eight modii to the price to be investigated.\" Quemque mode similaritatem proportionum fabricandi, although difficult for many who teach to calculate: more observe it now. For those who ignore a simpler method for forming right proportion and similarity, they seem inspired by merchants' practice: who value all things by money, and ask among themselves: what is the value of each merchandise? what is the value for each single one, what is the value for all? They make comparisons of other things mainly to money, not to the things themselves. However, whichever way is followed: the rule given will answer. If the comparison of proportions is made right or permuted, it will always be in agreement with the rule. We, however, wish to instill in young people the habit of following only the right proportion comparison in each of the following questions, where both extremes of the proportion belong to the same species: to make it easier to understand and more natural. In singular things: it is not necessary to insert anything that would cause displeasure to readers. Therefore, before all else, great caution is required, lest the similarity of proportions, which is considered according to the reason of numbers, be incorrectly conceived, or dissimilarity deceive us for similarity's sake. But once the correct proportion of similarity has been grasped through contemplation, the rest is easily explained by the given rule.\n\nFurthermore, the common people in their speech usually inquire about number first. For instance, if someone asks, \"How many modii of wheat will four modii be worth when they are sold for a hundred nummi?\" Considering this question, we immediately perceive two proportions. One between 100 modii and 4 modii, which is known to both extremes. The other between 1000 nummi and the price of 4 modii, which is still unknown to us. Therefore, the crowd, in their questioning, pays no heed to the order of the questioner, but rather whatever comes into their mouths, as with whatever is put into their mouths. We consider the proportion between one hundred and four modios, which the common people confuse: both extremes should be noted, and the former should be given precedence. The latter proportion, regarding a thousand nummi with an as-yet-unknown price, should be considered posterior. The former number is known, as it is greater. Therefore, we note the beginning of the former, which is one hundred. The smaller number, four, follows in the second place. In the third place, we note the beginning of the latter: it is one thousand.\n\nBy this reasoning, we arrange proportions in order according to the given rule by multiplying one thousand by four, yielding four million. This, when divided by one hundred, results in forty thousand, the number sought.\n\nIn this example, if someone prefers to compare proportions more closely by permuting them: they will compare them thus. \"As one hundred modii are to one thousand nummos, so four modii are to the price to be investigated. Let the middle number be the third, and the third number be in the middle. The difference is so small between the true proportion and the permuted one. However, if we wish to follow either of these comparisons, let us not look at the order of the vulgar. Rather, other tortuous comparisons, which can be brought back to the true proportion, can be sought.\n\nBoth examples have preceded in each proportion: the price follows, which was the question. Now let us take an example: when eighty nummos of wheat are bought for eight modii, how many can be bought for thirty nummos? Here are two proportions. The first is of 80 nummos to nummos. The second is of eight modii to the number of modii.\" Et quia in posteriori proportione, numerus modiorum, known as the greater one, is investigated. Which is greater than eighty numbers in the prior proportion. The smaller number of numbers, second, third, however, occupies eight modiorum numbers: which is known. The question is about the number of modiorum that is unknown. Arranged in order, the numbers are as follows:\n\nThirty multiplied by eight equals two hundred forty-five. If these are divided by eighty: the number of divisions will contain three. So, when numbers are reduced to eighty modios, thirty numbers will bring three modios. Therefore, if you compare the first example given with this: you will see that the places, both internal and external, are reversed in this example. The fourth number there was; here it is the first. The third number there was; here it is the second. The second number there was; here it is the third. The first number there was; here it is the fourth and last. quae illic proportio priora erat: hic posterior reperitur: ut facile cuius appareat res numeratae ratio numerorum et conversoru sequi: nihil que res numerentur, dummodo proportio illa, cuius in rebus numeratis ambo extremitas sunt nota, praecedit: illa que, cuius initium solum notum est, sequatur.\n\nAnd if anyone should experiment with it: whether an error has been admitted in the investigation of the fourth number, this method makes it easy to detect and, as in the proportion, convert the places and even the extremities of both. For example, in the given example, which has everything contrary at first. Or if you prefer. Change only the seats of eru, and from posteriore to priori, and from priore to posteriore. For, with proportions changed simultaneously in place, and even in only one place, if nothing is erroneous, it is necessary to follow the numbers of their seats changed. Therefore, if other numbers are discovered in these, the entire work must be repeated from the beginning due to the calculus error. The common practice of merchants with regard to this experiment, called the change in proportions among these, refers to the rule of three being converted: since there is only one rule, which can be used by anyone to pose the question at hand. If it happens that: when the number of things to be added in the third part is a divisor of the sum: the divisor does not leave a remainder: the smaller number should be multiplied by the number of parts it contains to ensure that the divisor's section is larger and can receive the increment. It is also necessary to consider: what kind of things the smaller number counts. For example, is it the number of pounds of silver or gold, which can be multiplied by the numbers in which they are valued? Or are larger weights involved, which are counted in pounds or ounces? Or are amphorae of wine or oil, which are multiplied in the number of measures such as anphorae, congeries, or sextaries? Or are iugera, which can be increased in the number of perticae or feet? Or whatever is similar. I will give an example of this. Centum oues emptae sunt aureis viginti: quantum constiterunt tres? Proportionibus recte in ordinem redactis, such is the case.\n\nSecond number, a triad, when drawn into the twenty-first, creates sixty. Since it cannot be divided by a hundred: let us divide the gold coins into smaller denominations. And since each gold coin (speaking of angels) is valued at eighty nummi: let us add twenty nummi to that number of nummi, which will yield four thousand eight hundred. This number is greater than the hundredfold: through it, cut in the number of parts, will yield approximately four hundred and eighty.\n\nSo when, 100. cows are bought. 20. aurei, 3. will constitute. 48. nummi. Not at all does that seem odd: {that} in the second proportion, the end does not begin, that is, the fourth place in the third does not respond: {that} this is nummi, he has aurei; since each place contains the same kind of thing, it does not contain money. However, even though money is greater in the smaller denomination, it is divided in order to take the divisor's section as a number to be divided. If the remainder of the VBI AVTEM division is anything: what the divider can no longer divide further: similarly, the counter is divided into its own parts, if it has any of this kind: another who is to be divided at his own discretion, is multiplied: so that the divisor, who is the denominator, can bear being made larger. Once this is done, the number of the partition will manifest: how many parts, whether noted or to be named, the remainder itself can sustain. As we said in the superior book about estimating the remainder. For example. If nine ells of cloth (as we call them in merchant terms) consist of three gold coins: how many ounces does one contain? Arranged in order: nine in the first place, one in the second, three in the third.\n\nSince unity multiplies by nothing: only what remains is: that the third place of the three gold coins is to be divided by the first number, nine. Since this cannot be done: the gold coins must be reduced to a lower currency, let us say, 80 numms. For they are estimated thus by the Angels. When nine units of gold are brought in the count of numbers, 240 will be produced. When they are divided into twenty-four parts according to a new method, there will be 26 parts, and the remaining ones will be 6/9. Since the known parts are quadrants, four of which are equal to one, we take six for four, and they will become 24. Again, this method produces two more parts in the division of one part by a new method, leaving 6/9 of one quadrant. Since smaller quadrants do not have known parts among us, we can name any parts we please, for instance, four; we take six, and the remainder is in four. These, when divided again by nine, will produce two. And the remaining 6/9 of one fourth part of one quadrant is still left. If one wishes to divide it further into smaller parts, it can be further divided. Therefore, when nine ounces of gold are bought, one ounce is made up of 26 numbers, two quadrants, two quarters of one quadrant, and six nonas of one quarter of one quadrant. When reduced to 6/9, they amount to 2/3. We have deemed it necessary to admonish this location: for, as anyone multiplies or divides unity, it neither increases nor decreases him in any way; rather, it retains its own place, requiring no division. For instance, if unity occupies the first place, and sixteen pounds of pepper are comprised of it, what will the price be for one hundred pounds? By arranging the numbers in order, the third number's price in the second centenary multiplication is completed without any division. For example, when unity occupies the second place, if a thousand gold coins redeem six captives, how many of them can redeem one according to that ratio, marked by the following numbers:\n\nAfter the third thousandth, only a division by the first sixth is required to complete the transaction. If unity holds in a third place, let it be made similar:\nIf proportions are arranged in order in any of the three places where minutiae intervene: the integers of that place, if any are mixed in:\nImmediately reduce it to minutiae. This is done: if the number of integers is multiplied by the denominator of the added minutiae: and the numerator is added to the product. With this increase, the denominator is subtracted below the line. As we showed in the superior book, regarding the resolution of integers into minutiae, we command. In that place, however, where no minutiae intervened: the integers, like minutiae, are written under the line as a unit: so that all places refer to the form of minutiae. Then, the multiplication of the second to the third and the division of the product is made: just as it is commanded regarding minutiae.\nAn example of such a thing will suffice. When three ulnae are worth six shillings, according to merchant terms, how many shillings and pence are three quarters? Integrating the first part, there are seven and two-sevenths. The second part contains only small change or simple small change mixed in, or whole coins offering themselves: before we begin the work, everything must be reduced to one form of small change. As we showed in the superior book. Then let there be multiplication and division: as it is customary in small change.\n\nIf anyone is going from one region to another far away to avoid danger from robbers, he may wish to give money of the nation from which he profits to the innkeepers there, so that he may know their estimation of the money in that region for his journey, and how much money he will have to spend there. This can be easily learned in the following way. First, convert a larger sum into smaller change through multiplication, but reduce a smaller sum into a larger one through division, almost manifestly. This is worth noting. A person desiring to exchange money first observes: how many smaller coins of that region, where the exchange takes place, does the messenger of one region carry with him for the journey. He must agree with him on this. Once he has observed this, he should reduce the entire sum of money that will be given to the messenger, whether it be in silver pounds or gold coins, to the same number of smaller coins as each part, for example, one pound or one gold coin, is valued. This will be done in such a way: if the total number of coins to be exchanged is the same as the number of coins of the same value in the region, the person should multiply the number. Then, he should count the number of gold coins he will receive from this by dividing the number of gold coins in the estimation of one region's coin, which he is preparing to carry, by the number of sections in the estimation. The reason for this will be clear to us through the rule of the four unknowns presented by the fourth party, as we will soon explain in an example.\n\nA person going to Rome gave a thousand gold coins to messengers, each one estimated at eighty smaller coins, so that they could return to him gold coins in Rome. In singles quinquaginta quattor numos, Romae singulos aureos ducatos redendi. One wishes to know how many ducatos are to be redeemed in Rome? By multiplying one thousand angels by eighty thousand numbers, which number is the representation of one aureus: eighty thousand thousand arise. This sum is easily converted into ducats: if one considers, What is the proportion between fifty-four numos, which are worth one ducat in a contract, and eighty thousand thousand, for which one thousand angels are valued? The same proportion should be investigated between one ducat and the number of ducats. Therefore, arranged in order by proportions, it is as follows.\n\nSince the second number, eighty million, is not augmented by the third unity's number: the division of the second only by the first quinquagennial number will suffice. quae in numero partitionis prodire faciet mille quadringenta octuaginta unum. Et viginti sex numi supersunt. Qui numerus est aureorum, Romae ex pacto reddendus.\n\nET QUAMVIS in pecuniarum permutatione, quae ab alia regione in aliam frequentatur: pecunia aliqua uilior plurimum quaeritur: qua utriusque nationis pecunia major secundum sui precium prius aestimetur: quam ad proportionem regulam iam datam numeri applicari possint. Veluti in exemplo proximo. Quis modus et facilis est: et experimentatus. Interdum brevior via reperitur. Quapiam quisquis argenti libraum, aut aureorum, aut cuiusque pecuniae numerus illius regionis, ubi fit permutatio: alicui vel libraum, vel aureorum, vel cuiuslibet pecuniae numero, in ea regione: in quam commutamus: ita ex pacto respondet. Ut nihil omnino vel desit: vel supersit. Tum absque ulla in uiliores numos reductione facienda, statim ad proportionem regulam numeros applicabimus.\n\nExempli gratia. Tres aurei angeli pro quinque coronatis gallicis valent. \"Considering proportions, we find that for every three golden angels, there should be the same number of coronated ones. Or, if you prefer, we can examine this proportionally: the same ratio exists between three golden angels and five coronated ones, as between one thousand angels and the number of coronators. Therefore, following the rule, we discover that one thousand angels yield one thousand six hundred and sixty-six coronated angels.\n\nThree merchants entered into partnership, contributing their respective shares of the money. The first brought ninety gold coins. The second, sixty. The third, fifty. And, eagerly engaged in trade, they earned one hundred gold coins in profit. Desiring to divide the profit among themselves, they are uncertain: which portion of the profit should each receive according to their initial investment.\" This text appears to be written in Old Latin, and it discusses the distribution of profits in a common fund. Here is the cleaned text:\n\nHanc quaestionem altius nobiscum considerantes, confestim animadvertemus id, quod in omni societate semper considerandum est: lucrum hoc commune centum aureorum ex communi omnium accreuisse pecunia. Et quemadmodum pecunia communis commune lucrum peperit: ita suam cuiusque pecuniam id attulisse lucro: quod ad quemquem propriam pertinebit. Quare, sicut omnium partes simul additae se habent ad singulorum partes, ita omnium commune lucrum ad lucrum singulorum se habere debet. Et permutatim. Sicut omnium partes simul additae se habent ad commune lucrum, ita partes singulorum ad singulorum lucrum se habere debent. Ita haec Harrisonem ad regulam quatuor proportionalium applicantes, statim eam explicabimus. Quod ergo primus locus omnes omnium partes in societate collatas simul aditas habere debet, secundus autem suam cuiusque pecuniae partem separatim aliam sub alia notatam, tertius vero commune lucrum. Post each number in the ordination, the money collected separately should be brought to the common fund. And the number from this should be divided by the total sum of all parts. Thus, the number of each partition will reveal the profit of individuals. This should be established in the fourth place, in the region of the partition concerned, as indicated by the following notices:\n\nIn all the rules of the society, it may be checked whether this has been correctly calculated: a sample may be taken from here. If the profits of individuals, when added to the common profit, yield the common profit's total sum, MINVTIAE also, if any remain: they should be collected and added to the number of the whole. And whatever is created from these should be added to the number of the whole. For example, if the first contributed forty, the second thirty, the third fifty, profits of forty-nine are made from the common money.\n\nAll minor details in this example are summarized into one whole. And the profits of individuals, when added, make forty-nine. Thus, fifty is produced, which responds to the common profit. The first person brought forty-four gold coins to the company and stayed for four months. The second person brought thirty-two gold coins and remained in the company for six months. The third person brought twenty-four gold coins and stayed for four months. In total, there was a profit of eighty gold coins. It is debatable: what proportion of profit each person should receive, according to the time and amount of money invested. In this dilemma and similar ones, another observer first notices this: we have stated that in every partnership, the superior rule is to consider the common profit. Indeed, just as all parts are proportionate to the parts of the whole, so each person's share should be proportionate to the profits of the individual. And just as there is a comparison of the whole to the common profit, so there is a comparison of the parts to the profits of the individuals. However, this comparison has a double aspect, both unequal in time and unequal in money. quorum utrumque ad cujusque lucrum excutium pari cogitationem requirit: nec alterum ab altero separari potest. Quandocas nipecunia cujusque sine suotempores, cujusque tempus sine suapecunia. Immo sicut omnium pecunia simul addita atque omnium tempora coacta, commune lucrum universis perit: itaque cujusque pecunia connexa cum suotempores, cujusque lucrum dedit. Itaque cum de pecunia et tempore cujusque, quae separari non possunt: gemina comparatio simul fieri debebit, tam ad aliorum pecunias et earum tempora: quam ad commune lucrum, et lucrum cujusque suum: pecuniam cujusque per suum ipsius tempos multiplicare oportet. Quo fit: ut numerus ex hoc productus proportiones partium suarum, eorum inquam numerorum: ex quibus in se ductis entas, comprehendat. Et enim secundum EVCLIDIS scitum, proportio inter duorum numerorum compositorum est ex lateribus suis producta proportionibus. Latera numerorum Evclides vocat, quorum multiplicatione numeri producuntur. Post multiplicatione cuiusque temporis cum sua ipsius pecunia separatim facta, quoniam singulae pecuniae partes cum singulis temporibus multiplicando coeuerunt: unusque numerus, sic de utroque factus est, ut de gemina comparatione proportio una virtutem utriusque complectens prodiverit. Numeri ipsi procreati sunt notandi, ut ad regulam sociatis applicari possint. Quippe sicut omnium pecunia cum omni tempore coniuncta se habet ad suam cuiusque pecuniam copulatam cum suo tempore: sic lucrum omnium communi primus locus detur. Secundus autem assignetur numeris separatim enatis ex unius cuiusque pecuniae ductu in suum tempus: ut separatim alius sub alio notetur. Tertius vero locus lucrum communi partum habeat. Deinde separatim secundi loci numerus cuiusque producatur in commune lucrum: quod tertii loci est. A number of men from this community were divided according to the first place number. As it was stated in the next rule of the society. And the number of the section will declare: how much each one is owed according to the amount of profit and the quantity of money. In the fourth place, in the region where the money is to be deposited, it is to be done in this way.\n\nFour merchants entered the society for a two-year term, pacifying: so that each one's profit would increase according to the portion of money contributed. The first one contributed thirty gold pieces at the beginning: and after eight months had passed, he took ten out of the middle. Again, at the beginning of the twentieth month, he added twelve gold pieces to the society. The second one brought twenty-four gold pieces at the start: and took eight after the sixth month. Again, at the beginning of the sixteenth month, he contributed fourteen. The third man entered the society immediately with twenty gold pieces: and after seven months had passed, he took all his money out. Again, at the beginning of the twelfth month of the eighth year, he added sixteen gold pieces. Quartus in the seventh month contributed twelve and eight gold coins for himself, and abstained from nine of them after the fourth month. Again, in the fifteenth month of the following year, he added fifteen to the partnership. Of all the money in the partnership, there were a hundred gold coins earned. It is uncertain how much profit each person received after two years. This uncertainty arises not only from the complexity of the matter, but also from the variation in time. For those closely examining the matter, it is clear that, just as the common money was combined with all the times, so too was each person's money combined with its own time, bringing its own profit. Therefore, the specific times when each merchant had the largest sums, whether they were communicated in the partnership or exempted from it, must be added to their own funds separately. Quem quampe conjunction of wealth with time, as it will be settled with each merchant: four proportional rules will regulate the entire matter as we explained more fully in the previous chapter. This will be sufficient for the learned. However, since the importation and exportation of money in society present a greater complexity in the foreground than in the background, and since various aspects of these transactions may cause fear due to their diversity: we will clarify this point by providing a guide, similar to what we have shown before, for those who may encounter such issues.\n\nA merchant, for instance, had thirty gold pieces at the beginning and eight partners in the society. Therefore, when the time came for the money to be combined with it, thirty-two hundred should be drawn. These should be kept separate until other transactions are completed.\n\nAfter the eighth month, ten gold pieces were subtracted, leaving twenty for the remaining eleven partners. Thus, twenty-two hundred should be added. Similarly, the following should be noted separately:\n\nAfter the twentieth month, ten gold pieces were subtracted, leaving twenty for the remaining eleven partners. Thus, twenty-two hundred should be added. In the twenty-second month, those who remained in the society: there are twelve. They were paid twenty gold coins. Adding these two sums together made thirty-two gold coins. These thirty-two gold coins were distributed among those who had completed five or six months in the society. Therefore, five of these thirty-two were taken out and distributed: one hundred and sixty were added. These sums, noted separately, were then added together: six hundred twenty were produced. This sum rose from the first mercators and the months that coincided.\n\nThe second merchant, however, joined the society at the beginning with twenty-four gold coins. Therefore, six were taken out and distributed: one hundred forty-four were noted separately. After the sixth month had passed and eight gold coins had been subtracted: sixteen were left. Therefore, nine of these sixteen were taken out and distributed: one hundred forty-four were noted separately. In the sixteenth month, fourteen gold coins were added to those already in the society. Sixteen more were added in the thirty remaining months. Therefore, nine of these were taken out and distributed: two hundred seventy were produced. quae ad reliquas summas seorsum notatas addita producet. 558. This is the sum for both time and money of the second merchant acting alone. The third merchant joined the partnership immediately from the beginning. 20. He brought in 20 aures for six months and left them in the partnership. Therefore, 140 would be produced if they acted alone. After the seventh month had passed, when he had paid off all his own money: he had nothing in common with the others until the tenth eighth month. In the beginning of this month, he re-entered the partnership. Seven months later, the sum multiplied by the number of months left produces 112. This is the sum for both time and money of the first merchant acting alone, noted separately. 252. What is the sum for both time and money of the third merchants acting together? The fourth merchant entered the partnership in the seventh month. 18. He brought in 18 aures and kept them in the partnership for four months. Therefore, 72 would be produced if they acted alone. After the fourth month, subtracting nine aures, they had 9 left until the end of the tenth seventh month. restabat meis bibliothecis sex volumis. Esta causa, a sexta eductionem receperunt. In novem, quae in societate restabant, seorsum et ipsa notanda. Decimo septimo me ipse incipiens, ad novem quae in societate restabant, adiunxi. Quae facit, per octo mensibus relictum, multiplicata educunt. 192. Quae ad alias summas seorsum signatas addita, perfecerunt. 138. Haec summa est tuis meis, tum pecuniarum quarti mercatoris. Iam vero singulorum mercatorum temporibus cum suis pecunijs coniunctis, superest: ut omnium summarium simul annotatae addantur. Inde surgent 1748. Quae summa ex omnibus tum pecunijs tum mensibus omnium mercatorum collecta primum locum tenebit. Secundum vero singulorum summis tum pecunis quisque tum mensibus complectentibus, seorsum aliam sub alis notatae. Tertium autem locum commune lucrum habebit. Deinde secundam regulam quatuor proportionalium multiplicatio et divisio fiat. Et lucrum in quarto loco patefiet, ad hunc modum.\n\nCentum pondo ab Alexandria in Angliam deportatum reddunt pondo septuaginta quinque. Quantus reddet Alexandriae two hundred and thirty-six pounds sterling? Those who have considered it can easily provide an answer. What is the proportion between 75 pounds sterling and 236 pounds sterling: the same as between 100 pounds sterling and the number we are inquiring about from the Alexandrians. And in reverse. What is the proportion between 75 pounds sterling and 100 pounds sterling: the same. 236 pounds sterling should therefore be placed before 75, then 236, and then 100 pounds sterling. According to this proportion, 236 pounds sterling should render Alexandria 319 pounds 15 shillings. How much profit in gold will be made in a century from 236 pounds sterling? This can easily be explained. For what is the proportion of a hundred-digit number to 79: the same as that of a twelve-digit number to the profit sought. Et permutatis, quae proportio est numeri centenarii ad duodecim: eadem erit de septuaginta novem ad lucro investigandi. Dispositis itaque in ordinem numeris, primum locum centum teneant, secundum septuaginta novem, tertium duodecim. Quattuor proportionalium regula demonstrat ex aureis: 79. lucrum fieri aureos novem, et 48/100, quae sunt 12/25.\n\nHoc lucrum per omnia par est usura centesimae. Quam minor apud Romanos leges non permittunt. De singulis namque centenis, duodecim quotannis in foenus usura centesima exigit. Quae ideo centesimam nomen habet: quod cum in menses usuram debentur, pars sortis centesima singulis mensibus in usuram venit, centesimoque mense sororem foenus aequat. Intra centesimam minores usuram interdum centesimas dominantem, interdum bessem, in menses reddunt. Quae etiam ipsas graviores existimantur. At mediocres putantur, quae eius semissem in menses pendunt. Quae vero non ultra centesimas trientem, aut quadrantem, aut sextantem in menses pariunt: civiles habentur, et humanas. \nMERCATOR EX AVREIS SEPTV\u2223aginta per menses tres lucri fecit quin{que}. qua\u0304\u2223tum lucri tredecim mensibus ex aureis septu\u2223aginta obueniet? Duae proportiones deprehenduntur. Altera de tribus me\u0304sibus ad me\u0304ses. 13. Altera de quin{que} lucri factis ad numeru\u0304 inuestigatum. Nam sicut tres me\u0304\u2223ses se habent ad menses tredecim: sic quin{que} lucri facta ad numerum quaesitum. Et permutatim. sicut tres menses ad quin{que} lucri facta se habent: sic. 13. menses ad numeru\u0304 in\u2223uestigatum. Quare quatuor proportionalium regula du\u2223ce, primo loco statuentes. 3. Secu\u0304do. 13. Tertio. 5. com\u2223periemus aureos viginti unum, et duas tertias. 13. men\u2223sibus lucri fieri. \nQVI PER MENSES QVATVOR EX aureis nonaginta lucri fecit quin{que}: scire cupit: quanto tempore ijdem aurei lucri centum red\u2223dent? Haesitatio haec proximae similis est: praeter{quam} {quod} hic de tempore illic de lucro fit inuestigatio. Simili igitur mo\u2223do soluenda est. Quippe du\u0119 proportiones apparent Five gold profits result in one hundred gold profits: thus, in four months, five gold profits have increased: this is investigated to the number of months. And step by step, five gold profits result in months. Four. Secondly, 100. Thirdly, the rule of four proportional gold profits. 100 from 90. profits are made in months. Eighty reveals this, which number of months creates six years and eight months.\n\nA merchant wishes to know, with three months out of ten, how many gold profits can be made from six hundred gold with the same ratio?\n\nNo hesitation is more frequent than this one among merchants: while they deliberate among themselves about which merchandise genre they should invest their money in, they ponder how much profit they can acquire in what time. \"This matter first requires consideration: what is to be done when in two proportions the difference in value of money is met with a certain difference in time, so that the money which increases in sum over time produces a complex number from both. This number then compares with profit through these parts. How this is to be done: so that the proportions can be reduced to order immediately, as we mentioned in the previous questions. For we have considered two proportions. In one, both extremes are known. In the other, only one. For instance, the proportion between ten gold pieces bound to three months and six hundred gold pieces bound to ten months and eight iugs: whose extremes are known: the same ratio of profit should be four gold pieces known for that profit, which is unknown in the other.\" The proportion is between ten connected gold pieces and a three-month period, and the known profit of four auctioneers: the same should be found between six hundred connected gold pieces and a ten and eight-month period, and that profit. By multiplying the number of gold pieces of the ten by their three-month period, thirty result. Again, if ten and eight are added to six hundred gold pieces, the number of their months will emerge as ten thousand and eight hundred. Four proportional rules will then explain this to us in the following transactions: if thirty is in the first place, one thousand four hundred and seventy-five gold pieces will result from ten months and eight profit. Therefore, we find that from six hundred gold pieces, ten thousand eight hundred and seventy-five gold pieces are made in ten and eight months.\n\nThis has been investigated: how much the profit increases.\n\nMerchants frequently turn over such a question: in order to avoid uncertainty regarding the time. If someone made a profit of four gold pieces in three months out of ten, they may want to know how many months they could make a thousand four hundred and forty gold pieces from six hundred. In this question, the profit in both proportions is uncertain. However, the time in one of them is still unknown. Therefore, multiplication of money by an unknown time cannot be done; it is necessary to apply both proportions to the four proportional rules once. So, another way must be tried. And those considering this should first look at the example of the profit of four gold pieces, which grew from ten gold pieces in three months: it seems necessary to find out how much profit of six hundred gold pieces would yield in three months. This can easily be discussed:\n\nIf we consider these two proportions present here, one is of ten gold pieces to six hundred, and the other is of the profit of four gold pieces to the profit we are seeking. Since ten gold pieces have the same ratio to six hundred as four gold pieces do to the profit we are seeking, we can find the profit by applying this proportion step by step. \"Just as ten gold pieces bring a profit of four gold pieces: so six hundred gold pieces should be investigated for profit. Placed first, second, and third. 10. Second, 600. Third, from the rule of four proportions, we will learn that three months' profit can be made from 600 gold pieces. 240. After this was discovered, it was continuously found out: how many months a thousand four hundred and fifty-five gold pieces can make a profit from six hundred gold pieces. For the proportion of two hundred and forty gold pieces to four hundred and forty gold pieces, which make a profit in three months, is the same as that of thousand four hundred and fifty-five gold pieces to the number of months, which was asked about. Therefore, placed first, 240. Second, 1440. Third, from the rule of four proportions repeated, we will be taught: thousand four hundred gold pieces.\" When the profit is made in months, it is twelve and eight. This period, which was to be investigated in detail, had to be determined so that the remaining part could be easily found. The rest of this kind of examples can be carried out in the same way. It is clear that there are four proportional rules, since one attempt is not sufficient: it must be visited twice: in order to bring everything to light.\n\nWhen a measure of wheat is sold for ten nummi: this is the primary note of the price of bread: which should have a high value: it will weigh forty-three ounces. Secondly, according to this ratio, what will be the weight of one loaf of bread for one nummus: when a measure of wheat costs thirteen nummi? It is necessary to observe where the received law of baking is: so that the price of the wheat is increased: and the weight of a loaf of bread is decreased accordingly. For just as the abundance of grain increases the price and size and weight of the loaves: so the scarcity of grain reduces the price and weight and size. Therefore, it is necessary to investigate how much the reduction in weight should be for a higher price ratio. For the given text, I will assume that it is in Latin and needs to be translated into modern English. Here's the cleaned text:\n\nAccording to the proportion, since one extremity is greater than the other: the larger one should be placed first, so that the question is about the smaller extremity of the other. However, when both proportions have a common denominator of ten numms, the one with the known extremes should be placed first. In the case of the question about the number of pounds to be investigated concerning forty-three unciae, the former should be followed. For in this matter, the excess of the number of numms thirteen over ten is equal to the excess of forty-three unciae over the number of pounds to be investigated. Thus, the same proportion holds. And in the same way, thirteen have a ratio to forty-three as ten does to the number of pounds to be investigated. Therefore, if thirteen occupy the first seat, ten the second, and forty-three the third, the rule of four proportionalities will fully illustrate the matter. By this rule, we will find out that a modius of tritici weighs thirty-three and one third pounds.\n\nIf one pound of wheat weighs forty-three ounces, how many pounds does a modius of tritici weigh when ten pounds make up?  quanti ponderis ad eam ratione\u0304 erit pa\u2223nis unius nummi, si tritici modius nu\u0304mis septem u\u0119neat? Hic quia tritici precium est imminutum: iuxta panificij le\u2223ges pondus panis augebitur. Quocirca qua proportione nummi. 7. superantur a. 10. eadem. 43. unciae supera\u2223buntur a pondere panis inuestigando. Et permutatim. si\u2223cut. 7. se habent ad. 43. sic. 10. ad numeru\u0304 ponderis in\u2223uestigandum. Quare in proportione priore de. 7. nu\u0304mis ad nummos. 10. extremum quod minus est praecedere, quod maius est: sequi debet: ut posterior proportio de pa\u2223num pondere, desinat in pondus maius. Ita digestis in or\u2223dinem proportionibus, per regulam quatuor poportiona lium co\u0304periemus unius nu\u0304mi pane\u0304 pendere uncias. 61 3/7\nquando tritici modius septem nummis constat. \nPRIVSQ VAM ad alias questiones progredimur: de lucro et damno quasdam regulas, operaeprecium pu\u2223tamus adiungere. Et ut de lucro primum dicamus. Mer\u2223catores ad hunc modum secum quaerere solent. Si caerae pondo centena aureis. 12. quispiam emat: at{que} ea deinde aureis 14. How much will the profit increase from hundreds? This is explained as follows. First, subtract the price at which the item was bought: from the price at which it is later sold. The remaining difference will indicate the profit: which is determined by chance. Of the twelve items, twelve will contribute to the profit. Two will yield how much profit: two hundred. We will leave the remainder in the number of hundreds, which is 16 2/3. Thus, such profit will increase from hundreds. For the proportion between the twelve items and the hundred, it must be the same as the proportion between the profit made from the twelve and the profit that should come from the hundred. And the proportion must be the same between the twelve items and their profit, which we are investigating. Similarly, one could inquire about denarii, ventises, trices, quadragesimas, quinquagesimas, or milia. A merchant, in his business dealings, particularly considered the ratio of profits and losses in hundreds. Therefore, we provide examples regarding hundreds in the following merchant's rules. In these merchant's regulations that we will present: we will arrange the comparison systematically, according to merchant custom, as the correct comparison of these will reveal themselves at once to the diligent.\n\nYou will find the calculation correct if you arrange the order of proportions in this manner. For instance, if 100 represent the total gain or loss, 16 2/3 represent the gain from one transaction, and 12 represent the loss from another transaction, what will 100 yield in total? We will explain the following rules to clarify this.\n\n116 2/3 represent the total gain and loss. If you subtract 100 from those who make this total, 16 2/3 remain as the gain.\n\nIt is up to your judgment, reader, which path you prefer to follow. Merchants themselves take one path or the other.\n\nIf 12 represent the total, they become 14 in total gain and loss. What will 100 yield in total from this amount? We will discuss the following rules to clarify this.\n\n116 2/3 represent the total gain and loss. If you subtract 100 from those who make this total, 16 2/3 remain as the gain. AD DAMNVM cognoscendum. This, which we have advised for profit: will suffice for the diligent. For nothing else matters, except that, just as the sale exceeds the purchase price, the profit grows; or, when the sale is reversed, it either lessens or exhausts. However, we will give an example of this as well. When a pound of saffron was bought for three ounces of gold and sold for two, what loss will there be from the hundred? Subtract the value of the lesser-seen thing from the purchase price, which is greater, and the remainder is the damage. Therefore, if three ounces of one thing remain, what will be left from the hundred? Bring two into the hundred, they become two hundred. If three of them secede, 66 2/3 remain. The difference between 66 2/3 and 100 is 33 1/3, which comes to the detriment.\n\nMerchants often calculate in this way when going to markets. Quanti podo cetena cuispia mercis puta terreni ertus emeda suet: ut aureis.\n13. postea vidita ex cetenis denarius in lucrari pariat? Huismodi quaeris expedies: si tecum consideres. Qui ex centenis denarius lucri volupt: is. CCC. augere et ad CCX studet. id, quod ex precio 13 efficere molitur. Itaque in illis 13 et sors et lucrus latet. Alioqui nisi esset lucrus: denarius ex centenis nolucrifieret. Ea propter cuus per venditione aureis facta, lucrus ex centenis destinatus sit acquisitus: sors quae latet in 13 per regulam de tribus notis ignotum proferentibus statim sic patefiet. Si CCC quae sortem et lucrum capiunt: prodeunt de sorte. CCC de qua sorte nascentur. 13 quod sortem et lucrum tenent. Multiplicare oporet. CCC in 13 et producentur. XXXI. quod si per SC centuris: numerus sectionis. XI 9/11 habebit. qui numerus sortium illarum demonstrat. Atque ideo tantae emeda essent terreni ertus podo. CCC ut postea 13. aureis vedita lucru proportionione denorum ad centena reddant. If we want to check an example: is it correctly assumed that: we should convert the order of proportions, as follows. If from 11 9/11, which are in pure chance: they emerge. 13. sortes et lucrum: what does 100. pure chance yield? we can test it by the rule. And we will find 110. from which the pure chance has been subtracted, 100. profit remains. If it returns more or less: there is an error. Another exception. How many pounds are a thousand sheep worth, which were later sold for 50 gold pieces according to the rule given, how much profit is made per hundred? Apply the rule as follows. If 112 were the profit before it was made, what is 100 according to that ratio? 50 were beforehand: with which profit was it made? The supposition according to the rule will teach you that it was made with 44 72/112, which are 9/14. Another example. What is the weight of a libra of crocus: when is an uncia worth nummis? 9 unciae are sold for a profit, what does it yield in proportion to denarii per centena? We find the unciae's worth. If 110 came before the profit, what were they worth? 100 we will find were worth 8 2/11. To determine the number of unciae in a libra: multiply the number of unciae in five. There are 8 librae and 2/11 unciae in a pound. Therefore, there are 98 and 2/11 unciae in the pound of crocus that was bought before.\n\nWho bought the gem: sold it later for gold. 100. And when he began to count his money: he realized he had lost. 10. From 100. How much did the gem cost at the beginning?\nConsider this carefully. Who lost 10 from 100. Reduced it to 90. Therefore, they should investigate the outcome. 100. In this way, he will think. If 90 were there before the loss. 100. What were they worth according to that ratio? Bring them forward. 10000. They yield 111 and 1/9. The gem was worth this much at the beginning.\n\nOn the opposite side, in reverse order, you can conduct an experiment: whether it was correctly calculated. If 111 and 1/9 are the only ones left: what remains of 100? If you follow the rule: you will find out. 90. If more or less was involved: the error is manifest. This text appears to be written in Latin, and it seems to discuss the distribution of profits and losses between different parts of a lot or a gamble. Here's the cleaned version of the text:\n\nAlia etiam via per lucrum aut damnum ad sortis partes redigere possimus. Si quis vendens nummis lucrat ducentis de centum, quid sit sortis pars: hoc modo expedietur. Ad quam partem sortis lucrum est? Manifestum est, si lucrum per sortem divisum in minutias solvitur, sic nascentur quae ad minima nomenclaturae redactae faciunt, ita qui ex centum lucrat decimam, decima sorits pars lucrat. Quod pars decima addita ad sortem fit undecima totius summae. Quisque de lucro et sorte unam decimam capiat: lucrum illius summa est. Quod quidem ab ea subtractum erit: sola sors restabit. De ducentis sumpto unam decimam invenies. Unam nonam et undecimam tantum inest lucri in exigua hac summa. Si autem a ducentis subtractum esset: restabunt pro sorte duodecim duas et octo.\n\nEa sors illa est. Ducenta. If you have this proportion: if you convert the order in this way, they become 18 2/11. What will increase from 100. in tetas: and you will find 110. subtract for chance. 100. remain for profit. 10. The calculus did not err.\nThe same way through damage, the dice itself can bring the parts to the share of the dice. For example, let us suppose: who with the number 20, says: to lose one-tenth from hundreds. He puts down 10/100. what he makes. 1/10. This should be considered. He loses one-tenth of 100. from his own share, 1/10. Therefore, they are saved for him 9/10. of which parts if you take 1/9. and add it to all the parts left out: you will restore the sum: which was before the damage: and you will have what you sought. And for this reason, in the given example, if one selling 20 numbers loses one-ninth of his own share, i.e., 2 2/9, and adds it to 20, they become 22 2/9. quem gladius emptor posed for certain price in the market: what if he had bought it for two gold coins less; or if he had sold it for twelve gold coins, he would have made a profit of ten denarii from hundred denarii. What was the value of the gladius at the beginning? You can answer such questions in this way. They draw twelve. Twelve in 100 produce. Two hundred and twelve, which secta through 110 returns. Ten and 10/11 would have been the lot: if nothing else had intervened. But since something was added: if he had bought it for two less gold coins; and sold it for twelve, his profit would have been greater in proportion to the cost. Ten and 10/11 he would have paid. And therefore, two should be added to make it complete. Twelve and 10/11 should be taken away in the beginning of gladius weight. One thousand constituted it. From which it is clear, if he had bought it for two less gold coins, it should have been given for ten and 10/11. That was the question of the quaesors: what profit could be made from selling it for twelve, with the price increasing denarius by denarius from hundred denarii. Fragmentum: emptor modios quaternos certum precio emit, quibus si pluris nummis quaternis quam sunt empti, constiterent et venditi fuissent nummis tricenis senis, denarii de centenis fuissent perdita. Quanti igitur constiterunt modij quaterni? Principio, sicut proxime in lucro factum est, sortem illorum. 36. erues, hoc modo te consideras. Si 110. erat, 100. quid secundum eam rationem ante erant. 36. ducantur in 100, et fiunt 3600. Quae deinde secunter per 110, et prodierunt 32 8/11. Tanti emi debuissent modij quaterni, ut uenditione facta nummis. 36. denarii ex centenis lucri fierent. Verum hic perdita sunt 10 ex 100. Atque ideo, sicut in iactura ante docuimus, sic considerabimus. Si 90 ante iacturam fuere, 100 quid secundum eam rationem ante fuere, 36 ducantur in 100 et surgent 3600. Quae secta per 90 reddent. 40 tanti emi debuissent, ut uenditi nummi. 36 danari addebat detrahens denarii ex centenis. Ceterum, quia dictum est, si 4 nummi amplius ad precium accessissent. The text reads: \"they sold that loss, which the hundred denarii were made up of, in coins. 36. It was followed by a loss of 40 coins: because they were bought for fewer. Therefore, it remains. 36. For how much the price of a modius of salt for seniors was sold: he makes a profit of eight from a hundred. What can be asked: if he sells that salt modius at the same price, how much profit will he acquire from a hundred? Many are deceived in such calculations: two give six, eight what will they give. Eight hundred, in eight, and the yield is produced through six. Six cutting off eight, they demanded. 10 2/3. What profit from a hundred was thought to come: if the price of a salt modius is muted from six to eight denarii. These were all mistaken, for that which they, as it were, put down as profit and loss, contains both. Therefore, wherever in thousands the question is asked about profit or loss, the first sort is found: what profit and loss it produced.\" If we draw lots and the dice, which contain six, are brought out: let this be done in the open as we have shown before. If there were 108 before profit: they existed. There were 100. What were there according to that ratio before profit? Six should be taken. Six should be in 100 and become 600. Sixteen will be what is drawn from the hundred and falls into the lot and profit. Five and 5/9 should be the extent of their share. Six lots, made up of six, should yield a sale of eight in a hundred, which profits from the hundred. But in this matter, if someone has polluted the coins with six: he has a desire in his mind to see with eight denarii: and he wants to know how much profit will increase from the hundred. This can easily be found out if the proportions are thus arranged. If five and 5/9 are the shares: eight will fall into the lot and profit. If you take eight from 100: divide the product by five and 5/9. This will be done so that they fall in the number of sections. 144 will be the extent of the small share. 100 will bring forth in the lot and profit. Therefore, if you subtract 100 from that sum, 44 remain for profit. tantu\u0304 lucri ex ce\u0304tenis accrescet: if each salt modius numbers 8. uendant\u0304, who were ante. 6. sold,\nOR, if you wish to act another way immediately: to ask for the sort, this will be done in this way. Add to the aforementioned 100. lucru\u0304 ex his accresces: summa\u0304 divide that beforehand, what lucru\u0304 that was lost. Or, that which in the number of sections was given: multiply it in the future by the price: in which it becomes a change. Thus, the number produced from this and the lucru\u0304 de 100. na tu\u0304 and sorte\u0304 ipsa\u0304. 100. co\u0304plectet\u0304. If from this lucro si sorte\u0304. 100. de\u0304mas: lucru\u0304 de ce\u0304tenis ex novo p\u0304cio obuenies restabit. Quod via non quam fallit. For example, in the given case, where the seniors' allowance of 100 numis was made from ce\u0304tenis octo, and lucru\u0304 introduced 8, add 108. Ea\u0304 summa\u0304 divide prius p\u0304cium 6. quo lucru\u0304 illud acquisitum est. Thus, in the number of sections, 18 proceed. Then, in the future, multiply it by the price: 8. in quod fit co\u0304mutatio: they should be taken. Thus, they will be produced. 144. quet et lucru et sorte ceternus capitunt. A quibus, si 100. pro sorte demas: resistent pro lucro. 44. Tantum lucri de centenis accrescit: si precium a nummis senis ad octos mutet. Siue aute de centenis, siue de alio numero, puta quinquagenis, millenis, aut quovis alio, certus lucrus quaeratur: regula huc invariata manet, ymmorchidze aute sua capit a multiplicatione et diuisione: quod a quatuor proportionalium regula supra data, et ab unitatis natura dependit. Porro in diuisione, sicut unitas se habet ad divisore: sic sectionis numerus ad numerandum. Et permutatim. Sicut unitas se habet ad sectionis numerum: sic divisor ad dividendum. Similiter etiam fit in multiplicatione. Nam sicut unitas se habet ad multiplicandum: sic multiplicandus ad numerandum producendum. Et permutatim. Sicut unitas se habet ad multiplicatorem: sic multiplicator ad numerandum producendum.\n\nTherefore. Have 1. in sorte: ea ratio. 108. Habereet. 18. Id quod sic exquiris. Si. 6. in sorte et lucro proceed (let us find). 1. of which are born. 108. which are in sorte et lucro. let it be brought. 1. in. 108. and thus the number will be produced. Then it is to be divided. 6 and the number of sections. 18. will declare: which of them, according to the same ratio, should belong to that [number]. 108. Now indeed, with the price changed, if you want to know how much they will give. 18. investigate in this way. If one is made from eighteen, how many from eighteen? Multiply. 8 in eighteen and the result will be 144. which lot and profit will hold for hundreds.\n\nDamnum ratio according to this rule will be executed: excepting that, just as gain is added to sort, so loss is subtracted from sort: and the number remains the same as before, which loss brought about: it will be cut off. Whatever number it finds in the division: two are required in the division: in which the change takes place. Thus the produced number and sort will indicate and the damage, whether it is required of hundreds, or of thousands, or of any other number of lots.\n\nFor example, who. 6. numis vendit: duodecena decem centis perdit: quid iacturae ex centenis sustinebit, si precio diminuto. 4. numis uendit de centum. Subduci debent centuris duodecim et restabunt 88. Quae centenis centuris prius illud precium sustinuerunt. 6. quod iactura perit et prodibunt in sectionis numero. 14 2/3. Ea autem multiplicetur in minus pretium. 4 in quod fit commutatio et 58 2/3 surgeat: quae sortem et iactura centenorum demonstrabunt. Quot ita ab eo numero usque ad centum desunt: totidem de centenis fuerunt. Frequentent etiam mercatores in merce quodlibet podere, numero, aut mensura videre solita, post copulas expensas terra marique supra emptio preciis factas, iuectilia, navula, victua, ceteraque id genus, certum sibi lucrum de quodlibet podere, numero, aut mensura, destinerunt: secuque deliberaverunt: quodna pcium statuat in singulis podere, numeris, aut mensuris: ut lucrum illud accrescat. Id quod fit ad eum modum: hoc exemplo demonstrabimus. Piperis podo mille nummos Lusitaniae. 10000 suet emptera. He put his vestige there, illic, where the pepper was pressed. 1000. Naulu\\_ inde there established a cost in Britannias. 300. Again, another vestige was exacted there. 500. A vehicle made of wood, numis, stood there. 200. Ministers were sent in this negotiation. 2000. The merchant brought merchandise, of pepper pounds, above the oes imposed numbers. 4. He was determined to profit: Yet he hesitated: how many pounds of pepper did the profit weigh: so that it might be added. Thus, all expenses, purchasing, vestige, another vestige, vehicle made of wood, ministers, all in one sum, must be considered. And it amounted to. 14000. Of which, four parts, we must add: we shall see the numbers. 4000. Thus it grew. 18000. That sum, in total, the merchant desired to receive for himself in the negotiation, both for sorte and lucro. Therefore, the four proportional rules, which concern the three known and four unknown, were presented above: one vestige will open up more than one pound. Na\\_sicut universum piperis libra advecta sed hent ad totam illam summam pciuatque lucrour oium collectete: Sic una piperis libra ad suum precio et suum lucro se debet. Quocirca si 1. ducatur in 18000, idem numerus produceret, qui si per numerum universarum piperis librarum, ut 1000. Secet numerus sectionis precium unum libra nummos. 18. Coctinebit. Ad euode modum pciuetque lucrour statuere licet: Hic, in his, numero aut mensura uidebant cosuert.\n\nSix merchants bought merchandise and loaded a ship. The first merchandise consisted of twenty-five. The second, forty. The third, fifty-six. The fourth, sixty-four. The fifth, seventy-five. The sixth, one hundred. Then, a grave tempest arose during navigation, and the ship, heavily laden, was on the verge of being overwhelmed by the heavy waves: The sailors, seeking their salvation, threw overboard the heaviest merchandise: which, for one hundred and twenty-five gold coins, would have been valuable otherwise. Quo factus est, ut seruata nauis reliquum onus in portum destinatum adueheret: mercatores, quorum erant eiectae merces, postulant, ut ceteri, quorum bonas servata sunt, communiter sustineant. Id quod omnibus aequum uisum est. Ceterum, ambigua quomodo pro rata cuiusque bonorum portione collatio ea fieri debet. Hanc haesitationem explanabis statim nobis occurrunt duae proportiones. Altera de mercium omnium in navim imposita summa ad cuiusque merces impositas. Altera, quae secundum illam adaptari debet: de communi omnium iactura ad cuiusque iacturam sustinendam. Nam ut omnium merces impositae ad cuiusque merces se habent, ita omnium iactura ad cuiusque iacturam se debet habere. Et permutatim. Ut omnium merces impositae ad communem omnium iacturam, ita cuiusque imposita merces ad iacturam cuiusque. Quocirca this hesitation can be easily explained through four proportional rules: first, let us establish an estimation for all goods as a whole. Second, let us note down the estimation for each good separately. Third, let us pool the common yield.\n\nFollowing the rules: we find the first merchant. 8 and 120/360 make \u2153. Compare. Second, 13 and 120/360 make \u2153. Third, 18 and 240/360 make \u2154. Fourth, 21 and 120/360 make \u2153. Fifth, 25 and 120/360 make \u2153.\n\nIf you want to know: whether the calculation was in error, add up all the individual comparisons. If they match the common yield, the subtraction was correct; otherwise, an error occurred elsewhere. If a debtor has debts to multiple creditors, the debtor should divide the good assets in proportion to the debts owed to each. This is so whether the debtor transfers the assets, converts them, or after the debtor's death, the assets are not reached by an heir other than the one entitled to them. The division should be made in the same way as we have said regarding the distribution of merchandise for sale. For example, let there be eight creditors to whom the debtor owes a thousand. The first owes fifteen, the second twenty-four, the third thirty-two, the fourth fifty-four, the fifth sixty, the sixth seventy-five, the seventh eighty-six, and the eighth one hundred. The goods of the debtor that are to be distributed among them do not exceed one hundred and fifty-five. We find two proportions immediately. One is of the total debts owed to all creditors, in relation to what each creditor is owed. The other is of all the debtor's goods in relation to the portion each creditor is entitled to. \"As debts owed to each individual are owed to all in regard to that debt: what is owed to one person, so their assets are to be applied towards the portion of payment that each creditor is to receive. And this is done gradually. As debts owed to all are to the assets of the debtor, so is that debt, which is owed to any one person, to that portion of their assets, which the creditor is to receive. Therefore, the four proportional rules will provide a swift resolution: first, let us place all debts owed in one place, due to all creditors. Second, let us note down that debt owed to any one person separately. Third, let us sum up the assets of all creditors. Then, let us apply the rule's instruction: we pay the first creditor an amount of 20/446, the second creditor 8 and 32/446, the third creditor 13 and 340/446, the fourth creditor 13 and 72/446, the fifth creditor 20 and 80/446, the sixth creditor 20 and 100/446, the seventh creditor 20 and 412/446.\" The testator left 833 and 282/446 parts. The following condition must be met, with no errors in calculation: if the person to whom parts come responds to all the debts of the debtor with his own property.\n\nA certain man, whose wife was pregnant, left 1,000 gold coins as an inheritance. Thus he made his will. If my wife bears a son: he shall be my heir. But if she bears a daughter: she shall be my heir. If my wife bears a son: the son shall be my heir. But if she bears a daughter: the daughter shall be my heir. The wife then gave birth to twin boys and a girl. It is uncertain: in what way the inheritance should be divided: so as not to contradict the wishes of the deceased. Those who have pondered this matter more deeply have first discovered the testator's will: that the son should have more than the wife, and the wife more than the daughter. Itaque cum filius partem matris duplam esse debet, quod is duas hereditas partes, mater tertiam sit habens, et mater partem ad filiam, quod ea duas consequetur, et filia tertiam: ita fiat, ut partis filii ad filiam portio quadrupla sit futura. Quisque numerus exigitur, qui simplicem, duplex, et duplex duplex, quod quadruplum ad simplicem est, habeat, ut totidem partes hereditas dividi possit. Minimus autem numerus, qui eas habet, capiendus est, quo facilior sit suppositio. Is autem est septenarius. Nam unum duplicatum duo profert, iterum duplicata quatuor, quae cuncta simul addita septem faciunt. Et quia mille aurei in septem partes dividendi sunt, quarum quatuor filius debet habere, duas matres, una filia. For the given input text, I will clean it by removing unnecessary symbols and formatting, and making the text readable in modern English. Here's the cleaned text:\n\nIf for each person their due shares are given:\nLet us execute them immediately, if we notice:\nWhat is the proportion of all parts added at once?\nThis is the proportion of the sevenfold number to each part:\nIt is necessary for one thousand gold pieces to be equal to that number for each person. And gradually.\nThe proportion of all parts added at once is to one thousand gold pieces:\nIt is necessary for the portion of each to be in proportion to the number of gold pieces to be investigated.\nTherefore, applying the rule of the four proportional quantities, we will explain the whole matter:\nFirst, if we place the number of all parts in the first place.\nSecond, the part of each person, marked as another.\nThird, one thousand gold pieces:\nThis is the entire inheritance, in this way.\n\nFollowing the rule, we will find that the sons take 571 and 3/7, the mother 285 and 5/7, and the daughter 142 and 6/7.\nIt will be evident, there is nothing wrong, if all the parts respond to the whole inheritance when added together. If a woman bore three sons and two daughters: a common occurrence in Egypt: women giving birth to twins: the division should be made accordingly: each son should have twice the share of his mother, and the mother should have the same share as each of her daughters. Therefore, the number to be taken into account would be: one that has simple twins, their doubles, and their triples. The minimum number having simple twins and their doubles is four: one who has triples: twelve. Added to these, there are sixteen. The inheritance was to be divided into how many parts: so that each son receives a quarter, the mother two, and each daughter her unique share. The rest is to be distributed according to the rule of the four proportional parts.\n\nA man making a will, who had sons as many as five and three million gold pieces as an inheritance: he made this kind of will. My firstborn son shall be the heir. My second son, the next in age from three. My third, from four. My fourth, from the fifth part. My fifth, from the sixtieth. Ambiguitur: how should an inheritance be divided? If we follow the words of the testator: the first son should receive half, five hundred and five hundred gold coins; the second son should receive a third, one thousand gold coins; then the third son should receive the entire fourth part, seven hundred and fifty gold coins; the fourth son, to whom the fifth part is due, and the fifth son, to whom the sixth part is due, will receive nothing at all. Therefore, the sons cannot inherit the designated parts from the father in their entirety: other smaller parts must be investigated, which respond to the designated parts as a share: so that each son may be satisfied. First, the designated parts should be sought out: they are easily found if the remaining sum is divided among the heirs individually by denominators. Therefore, the number of sections will contain a part by the designated portion from the father. To find a half, we will have five hundred and five hundred gold coins. A third, one thousand gold coins. A fourth, seven hundred and fifty gold coins. A fifth, six hundred and sixty gold coins. Six thousand. These numbers, which should be recorded one under another: so that the smaller numbers among them may be paid out in proportion to the sum of three thousand remaining in the inheritance. But if they had sons: they would be satisfied with what belonged to each one according to his wish, and they could not ask for more from this father's will. These numbers, collected together, make up four thousand three hundred and fifty, which the father designates as a whole. However, since the inheritance left behind is of a smaller sum than three thousand: it is necessary to consider how the partition should be made according to the testator's will. If, after these parts have been determined, we carefully scrutinize: we will find two proportions. One is from the parts added to the whole, which is designated for each son's share from the father's designation. The other is from the entire inheritance left to that part, which each son will receive according to the designation for his share. This should be formed according to that first one. The proportion between the parts assigned and added together, and the part of a son designated by the father: the same proportion should exist between the entire inheritance left behind and the part of the son entitled to it, according to the portion he received as his share. If one wishes to compare them item by item.\n\nJust as the designated and added parts of the whole inheritance belong to the entire inheritance left behind: so the part of a son designated by the father for that share of the inheritance, which a son would receive according to his designated portion: he should possess. After the parts of the entire inheritance have been first noted down: it is explained by the rule of the sum of the remaining proportions of the inheritance. From which it is clear that the first son obtains 1034 out of 4350, the second son 689, the third son 517, the fourth son 413, and the fifth son 344. Certain proof is rightly calculated to be: if after a heritage partition, all the children's shares amount to three million in gold. We have extracted this hesitation using common calculation methods. However, those who delve deeper into arithmetic: they find a shorter way. We will hand over this matter to the following question.\n\nFive associates divided three million profits among themselves according to the contract as follows: the first one has one part, the second one has one third, the third one has one fourth, the fourth one has one fifth, and the fifth one has one sixth. They hesitated, however, about how to make a fair division. In such cases, you must determine a certain number to answer this question: one who comprehends all the designated parts. This is the smallest number: the number that the part designers named. This is explained in the superior book. Therefore, in this question, the smallest number that contains all the proposed parts will be found to be sixty. In this number, each proposed part must be inquired into separately. \"This is divided separately through each of the sixty-seven sections of the number itself. To make this happen: the number of the partition should find a part of sixty-two. Third, twenty. Fourth, fifteen. Fifth, twelve. Sixth, ten. When all these numbers are collected together, they create the total of eighty-seven. For an example of these parts, from sixty: the portions of the individuals, from a sum of three million gold coins, must be made. For just as all the numbers of this section, which sixty divides into equal parts, have an equal addition to each of their parts: so the sum of three million gold coins, to that sum which each part is entitled, should belong. And each number of this section, to that sum which anyone acquires from the partition: should belong to it.\" Ite if all numbers in the first section of sixty number, let them hold the seats together: there are 87. Each one of them, separately, in their own year: The third part should be divided by three million in gold. Four proportional rules will extract the business. The parts coming from the section of each one will be responsible for the division of the sum, if the calculation is correct.\n\nQuod quartum amicorum habebat, primo legavit novem aureos in testamento, secundo viginti quattuor, tertio amplius quam secundo, quantum secundus supra primum habebat, quarto amplius quam tertio, quantum tertius supra secundum caperet. It is asked how much the third and the fourth respectively have. Considering the proportionality continuous in this, we find that, just as the first legacy is to the second, so the second is to the third, and again, just as the second legacy is to the third, so the third is to the fourth. According to the rule of the third proportion, as explained above, the second legate, which is repeated twice in the proportion, should be multiplied by itself; and the number resulting from this should be divided by the first legate. Therefore, the number of the third legate will be brought to light in gold. 64. And again, the third legate, which is repeated twice in the continuation of the proportion, should be multiplied by itself; and the number born from this should be divided by the second legate. This will make the number of the fourth legate manifest in gold. 170 16/24 which are 2/3 of one gold coin.\n\nThe division among the three gold coins of the hundred should be made in such a way that the first has three times more than the second, and the second has four times more than the third. Quonam modo, secundum pactu, iusta partitio? Facile id extricabitur: si tertio qui minima partem accepturus est, retrogrado ordine incipias, cui assignes: si libet, unitas. Et quia in quadruplo secundus tertium superabit, tertii partem quadruplice, et enascetur quatuor. Quaepars erit secundi. At quoniam primus triplo amplius quam secundus consequetur, secundi partes triplicanda est, et prodibunt duodecim. Ea pars erit primi.\n\nHumeri simul additi creant septemdecim. Secundae que summae propositae partitio fit. In totidem na partes aurei ducenti dividendi sunt: ut ex his Primus. 12. Secundus. 4. Tertius una habeat. Cumquidque aureorum numerus cui cederet, secundum regulam quatuor proportionalium exectiemus, si primo loco partes omnium simul additas ponamus, esunt. 17. Secundi cuique partem, aliam sub alia seorsum annotatam. Tertio summam. 200. aureorum dividendam. The proportion of all parts being added at once is the same for each part in relation to the total, and for the total in relation to each part, whichever part one is to have. This proportion is the same for the total and for each part in relation to it, for whoever will acquire it.\n\nTo prove according to the agreement, divide the first part into three. The number of sections of the second part will follow suit. Then divide the third part into four, and the number of partitions of the fourth part will reveal it.\n\nThree among them want to divide one hundred gold coins. The first one demands to have one third and one fourth. The second one demands one fourth and one fifth. The third one demands one fifth and one sixth. What share does each one get? To answer these questions, first inquire about the smallest number required: the one who can capture all the denominators at once. This number is sixty. From the given number, how many unique parts should come from the partition: if it is divided, it should be investigated first. This is easily found by dividing the number itself into its six primary factors, separately for each divisor. Then, the parts from the number of sections are to be found, to be added separately: so that we may know how much each part contributes. Thus, the number of the partition is from sixty, the third part. 20. the fourth part. 15. which are produced by addition. 35. The first part is revealed. The fourth part is 15. And the fifth part is 12. which are produced by addition. 27. The second part should be. The fifth part is 12. And the sixth part is 10. which are produced by addition. 22. The third part should be.\n\nTo the sixty-two parts, after the numbers have been collected from these: each portion will easily follow from the total of a hundred. Indeed, if the numbers, from which the portions of each section have already been collected and added, are placed first: 35, 27, and 22. mox omnes in unum rursus statuantur: ut sint. 84. Secundo, singuli illi separatim per additionem numerorum sectionis collecti, alius sub alio seorsum annotati. Tertio, summa centum: quod dividenda est: facile per quatuor proportionales regulas res tota extricabitur. Nam ut in uno coactis omnes illi, cuiusque numeri, qui per additionem numerorum sectionis seorsum collecti erant, habeant ad ipsores singulos: sic cetus, quod dividenda sunt, ad illas summas, quam cuique parte assignabit, debent habere. Et permutatim. Ut in uno coactis omnes illi, cuiusque numeri, qui per additionem numerorum sectionis seorsum collecti erant, habeant ad centum: quod sunt dividenda: sic singuli cuiusque numeri, per additionem numerorum sectionis seorsum collecti, ad illam summam, quam cuique partitio assignabit, debent habere.\n\nQuattro desiderano dividere seicentos aurei cos\u00ec: che il primo abbia due terze e nove aurei. Secondo ha tre quinte e otto aurei. Tertius quintus sextas et aureos septem. Quartus septem octavas, et aureos sex. Amor, quantum cuilibet confert partitio. In primis minimus numerus, qui omnes has denominationes capiat: inquirendus est. Is autem est centum et viginti. Quem numerum si per singulas denominationes secas: invenies ter tertiam eius partem, quadraginta. Quinta viginti quattuor. Sextam viginti. Octavam quindecim. Ceterum quoniam primus duas tertias habebiturus est: quadraginta illa, quae partem tertiam faciunt: duplicare oporet. et surgent octoginta. Qui numerus in duabus tertiis reperitur. Et quia secundus tres quintas habebit: viginti quattuor, quae quintam faciunt: triplicanda sunt. Et fiunt septuaginta duo. Qui numerus in tribus quintis reperitur. Et quia ultra tres quintas octo aurei secundo debentur: adduntur octo: quod creant 80.\n\nThis text appears to be written in Old Latin, and it describes a method for calculating the sum of money owed to two individuals based on their respective shares of certain denominations. The text mentions the numbers six, seven, eight, ten, twenty, and forty, and it instructs the reader to multiply certain numbers by specific factors to arrive at the total sum. The text also mentions the numbers 80 and 89, which may represent the total sum owed to the first individual and the total sum owed to both individuals, respectively. The text appears to be complete and does not contain any meaningless or unreadable content, so no cleaning is necessary. Therefore, I will output the text as is. The sum total of those following: what the second seeks. In addition, because the third is to follow the sixth: twenty, which complete the sixth: five are needed to carry. And they will amount to one hundred. The number of these in the sixth is found. Since the third asks for seven gold coins more: seven should be added. And there will be one hundred and seven. What is the sum of all these: what the third is to pay in thirtieths. But the fourth, since he demands seven eighths: fifteen, which make seven eighths: seven should be added. And they will amount to one hundred and five. The number of these in seven eighths is found. To this number (since the fourth still requires six gold coins) six should be added. And there will be one hundred and thirteen. What is the sum of all these: what the fourth will have. Here are the numbers: which whoever has. If all are added together: three hundred eighty-seven will result. According to this example of numbers collected in such a way, a six hundred and twenty gold coin division should be made. Just like all numbers that appear in designated parts by themselves: when joined with any given gold addition, they hold a relation to each individual number. For instance, 600 coins, which are to be divided: they should have the number assigned to them. And each of those numbers, which are among the designated parts and the gold addition, hold a relation to 600 gold coins: each number of the individual numbers should have that relation. Therefore, the numbers, which are among the designated parts and the gold addition, should be placed first in a group. These numbers, which arise from the designated parts and the gold addition, make up the total. 387. Next, the individual numbers, each marked separately. Third, the sum of 600 gold coins: immediately, the amount that each part will increase will be shown by the rule of four proportions. And if all the parts, which are increasing, are collected, the sums to be divided will be equal. Therefore, there is no error. This text appears to be written in Old Latin, and it seems to discuss a method for dividing a sum of money among three people using a specific rule. Here's the cleaned text:\n\n\"ESTO, inter tres ea lege diuidendi sint: ut quoties primus quinos sequitur, secundus senos capiat. Quoties secundus septenos habet, tertius sumat nouenos. In hac partitione ambiguitur: cum pro quibusque primis secundus senos habet, quanto debetur tertio. Rem inuolutam sic extricabis. Considere, si septeni aurei secundi tertio nouenos pariunt: facile sciri potest: quot aureos tertio seni secundi referent. Nam sicut septeni secundi se habent ad senos eiusdem, sic noueni tertij ad summam, quam is habiturus est, se habeant debere. Et permutatim. Sicut septeni secundi se habent ad tertij nouenos, sic secundi seni ad summam, quam tertius est consequuturus. Quar in hac partitione, quia alioqui exercitati labi non vidimus, iuvenes esse duximus, ne errarent: ut postquam primi et secundi numeros in centum multiplicatos per divisorem separaverint, cum ad tertium numerum uenit, integra in minutias eis adiunctas statim redigant, ut fiant.\"\n\nTranslation:\n\n\"According to this law, the shares for the three of us should be determined as follows: whenever the first one follows five, the second one should take six. Whenever the second one has seven, the third one should take nine. In this distribution, there is ambiguity: since for each set of first and second numbers, the amount due to the third one is uncertain. You should extract the hidden part as follows. Consider, if the sevenths of gold for the second person amount to nine for the third person: it can easily be determined: how many gold pieces the third person is entitled to from the second person's sevenths. For just as the sevenths of the second person are equal to their own sixes, so the nines of the third person should be equal to the total amount that he will have. And in reverse order. Just as the sevenths of the second person are equal to the sixes of the third person, so the total amount of the second person should be equal to the nines of the third person who follows. Therefore, in this distribution, since we have seen that the trained ones did not err, we assumed that young people should be warned: after the first and second numbers have been multiplied by one hundred and separated by a divisor, when the third number comes, they should immediately reduce the fractions to their smallest parts, so that they may become whole.\" quas deinde in centum multiplicent, thus arise 5400/7. In this place, since the details to be divided and the divisors have the same nominator, the greater numerator of those to be divided is the divisor's minorem, integrated, and the divisor is the dividendus: so that the number may be reduced to the third. Otherwise, since the same integrorum minutias are numbered by both divisor and dividendum, namely sevenths, if we put the divisor's denominator, that is seven, in the dividendum's numerator, and the divisor's denominator in the divisor's numerator, both numbering the same integrorum minutias, we would reduce the divisor and dividendum to the form of a cross: then the divisor and dividendum, which contain the integrorum septimas, would be reduced to septimas septimas. And when we have settled the first two sections concerning simple minutiae, namely the integrorum septimas, the third, which should agree with them, would be concerning the minutiarum minutiae: we would change the divisor for the whole partition. Four among them desire gold coins in this way: the one who takes three should have as many as four. The second, as many quintans as the second. The third, as many sextans as the second. The third, as many septenans as the fourth. It is uncertain how this fair division will be made. We have seen many trained men apply themselves to this stone, yet none of them, upon discovering the solution, were able to divide the sum proposed according to their shares. The whole matter will be settled in this way. At first, the shares of each are sought after equally. The one we have asked for next has been shown to you. As for the second part, which is to determine the portion for making the first, it is easily recognized: one accepts three, another accepts four. But the third portion, which is to have as many sextans as the second, does not present itself readily: since the second does not take five, but four. Therefore, what should be considered is: if sextans coins bring five gold pieces, how many gold pieces will the quaternions yield. et per quatuor proportionalium regulam, tertia portionem aureorum prendere 4 et 7/12. Quarta pars, quae tot octonos debet habere: quot tertia septenos sumit: cum tertia portionem 4 et 7/12 non septenos, capiat: ad hunc modum exquiritur: ut uidendum sit: si septeni pariunt octonos: quot ad eam rationem ex 4 et 7/12 obvenire debent. Quatuor proportionalium regula monstrabit quartae portionis: 5. et 17/35. Demum, postquam exquisitae sunt singulorum partes: omnes in unam summam per additionem colligantur et prodibunt 605/35. Quae divisoris fungentur partibus: et primum locum tenebunt. Secundum singulorum partes aliae sub alis seorsum annotatae. Tertia summa aureorum: quae dividenda est. Caetera per quatuor proportionalium regulam expedientur. Cum autem ad quartae portionem uenit: postquam ea in centum multiplicata est: quia minutiae productae denominatorem habebunt eundem: quem divisor: ideo numerator earum solus per divisoris numeratore, integrorum ritu, secetur. The legion has six thousand one hundred pedites and one hundred equites septingentos. Let's find out how much each pedite and eques receives in stipendium per day. The pedites receive four nummos sestertios, and the eques receives nine. If the same ratio is to be followed for the two million aureorum prize, how many pedites and equites will follow?\n\nWe must first multiply the number of pedites by the number in the stipendium, which results in 6,534. This sum must be recorded separately. The same must be done for the number of pedites, which results in 24,400. These two numbers, when added together, will yield a total of 30,934. This sum, if discovered, could not be divided between equites and pedites without any business dealings. Therefore, from these two sums, this is what is proposed: the sum of the two million will be divided. The proportion is the ratio of the number of producers to each other, derived from the number of knights in their stipends: the same ratio will be found between the number of two million proposed and the number of knights born from their herds. Similarly, the ratio of the numbers of producers to the proposed number of two million: the same will be the ratio of one of the knights born from their herds to the number of knights who will acquire. Therefore, to first deal with the knights: we should first add the two numbers produced together. Second, the number produced from the knights' stipends. Third, distribute the two million. Then, using the proportional rule, the number of knights required will be found to be 422 13852/30934. That is, 6926/15467. This sum is what all knights together will acquire. If this sum is later divided among the knights: then each knight will have to take his share accordingly. Et quo minore ea est: quam ut per equitum numerum secari possit: in uiliorem pecuniam multiplicanda est: ut sectionem subire valeat. De peditum portiones investigandae, similiter agendum erit. Nam quae proportio est amborum producerum numerorum, ad alterum ex ductu peditum in sua stipendia natum: idem reperietur propositi numeri duorum millium, ad eum numerum, quem pedites ex prada consequentur. Et permutatim, quae proportio est amborum producerum numerorum ad numerum propositum duorum millium: idem erit alterius ex ductu pedium in sua stipendia nati, ad numerum, quem pedites ex prada habuerunt. Caetera eodem modo expedientur, quo de equitibus dixi. Et peditus portiones erit 1577 17082/30934.\n\nAdequatus modus distributio fiat: si inter canonicos et alios quavis templi sacerdotes pecuniam dividere oporteat: ut quoties inferiori sacerdoti tria obveniunt: canonico accrescat quinque. In a mill, there are found thirteen, twelve of which grind thirty-nine modii of wheat in twelve hours, sixteen in the same time, and eight. A peasant comes to the mill with twenty-four modii of wheat, wishing to add them all at once so that each one may finish its task in one moment, and he wants to know from the miller how long it will take, how much each millstone should be subjected to, and when they will all stop. The first question is easily explained: if we add all the modii together, there are thirty-nine. Since two of the millstones grind, the proportion of one millstone to the other is 39:24. mo dios a rustico alleys: the twelve hours should have that number of hours in which business can be concluded. And they are changed, just as they are. 39. modij are added for the twelve hours: so. 24. modij for the number of hours in which business will be finished. In this way, the rule of the four proportional parts will be discovered, namely, that 20 modios can be expended by the three millstones in seven hours and fifteen thirty-ninths.\n\nAfterward, the question was asked about the time in which 24 modij can be concluded: it was investigated. Then the question was continued about the number of modij that should be subjugated to each of the three millstones: it can be continuously extracted: since, as 12 hours have to the hours, so 18 modij, which are ground by the first millstones, should have that number of modij for the same number of hours. And they are changed, just as 12 hours have to modij, 18 which are ground at that time: so hours. 7. et 15/39 to the same measure, the number of which they have been subjected to during that time. If you make a comparison with the second and third measures: what was previously unclear will be made clear by a rule of four proportions. This will be done immediately: if we place hours. 12 for the first, 7 and 15/39 for the second, and the number of measures that can be made from each twelve hours, marked separately.\n\nWe will find that the first measures contain eleven and one modius of one tenth. The second measures contain eight modii. The third measures contain four and twenty decimas of thirds. The sum of these measures amounts to twenty-four modii brought in by the rustic.\n\nA cistern full of water which has three outlets: if the largest one is opened, the entire volume of water flows out in one hour. The second outlet, which is smaller, discharges water in two hours. The third outlet, which is the smallest, discharges water in three hours. I am assuming the text is in Latin, as it appears to be a Latin passage. Here is the cleaned text:\n\n\"Iam si excludendis epistolis, omnes fistulae patere: quanto tempore totum aqua effluat? In hac haesitatione explorandum est, quia una hora integra, maxima fistula totam aquam fundit; minora duabus horis; minima tribus; minimus numerus in primis inquirendus; quem hi numeri, unum, duo, et tria, numerant. Quisquam quemadmodum sit investigandus, superiore libro docuimus. Is autem comparetur numerus senarius: qui numeros hos, a quibus numeratur, partes suas habet. Et quia intra se complectitur numeros illos omnes: a quibus numeratur, qui tempus aquae numerant per singulas fistulas seorsum exeuntibus, dividendus erit, uti mox dicemus. Postea considerandum est: sicut maxima fistula una hora tota aqua exhaurit, ita quae duabus horis id facit, una hora dimidium aquae effundit; quae tribus horis id peragit, una hora tertiam aquae partem emittit.\"\n\nTranslation:\n\n\"Once all the letters have been excluded, all the pipes should open: how long does the entire water flow? In this uncertainty it is necessary to investigate, because a major pipe discharges the entire water of an hour; a smaller pipe, two hours; a very small pipe, three; the smallest number is to be sought first; these numbers, one, two, and three, name them. Whoever investigates how it is to be done, we have explained in the superior book. However, this number, a senarius, should be compared: it has parts from the numbers by which it is measured. And since it includes all those numbers within itself: from which numbers the time of the water is measured in individual pipes, it will be necessary to divide it, as we will soon say. Furthermore, it is necessary to consider: just as a major pipe discharges the entire water of an hour, so a pipe of two hours discharges half an hour of water; a pipe of three hours discharges a third part of an hour of water.\" Since all pipes are open at once: all numbers included in that whole and its parts, which numbers count it: they must be added together at once. This will be done: if the whole is divided by the parts separately. For example, if six are divided by two, you will find three. If by three, two. Since the numeral sixteen is the smallest integer: which one, two, and three did not count: whoever has a part of the second and a part of the third: he himself must be considered an integer. Its parts, three, contain two each. These numbers, six, three, and two, added together, produce eleven. The numbers expressing the time of water flowing through all the open pipes, take in all the numbers coming out from the whole and its parts, added together at once. Here the separator will operate in this matter. Since twelve cannot be divided by six: in order that the number may remain an integer: Therefore, the number of hours, being a senarius (a multiple of six), is divided into sixty minutes, which are the parts of one hour: This number must be multiplied: in order to make the division. Thus, the number containing the time of water flowing from each individual spout separately: is divided by the number of moments of water simultaneously flowing from all the spouts. Whenever this number is found in the division of the hours, that many moments of water flowing from each spout will be discovered.\n\nFurthermore, this can be understood in this way. Since the senarius (a number consisting of six) completes every time period of water flowing separately: one spout, which draws off an hour of water, would do this six times in an hour; a spout that emits water for two hours, would do this six times; a spout that empties itself in three hours, would do this six times; and in this way, all the spouts would draw off eleven hours' worth of water in six hours. The proportion of water that must be drawn from a cistern at a time to make the ratio of water drawn to the total amount equal to 11 to 1 is the same as the ratio of 6 to 1 and the time required to draw the water. This applies according to the rule that four proportions divide the hours into 11 parts, with 6 parts in the first 60 moments, as we have previously mentioned. We will find that all cistern faucets open wide emit water in 32 and 8/11 parts of an hour.\n\nAlternatively, if you'd like, we can divide 6 into 11 parts in minute details. We take 6 parts of one hour, which the times indicated show as the time for emptying the water. The method of drawing agrees with the previous one, since there are 60 minutes in an hour, of which the upper part, 5 and 11/60, takes 32 and 8/11 parts. The lower 5 and 6/11 parts accumulate 30 parts, and the aggregated 6 and 11/30 parts yield 30 and 2/11 parts, which together make up the integral parts and 8/11. If one question has a single issue, it will be much easier to explain. There will be no need to divide the number into its parts, whenever a larger divisor is found. For instance, if a cistern has three outlets: if the largest is opened, water will flow for four hours; if the middle one, for six hours; if the smallest, for eight hours. It is uncertain: if all the outlets open at once, how long will it take for all the water in the cistern to flow out. The smallest number should be investigated first, which is counted as four, six, and eight. This number will be twenty-four. Each of these numbers has parts that count the time the water takes to flow through each outlet separately. This number is called \"secus\" in order to extract what is sought. Then it should be considered that, just as an outlet drains the cistern in four hours, one fourth of its hourly output flows out in an hour; similarly, six hours require one sixth, an hour leaves one sixth empty, and eight hours take one eighth away. Since all pipes flow at once: all numbers are contained within its integral parts, which are called its parts by those who count them: one should be added. So that the whole, which includes these parts, may be divided. And since 24 contains six in the fourth part, six in the sixth part, three in the eighth part, six, four, and three together, and these numbers are added: there will be 13. This number is the divisor of the number. 24, which shows the time of water flowing through each pipe separately, should be cut within its own parts and include the added numbers. Indeed, since the number 24 contains all time for water flowing separately: a pipe, which fills in six hours, should fill in six hours. 24 should do this six times. The pipe that pours water for six hours: should pour water for 24 hours. 24 should complete this four times. The pipe that educates water for eight hours: should educate water for 24 hours. 24 should emit water three times from it. And thus, all together, water should be educated for 24 hours by all the pipes. The ratio of water that must be drawn from a cistern at one time is the same as the ratio of 13 to 1 and of 24 to the time it takes to empty the cistern. The same ratio holds between 13 and 24, and between one and the time it takes to drain the cistern. According to the rule of the four proportions, we find that with 13 parts cutting off 24, all the faucets being open, the cistern will discharge an hour and eleven-thirds parts of water.\n\nSuppose the cistern has three faucets: the largest one empties the cistern in an hour; the middle one in half an hour; and the smallest one in a certain space of time. If all the faucets are open, how long will the cistern remain empty? This problem is similar to the one above, but here we are asking about the parts of an hour instead of whole hours. If one finds pleasure in minute details of life: as most do, to the extent possible: here those of integrity can complete everything, making a quarter hour seem like a whole. A half hour, which holds two quarters: for two wholes. An hour, which contains four quarters: for four wholes. Therefore, the smallest number, which they number one, two, and four: investigating, we find. Eight is the number that governs all time spent filling water. Next, we consider the pipe that pours water through a quarter hour: pouring it eight times: that pours water a quarter of the amount eight times: that pours water through a half hour: pouring it eight times: that drains water through an hour: drawing it out twice: and thus all pipes open, pouring ten and a quarter waters. But since in the matter at hand, water is not to be poured ten and a quarter times but once: what is the proportion between fourteen and one? It will be the same as between eight. Et aquae exhauriendae tempus et permuta. Quid proportio est inter 14 et 8? Et tepus aquae effundendi, 14 ergo numerus divisor erit. Et quia 8 numerus dividendus, qui horas quadrantes numerat, minor est quam divisor: propterea multiplicetur in horarum momenta totidem: quot horae quadrantem efficiunt. Ita surgent 120. Si per 14 secentur, partitionis numerus proferet 8 8/14. Totidem momentis cisterna vacua fit.\n\nAqua ductus per salientem implet cisternam quartuor horis. Obstruso vero saliente, fistula in cisternae fundo, si tollas epistomium, undecim horis emittit aquam. Si uas cum cisternam patet factus saliens impleat: atque per idem tempus aperta fistula exhauriat: intra quantum temporis plena erit cisterna? Ex hoc labyrintho sic te extricabis. Since the text is in Latin and there are no apparent OCR errors, I will translate it into modern English while maintaining the original content as much as possible.\n\nQuia cisternam implendi tempus, quod quatuor est horarum: cum evacuandi tempore, quod est horarum undecim: concurrit: quatuor in undecim ducere oportet: ut unus numerus de utroque fiat: qui partes omnes utriusque temporis intra se complectatur. Is autem numerus erit. Et quia tempus implendi celerius est: et evacuandi tardius: observare oporteret utriusque temporis inter se distanciam. Eam autem, si alterum tempus ab altero subducas: numerus restans indicabit. Qui reperietur septem. Ita si numerum ex utriusque temporis multiplicatione procreatum dividas per eorum inter se distanciam: numerus sectionis patefaciet: quot horas, effluentis aqua tarditatem celerritas influentis assequitur.\n\nHoraegitur. 44. Divisae per septem, in numero partitionis educunt. 6 horas et 2/7 intra quas cisterna plena erit. Sic omnes horas et implendi et exhauriendi per multiplicationem in unum numerum collectas earum inter se differentia secabit: ut quaesitum explices.\n\nTranslation:\nSince the time for filling the cistern is four hours, and the time for draining is eleven hours: the four must be led into the eleven: so that one number may be formed that includes all the parts of both times within itself. This number will be. But since the time for filling is faster, and the time for draining is slower: it is necessary to observe the distance between the two times. If you subtract one time from the other: the remaining number will indicate [it]. Seven [parts] will be found. Therefore, if you divide the number produced by the multiplication of the times by their distance between them: the number of sections will be revealed: how many hours, the slowness of the outflowing water is equalized by the swiftness of the inflowing water.\n\nHours are generated. 44. Divided by seven, they bring forth the partition [of the times]. 6 hours and 2/7 are within which the cistern will be full. Thus, all hours, collected into one number by multiplication, and their difference separated by division: will explain what is sought. Ita can be determined from this consideration: quod saliens quatuor cisternas implens una hora quartam eius partem reddit plenam. Fistula autem, quae undecim horis aquam educit: una hora partem undecimam eiusdem quartae partis exhaurit. Therefore, to know how much one hour contributes: una quarta parte pars undecima subducatur, and supersunt 7/44 remain. Quae minutiae restantes indicant: quot partes cisternam una hora implent. Nam sicut septem cisternas partes habet ad 44, sic una hora, quae septem cisternas partes implent, ad illum horarum numerum, quo tota plena erit, se debet habere. Et permutatim, sicut septem partes cisternam habet ad unam horam: ita 44 partes cisternam habet ad eum horarum numerum, quo poterit impleri. TVRRIS EXSTRVENDA EST: Cavius height will be forty cubits. Each exterior side will have a width of ten cubits. But the interior dwelling will open to the public with fifteen cubits on each side, and the redemption worker, a single gold piece for each cubit.\n\nMURVS ERIGENDVS EST: Which will be extended in length by one thousand cubits. It will rise in height to forty cubits. Its thickness will be five cubits. It will be made of hewn stones: each of which will have a length of half a cubit, a third part of a cubit in width, and a fourth part in thickness.\n\nHow many such stones are needed to build the wall? This calculation will be made as follows. The wall's length, which is a thousand cubits: in height, which is forty cubits: we must multiply. And it will rise to forty million. The number produced will again be brought back to thickness, which is five cubits: and more will come forth, five million. The entire wall's square contains this many cubits. Similar to the measurement of stones, which is based on small units: it should be multiplied within itself. It is one twenty-fourth of one cubit. The dimension of any stone that has this measurement is what is needed to determine the number of stones required for the wall. To know how many stones a wall will consume, the wall's square measurement should be divided by the stone's measurement with a fraction of 1/24. This results in 20,000. Therefore, four and a half million stones will be created. Since the square measurement of the stone is found in the walls that many times, this is why that number of stones is sufficient for building the wall.\n\nFour architects are summoned to build the praetorium. The first one promises to complete the work in one year. The second one in two years. The third one in three years. The fourth one in four years. If all of them work together on the project, how long will it take to complete the praetorium? This question is similar to the one about the cistern that has three outlets: it should be solved in the same way. The smallest number among the first should be inquired about: these numbers, one, two, three, and four. This text appears to be written in Latin and is likely related to the calculation of the duration of various projects based on the number of years. Here is the cleaned text:\n\nHunc inueniemus, duodenarium, qui numeros illos omnes, a quibus numeratur, separatim exstruendi operis tempus numerantes, intra se complectitur. This one we will find, a twelve-year man, who counts up all those numbers from which he is numbered, separating the time for each task within himself. He should be noted for this reason: so that he may join a section. That which asks for an annual payment: will give a complete and absolute remuneration with an entire year. He who asks for two years: will complete one half of the work in the first year. He who asks for three years: will erect one third in the first year. He who asks for four years: will build one fourth in the first year. Since all architects bring their works together into one: when all the numbers are taken into account in their entirety and in their parts, which are called these parts, they have been examined and added. This will be done: so that when the smallest integer number, which is numbered from one, two, three, and four, and has \u00bd, \u2153, and \u00bc, it itself should be taken as the integer number for one year. And when its half contains six, four, and three, these six, four, and three, which are called these parts, should be added, and they will make 25. This text appears to be written in Latin, and it seems to discuss the number of years required for different architectural projects. Here's the cleaned version:\n\nqui numerus tepus numerat: quo omnes architecti simul operas conferunt. Hic divisor erit et numerum illum. 12. in se atque in suis partibus numeros additos includentem, separatim aedificantium tempora complectentem, secare debet. Ita tempora separatim extruendi operis uno anno comprehensa, per tempus iunctim erigendi dividetur. Nam cum numerus duodecimaris complectitur omne tempus operis separatim extruendi: architectus, qui uno anno pr\u00e6torium edificabit: duodecim annis duodecies id absolveret. qui biennio id compleverit. 12. annis sexies id efficeret. qui triennio 12. annis quater opus erigeret. qui quadrennio 12. annis ter opus consummaret. atque ita una omnes architecti 12. annis uicies et quinquies opus absolverent. Sed quia in questione proposita opus tantum semel non uicies et quinquies fieri debet: ideo quae proportio est inter 25 et 1, eadem erit inter 12 et tempus finiendi operis. Et permutatim, quae proportio est inter 25 et 1, eadem erit inter 1 et 1. et operis finiendi tempus. Itaque per 25. secare requisitum est, et quoniam numerus dividendus divisore minor est 12, quae annos numerat: in 12 menses, quae partes sunt unius anni: ducantur et prodibunt. Quae secta per 25 reddent menses, 5 et 19/25, in quo tempore praetorium omnes consummabunt.\n\nQuaestionem facilius explicari potuisset, si unum integrum non esset admixum. Sicuti in fistularum questione diximus.\n\nDuo architecti suscipiunt opus faciendum: quorum alter triginta dies solus id perficeret, alter quadraginta. Hi, quo celerius rem finiant: tertium architectum accersum: et omnes una conferentes opera, diebus quindecim totum opus conficiunt.\n\nQuaeritur, quot diebus tertius accersitus id solus effecisset? Ut rem abstrusam eruamus, principio excutere oportere: quantum illius operis uno die, primorum duorum uterque pro parte sua fecerat. If it is easy to know this: if they had spent not more than fifteen days out of all the working days, during which only he had completed the task; let us divide them separately. So, if we subtract fifteen from thirty for the first thirty days, since the greater number is the dividend and the smaller the divisor: these details must be done. The divisor, made into a numerator, and the dividend into a denominator, will create the fractions. For instance, 15/30. They reduce this to the minimum nomenclature as \u00bd. He completed only one day's work on the first task. Similarly, on the second day. 40. Cutting fifteen from forty, we have 15/40, which makes \u215c. He completed only one day's work on the second task. Now, if \u00bd and \u215c are added together: they total \u215e. Together, they completed this work in one day. However, the amount this is: you will know: if you subtract from the day on which all the work was done, the work done by the first two. Namely, \u215e. Therefore, \u215b remains as the remaining details, indicating that one day's eighth part of the work was done by the third. Quod cum est compertum, uidendum est: quoties in quindecim illis diebus minutiae illae, quae tertio opus demonstrant, contineantur. Id quod ex uestigio per sectionem patefiet. Nam si per \u215b quindecim secentur, numerus partitionis proferet centum uiginti, quo dienum numero tertius solus opus consumpsisset.\n\nTVRRIS CVIVSPIAM PARS TERTIA sub terra latet. Quarta pars sub aqua demergit cubiti in terra sint: quot in aqua. Minimum numerum exquirere oportet: quem partium illarum sub aqua et terra tectarum denominatores numerant. Si denominator alter in alterum ducatur, invenietur. 12. Qui numerus non solum latentes illas turris partes intra se complectitur, sed totum etiam turris corpus: cuius illae sunt partes. Qua ratione partes quoque caeteras eminentes uniueras capit. Ita si a tertia et quarta turris parte reliquas separare volumus, subducamus a. 12. Ambas illas sub aqua et terra latentium partium denominationes, quae faciunt. 7. Et restabunt. 5. Partes reliquas de. 12. These parts exist above the water, which contain cubits, 60. Now, indeed, since we have learned that all the parts of the tower are, there are 12, and each of them can contain cubits, 60. It is easy to know: how many cubits there are in the whole tower. For just as the parts of the tower have a relation to their parts, which are all universal, so do the cubits have a relation to the number of cubits that is contained in the whole tower. And just as the parts of the tower have to the cubits that they contain, so do the parts, which are universal, have to the number of cubits that the whole tower holds. Therefore, designate an order for them in proportion, first, second, third, and fourth. The number of proportional rules in the whole tower is 144.\n\nThe number of cubits that a third part of them buries under the earth can easily be found: if you divide the number of cubits of all the tower's cubits into three. For the number of sections, 48, presents itself, which are hidden under the earth. Eadem facile invenire quos quarta parte sub aquis mergos tenet, si universos turris cubitos in quatuor secures. Nam numerus partitionis educet. 36. Idem si universorum cubitorum numerum, in partium universarum numerum, 144, dividas: in numero sectionis 12 surgent. Ex quo scire licet, cubitos 60, quinquies enim 12 creant.\n\nRomam ex Britannia quavispiam proficiscens, singulis diebus viginti millia passuum conficit. Septimo post die alter, iter ingressus, triginta tria passuum millia quottidie progreditur. Quot diebus illud praecedentem hic sequens assequetur? Ut rem confestim explices, spacium in primis annotare oportet, a priore coactum: id est centum uiginti millia passuum, sex diebus confecta. Quae summa dividenda est. At illud spacium, quod quotidie posterior amplius quam prior conficit: divisor nobis erit. Id autem est tredecim millia passuum. I. One hundred and twenty-three sections, making fifteen thirty-sevenths, are in the number of a league. We find nineteen days and three thirty-sevenths in this time span, with the latter following the former.\n\nA ship sailing from Britain to Palestine covers seventy thousand paces each day. However, when the night wind is against it, it retreats fifteen thousand paces.\n\nCURSOR departs from Eboracum for London in five days. But he departs from London for Eboracum in three days. Given that they both leave at the same time, how long will it take for one to catch up with the other? To determine this, we must find the time difference between the two runners by multiplying the time taken by one against the other. Thus, we get: three times five, fifteen. This is the number to be noted to determine the section. Since both parts are made at once: the time for each must be added together. And from this total, the number of sections will be determined. Therefore, we should consider this.\n\nWhoever completes the journey in five or fewer days: one day completes one fifth of it. Whoever completes it in three days: one day completes one third of it. Therefore, the numbers included in the parts of the journey, which are completed by whoever completes one day's portion, must be investigated and added.\n\nTo make this happen: so that three parts of the fifteenth are taken as three, and three and three are added, and eight result. This sum will number the time for both parts of the journey.\n\nThis divisor will be. For the number containing within itself the separate times for performing the individual journeys is cut through by the number of the single journey to be completed. And whenever this is found in that: so many days are required for the occurrence. I. Fifteen are divided into eight: the number of parts will give one day and seven eighths. At what time on the journey will they encounter each other?\n\nII. Another consideration makes it easy to see this: since in the fifteen days, each cursor (traveler) passes through itself five times, one quintupled, the other tripled, they both complete eight iterations in that time. However, since in the matter at hand, they are only making the journey once, not eight times, what is the proportion? The proportion between 8 and 1 is the same as that between 15 and the time of encounter. Therefore, according to the rule of the four proportional parts, by dividing 15 by 8, we find 1.875.\n\nIII. SPACE IS: WHAT THE FASTER traveler covers in one day, the slower one traverses in two. If one encounters another at the same time: when they meet: If you consider the matter carefully: since one day comprises the entire time for traveling for both parties within it: one day should be divided for the time for both simultaneously. This makes five parts. From this section, one day's fifth part will be indicated by the occurrence of time. This will be clear from the fact that one covers three parts of a day to complete what one does, while one completes it in half a day if one does it twice in one day. According to the rules to be sought next: if half is carried for one third, one exits with five sixths of a day: in which one spends half of one's own itinerary, and the other one third. Since they both make the journey together: their itineraries are added together: this is half to one third. The total will be five and a half. In which both complete five sixths of their own itinerary in one sixth of the day. Centerum quia in questione proposita ambos tantum semel in quarta partes itineris in quarta diei parte sunt facturi: quae proportio est inter quintum et unitatem: idem erit inter quartum. Et tempus, quod inter se occurrent. Secundum regulam quatuor proportionum, per quintum secantes quartum, in numero sectionis deprehendemus 6/30. Quae ad minimam nomenclaturam redacta faciunt quintus. Quae diei parte ambos inter se occurrent.\n\nPaterfamilias ministro quadravos aureos tradit: iubet et emi piperis, zingiberis, amydalarum, et saccharum tot libras aequali numero: quot ea pecunia suppeditare potest. Minister cum pharmacopola haelitat: quot libras de singulis generibus sumi debet. Negocium hoc sic extricabis. In primis animadvere: quanti una libra generum singulorum in pecunia minori constabit. Ea ipsa unius librae singulorum precia simul addere. Is numerus futurus est divisor. In his smaller quantity of money, the price of one ounce of gold was usually multiplied by four to get the number. By dividing this number, the entire sum will be revealed. For example, if pepper is sold for sixteen numbers, ginger for ten and eight, almonds for two, and saccharum for four, the prices add up to forty. Since one ounce of gold is valued at one hundred numbers, four ounces make up one hundred numbers, yielding four hundred. When divided by forty, this gives ten. The number of pounds for each type of goods is equal to four ounces of gold. Therefore, the ratio is clear. Since the prices of individual pounds of each type are divided by a common divisor, it is necessary that whenever this divisor is encountered in division, the same number of pounds of each type results. Avri libram quispiam minuto transfers to a lower amount from the moneychanger through exchange, and when a gold pound costs forty eight ounces of gold: an ounce should take twenty five and a half denarii; denarii fifty; sestertii one hundred; from singuliars an equal number, as much as can be had: let him be allowed to refer to himself. The minister deliberates with the moneychanger: how many of which kind of numismata should be taken in order to maintain equality. This hesitation, which we have just explained, is similar: and it should be solved in the same way. It will be necessary to find some money of lower value: what are the numismata in which exchange is made? And in each kind of numismata, which are worth more, they must be reduced to that value. If no lower value numismata are found: let us imagine a lower value: and in those nummismata which we want to be returned, let us reduce them. And since among the ancients, a nummus sestertius, which was the smallest of the desired ones, was valued at twelve copper quadrantes: let us resolve all nummismata into quadrantes. From a golden coin, four parts emerge: 1000 from a denarius, 40 from a quinarius, 20 from a sestertius, 10 from a nummus. Then all these numbers must be added together: and the total is 1070, which is the number of one kind of coin that a pound of gold contains. This separator will be used. Next, a pound of gold is divided into these quarters: and there will be 48,000. If you divide this number by the separator, the number of sections will be forty-four. For as many and no more coins of each kind can refer to a pound of gold with an equal number of minims. However, there are still some minor details, namely twenty-odd quadrantes. Since the number of each coin cannot provide an equal number, they must be exchanged at the discretion of the one receiving the payment. I take a golden coin: such were the ancients Romans. Many of whom still exist: and they are valued like gems in price. The Nobles, our satisfied ones: who respond to the ancient Romans in power: a certain amount of verification of gold was taken away: when our ancient Eduards and the verification and power of the ancient Romans were referred to.\n\nThree entered into society in a year. The first contributed sixty gold coins: he stayed in the society for six months. The second brought the sum of the gold: he was common property for seven months. The third put his gold in the middle and communicated it to the others for five months. The profit from these was divided equally. It is asked how many gold coins the second and the third brought into the society. Solve this problem in the following way. Let the amount of money each one has be the number of months they spent in the society with the gold: the first had sixty gold coins for the six months they spent. Multiply this. The result is 360. Then the profit, divided equally, is also indicated for other gold coins, multiplied into their months, which produced the profit: they made the same total: which was the amount of the first months and money. If one of the gold-bearers among them had a sum of gold greater than the number of their months in the common treasury of the gold-bearers, and each one of them did not know the number of his own gold, but the first one knew that both had two hundred and fifty and fifty gold pieces, the second one knew they had two hundred and forty, and the third one knew they had three hundred and twenty-six; when they opened the sack and counted the number of gold pieces in it as a large sum of foreign money, they hesitated among themselves: each not knowing his own share, nor how much each should ask for individually. It seems difficult to explain at first sight. But upon closer examination, the matter reveals itself. Since there are three of them, and each one knows the number of the foreign money, but not his own, the share of the money that belongs to each one is known to the other two. Therefore, if we add up all the numbers known to each one, the total number of the gold pieces will be held by that one number. All these numbers together will amount to one hundred seventy-two. This number can hold the total sum of all gold: it is necessary. This is accomplished by showing the half of this gold's number, which was poured out from the bag. This number is 358. This sum represents the total amount of gold. If you want to know how many gold pieces reached the first one, subtract the sum that the two remaining ones were known to hold from this number. The remaining number will indicate the number of gold pieces of the first one. This number is 208. If the number of the second one is subtracted from this same half number, which the third one was known to hold, thirty-two will remain. There were as many gold pieces of the third one. It is certain that the argument is correctly calculated: if all the numbers remaining after subtraction are added up, they will equal half the total sum of gold and the subtraction explains itself. If there had been a question about which of the four proposed matters: whoever knew the money of another would have been ignorant of his own. Since the share of money that one might have noticed belonged to him was ignored by that person, but known to the other three, the total amount known to all would have been added together three times, making the sum of all the money collected. And so the number of collections would have been divided into three sections, to show the amount of money in each section. The same would have been true if there had been a question about five. Since the share of money that one might have noticed belonged to him was ignored by that person, but known to the other four, the number of known sums would have been grouped together and held by four, to be divided into four parts. In the same way, if there had been six, it would have been divided by five; if there had been seven, by six; and so on indefinitely. querendae que essent per subductionem partes singulorum eo modo, quo dixi. The priest holding the purse gave money to three physicians for the poor. Moved by their poverty, he gave each one half of what was in the purse: first, he gave a coin and two more. Secondly, he gave them the remainder, half, and three more. Thus, only one coin was left. It is worth investigating: how many coins were in the purse. It is much easier to enumerate this: as it appears at first sight, if you begin to count the numbers backwards from the last coin left. For if you add four to the last coin remaining, it becomes five. These three coins, given secondly and more than half, must be doubled by the priest for him. Why is this so? His return you shall add two nuts more than half the first given: you will create 28. Which even these you must also duplicate due to the first half given. Thus it will be. 56. These numbers, which were in the pouch, you will be able to prove as follows:\n\nIf you distribute that number in this order as the priest did:\nIf you first give half, there are 28. From these, if you give him two more: there remain 26. Again, these for the second half: and they remain 13. Of these, three he left over. 10. These again for the third half: and you have the rest. 5. Of which, if you give four to him: one coin remains.\n\nWhoever had stolen two aurei from the royal cubicle: suspected by the host, and driven out, he gave half, to escape: he was given back the aurei. The second host, suspecting him, prevented his departure. He offered him the other half, which remained. The third host, also suspecting him, gave him 4. Himself innocent, he gave 4 aurei to him. Rursus illum exeuntem palacio portem comprehendit ianitor. He also took half of his remaining bribe: a dog offered him food instead. But he dismissed the others benevolently and returned six gold coins. Thus, the thief escaped with only the gold.\n\nInvestigation: How many gold coins did he steal in total? This can be determined by proceeding in reverse order, from the last to the first: except for the additional gold coins required due to the surplus of stolen gold, subtraction is necessary at each individual instance. Therefore, if we subtract the six gold coins returned by the second guard from the twelve stolen, some will remain. Six, since he also gave a half bribe to the third guard: these must be doubled.\n\nThus, from the twelve gold coins stolen from the first guard, we subtract the six returned by the second guard: some remain. Six, since he also gave a half bribe to the third guard: these must be doubled.\n\nTherefore, we subtract the four gold coins returned by the second hostelier from the eight remaining. Thus, these twelve coins: two must be deducted since the first guard returned two. Qua ratione relinqued. 14. Et quae prima hostiario furti dimidium oblatum est: ea duplicentur. Et prodibunt. 28. Qui numerus universos aureos surreptos capiat. Ut autem probemus non errasse calculo: converso ordine numerum hunc ad furis exemplum distribuamus. Et de 28. primo hostiario dimidium, quod est 14, offerimus. Is nobis reddat. 2. Et fient 16. Horum dimidium secundi hostiario demus: nempe 8, qui nobis 4 restituat. Sic habentur 12. Eorum deinde dimidium quod est 6, tertio ianitori offerimus. Is 6 reddat. Et rusus fient 12. Quae ex universo furto restabant. Ad eundem modum res explicabitur: si quis libet investigare, quanto sibi quisque hostiarius quantum diametrum, quam Vando Rotae lineam, quam Greeks diametron vocant, septem pedes habet: quantus erit ambitus apsidis extremi? Et quoties in mille passibus rota circuvolvetur? In primis observare oportet Geometrica ratione ab Archimedes deprehensum esse, diametrum tertiam circuli partem et tercium paulominus septimanam comprehendere. Nam if someone cuts it into the seventies: he takes the third part of the circle's circumference, less than ten seventies: which make up the seventh part. But if it is divided into seventies beyond the primas: it takes more than three parts of the circle beyond the decem septuagesimas primas: which are the smaller seventies; they do not fill the seventh part. Archimedes could not get closer, despite many attempts, to find the ratio of the diameter to the circle. If this had been discovered, the quadrature of the circle would have been found. Most philosophers, including Aristotle himself, had given up on it. No one has been able to penetrate beyond Archimedes. Since every circle is three times its diameter and a little less than seven times its diameter encompasses it: let us make what mathematicians are accustomed to do: count the seventh, which is a little short, as if it were an integral seventh. That is, if we take away the third part of the circle and the third part of the seventh takes it. To find the circumference of the extremes of the great circle's boundary, we must multiply by seven. This number is the diameter's. Add one seventh to this product. Thus, it will be 22. This will be the ratio of the circle's circumference to the diameter. One revolution will take this many feet. After this was discovered, it was found that the number 22 should be divided into the number of feet that a thousand paces contain. This is five thousand. For this reason, the number of sections will be 22 divided by 6/22, which equals 3/11. The wheel will revolve around this number of times in a thousand paces. It is difficult to know this exactly due to an uncertain ratio of diameter to circle. And it is hardly possible to grasp the error in such immense space.\n\nWho would not be put to shame by the rule of making and delegating the rules to artisans? But the entire ratio of making rules depends on numbers. As Pythagoras is said to have discovered. This is of such a kind. If there are three rules, and one of them is about the feet: one rule should be three feet, another four, and the third five. When these rules are combined at the highest points, they will touch each other and form a triangle, making it regular and properly formed. According to Euclid's rule, where a side of a triangle is produced from one triangle's side to itself, an equal side is found in two squares described from the other two sides. Therefore, the angle whose side is opposed to it is a right angle. Similarly, if the sides of this triangle are themselves multiple, you can create the numbers of the squares. The largest of these will contain as much as the sum of the two remaining squares. For three taken together, they produce nine. Four multiplied by itself produces sixteen. When these numbers are added together, they total twenty-five. If five is taken by itself (which is the largest side), twenty-five will result. This number is equal to the sum of the two remaining squares. When Pythagoras discovered [something]: not hesitating that he was helped by the Muses in this discovery, he was commanded by them to sacrifice animals. From him, therefore, a certain rule was devised for making measurements: before him, craftsmen had great difficulty in bringing things to the true measure. It makes no difference: whether they measured by ounces, palms, feet, cubits, or anything else; the summits of these measurements, when brought together, distorted the true measure. Now, since these primary numbers, three, four, and five, in which the true ratio was first discovered: they will perform the same function for all larger numbers, having the same ratio. Which ones are these: thirty, forty, fifty. Which one is the greatest: when drawn in a square, it will produce an equal square of the two remaining sides.\n\nEND OF BOOK THREE ON VARIOUS NUMBER QUESTIONS TO BE EXPLAINED. FOLLOWS BOOK FOUR ON PROPORTIONS. If you want to penetrate the secrets of numbers, you should give your effort to proportions. Their desire is not only in numbers but also in the entire nature of things: so that without it, nothing useful for human needs or pleasing to the eye can be found. Whatever you look at, you will easily understand this. At the beginning, whatever has been made for our dwelling: unless they have a suitable length and height, they are considered both unattractive and inconvenient by all. If they are too bright: besides the fact that we see them under the sun: they are considered useless for warming off cold temperatures and driving away harsh weather. If they are too narrow or rare: they resemble prison cells. If the walls are too thick: they look like castles. Conversely, if they are too thin: we do not consider them secure enough. Therefore, unless each part of the buildings responds to its share: the whole perishes. The entire art of painting, discovered for human delight, was held in high value among the ancients and had a great price: nothing was given for free if it was to be illuminated; if it was to be darkened, let it become illuminated; if the limbs in a body were to be depicted as disproportionate, neither agreeing with each other. For surely laughter is not stirred by the ungraceful labor. But if colors are tempered with austerity, the whites are bright, the shining ones are contrasted with the dark, so that the distant ones appear to have receded, and the near ones to emerge, so that they can be seen to exist: if all the lines agree with their proper proportions, no one will be able to resist the charm of this art. Those who give their services to military discipline seek weapons for bodies and tortures for souls. Those who build fleets prepare armaments in proportion to the size of the ships. Cloth makers strive to adapt their work to the body, so that it may serve to maintain symmetry. The same is true for all other crafts. What do physicians give the sick with medicines? Is it not of great concern: that they seize the power of disease with keen and alert intelligence, and then mix the warm with the cold, the dry with the moist, so that the body of the sick may come to a balance? This cannot be done if you are ignorant of the proportions. Our own food condiments, when we are healthy, are not pleasing to the palate, nor tasteless, unless they are salted for the recent, sharp for the sweet, austere for the lenient. Neither can the healthy and unharmed body be preserved or restored without knowledge of proportions, as it is not to be supposed that our bodies consist of the four elements any more than of these in their proper proportion. Moreover, the omnipotent God, the creator of all things and of all things in them, gave it that form, so that all things might hold symmetry with each other. This is evident in each individual: whether you look at the heavens or the earth. About the sacred letter, measure, number, and weight, in which proportion they most prominently appear: God is said to have disposed of all things. And let us not linger longer in confirming these facts.\n\nTwo kinds of quantity are found. The first is continuous: which is called magnitude. The second is discrete: whether it is called multitude or number. Magnitude itself has two species. The first is immobile: which geometry treats. The second is mobile: concerning which astrology considers. Two kinds of multitude also occur. One is considered in itself: which differs from arithmetic. The other refers to something else: which music deals with. Number, however, is the collection of units.\n\nPROPORTION IS THE RATIO OF TWO THINGS: WHICH ARE OF THE SAME KIND OF QUANTITY, A CERTAIN RELATION OF ONE TO THE OTHER, AND THEIR HABITUD. Given these two quantities, it is necessary that one is either larger, equal, or smaller than the other. Illa habitutdo is that which mutually reflects itself when equal, and proportion is what it is called when one is either greater or lesser. The comparison of quantities of the same kind falls under this category. For example, two numbers, two lines, two surfaces, two bodies, two places, two times. Neither line nor surface nor body nor time is greater or lesser in one kind. But line to line, surface to surface, body to body, and time to time are comparable. Proportion is a certain habit. For instance, it is determined as this and not otherwise. Not every proportion needs to be known to us or from nature. Among the ancients, proportion is divided into three kinds. One of these is that of discreet quantities, namely numbers, which is called arithmetic. The other is that of continuous quantities, which is called geometric. All quantities are compared to others and found to be either equal or unequal. A quantity equal is that which neither exceeds the one compared to it nor is exceeded by it. For example, a cubit to a cubit, a foot to a foot.\n\nThe species of rational proportion is as follows. Every quantity is compared to another and found to be either equal or unequal. A quantity equal is that which neither exceeds the one compared to it nor is exceeded by it. For instance, a cubit to a cubit, a foot to a foot. The number four to four. However, the quantities compared to each other are unequal: the proportion to those is greater: for example, a cubit to a foot. The number four to two. If it exceeds that, it has a smaller proportion to it. For example, a foot to a cubit. The number two to four.\n\nThere are two kinds of greater inequality: among which are three simple ones: namely, compound, superparticular, and superpartient. But the two remaining ones are formed from the first and the two remaining: namely, compound superparticular and compound superpartient.\n\nNow, regarding the first five:\n\nA quantity is multiple to another when it contains it several times. And if it takes it twice: it is called double. If thrice: triple. If quadruple: quadruple. And so on in infinite varieties.\n\nIt should be understood: how each individual species is generated - made from the natural properties of numbers, such as: 1, 2, 3, 4, 5, 6, 7, 8, 9. Every generation of these kinds is preceded by this one in the first place. For instance, the first even number, which is binary, is two more than one. The second even number, which is quadruple, is two more than binary. The third even number, which is sextuple, is two more than triple. The fourth even number, which is octonary, is two more than quaternary. The fifth even number, which is denary, is two more than quinary. And so on, an infinite progression of doublings can be made in this generation.\n\nTriplorvn generation is observed in this way in the natural number series after triple, which is three times greater than one. By omitting two numbers, for instance quaternary and quinary, the septenary, which follows, is three times the second number in the natural series. Again, by omitting septenary and octonary, we arrive at nonary, which is three times the third number. And again, by omitting ten and eleven, duodecimal is three times the fourth number. ad quem modum sine fine progredientes, triplos imprimis Quadrvplorum generatio incipit. If one intermitts three numbers after the fourth, such as quinarium, sesarium, and septenarium, the fourth encounters octonarius: which is found to be the second power of two. Again, 9, 10, and 11 are intermitted, and the twelfth number, which is ternary, will be quadruple. And so on. Each quadruple number is even, as it is in duplis.\n\nSimilarly, if four numbers are intermitted, quintupli are found. If five, sextupli. If six, septupli. And thus, the denomination of proportions always exceeds the numbers to be omitted by a unity.\n\nOf all the composite numbers, whose denomination is odd: one is even and another odd alternately. But of those whose denomination is even: all are even.\n\nQuantitas superparticularis is called ad alia dicitur: whatever contains it once and a part of it. And if it and its half are contained: it is called sesquialtera. si eam et eius tertiam: called sesquitertia. if you add eam and its quartam: will be sesquiquarta. And so on, you can generate names in this way indefinitely.\n\nIn comparison, larger numbers are called duces, while smaller ones are comites. Superparticular ratios are generated in this way.\n\nFirst, let's discuss the origin of sesquialtera. With numbers arranged in a single line, those having a third part consist of numbers that proceed indefinitely through continuous addition of thirds. For example, 3, 6, 9, 12, 15, 18, all duces of the sesquialtera proportion will be produced. If each of these numbers is paired with the preceding one, they will all be comites of the same proportion: if the first is to the first, the second to the second, and so on.\n\nSimilarly, with numbers having a quarta part, those produced through continual quartic addition are called duces of the sesquitertia proportion. All numbers having triple or teria parts will be their comites. Compare the first with the first, and the second with the second. et sic deinceps.\n\nIf a proportion of the sesquiquarta is generated: if singles quadruples are compared to singles quintuples.\n\nIt happens that a property pertains to proportions of this kind: that in some numbers of this species, the first exceeds the first by only one, the second exceeds the second by two, the third exceeds the third by three, and so on in order. A table can be painted, in which the ratio of as many multiples as we wish of this species of proportion can be generated. We described its form in the first book on the multiplication of numbers. Here we wish to present it to the eyes once more, as a precaution.\n\nIn this table, the progression of numbers is the same for both length and width. Therefore, if the second order, whether in length or width, is compared to the first: a continuous generation of the first multiples of this species occurs, namely, of doubles. For if 2 is to 1, or 4 to 2, or 6 to 3. et ita ulterius comparemus: prouenit ubique dupla proportio. Undeniably, in this proportion's form, the first dux surpasses the primus comitatus only through unity. The second secundus through binarium. The third tertius through terarnum, and so on in order. Similarly, if the third ordine ad primum conferamus:\n\nAccidunt autem predicatae formulae proprietates admirandae. The first property of these numbers is as follows. Consider the numbers of this formula descending from unity at an angle, secondly, according to the square's diameter. That is, 1, 4, 9, 16, and so on, they are perfect squares. Moreover, to each square, numbers arising from the numbers' progression can be reached. The second property of this formula is that its numbers are longilateri. These are numbers produced from the ductus of two numbers, one number surpassing the other only through unity. For instance, around 4 are 2 and 6. Their binarius is produced from the ductus of unity in 2 and 2, and their senarius from the ductus of 2 in 3. Similarly around 9. Sixth property: In order to find all long numbers, count around the angular numbers. The third property of this formula is: if two numbers adjacent to a angular number are added together, a square number will result. For example, 2 and 6 produce 16. Fourth property: if two square angular numbers are added together and the result is double the number surrounded by them, a square number will be born. For instance, 4 and 9 produce 25. Fifth property: wherever a square figure is found in this formula, a line drawn from the angles opposite each other will produce a square. For example, if 4 is drawn in 9, or 6 in 6, the same result will be obtained. Many other things can be found in this formula of the table, which we showed in the first book regarding multiplication and division. But setting these aside, we will return to the topic at hand. Quantity is called a superpart, which contains it entirely and some of its parts, such that the sum of these parts does not form a greater part. This species is more abundant in number of parts than it is varied in denomination infinitely. From the number of parts. For example, if a larger number contains a smaller one and two of its parts, it is called superbipartite. But if it contains it and three of its parts, it is called supertripartite, and so on. Similarly, from the denomination of parts. For instance, if a larger one contains a smaller one once and two parts of which each makes a third, it is called superbipartite of thirds, or more correctly, supertertius. And if it contains the smaller one and three of its parts, of which each is a fourth, it is called superpartite of fourths, or more correctly, superquartus. No status will be in such progress. This text appears to be written in Latin and discusses the ratios of numbers in two series. Here is the cleaned version:\n\nIf such a proportion exists, where both the number and its denomination vary: it can be continued: let there be one series of numbers starting with a ternary, and let it proceed in natural order as far as desired. Then let there be another series: starting with a quadrate number, let odd numbers follow in order. If we compare the first number of the lower series with the first number of the upper series, the proportion will be 3:2, or 3:4 and so on without end. If we double each number in both series, the second numbers will be in the same proportions: for instance, 10 and 6 are second numbers, and the proportion between them is 3:2. Similarly, if we double the second numbers in both series, we get 8 and 14, which are second numbers, and the proportion between them is 2:3 and so on. If we triple each number in both series, the third numbers will be born in the same proportion: for example, 15 and so on. ad 9. 21. and so on indefinitely through each species. From this, the Arithmeticians infer here that Boethius, the author, dozed off: who in the generation of such numbers says \"according to the leader and according to the count,\" among whom is the first proportion of the fourth: should be multiplied by three. And the third leader and third count, among whom is the proportion of the fourth power: should be quadrupled. And so on. Thus, second numbers are generated under the same proportions. Boethius also warns poorly here that tripled numbers should be tripled again: quadrupled numbers should be quadrupled, so that other numbers are generated under these proportions. Furthermore, Boethius only warns about superpartients regarding those in whom the denomination of the parts exceeds the number itself by a unit alone. For example, of superpartients of the third, fourth, fifth, and the like. He does not teach how the superseptimas or others of this kind are generated. Etido we learn about any species generation: let us take the number representing the parts of the proposed proportion. This number will be the first come of that proportion. To this number, we then add a counter for the proportion of the same: thus we will produce the first leader. For example, if the proportion of the supertriseptate is to be generated, we take 7. To this we add 3, and the result is 10 and 7. These are the smallest numbers among which the supertriseptate proportion is found. By duplicating these numbers, the second numbers of the same proportion emerge. By tripling them, the third numbers result. This is clear because the multiplicative and submultiplicative proportion is always the same.\n\nQuantity complex over particular is called: which contains it more than once and some part of it: and this varies both from the multiplicative part and from the superparticular part. For instance, the double sesquialter and the triple sesquitertia. When the denomination is given in terms of the whole for various species, it varies only in the superparticular part. For instance, in the case of duplas sesquialteras and duplas sesquitertias, generation will be made in this way. Arrange the numbers in a binary second order, to which each impair number is subscribed. Compare the first with the first, we will have a double sesquialterate proportion. Compare the second with the second, a double sesquitertian proportion. Compare the third with the third, a double sesquiquartan proportion.\n\nFor the generation of triplos superparticularia, let the first order be as before, and in the second order, after the septenary, take numbers continuously by addition of ternaries.\n\nHowever, for the generation of quadruplos superparticularia, with the previous order standing, take numbers in the second place starting from 9, and let them increase continuously by addition of quartaries. And so on, as much as we wish: we will proceed accordingly. But after the numbers have been established in their proper ratios in the aforementioned orders. If we duplicate the numbers, new ones will be born in pairs. If we triple them: the third will be proportionate.\n\nA quantity is called complex when it can be divided into more parts than one, and contains some of these parts, none of which forms a greater quantity by itself. This species varies infinitely in both the part and the whole.\n\nHowever, the generation of this proportion occurs in the following way. Given any such proportion, take the number representing the parts of the smaller number in the same: let it be the first term. Then either double or triple this number according to the multiplier in the proportion. Add units to this number according to the number of parts in the proportion, and the first term will be born in this proportion, compared to the first term.\n\nFor example, in the triple superquintiseptime proportion, 7 is taken as the first term. If we triple 7 and add 5, we get 26. Therefore, 26 and 7. The first numbers, in the proposed proportion, are formed. By doubling these numbers, second numbers are produced: tripling, third; and it is possible to obtain all numbers consistent with this proportion. However, it is important to note that the five species of these, as written in the square formula above, can be found to exist in infinitum. For the first order, the remaining singles compared to each other will produce multiple species. Similarly, if we compare the third order to the second, and the fourth to the third, and so on, a thousand superparticular species will arise. If we compare the fifth order to the third, the seventh to the fourth, and the ninth to the fifth, and so on, we will have singular species of superparticulars. If we consider the second order in relation to the fifth, and to the seventh, and the ninth, and so on, various multiple superparticular species will emerge. At multiplicium superpartes will produce species: if to the third order eight, eleven, and the fourth tenth, and also the second in the same ratio, other species, which are opposites to these: manifest these. For these species, which have already been described: other five, which are opposed to them: this is the only difference: that in these, as we have said, a greater number is compared to a smaller one; in them, however, the opposite occurs: that a smaller is compared to a greater. And therefore, sub, preposition, is added to each of their names. Furthermore, their generation is the same.\n\nPROPORTIONALITY is the similarity of proportions: This, however, is divided into two. One proportionality is continuous. The other is discontinuous: it can be called separated.\n\nCONTINUOUS PROPORTIONALITY is: for any quantity of the same kind, which the first precedes the second in the same proportion, the same one precedes any other following one. An example in numbers: 8 to 4, 4 to 2, and 2 to 1. Every proportion of continuous, commensurable quantities has a preceding and following term. That is, it supplies the place of the first extremity in the proportion of the first, and of the second in that of the second, except for the first, which preceded alone, and the last, which follows only. And in this very proportion, because of the continuity of proportions, it is necessary that all quantities be of the same kind; since the ratio of different kinds of quantities does not exist. This proportion, however, cannot be found in fewer quantities than three. Moreover, the middle quantity is repeated twice in them. It is superseded by the fourth.\n\nHowever, in proportions, the extremes themselves, whether they precede or follow, are not called terms by the mathematicians.\n\nBut quantities whose ratio is similar are called proportional.\n\nEvery proportion, whether continuous or separate, can be varied in six ways among arguments. Quantities, among which there is a similarity of proportions, are similarly proportional to one another. The comparison, which was made from the preceding quantity to the following, can be reversed. This reversed proportion is called the \"reciprocal\" proportion. In quantities with similar proportions, when their preceding parts are compared to their following parts: the similarity of proportions entirely disappears, even if the quantities of one preceding part are interchanged with those of another preceding part, and those of one following part with those of another following part. This reciprocal proportion is illustrated in numbers. For example, as 8 is to 4, so 6 is to 3. The same similarity will also hold in reverse: as 4 is to 8, so 3 is to 6. In quantities, the reciprocal proportion is formed when the preceding part of the second proportion becomes the following part of the first, and the following part of the first becomes the preceding part of the second. Example in numbers. For example, as 8 is to 4, so 6 is to 3. Quare etiam permutatis, such as: 8 is to 6, 4 is to 3. From this, it is clear that in permuted proportions, both extremes of the first proportion become antecedents, and both extremes of the second become consequents.\n\nConnected proportion is called: when the quantity of one proportion's antecedent is connected to its consequent, and similarly, the quantity of another proportion's antecedent is connected to its consequent. This connected proportion is inferred from the similarity of proportions. An example in numbers: 8 is to 4, 6 is to 3. Therefore, just as 8 and 4 are connected, 6 and 3 are connected, and 8 and 3 are connected to 3.\n\nDisconnected proportion is: in which, when the quantities are compared and their ratios are taken to their consequents, comparisons are also made to the disconnected parts. An example in numbers: 8 is to 4, 6 is to 3, 8 is to 3. Proportionality is: when, as one proportion's quantity preceding is connected to its consequent in relation to its precedent; so too is the quantity preceding another proportion connected to its consequent in relation to its precedent. This reversed proportion can be inferred from the similarity and conjunction of proportions. For example, in numbers: as 8 is to 4, so 6 is to 3. Therefore, 8 and 4 should be to 8 as 6 and 3 are to 6.\n\nEquivalent proportionality is: when several quantities are taken together in one proportion by one part, and others of the same kind or different, are applied to another part according to the same proportion; and a middle term, equal in number, is removed, a similarity of proportions is imposed on the extremes. For example, in numbers: as 8 is to 4, and 4 is to 2, so 12 is to 6, and 6 is to 3. Therefore, as 8 is to 2, so 12 is to 3. Ideas are subject to change if the order of cooperation is reversed and intermixed. For example, it may be said: as 8 to 4, 12 to 6, and 4 to 2, are 6 to 3. Therefore, as 8 to 2, 12 to 3, there is another division of proportions: which will become apparent from the middle terms.\n\nMedietas is a certain connection between extremes through a relationship to the middle. For instance, in arithmetic, the middle is: when the difference between the greatest and smallest is equal. That is, where the proportion is disregarded, the difference between the terms is kept equal: whether in the natural order of numbers, such as 1, 2, 3, 4, or equal numbers in continuity, such as 1, 4, 7, 10. If one is omitted, there will be a difference. 2, 2 will have a difference. 3, 3 will have a difference. 4, and so on. Difference, however, refers to the excess of one number over another.\n\nThe property of this median is: Maximus to the middle is not the same ratio: the middle to the minimum is always greater: for the reason that in smaller numbers the ratio is always greater in larger ones. For example, the sesquialter ratio of 8 to 4 is the middle number in this proportion. The binary number makes the difference in both. However, the proportion of the middle to the minimum is greater, for instance, 6 to 4 is sesquialter than the maximum to the middle, and 8 to 6 is sesquitertia.\n\nAnother property of this is, if such proportions are arranged in three terms, the extremes added together will make twice the middle. For example, 6, 4, 2, and twice 4 makes 8. Similarly, 6 and 2 make 8.\n\nIf placed in four terms, the two medians combined by addition will make as many as the two extremes joined together. For instance, if 8, 6, 4, 2 are placed, they make as many as 8, 6 combined. Therefore, 8, 6, 4, 2 make twice as many as the middle number. From this proprietary, a general ratio can be derived. Reversely, another property of it is this: if this middle is established in three terms before, the extremes multiply: then subtract the middle itself: whatever results from the middle's lead in itself: will exceed what results from the extremes' multiplication: by how much it will be produced from the other difference in the other lead. An example in these terms: 2, 4, 6. Just as 2 relates to 4, so 4 relates to 6: 2 multiplied by 2 makes 4, 4 multiplied by 2 makes 8, 12 is quadrupled, 16 exceeds 4 by a factor of 4. This excess also comes from the difference's lead in itself. For both differences are the same. But 2 multiplied by 2 makes 4 again. In the same way, if terms 4 are placed and the extremes are multiplied, whatever results from one middle in the other lead: will exceed what results from the extremes' multiplication: by how much it is produced from the multiplication of the two differences: what are the intermediates between the extremes and the middle. For instance, if there are 2, 4, 6. 8. bis make. 8 creant. 16 quater. 6 faciunt. 24 quod per. 8 exceed. 16 Idem autem excessus differ, which are between the intermediate and extreme: is produced. For if from two mediators, that which is earlier: take; namely, the difference between the earlier extreme and the middle is. 2 and the difference between the middle and the later extreme is. 4 thus the differences themselves. 2 and 4 are drawn from themselves and produce. 8\n\nGEOMETRICA: the mean is that ratio where the maximum is to the medium, is equal to the medium to the minimum. In this mean, the difference, neglected in equality, observes proportion equality. As can be seen in these numbers. 4, 6, 9. Where the middle number is. 6. Secudum Arithmetica, the concept of proportion in this median is called more appropriately: according to Arithmetic. For in it, proportion is understood in terms of equal differences, not similarity of proportions, as is the case here. The ratio is always equal in larger and smaller terms. For example, the ratio is the same between 30 and 10, as it is between 6 and 2.\n\nAnother property of this median is that in continuous proportions, the ratio of the differences is proportional to the ratio of the terms: the ratio of the terms being the same as the ratio of their differences. For instance, as 8 is to 4, so 4 is to the difference between 4 and 2, which is 2.\n\nReverseely, another property of the geometric median in continuous proportions is this: in doubled numbers, the smaller term exceeds the larger one by itself. For example, 1, 2, 4, 8. In tripled numbers, the smaller term is surpassed by the larger one. For instance, 1, 3, 9. 27. In a druple, the third part is tripled, just as the smaller term exceeds the larger. Another property is this: if this median is established among three terms, whatever is produced from one extremity to another will be equal to what is in the other by nature. For example, if there are 8, 4, 2, twice 8, which is 16, 16, 4, 2, twice 16, which is 32, and 8, the same. In four terms, whatever is produced from one extremity to another by multiplication will be equal to what is produced from one median to another. For instance, if you take 16, 8, 4, 2, twice 16, which is 32, and 16, 8, 4, 2, twice 16, which is 32, 8.\n\nThe arithmetic mean is this: when three terms are placed, as the greatest has to the least, so the difference between the greatest and the median is to the difference between the median and the least. This middle part admits both differences and proportions: it is not composed of equal differences, nor of equal proportions. For instance, if 6 and 2 are placed in the middle, there is a triple proportion between the maximum and the minimum: that is, 6 to 2. The same proportion holds for the difference between the maximum and the median, which is 3 to the difference between the median and the minimum, which is the unit, since 3 to 1 is also a triple proportion. From this middle value, musical consonances are derived.\n\nOne property of this middle value is that in larger terms, the larger is the proportion: and in smaller terms, the smaller. This is contrary to arithmetic. For example, 6, 3, 2 have a compatible proportion in the first place. But 3 to 2 is sesquialtera, which is less than double.\n\nAnother property of this middle value: in how many parts of the smallest term does the smallest term itself exceed the median? In as many parts does the largest term surpass the median itself. For example, consider 6, 3, 2. Two are taken from the third part of a binary: which is one. Similarly, six are taken from the third part of a hexadecimal: which is three. Again, this belongs to it. If the extremes of the proportions are joined together and the whole is multiplied by a medium, what comes from that will have a double proportion to what is produced by multiplying the extremes by each other. For example, take 6, 3, 2. Add 6 to 2. They will be multiplied by 3. Which are in the middle. Also, if you multiply the extremes by each other, that is, 6 by 2, they will produce twelve, which makes half of 24.\n\nIt should be noted that these three medians can be found between two terms, for instance between 10 and 40. Indeed, if you place a median between them, arithmetic, geometric, and harmonic medians will respectively emerge.\n\nTo find the mean between two terms using arithmetic mean, subtract the smaller number from the larger and divide the result by two. Add half of the smaller number to the result.\n\nTherefore, to find the mean between two terms using arithmetic mean: subtract the smaller number from the larger, and divide the result by two, then add half of the smaller number.  ita quod inde ex\u2223ibit: medium proportionale erit. Exempli causa inter. 10. et. 40. subductio deprehendit reliquum esse. 30. horum dimidium. 15. adde minori numero. 10. et prodeu\u0304t. 25. quod medium est quaesitum. Vel si libet facilius. Coniun\u2223ge utrum{que} extremum: et totius coniuncti dimidiu\u0304 pro\u2223portionale Arithmeticu\u0304 erit. Nam si sumas. 10. et. 40. adde ea simul: fiunt. 50. Ea si dimidies: habebis. 25. id medium est Arithmeticum.\n{QUOD} SI inter eosdem terminos. 10. et. 40. mediu\u0304 pro\u2223portionale Geometricum eruere uoles: alterum in alteru\u0304 multiplica. uidelicet. 10. in. 40. et procreabis. 400. cu\u2223ius numeri producti radicem quadratam extrahere opor\u2223tet: quae medium proportionale Geometricum erit. radi\u2223cem autem eam. 20. inuenies.\nAT SI medium proportionale Armonicum inuenire\nlibet. Primum iunge extrema ipsa, uidelicet. 10. et. 40. sic surgent. 50. Deinde subtrahe minorem de maiore. et reliquum erit. 30. differentia utrius{que} numeri. quam si in minus extremum uidelicet. 10. ducas: nascentur, 300 Eum numerum productum dividere per summam utraque extremi conjunctam: id est. 50. et 6. exibunt, quae adde minori termino 10. et prodeunt 16. This is the proportional mean between 10 and 40.\n\nArithmetic mean of the republic is represented: which is ruled by a few: since the proportion in its smaller terms is greater. Armonic mean of the best republic is compared: since the proportion in its larger terms is greater. But the geometric mean, which keeps proportion equal in larger and smaller terms, in a certain way represents a popular republic: in which all citizens, magistrates, and plebeians have equal right.\n\nPraeter these three principal means given by the ancients, IORDANUS added eight others, which he calls collaterals: whomsoever wants to know them, let him ask him. We, who only concern ourselves with the proportions necessary for learning calculation in life: tasting; have decided not to reach all proportions or rules, or properties. We will omit the intermediate proportions from the posterity for the sake of the variety of proportional relationships. Those that do not pertain to our institution we have decided to omit.\n\nWith the two proposed terms, it was necessary for one to be either greater, equal, or smaller than the other. Therefore, if they are equal: equality is the proportion: which no one can deny. But if one is greater or smaller: whatever the proportion of inequality is: it has many species: easily it will fall into which species the relationship will be. The greater term is to be cut continuously by the smaller. This will be done: so that what exists in the number of partitions may denote the proportion between the two extremes. If this text is about proportions in mathematics, here is the cleaned version:\n\nIf a greater term is in comparison to a smaller term, it makes a proportion with a greater inequality. For instance, if you seek the proportion between 8 and 4, divide 8 by 4 and in the number of quotients 2 remain, which shows that the proportion is double. Similarly, if you seek the proportion between 3 and 2, 3 divided by 2 yields a quotient of 1.5, which indicates a proportion in the form of a superior fraction. However, if the smaller term is in comparison to a greater term, the term \"sub\" should be added before the denomination of the proportion from the number of sections. For example, if the proportion between 4 and 8 is sought, since after the section of the greater number, 2 appear in the number of quotients, the proportion will be sub-double, which is contained in the submultiple form. Similarly, if it is between 2 and 3. proportion you ask for: because a smaller number precedes in comparison, and a larger section is made from the smaller number in the partition. 1\u00bd. The subsequent proportion will be less than that, as has been shown in the case of sub-superparticular ratios. But if someone wants to know which senior term this proposition refers to, in the case of unequal proportions of the lesser, it is necessary to observe this. Just as the major term in each species of unequalities contains the minor term more than once: so in each species of the lesser unequalities, the difference between the lesser and greater is the amount needed to make up the whole. Therefore, since this proportion of the lesser to the greater produces fractions: the ancients added the signifying \"sub\" to each species of major unequalities to indicate that the whole must be made up: only as much is lacking as the major term exceeds the minor. You are asking for the cleaned version of the following text: \"Veru\u0304 si exquirere libet: quas minutias hae proportiones pariunt: continuo id cognos\u2223ces: si minorem terminum, qui in proportione praecedit: in minutias frangas, secans per maiorem, eo modo: quo in minutiarum libro admonuimus. ita ante oculos appa\u2223rebunt minutiae, in quauis minoris inaequalitatis specie occurrentes. Exempli gratia. si proportionem inter. 4. et 8. qu\u0119ris. 4. per 8. diuide. et fient 4/8. qu\u0119 minuti\u0119 signi\u2223ficant. 4. partes de. 8. adhuc deesse ad integru\u0304 compo\u2223nendum proportionem{que} aequalitatis creandam. Qua\u2223tuor uero octauae ad minimam nomenclaturam redactae faciu\u0304t \u00bd. Ita adhuc ali\u0119 4/8, qu\u0119 sunt \u00bd. desunt: ut fiat unu\u0304 integrum. Quo fit manifestum, 4/8 facere dimidium illius proportionis, quae aequare deberet. 8. Sicut e contrario, si. 8. ad. 4. comparentur: dupla erit proportio: {quod}. 8. in se capiant bis. 4. Similiter si. 3. ad. 9. co\u0304parentur. 3. per 9. secta proferent 3/9: qu\u0119 faciunt \u2153. Et 6/9 quae creant \u2154. desunt: ut aequalitatis proportio ad. 9. producatur\"\n\nHere is the cleaned version of the text:\n\nYou wish to know what details these proportions yield: continue to examine them closely: if a smaller term precedes in the proportion, break it down by the larger term, in the same way as we have advised in the book of details. Thus, the details will appear before your eyes in any form of minor inequality. For example, if you inquire about the proportion between 4 and 8, divide 4 by 8 and the parts will be 4/8, which represent the details. Four parts of 8 are still lacking to make up the proportion of equality. However, when the four eighths are reduced to the minimum, they make up half. Therefore, three more parts of 4/8, which are half, are still missing to make one whole. It is clear that 4/8 makes half of the proportion that should be equal. On the contrary, if 8 is compared to 4, the proportion will be doubled: 8 contains twice 4. Similarly, if 3 is compared to 9, 3 parts of 9 will be produced: one third. And two thirds are still missing to establish the proportion to 9. In this third part, they make up one third of the proportion: 3/9. It should be 9 equal parts. If you want to know the proportion of 16 to 5, the larger section is found in the number of partitions. 3 5/12. This number, with the smaller ones, indicates which species of superparticular ratio falls. Conversely, if you seek the proportion of 5 to 16, 5 divided by 16 creates 5/16, which shows that only five of the sixteen parts are found. Thus, eleven-sixteenths are missing to make a whole. In the same way, it is permissible to investigate the proportion between any numbers. These words are extracted from Euclid: in the seventh book, he says that the denomination of the smaller number's proportion to the larger is called a part or parts of the larger: which are in the larger. Conversely, the larger, in relation to the smaller, is the whole or the whole and a part, or the parts, and the larger exceeds the smaller. With these words, the denomination of the proportion, which is taken from the smaller number, is said to be a single part: such as 1/2, 1/3, 1/4. Partes plurales: ut 2/3, 3/4, 4/6. Find them in this way, as stated. Moreover, the denomination of the greater number to the smaller, he advises, should be either the whole or: as in a fraction. Or the whole and a part: as in a superparticular fraction. Or the whole and parts: as in a fraction with multiple parts. It is clear that, in the case of an inequality in the minor's form, you must choose which way to follow: reduce the term to the same level: either to avoid the unequalities in the denominations of the greater proportions, or to express the minutiae themselves with their own denominators.\n\nSimilarities are called proportions: which receive the same denomination. Arithmeticians also call those between one proportion, not only its own. The greater proportion is: which has the greater denomination. The lesser: which has the lesser. However, every denomination is said to be as heavy as: the number that denotes it. If we seek to determine which of the triple and quadruple numbers is greater, since the greater number is denoted by the quadruple, which indicates a multiple of four, while the triple indicates a multiple of three: therefore, the quadruple will be greater than the triple. And the quintuple greater than the quadruple, the sextuple greater than the quintuple, and so on indefinitely. Furthermore, sesquialtera is greater than sesquitertia, because sesquialtera is denoted by 1 \u00bd, and sesquitertia by 1 \u2153, and in both numbers the integer part is equal, but the fractions are unequal in magnitude. Moreover, the minor fractions in sesquitertia are smaller. For example, in fragments (as we mentioned earlier in the second book), the smaller the denominator, the smaller the fragment. Therefore, \u00bc is smaller than \u2153, just as \u2153 is smaller than \u00bd, and therefore sesquialtera is greater than sesquitertia. And sesquiquarta greater than sesquina, sesquisexta greater than sesquiseta, and so on indefinitely: where fractions occur in the denominator. It is clear that the triple is less than sesquialtera, for example, 7 2/3. If greater, it is: more than triple sesquitertia. Yet triple sesquitertia is greater: than double sesquitertia or double sesquialtera. This is not due to the addition of smaller parts, which are greater in sesquialtera, but due to the denomination from the integers designated. For instance, triple is denoted as 3, while double is denoted as 2. However, no single integer can equal the sum of these parts denoted by sesqui. Therefore, the proportion of triple sesquitertia will be less than that of double sesquialtera due to the greater denomination.\n\nRegarding continuous proportionalities, before discussing disproportionalities, consider how you can recognize when a proportion is composed of more than one. Furthermore, regarding continuous proportionalities, Euclid advises us about this in the following manner. If there are three continuous proportional quantities, the proportion of the first to the third is called the proportion of the first to the second, multiplied by two. He teaches this with the following words:\n\n\"If there were three continuous proportional quantities, the proportion of the first to the third is called the proportion of the first to the second, multiplied by two.\"  si fuerit proportio primi ad secundum: sicut secundi ad tertium: tunc {pro}portio pri\u2223mi ad tertium erit, sicut primi ad secundum, duplicata: id est, ex duabus talibus composita. qu\u0119 compositio cogno\u2223scetur per multiplicatione\u0304 denominationis ipsius {pro}por\u2223tionis in se. Verbi gratia. Sint tres numeri continuae pro\u2223portionales, dupli. veluti. 2. 4. 8. in his proportio inter primum et secundum dupla est: si a maiore incipiat com\u2223paratio. alioqui si a minore: subdupla est. Denominatio autem dupl\u0119 uenit a. 2. Igitur si. 2. in se ducantur: nas\u2223centur. 4. quae inter primum numerum et tertium qua\u2223druplam esse proportionem denotabunt, ita demum: si a maiore incipiat comparatio. alioqui si a minore: subqua\u2223drupla erit. Ipsa autem quadrupla est dupla duplae: quia constat ex duabus duplis. Rursus de. 4. quantitatibus EVCLIDES admonet, inquiens. Si fuerint quatuor quantitates continuae proportionales: proportio primae ad quartam dicetur proportio primae ad secundam tripli\u2223cata. Quorum uerboru\u0304 sensus hic est If there are four quantities in continued proportion: The ratio of the first to the second will be in the same proportion as the first to the third, multiplied by three, both in magnitude and in product. For example, if there are four numbers in continued proportion: such as 1, 3, 9, 27. The ratio between the first and second is tripled:\nIf the comparison begins with the larger: which is called a. 3. The three taken together produce 9. The same 9, when taken together, produce 27. Which denotes a twenty-sevenfold proportion between the first and fourth: if the larger precedes in comparison. And this is triple of triple: because it consists of three triples. It is the same in this way: as if it were said. The ratio of two quantities is a simple interval: and it has the nature of a simple dimension, like a line. But the proportionality in three, is double the interval, having one middle term: and it refers to the nature of double dimensions, like surfaces. Proportionality in four parts is a triple interval, having two medians. It refers to the nature of three-dimensional dimensions, like solids. Similarly, if five terms are proportionate: the proportion of extremes will embrace the proportion of the first four. If six terms: five times. If seven: six times. And so on. Proportion of extremes always contains the proportion of the first ones: as many as there are terms minus one. Since proportions are intervals: an interval, however, can only exist between two extremes: it is necessary that the terms themselves enclose the intervals, with more than one interval. Therefore, if there are four terms: the proportion of extremes, for instance, 8 to 1, will be octuple. And since the proportion of the first ones is double: and there are three intervals: the octuple one will consist of three doubles. If five terms: for instance, 1, 2, 4, 8, 16, the proportion of extremes, 16 to 1, will be sixteenfold: since the proportion of the first ones is double: and there are five intervals. The text appears to be written in old Latin, and it seems to be discussing proportions. Here's the cleaned text:\n\n\"sunt intervalli: constabit ex quatuor duplis. Qua ratione proportio debita erit ad 32. Ad quem pateat, composita erit ex quinque duplis, et sic in infinitum. Manifestum fit, in proportionalitate continua proportionem extremis producere omnibus mediis. Vel si alia via scire voles, pluribus continuis proportionalibus propositis, quid sit inter extremas, proportio observabis: ut in exemplo proxime datum: ubi dupla erat inter 2 et 1. In qua denomiator eius in se ductus, et surget. 4. Quid inter extremos trium terminorum, sicuti 1, 2, 4, quadruplam proportionem designant. Deinde si denominatorem illum, hoc est 4, iterum ducatur in denominatorem primae proportionis, hoc est 2, nascetur proportio inter extremos quatuor terminorum, ut puta 1, 2, 4, 8, quod est octuplum. Bis enim 4 sunt 8. Rursus si octo in denominatorem primam proportionis 2 ducatur: fiunt 16, quod inter extremos quinque terminorum, ut 1, 2, 4, 8, 16.\" The text reads: \"sedecuplam shows sixteen in the first denominator, that is, it is divided into 16 equal parts. The first denominator is multiplied by 2 to produce 32 parts. Between the extremes of six terms, that is, 1, 2, 4, 8, 16, 32, they will be called denominators. And so, the last denominator's product can be obtained by multiplying the last denominator by the first denominator. Thus, in a long series of continuous proportions, all smaller multiplications will be even through the binary number. As the terms themselves are proposed, all multiplications will be as they are. However, I would like to add a few words about improportional terms: those terms are improportional where the proportion of dissimilarity falls. For instance, dissimilarity is called improportionality, which can be continuous or separate. Improportionality, however, is continuous: either because the proportion of the first to the second is greater than that of the second to the third, and so on, or because it is smaller. For example, between 1, 2, 6, 24.\" ubi tres sunt proportiones dissimiles: quarum prima duplica est, inter 2. et 1. secunda tripla, inter 6. et 2. quae maior est, quam duplica. tertia quadrupla est, inter 24. et 6. quae maior est, quam tripla. Improportionalitas vero separata aut incontinua est: aut quia major est proportio primi ad secundum, quam tertij ad quartum: utpote inter 10. et 2. ac 6. et 4. aut quia minor. Huius autem inter proportiones dissimilitudinis, continua sit aut separata: duae species habentur. Altera est: quando major est proportio primi ad secundum, quam secundi ad tertium in continuis: vel quam tertii ad quartum, in separatis. quae maior improportionalitas nuncupatur. utputa inter 8. et 2. ac 6. et 3.\nAltera est, quando minor est proportio primi ad secundum, quam secundi ad tertium, in continuis: vel quam tertii ad quartum, in separatis. sicuti inter 6. et 3. ac 8. et 2. quae minor improportionalitas appellatur. This text appears to be in Latin, and it seems to be a passage from Euclid's work. I will translate it into modern English and remove unnecessary formatting.\n\nHere is the cleaned text:\n\nTo this point, most people define and distinguish between proportions and non-proportions, and determine their continuity and degree of dissimilarity. Euclid advises us, saying: When there are multiple proportions, whether they are the same or different, the proportion between the first and last is taken from all of them. Therefore, if there are more proportions, whether similar or dissimilar, that intervene between the first and last terms, the proportion between the first and last terms must include all the intermediate proportions, however many there may be. The name of this proportion is derived from the product of the denominations of all the proportions, namely the first two, the second in the first, and the product of any number of continuous denominators up to the last. The ratio is said to be composed of two ratios: when the denominator of the former produces the denominator of the latter through the denominator of the intermediate ratio. Similarly, when several ratios are composed, it will be said to be done when the denominator of the former produces the denominator of the latter through the denominator of the next one, up to the end. It makes no difference which hand you begin multiplying the denominators from, as long as you follow the correct order and pass through all of them. This is true both in continuous proportionality and in any improportionality perpetually.\n\nRegarding continuous proportionality, as can be seen from what has been said: we will not hesitate to give one example. Let these five numbers be given: 1, 2, 4, 8, 16. Starting from the first two, we find a double proportion between 16 and 8. Similarly, advancing between 8 and 4, we find another double proportion. Duae illae denominations altera in altera quadruplam demonstrating, 16 and 4 have a fourfold proportion between them. Proceeding further, we find another double proportion between 4 and 2, which, when added to the previous quadruple proportion, makes sixteen and two octuplum. Progressing again, we find a double proportion between 2 and 1, which, when added to the previous productions, makes twenty-one. The proportion between 16 and 1 is found to be encompassing all the medians. The same will hold true in every other proportional relationship: however many proportions are connected in it.\n\nNow let us give examples of disproportion: in what way do dissimilar proportions intervene? For instance, let us take these four numbers: 1, 2, 6, 24. Among these four numbers, three are proportions. The first, between 2 and 1, is double. The second, between 6 and 2, is triple. The third, between 24 and 6, is quadruple. If you want to know what the ratio is between 24 and 1, investigate the first ratio between 2 and 1 and multiply it by 2. Then investigate the ratio between 6 and 2, and you will find that the names taken in turn create sextuple ratios. This will be the ratio between 6 and 1. If you progressively ask for the ratio between 24 and 6, you will find it to be quadruple. If this quadruple ratio is taken as the common denominator of the previous ones, it will result in 20 times quadruple. You can see that the composition of ratios is similar: even if all the ratios are from whole numbers or only from small ones.\n\nFor example, let's take these four numbers: 4, 6, 8, 10. Among these three proportions, there are three different ones: the first is between 6 and 4, the second is between 8 and 6 in a sesquialterate ratio, and the third is between 10 and 8 in a sesquiquartic ratio. Let us take the denominators of these ratios. For the sesquialterate ratio, it is 1 \u00bd, for the sesquitertian ratio it is 1 \u2153, and for the sesquiquartic ratio it is 1 \u00bc.\n\nIf we bring the first two earlier ratios one into the other in the same way as we showed in a small book, we will obtain the denominator of the proportion from both. Then, taking the product of the denominator from the third proportion and multiplying it by the denominator of the third proportion twice, we will create the denominator of that proportion.\n\nTherefore, if we start with 1 \u00bd in relation to 1 \u2153: two new ratios will emerge. They demonstrate that the double ratio of sesquialtera and sesquitertia is formed. Then, if we add 2 to 1 \u00bc, we will obtain 2 \u00bd. This number will be called the proportion composed of the three different ratios mentioned above, which will be between the extremes of all the proportions, i.e., 10 and 4. In this example, a greater disparity continues: if you begin on the left and compare larger terms to smaller ones on the right, as we have done. Here is an example: when disparity, whether separate or continuous, occurs. EVCLID'S rule, which is like an oracle, does not deceive anyone about this. It also states the same thing. Let us compare these four terms: 8 to 2 for the prior proportion, and 6 to 3 for the subsequent. The first pair is quadruple: which is greater than the second. The second pair is double. And this disparity is found in the former. Yet, the proportion between the first term of the prior proportion and the last term of the subsequent proportion will be found among all the middle proportions. And also the extremes: which separate the two middle terms. This is between 2, which are in the end of the prior proportion, and 6, which are in the beginning of the subsequent. This will be made clear by the rule given. For if anyone, starting from the left, makes a beginning and moves to the right, the disparity between: If we move on to the eighth and second terms, the first issues that arise: the eighth will have four parts. Then, progressing between the second and sixth terms, since the smaller end precedes in comparison: it will be less than three parts: the ratio of the smaller inequality is: and for this reason, as we showed earlier: it is named by its parts, namely 2/6, which are one third. If one third, which denotes this proportion, is drawn into the fourth term, which denotes the denominator, four-thirds will be born, making one whole and one third, resulting in a sesquitertian proportion between the eighth and sixth. Repeating the progression between the sixth and third terms, a double sesquitertian proportion occurs. Since the second is denoted by the denominator, two and one third should be taken from the product of the previous multiplication denomination, that is, 1 1/3, and they will be found in the product of eight and three as 2 2/3. These manifest the double sesquitertian proportion between eight and three. We explained earlier how this proportion is composed of the three middle terms according to Euclid's rule. We made a few small adjustments to make it clearer: we wanted to show that there is no difference between those in the middle and those at the extremes, even if the latter are larger or smaller. Those who think this is not the case deny that proportions in the middle will not appear as often as those between the first and last, and claim it is impossible for a part to be larger than the whole. However, this will not prevent the proportion between the first and last from being gathered from all the middles. It will not happen that a larger proportion in the middle will intervene between the one between the first and last, unless another proportion of lesser inequality is also found in the middle, which, when coupled with the larger one, would reduce the former too much and make the creation of an equal proportion between the first and last from all the middles difficult. Regardless of which side you take, it presents: Center the two parts that have a greater or lesser proportion, and the two parts with a smaller proportion are combined: the former is smaller than the latter in terms of the greater inequality, and the latter is greater than the former in terms of the smaller inequality. This results in the fact that not every collection of proportions will always increase. And so, according to mathematical tradition, the proportion that exists between the first and last is said to be constituted by the middle terms of all proportions. Any single middle term is not always a part of this proportion, as the combination of these proportions can detract from it. Whatever is taken away from each individual term will be replaced in the proportion that exists between the first and last. Let us illustrate this with an example. In which we place a larger term at the greater extremity and a smaller term at the lesser extremity. Let us take these four terms: 2, 1, 20, 16. Comparing 2 to 1 and 20 to 16, the greater term will be separated by an improportionality. Nihil refert ab utra manu multiplicatio initium sumat, dummodo ceptum ordinem servet. Therefore, from the left, let us arrange. When we find the duplicatio 2 and 1, and the proportion between 1 and 20 is the next one approaching, it is expressed as 1/20. Therefore, we denominate the numbers with these two ratios one in the other: for example, 2 in 1/20 and 2/20 results in 1/10 and subdecimal proportion between 2 and 20, as indicated by the former ratios. Again, the proportion sesquiquarta occurs between 20 and 16. Therefore, we denominate the number named by it, that is, 1 \u00bc, in that denomination, namely 1/10. And it produces 5/40. What they make is 1/8 and suboctal proportion, between 2 and 16, as shown by all the intermediate proportions. Vides nihil referre qui termini inter primum et ultimum mediis intervennant, dummodo multiplicatio ab altero extremo ad alterum progressa, cumque ordinem servans, nihil errat: neque minoris inequitalitis proportio pro maioris sumatur. Hoc id est: si proportiones eaeidem sive diversae fuerint. Iuxta regulam ab Euclide data, proportiones inter primum et ultimum ex mediis omnibus constare: multo magis in separata proportionalitate verum erit, in qua proportiones inter primum et ultimum partim similes, partim dissimiles sunt. Velut in 4. ad 2. et 6. ad 3. his terminis tres proportiones habentur. Quarum duas similes, tertia media dissimilis disiungit. Hic 4. ad 2. collata duplam creant: ea prima est proportio. Iterum 2. ad 6. collata, subtriplam proportionem mediam, quam duas similes utrimque disiungit: ostentant. Quocirca. Two out of three are carried forward, two-thirds being the proportion between the fourth and sixth. Again, a double proportion is found between the sixth and third, which is the first similar one: it is two-thirds in the product of the subsequent proportion, and they produce a proportion of 4/3. These proportions reveal the relationship between the first extreme term and the last term of the second. It is clear that the third proportion, composed of the two extreme and the middle ones, is different. Let us proceed in the same way: how many and which proportions, whether similar and the same, or dissimilar and by how much they intervene between the first, third, and last terms.\n\nPerhaps someone may ask: Where was the composition of proportions first demonstrated, in continuous proportions? Then in discontinuous proportions? and again in separate proportions? One rule could have sufficed for all. A reason for this is at hand. We have followed Evclidem: he who first taught about the composition of proportions in quantities that are continuously proportional, and who wanted to prepare students' minds for more advanced topics through easier methods, rather than explaining the rule with its many intricacies and turns.\n\nWhoever proposes a proportion, can say how many and what other proportions make it up: it fell into chaos for the uninitiated. Six hundred, or even infinite proportions can be sought: from which any can be combined, as will soon become apparent. Therefore, no one can enumerate all the proportions that can be formed: since they are infinite. However, many can be found: from which any proportion consists. For any proposed proportion, it necessarily has two terms. Between these terms, if you do not place a middle term, these three terms will form two proportions: from which the one between the extremes will be entirely composed. Porro cum multis modis terminus medius variari potest: multum et variae proportiones peridere possunt. Id quod hoc exemplo fit manifestum. Inter 16 et 1, sedecuple proportio est. Inter quae, si medium terminum in 6 interponamus, per hoc inter 16 et 6 proportio dupla superbitertia habetur. At inter 6 et 1, sexuple est proportio. Undelegit proportionem sedecuple, quae est inter extrema: ex his duabus mediis constare. Id ita habere probabis, si duas illas medias denominationes alteram in alteram ducas. Nam 6, quae sexuple denominant, ducta in 2 2/3, denominet alteram 16, quod producunt quam sedecuple proportionem inter extrema demonstrant. Quod fit manifestum, duplam superbitertiam et sextuple compositam sedecuple esse. Si inter eadem extrema alium terminum medium ponas: aliae proportiones orientur. Utpote si 8 interseras, eo fit ut inter 16 et 8 dupla oriatur, at inter 8 et 1 octuple habebitur. Quae denominationes, altera in alteram ductae, sedecuple iterum creabunt. Qua ratione sedecuple conflata erit ex dupla et octupla. If more extremes are interposed between them, more proportions will ensue, which should encompass all that is between the first and last. For instance, if we place 5, 6, 8 between 1 and 16, among these terms there are four different proportions: quintuple between 5 and 1, sesquiquinta between 6 and 5, and sesquiteria between 8 and 6. Therefore, the quintuple ratio is 1 \u2155 of the first, the sesquiquintam 30/5 of the second, and the sextuple proportion is indicated between 6 and 1. Then, advancing between 8 and 6, you will find sesquiteria. If 1 \u2153 is multiplied in 6, which is derived from the previous ratios, it produces 24/3. Between 8 and octuple, they designate. Again, advancing between 16 and 8, you will find dupla. quae ex priorum denominationum multiplicatione proxima producta sunt: ducas tandem producunt. 16. quae sedecimplem proportionem inter. 16. et. 1. denominant. Quo fit manifestum, sedecimplem proportionem ex quintupla, sesquiquinta, sesquitertia, et dupla compositam esse. Igitur cum terminis inter extrema interponendis innumeris modis per minutias, variare licet: manifesto liquet, proportionem propositam nequaquam explorari posse: quot et quae proportiones eam possint composere: cum infinita varietas una et eandem proportionem producat. Id quod per exempla iam data uidere licet. Nam ut multiae proportiones, ex quibus quaequam constat: facile invenientur: ita omnes exquirere, quae eam formare possint: infiniti operis erit.\n\nTranslation:\n\nThe terms next in succession to those given by the multiplication of previous denominators are: ducas, tandem producunt. 16. The proportion between 16 and 1 is 16 and a half. It is clear that the proportion of 16 and a half is composed of quintupla, sesquiquinta, sesquitertia, and dupla. Therefore, when terms are to be interposed between the extremes innumerable ways, through minute details, in which each can be broken down: it is permissible to vary: it is manifest that the proposed proportion cannot be fully explained: how many and which proportions can compose it: since an infinite variety produces one and the same proportion. This is already evident from examples. For just as many proportions as there are, of which any one is a constituent: they can easily be found: so it is an endless task to inquire which ones can form it. Two extremes have every proportion: the one that is greater we recognize by the name of the proportion itself. The division of the known extreme is called the denominator of the proportion, and the lesser extreme, which is unknown, will not reveal itself in the number of partitions. For example, let a double proportion be proposed. Let us make the greater extreme known to us. If we wish to extract the lesser extreme. We will find it by dividing it into two, the denominators of the proportion. The lesser extreme will then come to light. But if we know the denomination of the proportion, the lesser extreme will be unknown to us, and if we wish to inquire about the greater, we will bring the lesser extreme into the denomination of the proportion, and the resulting number will be greater than the greater extreme. For instance, if we know that the proportion is triple, we will call it three. By adding three, a greater extreme will be born. That is all. If we know the name of the proportion, we do not have either extremity known to us: we can add any numbers we please that form the proportion.\n\nCicero on the Origin, Kinds, and Composition of Proportions. Now remains: that we speak of Addition, Subtraction, Multiplication, and Division. As we promised the diligent reader at the beginning of this book.\n\nThe Addition of proportions, whether continuous or separate, similar or dissimilar, must be done according to the method explained a little above. For instance, in a long series of proportions, the denomination of the first two terms is successively taken as the denominator of the following term. And thus the product of each denomination is multiplied by the next, and this process is continued until the end, when the final product becomes the sum of all the proportions. Therefore, the last product, as it were, a kind of sum, will demonstrate that it encompasses the entire proportion. Esto, these are the seven numbers proposed: 30, 24, 12, 6, 4, 2, 1. Between these seven numbers, there are six intervals, and the same goes for the proportions. Starting from the left and comparing each larger number to the next smaller one, we find the first proportion to be the fourth power of two: the second and third are both doubled: the fourth is the fourth power of two multiplied by three-halves: the fifth and sixth are both doubled: we label all these ratios as follows. The first ratio is the fourth power of two between 30 and 24. The second ratio is doubled between 24 and 12. We produce 200 of the fourth power of two as the third ratio between 30 and 12. Then, we multiply this third ratio by three-halves to obtain the fifth ratio between 30 and 6. Finally, we multiply the fifth ratio by three-halves to obtain the sixth ratio between 6 and 4. \"30. If the septupla sesquialtera is brought close to dupla twice between 4 and 2, a quindecuple will be produced between 30 and 2. Multiplying the quindecuple by 2 in the last place, we obtain trigintuple between 30 and 1. By forming the ratio of the first and last term in this way, all the proportions of the numbers between them are contained as if in a sum. Proportions, which are the relationships between numbers, are most conveniently collected into one sum through addition. Some prefer to add proportions in another way: such that none of the terms to be added are under another, but the terms to be added are opposed to each other from the right. Then, they multiply the first term of the first proportion by the first term of the second proportion, and set the resulting number as the composite of the first term of the first proportion and the first term of the second proportion. Then, they bring the last term of the first proportion to the last term of the second proportion. And they make the product similarly the composite of the last terms.\"  Porro si proportiones plures adde\u0304\u2223dae fuerint: tertiae proportionis priorem terminum in pri\u2223orem terminum ex primis duabus compositae ducunt: et productum statuunt terminu\u0304 priorem ex tribus illis com\u2223positae. posteriorem{que} terminum terti\u0119, in posteriorem ex duabus primis compositae ducentes, productum termi\u2223num apponunt posteriorem ex tribus illis compositae. Et sic ulterius, priorem terminum cuiusque proportionis composit\u0119 in terminu\u0304 priorem proxim\u0119 cuius{que} propor\u2223tionis\naddendae, et posteriorem cuius{que} composit\u0119 in po\u2223steriorem cuius{que} addend\u0119, ducunt, us{que}, in finem. Quo fit: ut inter terminos postremo productos, proportio enasca\u2223tur omnes priores complectens. Exemplum afferamus.\nSint hae tres proportiones addendae: prima sesquitertia inter. 4. et. 3. secunda sesquialtera inter. 3. et. 2. tertia dupla inter. 2. et. 1. Si primae proportionis priorem ter\u2223minu\u0304. 4. duces in. 3. priorem secundae, creabis. 12. qui terminus erit prior proportionis compositae. Et si. 3 posterior terminus primum duces in. 2. posteriorem secundas: producet. 6. qui terminus posterior compositae proportioni erit. Ita producuntur illi termini. 12. et 6 duplam proportionem habentes indicant ex sesquitertia et sesquialtera simul additis, fieri duplam. Id quod etiam denominaciones ipsa altera in alteram, ductas probant. Iam vero priorem terminum proportionis compositae, scilicet 12, in tertiae proportionis priorem terminum 2, multiplicia producunt. 24. qui terminus erit prior proportionis ex tribus compositae. Et si 6 posteriorem terminum compositum in posteriorem tertium 1 ducis: nihil nisi ea ipsa 6 habebis. qui terminus erit posterior proportionis ex tribus collecta. Ita proportio inter productos terminos 24 et 6 monstrat ex sesquitertia, sesquialtera, et dupla simul additis, nasci quadruplam. Nam, ut inquit Euclides. Omniorum duorum numerorum compositorum proportio unius ad alterum, est ex laterum suorum productis proportionibus.  Latera autem numerorum appel\u2223lantur: quorum multiplicatione numeri producuntur.\nId quo{que} uerum esse, proportionum denominationes, alia\nin aliam, sicuti praecepimus: ductae, manifesto probant. Hic proportionum addendi modus per earum terminos multiplicatos, uti diximus: productorum terminorum mutuas habitudines obiter commonstrat. Caeterum mo\u2223dus ille superior denominationes proportionum, alias in alias ducendi, multo apertius indicat collectam earum summam.\nSVBDVCTIO proportionu\u0304 monstrat: quando aliae proportiones ab alijs subtrahuntur: quae proportio\u2223nes restabunt. Ea autem tum demu\u0304 fieri potest: si mino\u2223res sunt: quae subducuntur: {quam} a quibus fit subductio: uel si sunt eis aequales. Ne{que} enim per rerum naturam, quod maius est: ex minore demi potest. Quamobrem plurimu\u0304 iuuabit meminisse, proportiones eas alijs maiores esse: quae maiores alijs denominationes habent: minores au\u2223tem, quae minores. id quod superius admonuimus. Porro modus subducendi proportiones longe facillimus hic est Denominations of ratios to be subtracted: and of those from which the subtraction is to be made, are to be marked by the numbers themselves. Then, the number denoting the ratio to be subtracted, from which the subtraction is made, is to be divided by the number that denotes the ratio to be subtracted. And the number of the section denotes the ratio remaining from the subtraction. For example, the sesquialterate ratio is to be subtracted from the double ratio. 2.\n\nWhich ratios denote the double ratio, divided by 1 \u00bd. The sesquialterate ratio is called by this name: in the number of the section, they present 4/3. When reduced to the integral, they make 1 \u2153. Thus, they demonstrate the sesquitertian remaining from this subtraction. A certain example shows that the ratio left is correctly subtracted: if the left ratio, added to the subtracted ratio, restores the proportion from which it was taken. This is how it works in numbers. The sesquitertian, added to the sesquialterate, by another method, forms a double in the number of the section. Another example: we subtract the sesquiquartan from the double sesquialterate. Two and a half denarii represent a double sesquialter, divided by 1 \u00bc, which are called sesquiquartas: in the section number they show 20/10. There are two integers: and the double proportion remains from that subtraction. Again, if you please, take an example: add the remaining double to sesquiquarta, leading one into the other: and in the section number, double sesquialtera returns again.\n\nAnother method of subtracting proportions through these terms is given by some. The proportion to be subtracted, as well as that from which it is to be subtracted, and the subtraction itself, should be noted with small numbers: so that the work may be more easily done. They write the proportion from which the subtraction is made above: but the proportion to be subtracted is written below: so that the terms respond to each other, prior to prior, and posterior to posterior. The ratio of the first term to the second, from which the subtraction begins, is led to the second term in subtraction, and the first term in subtraction is led to the second term in the subtraction, in the form of a cross. And they establish the terms produced from these two multiplications as the terms of the ratio. For example, the ratio of 3 to 2 is subtracted from the ratio of 4 to 3. In this subtraction, 6 is taken from 3, and 2 is taken from 4, and they produce 8 between these terms. Between the terms produced, that is, 18 and 8, a double sesquiquartic ratio is found: which is left over from the subtraction of the sesquitertia ratio of 3 from 4.\n\nYou will prove this subtraction to be correctly made: if you add the remaining ratio's denomination to the subtracted one, as we have said. Or if you prefer: add the remaining ratio to the subtracted ratio, through the terms of one in the other. As we will explain in the next chapter. When that ratio returns, the proportion is as follows: If several proportions must be subdued, add them all first to one proportion, or to a sum proportion, and then proceed to their subduction, provided it is not greater than the one from which the subduction will be made.\n\nSome subdue them singly, but one who gathers all to be subdued beforehand in a sum makes one subduction easier. Proportion is thus subtracted in two ways. The first through denomination and section. The second through their terms obliquely, as we have said, making them complex. A third method is also found among these two, which presents all three - the remaining proportion, the subtracted one, and the one from which the subtraction is made - before our eyes. That is the method in question.\n\nNote that ratio - from which the subtraction is made - should first be noted in its numbers. When the greater and the smaller third number are placed next to each other: to which the greater term should be related, what is this ratio, and what should be subjected to it. Therefore, the ratio between the mean number and the smaller term will be found. This ratio, which will remain stable after subtraction, is what we are looking for. For example, a triple proportion between 6 and 2: we subtract sesquiteria. After this has been noted, the number to be sought is 6 and 2. The major term, as we can see, is related to this proportion in a ratio of sesquiteria: this number can easily be found if you remember the rule we mentioned earlier. What is this kind of rule? When the greater term of any proportion is known, along with its denomination, the smaller term, which is ignored in the division by the section number, can be found from the quotient of the greater term by the denomination of the proportion. Therefore, 6, which we have determined to be the smaller term of the sesquiteria proportion we have examined, is related to the greater term in a ratio of 1/1.3, and the number of the partition will be 18/4, which equals 4.5. Is the minor term the proportion's sesquitertia between 6 and 4.5? After finding the proportion to be subtracted between the larger extremes and the intermediate number, compare the intermediate number to the smaller extreme and find the proportion left, which will be twice sesquiquarted. This will be true if you add the remaining double sesquiquarted denomination to the subtracted sesquitertia, as we have shown. Furthermore, a triple proportion will be born through this subtraction, allowing us to see three proportions at once: one that we subtract, another that we subtract from, and the third that we leave. The difference in these proportions, which arises from subtraction, is revealed in this way.\n\nTherefore, the addition of proportions seems similar to the multiplication of numbers. However, their subtraction is significantly different, especially in small quantities. \"The things stated in this head and the next: they make it clear.\nMultiplication of proportions is greatly different from that: which is done in numbers. A number, indeed, is a collection of units. But a proportion is not a number: it is rather the relationship and position of numbers: which can only exist between two or more numbers. Therefore, the term \"proportion\" is not properly multiplied, according to common sense. However, when a proportion is added to a proportion, the numbers themselves or their terms can be multiplied by other names of numbers or terms, as was said above about their addition. The multiplication of names of numbers or terms is the addition of their relationships. Therefore, the public thinks that the collection of various proportions can be made into one sum through multiplication: whereas it is an addition. Therefore, the multiplication of other proportions can be done in no way similar to the multiplication of numbers.\" If the numbers involved in your proportion are similar and continuously generated from one stock of numbers, in accordance with the given proportion of the smallest: it will be of this kind. The proportion that you wish to make of several similar numbers: note the smallest numbers, which have this proportion. Then, multiply the first term in itself: the resulting number is marked. Repeat the same term is drawn for the next one. The third posterior term is drawn into itself. The third product is placed in the third position. And thus, between these three products, two intervals will be present: and the two proportions of the stock from which they were taken will give birth to similar proportions. And this proportion, which is between their extremes, encompasses two intermediate proportions. For example, we mark the sesquialteral proportion with these numbers. 3 and 2 we take. 3 multiplied by itself, and 9 result. Repeat, 3 multiplied by 2. We multiply them: they will become many. Then, among those produced, three numbers are in a sesquialteral proportion. The first and last of these three proportional terms are contrary to each other, so the three numbers in the middle, according to the given proportion, are the three smallest. According to EVCLIDES, if any number of proportional numbers have two extremes contrary to each other, all of them, according to their proportion, must be the smallest. If we want to produce four numbers in the same proportion: we take the first term of the series and multiply it by each of the three produced numbers; then we notice the new numbers produced. Next, we multiply the third produced number by each of the three. multiplicantes, producemus. 12. According to the term of the stem, that is, 2. in the third number before it is produced, that is, 4. Let us make four, and among these four numbers there are three proportions in the sesquialterate ratio. And the proportion between the first and the last, 27 and 8, is three times the fourth part of the third, 27 and 8 being the extremes of the four continuous proportional numbers. Therefore, these four are minims against each other. In the same way, multiplying the first stem by all four numbers produced last, and then multiplying the last stem by the fourth number produced last, we will form five numbers and four continuous sesquialterate ratios. And in the same way, any number can be formed using this proportion continuously. As the note below shows.\n\nCleaned Text: multiplicantes, producemus. According to the term of the stem, produce in the third number before it is produced, let us make four. Among these four numbers, there are three proportions in the sesquialterate ratio. The proportion between the first and last is 27:8, which is three times the fourth part of the third: 27 and 8 being the extremes of the four continuous proportional numbers, these four are minims against each other. Multiply the first stem by all four numbers produced last, and then multiply the last stem by the fourth number produced last, we will form five numbers and four continuous sesquialterate ratios. Any number can be formed using this proportion continuously. If someone wants to multiply a proportion by a certain number: then the denomination of the proportion or some other thing will be considered as a number to be multiplied. And therefore, as many units will be found in the product as there are proportions of that denomination marked by that number.\n\nFor example, if we want to multiply one triple by three: there will be 3 triples. If we take two triples and put them into three: there will be 6 triples. Similarly, if we multiply one quadruple by three: three quadruples will result. If we take two quadruples and put them into three: there will be 8 quadruples. And so on.\n\nThe division of a proportion is made: when an extremity or a middle term, or several terms are interpolated. This is how it happens: so that the proportion is divided into as many other proportions as there are intervals between its extremities from that interpolation. As we have already said above: how to find proportions from any given proportion. If we wish to divide a proportion into two parts, a unique term must be inserted. For instance, if you insert it in a proportion of sixteen to one, you will have eight parts, one of which will be double the first, while the other will be eight times the last. However, if you insert three terms between the extremes of a proportion of sixteen to one, such as eight, six, and five, the division will result in four proportions: the first will be double, the second sesquiteria, the third sesquipenta, and the fourth quintuple. This applies similarly if you add any number of middle terms. The division will be made into the same number of proportions as the number of intervals between the first and last. It makes no difference which middle terms are inserted, whether they are larger or smaller than the extremes, as we explained more fully earlier. A division is made in proportions in this way, and there is no other way, for in these proportions the division is found. To find what proportion a given number bears to numbers according to that proportion, follow this method: First, ensure that the numbers of the given proportion are mutual primaries, so that no number counts them except in unity. If they are of this kind, they themselves have that proportion in common. However, if another number counts them beyond unity, the greatest common number is to be investigated. And through it, the numbers to be divided, both preceding and following, are found. This is done so that the smallest numbers of the same proportion appear in the numbers of both sections. The greatest number that counts both numbers in common counts them through the smallest numbers of the proportion. But how the greatest number that counts both numbers in common is to be investigated:\n\nWe have explained more fully in the second book: When we discussed: How parts are reduced to the least possible nomenclature. Quare quidem illic de numeratore et denominatore partium sunt dicta: hic de numero antecedente et numero consequente cujusque proportionis repetita intelligentur. Nam idem utroqueque modus est et partes ad minimam sui nomenclatura, et proportiones ad minimos numeros ipsas habentes, reducendi.\n\nExemplum demus in numeris contrariis primis. Inter 7 et 3, proportio duplex sesquitertia est. Nec minores numeri haberi possunt: inter quos sit ea proportio.\n\nExemplum demus in alijs numeris communicantibus: quos communis aliquis numerus alius quam unitas metitur. Veluti inter 35 et 15, inter quos proportio duplex sesquitertia ex sectione maioris per minorem reperitur. Et quia utraque illorum numerorum numeratur a 5, qui maximus numerus est, eos communiter numerans: utrumque per 5 dividamus. Et ex sectione 35 facta per 5 in numero partitionis 7 inueniuntur. Ita ex duabus illis sectionibus minimi numeri 7 et 3. The following proportions, when separated and disconnected, can have a proportion of two to three. To find a proportion of any given proportion, separated and disconnected, follow this method:\n\nLet any proportions be noted, with their own minimum terms. Then find the smallest number: the one that follows the extreme of the first proportion, and the extreme of the second. This number, when joined with the first proportion, will produce the result. The number preceding this should be added to find: if we multiply the number of times the first extreme follows in the first proportion, by the number of times it is multiplied in the second proportion, we find the smallest number produced by the first, and the number preceding the second.\n\nTo find the second proportion's continuation: if we multiply the number of times the second proportion's continuation is multiplied, by the second proportion's disconnected part, we produce the number created by the preceding term of the second proportion, and the number following the first disconnected part. If the preceding term of the third proportion is disconnected, the following term continues the second connected term: by the same number, the following term multiplies the disconnected third term; and produces the third proportion's third term, which must be continued with the two preceding terms. However, if the third term preceding is not a multiple of any number of the second term's continuity, the smallest number must be found: which the following term counts and the preceding third term counts; this number is found, and the preceding third term will be a multiple of the third proportion's continuity. Then, if you multiply the following term of the third disconnected term by that number, the product of the following term multiplied by the preceding term will produce the minimum number counted by the following term of the third proportion. Finally, if you multiply the remaining preceding terms by that number, the following term of the second continuity produces the minimum number from itself and the preceding proportion's third disconnected term. When the three proportions are proposed separately, they will be combined. In my smallest terms, and the two first proportions were once connected in smaller terms; yet, we cannot combine these three into smaller terms as one. Furthermore, any proportions can be continued to the same degree. I will give an example with three proportions for clarity: let us reduce these proportions to their smallest continuous proportional numbers, namely the first, the triple, and the superquarter. Secondly, the subquadruple sesquialter, and thirdly, the sesquialter. Moreover, the smallest triple is found between 15 and 4, the smallest subquadruple sesquialter between 2 and 9, and the smallest sesquialter between 3 and 2. Let us arrange these numbers in order: 2, 1, 4, 2, 3, 9. To find the minimum number that continues the first two proportions proportionally, we inquire about the number that follows the first and precedes the second. Since these numbers are commutative, we inquire about the smallest numbers of the proportion to which they belong. These numbers are 2 and 1. Every two numbers have between them two smallest numbers, either greater or smaller, when multiplying, the smaller number is produced by the numbers themselves. Therefore, we note the numbers above the first proportion and the second annotation. Next, we investigate the preceding number: since the fourth follows the first, the fourth minimum number is multiplied by the number following the first and the unit. Multiplying the preceding number of the first by 15 and the number itself, we get the numbers above the line that will be equal to the fourth number marked. Then, to find the following number of the second proportion, we note that the second preceding number of the second disjunct is multiplied by 2. Therefore, the fourth minimum number is multiplied by the number following the first disjunct and the preceding disjunct: multiplying by 2, we get 9, and producing 18, which are the numbers above the line and will follow the second proportion. From two disconnected proportions, we derived two continuous proportions: with which we coupled a third disconnected one. Since the preceding term of the third disconnected proportion counts the following term of the second continuous proportion, that is, 18 multiplied by 6 of the third disconnected proportion, we obtain twelve, which will be the following terms of the third continuous proportion. Thus, the four numbers 15, 4, 18, 12 will continue as proportional numbers, according to the three disconnected proportions, namely, the tripla, subquadrupla, and sesquialtera. The smallest continuously proportional numbers are those in which these proportions can be found according to these disconnected proportions, because no smaller numbers can do this.\n\nHowever, if the third term of the disconnected proportion does not count the following terms of the second continuous proportion through any number, we must find the smallest number: which the following terms of the second continuous proportion and the third disconnected term count. For example, if sesquialtera is between 4 and 1. 3. et. 2. wish to continue with a triple intersection of the three: we will find this out in the same way we have said: the first two proportions are quadruple and sesquialter to these three minimal terms. 12. 3. and 2. are to be continued according to the given proportions. Which third term is to be joined later is to be determined. And since the preceding term of the third term disjoined does not continue the sequence, the smallest number among these numbers is to be found. And since 3. and 2. are numbers opposed to each other: when one is added to the other, the result will be produced. 6. what will be the antecedents of the third term to be continued: and above the vertex of the consequent second term, the continuation is to be placed. Then, since 3. added to the disjoined third term produced 6, similarly through these very same 2. we multiply the following term disjoined from the third term, that is, 1. and 2. will result from the continuation of the third term. However, since the third proportion continued between 6 and 2 has greater terms: {than} so that the preceding terms may agree with it. et consequens secundae continuatae. In the second place, they produced a minimum number from themselves. We will multiply by these. The antecedent of the second continued sequence, namely, they also proceed. Nine will be the antecedents of the second continued sequence that will follow: they will be placed above the vertex before. Again, we will multiply by these. We will also multiply the antecedent of the first continued sequence, as you see. Twelve and thirty-six will be the antecedents of the first continued sequence: they will be placed above the vertex before. Nine will be placed back. Thus, in these four terms, there are three proportions, namely the quadruple, sesquialtera, and tripla, according to these three distinctly given ratios. From the last term before the continuation of the multiplication by the same number, this is done so that the third continued sequence will agree with the proportion. These same proportions are produced: which were the continued sequence in smaller terms. According to Evclidem, if one number is led into several: the producer of the numbers following will be the same ratio of multiplicators.\n\nBy this reasoning, the fourth proportion is thus distinguished, and it is allowed to connect any other proportions continuously through the terms. For the first proportions that need to be handled, it is necessary to deal with them first, and then move on to others.\n\nAmong geometrists, a certain kind of proportion is found, which is very different from the ones mentioned above. Evclid\u00e9s calls this proportion the one having two middle and extreme terms. This proportion cannot be had in fewer terms than in three, since other proportions consist between two. In this respect, the similarity of proportionality refers to it. And to some body's dimension, which cannot be understood without it, Geometrists bring much benefit. As Evclid\u00e9s explains in his book 13, with many teachings, and in proposition 29 of book 6, according to this proportion he teaches to cut a line. In numbers of this kind, there is no such proportion: it does not apply: we shall leave it to the Geometers.\n\nThe progression of proportion, which is called Geometric, is the collection of more than one number in a sum, which affords a compendium for numbering those numbers. Among them, the proportion is continuous and equal. Regardless of the differences in number, they are unequal. Since the forms of proportions are various, various rules are given for these progressions. Therefore, from the species of multiplication, in which the first duplicates occur, we shall begin.\n\nIn continued numbers with doubled terms, the first number should be subtracted from the last one in a long series, and what remains of the last one should be added to the whole last number. Thus, the sum that arises will contain all the units of the series. It makes no difference from what number the series begins. Nor does it matter whether the numbers are proportional to integers or fractions.\n\nLet us give an example with these numbers: 3, 6, 12, 24, 48. From the last one, subtract the first, and 3 remains. When numbers in a triple series are continued: subtract the first number from the last, and add half of what remains to the last number. Thus, the universal number will emerge: from whatever number the series begins, even if minor interventions occur. For example, these are the numbers: 1, 3, 9, 27, 81. Take away the first number. 1, 81 remain. Half of 81 is 40. Add 40 to 81 and you get 121, which is the sum of all. Similarly, it will happen if you start with 2. 2, 6, 18, 54. Take away 2. 52 remains. Half of 52 is 26. Add 26 to 54 and you get 80, which is the sum of all. In numbers that have quadruple proportional continuations, take the first from the last, and keep the third part of what remains, which is left over. Add this to the whole last number. The sum of all will be held. For example, in these numbers: 1, 4, 16, 64. Subtract the first from 64, and 63 remains, whose third part is 21. Add this to the last, and they total 85. The same will be in 2, 8, 32, 128. Take away 2 from 128, and 126 remains, whose third part is 42. Add this to 128, and they total 170.\n\nIn numbers that have quintuple long series connected, if the first is taken away from the last, and the fourth part of what is left over is added to the whole last number, the sum of all will be produced. For instance, in these numbers: 1, 5, 25, 125. Subtract 1 from 125, and 124 remains, whose fourth part is 31. Add this to 125, and they total 156. In infinitum, this order proceeds in multiples: for if the progression is in sextuples, after the subtraction of the first from the last, the fifth part that is left over is joined to the last. If in septuples: after the subtraction of the first from the last, add the sixth part to the last. If in octuples: add the seventh part after such subtraction. If in nonuples: add the eighth part. And so on, after the subtraction of the first from the last made, add that part which is called the unit less than the denominator to the last. And the sum of all will result.\n\nRegarding the progression of numbers of a superparticular ratio, rules are also given. And first, concerning sesquialter numbers. In continuing sesquialter numbers, if the first is doubled and subtracted from the last tripled: what remains will show the sum of all the numbers. For example, 2, 3, 4.5, 6.75 are doubled, yielding 4, 12, 28.32; and tripled, yielding 12, 36, 84.368; then subtract 4 and 12, and the remainder, 16.326, is the sum of all.\n\nIn sesquitertians, the first number, tripled, is to be subtracted from the last, quadrupled: and the remainder will indicate the sum of all. IN sesquiquartis, take the first quadrupled and subtract the last quintupled. The remainder will show the sum.\nIN sesquiquintis, take the first quintupled and subtract the last sexupled. The remainder will show the sum.\nIN sesquisextis, take the first sextupled and subtract the last septupled. The remainder will leave the sum of all. And so on, the number obtained from multiplying the first in the proportion of the denomination should be subtracted from the number obtained from multiplying the last to the number one greater: whatever the denomination is. Thus, the remainder will show the sum of all. It makes no difference from what number the progression begins.\nSIMILARLY, rules can be given for other proportional forms, if one wishes to observe. However, unless they are simple: it is much better to collect the sum of their numbers themselves rather than to seek a concise summary through winding paths. If from where does progress come: if no compendium brings it? Now: in order to open a way: through haphazard conjectures, and indeed false ones, and to the errors that follow: the truth of the proposed question is explained. The Arabs and Phoenicians were famous for trade, and from whom arithmetic is believed to have originated: they called the art of finding truth by a barbaric name, Cathay. The Latins call it falsarum positionum or falsarum collectionum rules. Through these almost all things that pertain to negotiation can be solved, and many other things as well: provided that some part of the matter is certain and known. For through these miraculous means, all other things are unknown. When riddles present themselves: they are continuously investigated and brought to light. It cannot be said how divinely inspired the first discoverers of this science were: whose cleverness it was that brought what seems incomprehensible into the light of day. \"Many questions concerning numbers often arise for us, in which one part is known to us, but another, which we desire to know most, is far removed from common sense. We can make this part congruent with the known part at our discretion, and if our conjecture has come close to the truth or has strayed far from the goal, we can conjecture again. Those that are found in a single position can be resolved in two ways. However, the converse will not be the same. For not everything that two positions explain will be provided by a single one. And what is most remarkable about this is that it makes no difference if our conjecture deceives us greatly. For through these very errors, rules will soon be given to lead us to the truth. We will speak of these matters first, indeed, concerning that which can be resolved in a single position, that is, the thing that was sought.\" In numerorum questions, where part is known: part is unknown: it frequently happens: that a single hypothesis, even if it deviated from the truth: is evident. For, as we observe: what the hypothesis itself produces: we note the proportion: which is between what we assume and what follows from it: a similar proportion can be produced between the known and the unknown parts, provided that the four proportional rules, which bring forth the fourth unknown from the given three, assist us. Indeed, since three are known: the first, what we assume: the second, what follows from it: the proportion between them: the third, which is known about the proposed question: certainly the fourth, which is unknown: and the third should respond: it cannot hide. Thus a single position creates the first proportion for itself. An example of this rule of the four proportionalities succeeding, and forming a similar proportion, brings the sought-after thing immediately into light. Exempla reveal more. A traveler finds six golden coins on the road: making the second, third, and fourth parts equal. What was the total amount found? Choose any sum: what parts does it consist of? And see if the named parts make up the total. If not, you will have to investigate further in this way. Imagine a sum found: what parts does it have? Let it be one-twelfth, six-twelfths, four-twelfths, and three-twelfths. You ask which parts make up the total. But you are mistaken. That very error on the road will lead you back: if you consider the matter well. For just as the parts of a sum, placed by conjecture, are equal if added together. They have the same relation to the total sum: what it is. The parts of the found sum, when added together, have the same relation to the sum found, which is unknown. Therefore, do not apply the rule of three notices to such a case. Yes. If the parts added together do not make up the total: they come from. Twelve. male a te posito: cuius summates illae erant: quae completent. 50? Sequens regula multiplicare. 50. per. 12. et surgent. 600. ea per. 13. dividere: et prodibunt. 46 2/13. quae summa aurea reperta erat. cuius dimidium est. 23 1/13. tertia pars. 15 5/13. quarta pars. 11 7/13. quae simul additae faciunt. 50.\n\nSimilarly, this will be the case for other sums: quae cumquae alia summa quam. 12. per coniecturam ponere: modo partes eas habeat. Vides ita per adiumentum regulae de tribus notis una coniectura fortuita rem expediri: nec duabus positionibus esse opus.\n\nPER THIS rule, examples following: explicari sunt. quae subijcienda duximus: ut ad ea iuvenes exercetur.\n\nINVESTIGETVR numerus: in quo sint 5 2/3. Pone numerum: quem uoles: qui partes eas habeat. ueluti 6 et uide: quantum capiant 2/3. et invenies. 4. at tu quaeris. 5. Scrutare igitur: si 4 faciunt 2/3 de 6 de quo. 5 complebunt 2/3? tenta: et invenies. 7 1/2. EXQUIRATUS number: since one third, one fourth, and one fifth have been subtracted: there are still some left. 24. Place a number there: which has those parts. For example, 60. Afterwards, divide those parts: and see what remains. You will find it. 13. Behold how far off you were. Did you ask for 24? But you found less than that. 13. Therefore, reason acts this way with you. Yes, 13. After subtracting the parts, there will be some left of the number. 24. Try it according to the rule of the three known parts: and you will find it. 110 10/13. That number is the one whose third part is it. 36 12/13. Fourth. 27 9/13. Fifth. 22 2/13. All of which added together make up. 86 10/13. And if they are subtracted from 110 10/13. they will remain. 24.\n\nIf anyone orders to find a number: from which the third, fourth, and fifth parts have been subtracted: the remainder will be. 24. I continually respond, it is not possible for that to happen. For whatever number is taken: which has those parts: it will always be smaller than the sum of its parts. How can it be made: so that even those cannot be subtracted, let alone remain. Three types of numbers exist. The first type is called \"abundant numbers,\" which have a larger sum of their parts than they themselves possess. An example of such a number is 12, as its half is 6, one third is 4, one fourth is 3, one sixth is 2, one twelfth is 1, and the sum of all these parts exceeds the number itself by 16 and surpasses its own value. Another abundant number is 24, as its half is 12, one third is 8, one fourth is 6, one sixth is 4, one twelfth is 3, one twenty-fourth is 2, and the sum of all these parts exceeds the total number by 36.\n\nThe second type is called \"deficient numbers,\" which have a smaller sum of their parts than they themselves possess. An example of such a number is 8, as its half is 4, one fourth is 2, and the sum of all its parts is less than the total number by 2. Another deficient number is 14, as its seventh part is 7, one fourteenth is 2, and the sum of all its parts is less than the total number by 10.\n\nThe third type is called \"perfect numbers,\" in which the sum of all parts collected is equal to the number itself. An example of such a number is 6, as its parts 1, 1, and 4 sum up to 6. Another perfect number is 28, as its parts 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 sum up to 28.  Hi, quia nec excessum nec defectum habent: perfecti uocantur. cuiusmodi numerus est. 6. nam eius dimidiu\u0304 est. 3. pars tertia. 2. sexta. 1. qu\u0119 partes compositae reddunt etiam. 6. Item. 28. numerus est perfectus. nam dimidium habet. 14. quartam partem 7. septimam. 4. quartamdecimam. 2. uicesimam octa\u2223uam. 1. quae partes coaceruatae suo toto numerum aequa\u2223lem faciunt. 28. Hi numeri perfecti, {quod} rari sint: uiris bo\u2223nis assimula\u0304tur. Ita{que} ut ad rem redeamus: minimus nu\u2223merus, qui \u00bd \u2153 \u00bc habeat: est. 12. huius dimidiu\u0304. 6.\ntertia. 4. quarta. 3. partes h\u0119 collectae proferunt. 13. qu\u0119 a. 12. subduci nequeunt. {QUOD} si numerus maior partes easdem habens sumatur: partium summa maior redu\u0304da\u2223bit. Ita nun{quam} fiet earu\u0304 subductio: nedum quic{quam} restabit. Quare propositis huiusmodi qu\u0119stionib{us}, continuo, pri\u2223us{quam} ad illas respondemus: nobiscum meditemur: an pos\u2223sibilia suggerant: ne labor irritus frustra suscipiatur.\nSI. 4. ESSENT. 6. quis numerus esset If you are asking for a cleaned version of the given text, I will do my best to remove meaningless or unreadable content, correct OCR errors, and translate ancient English or non-English languages into modern English, while staying faithful to the original content. However, I cannot remove the text itself, as it is the only thing provided as input. Here is the cleaned version of the text:\n\nIf you ask me this, ask the questioner what he understands by such ambiguous command. Does he want to increase or decrease? For instance, if they increase according to the same ratio, they will increase to 15. If you lead them for six, it will grow. Sixty who separate from four advance. Fifteen, on the other hand, if you want to decrease them by six, make it so. If they become six, what will they be? Ten are led in and rise up. Forty are divided into six and exit. Six and a third are those that need to be decreased.\n\nLikewise, if someone asks in this way. If half of five were three, what would be the parts of the number that are a quarter? It is necessary to find out before anything else what he wants for himself. For instance, if half of five, which is 2.5, should increase to 3, it increases in the same way. Five, according to the rule of three notices, will increase to six and be a quarter of 24. However, if three should decrease to 2.5, it decreases in the same way. Five, according to the rule of three notices, should decrease to 4.16667 and be a quarter of 16.6667.\n\nIf three were half of seven, what part would three be of eleven? If three should increase to 3.5, consider what follows according to that ratio. Four should be. et per regulam de tribus notis find you that it grows to 4 \u2153. Now it is necessary to consider. 4 \u2153. What part are 11. The thing you should know: if 11. through 4 \u2153 sections, they will yield. 2 7/13. That part will be 4. There will be 11. According to this theme, 11. remain unchanged. It is necessary to pay attention: what it is that is being sought, so that the response does not deviate from what was asked. Furthermore, if half of 7 should be reduced to 3, and the ratio of conversion was to be reversed as stated in the previous passage, it would be as follows.\n\nThe Roman numerals two: in which half and third are one fourth and one fifth of another. Take some number: which has one second and one third. For example, 54. Whose fourth and fifth are one fourth and one fifth: you can determine this by investigating a single number in the following way. Place any number that has a fourth and fifth: for instance, 60. Then see which number is the fourth and fifth of that number. And you will find that it is 27. But you were asking about 45. Therefore, through the rule of three notices, this is how the ratio is calculated. If 27 are fourth and fifth of 60, what number is it? \"Examine the following rule: you will find it to be so. 100. Thus, there are 54 numbers to seek. Another way, which is indeed more succinct, can be followed to this end. These minute details, which pertain to one number each: should be added together to make one set. This will result in 3/4 and 5/12. However, 1/4 and 2/5, when added, will result in 9/20.\nNext, these minute details should be arranged separately, with each denominator being made the denominator of the other's numerator and placed in a cross form. The resulting products should then be arranged next to their respective denominators: from which they were derived. Thus, from this multiplication, 54 will be in the ratio of 3/4, and 100 will be in the ratio of 9/20.\nHowever, numbers that communicate unequally from the oblique products of the multiplication: are those that arise from the mutual multiplication of the denominators. For example, 45 are 1/2 and 3/4 of 54. Similarly, 45 are 1/4 and 2/5 of 100.\"  Hoc compendium inueniendi numeros: qui partes ad hunc modum inaequaliter commuicantes habeant: ad multos in numeris nodos explicandos plurimum iuuabit: si in promp\nMERCATOR argentilibram, quae de nota at{que} in\u2223dicatura. 8. scrupulorum erat: aureis. 9. emit. qu\u0119\u2223ro, si secundum eam rationem uelit de nota. 10. scrupulo\u2223rum emere libras. 20. quot aureis constabunt? Pone: ut libet: puta constituras aureis. 80. ut singulae librae u\u0119ne\u2223ant aureis. 4. Vide quid secundum eam rationem una li\u2223bra\nde nota. 8. scrupulorum constabit. id, quod facies sie ratiocinando. Si argenti nota de. 10. profert unius librae precium. 4. quod precium unius libr\u0119 dabit nota scrupu\u2223lorum. 8. Sequere regulam de tribus notis: et inuenies ea\u0304 dare. 3\u2155. at tu scis cam dare. 9. Quamobrem sic suppu\u2223tabis. Si. 3\u2155 prodeunt ex. 80. quae coniectura posuit: de quo prodibunt. 9? Duca\u0304tur. 9. in. 80. et surgent. 720. quae si seces per. 3\u2155. in numero partitionis prodeu\u0304t. 225. tot aureis librae. 20. de. 10 scrupulorum nota constabant: quarum unaquaeque precio erant aureis. 11\u00bc.\nQVI ARGENTI libra de. 10. scrupulorum nota, aureis. 11. emit: postea libras argenti. 25. de alia nota aureis. 200. mercatus est. Quia earum unaquaque constituit aureis. Sic rationem considerare. Si. 11. aurei reddent notam. 10. quam notam reddent aurei. 8? Scrutare et invenies. 7 3/11. Ea nota et indicatura librarum. 25. erat. Hic propter facilitatem nulla posione fuit opus.\n\nComplures huiusmodi quaestiones, et multae etiam ex his, quas supra libro tertio explicavi, mus: per una positionem, adiuvante regula de tribus notis, solvi possunt.\n\nInnumerae quaestiones in numeris occurrunt: quae, tam et si earum pars sit cognita: pars sit ignota: per unicam coniecturam explicari nequeunt. At hae per duas positiones statim expedientur. In quibus ante omnia observari oportet: quam prope utraque posito vel accedit ad verum: vel a vero recedit. Item ipsorum errorum differentia magnopere notanda est. According to the observation of both positions in relation to the true one, and through the difference of errors that follow: truth will come to light. This can be found in two ways: either through the rules of the greater and lesser [things], or through the observation of differences. Therefore, these four principles should be learned first and foremost.\n\nFirstly, if both positions exceed the true one: one should subtract the greater from the other.\nSecondly, if both positions fall short of the true one: one should subtract the smaller from the larger.\nThirdly, if the first position exceeds and the second falls short: both plus and minus should be added.\nFourthly, if the first position falls short and the second exceeds: both minus and plus should be joined together.\n\nThus, with these four methods, the estimation will vary. The first of these, if either estimation is greater or lesser, requires the subtraction of one from the other.  At postre\u2223mis duobus, uidelicet quando prior positio plus posteri\u2223or minus conijcit: uel e conuerso: pluris et minoris addi\u2223tio necessaria est. Et quo facilius has praeceptiones studi\u2223osi\nmemoria tenerent: MORVS, rogatu nostro, redegit eas in hoc carmen.\nA plure deme plusculum.\nMinus minori subtrahe.\nPluri minus coniungito.\nAt{que} ad minus plus adijce.\nHARVM primae regulae, in qua utraque positio plus affert: sensus est. Si per priorem positionem at{que} ite\u0304 per posteriorem plus {quam} ueritas exit: tunc alter error plus afferens ab altero plus afferente subducendus est. et id, quod restat, diuisor totius operationis erit. Postea prior error ducendus est in positionem posteriorem. Item{que} er\u2223ror posterior in positionem priorem ad forman obliqu\u0119 crucis. et numerorum ex his duabus multiplicationibus productorum alterius ab altero fit etiam subductio. De\u2223inde id, quod reliquum est: per errorum differentiam di\u2223uiditur. At{que} ita numerus partitionis ueritatem profert. Exemplum prebeamus Three merchants, having gained a profit of one hundred gold coins, distributed them as follows: the second merchant gained more gold than the first, and the third gained more than the second. It is worth investigating: how many gold coins did each possess. In the first place, there is a great oblique mark, shaped like the X letter in Greek, which the common people call the cross of St. Andrew: it should be painted on the abacus. After this, let us approach the collections. Let us first assume that the first merchant received 33 gold coins. This would make it so that the second received 36, and the third 40. What is the total amount collected? But there were only 100 coins in all. We have therefore made an error of 9 coins, which arose from their false positioning. Therefore, let us mark the oblique mark on the left side of the cross as a sign, and let us note the error at the left foot of the cross. And since our conjecture exceeds the truth, let the letter P be written in the innermost part of the cross, between the left oblique and the left foot. Deinde, since we made little progress with the first hypothesis: let us assume the first merchant had gold. Thirty-one had it. The second had. Thirty-four. The third had received it. Their total was collected and amounted to 103. But only 100 were distributed. Therefore, the second hypothesis also brought more than one thing: it erred in number three.\n\nWhy the second hypothesis should be placed second, starting from the thirty-first, is indicated by the error that followed. An \"L\" should be added to the right foot of the cross. And the letter \"P\" should be placed closer to the inner bend of the right foot to signify more.\n\nNext, correct the smaller error. Three should be subtracted from the larger one. And the remaining six should be noted as the space between the two errors.\n\nThen, bring the first error back to the second position. Thirty-one will result. And the second error will be brought to the first position. Thirty-three will be produced. Ninety-nine will be the sum of their products. Ninety-nine should be subtracted from the larger one. Three hundred twenty-nine will remain.\n\nIf this difference in errors, that is, six, is separated: the number of the partition will be revealed. Thirty. quae vera summa primum erat. Primusque 30. secundus 33. tertius 37. quae summae coactae factae sunt, sicut thema proposuit. Ita demum veritas per duas falsas positiones inventa est in medio spacio inter crucis vertices, in quibus hae signatae sunt: annotanda est.\n\nSecunda regula, in qua utraque positione minus affert: sensus pene primae similis, hoc est. Si per priorem positionem, atque item per posteriorem minus quam veritas habetur: alter error ab altero subduci debet. Et reliquum erit divisor. Tum prior error in posteriorem positionem, et error posterior in positionem priorem ducatur. Productorumque numerorum alterius ab altero fit subductio. Et quod reliquum est per errorum differentiam secatur. Sic in numero partitionis veritas apparerebit.\n\nExempli gratia, in themate modo dato probemus. Et ponamus primum mercatoriem habuisse. 27. secundum. 30. tertium. 34. quae summae collectae factae sunt. 91. At. 100. sunt quaesita. In 9. igitur erratum est. Ita position 27. in sinistro crux vertex: error. 9. In sinistro pedes statuatur. Since the error is less pronounced, let M letter, which indicates less clearly, be placed in the sinistrum recessum, between position and error. Let us try the hypothesis again. And let us suppose that the first merchant received, 29. the second, 32. the third, 36. and all the rest, 97. The goal was, 100. thus. 3 of the number designated are missing. Therefore, the posterior position is marked, 29. in the dexter crux vertex. And there is an error in the posterior. 3. in the pedes dexter. Also, M letter, which is in a less pronounced error: in dexter recessum, between position and error, should be placed. After this, a smaller error, 3. is to be subtracted. And let the remaining 6 be placed in the middle between errors. Then, the prior error, 9, is to be moved to the posterior position. 29. and will be born. 261. Similarly, the posterior error, 3, is to be moved to the prior position. 27. and will be produced. 81. Of these numbers produced, if a smaller number is subtracted from the larger: the remainder will be left. 180. which, if the difference between errors is considered, amounts to 6. centur: a number of a partition, the first being 30. According to the third rule, where the prior position adds more, the posterior less: this is the meaning. If truth is revealed by the prior position adding more, or the posterior position subtracting more: one error should be added to the other, so that both become a common divisor. Furthermore, after the prior error is in the posterior position, and the posterior error is in the prior position, they will be limited: numbers produced from these multiplications should be added together. This will make it so that if their sum is taken through the sum of the errors: the number of the partition reveals the truth. We give an example taken from those treating the theme. Let us suppose that the first merchant received 32, the second 35, the third 39, and the sum of these is 106. Six redundant: when 100 are sought. The position is therefore: 32 should be noted to the left of the crucifix crosspoint. And the error, 6, to its left foot. And the letter P, which indicates an error in the plus, should be written between the position and the error in the left crucifix sinus. Et quia coniectura nimium attulit: paulo infra tentemus. Primo habuimus singulum mercatorem. 29. secundus fuit. 32. tertius. 36. qui numeros collectos proferunt. 97. atque hi tres desunt de numero. 100. destinatum.\n\nEa propter ipsa secunda posito: 29. ad dextrum crucis uerticem. error autem. 3. ad pedem eius dexterum signetur. Et M litera, quae in minus errat, monstraret: in crucis dextero sinu inter positionem et errorem replaces.\n\nDeinde post ambos errorum additionem, quae facit: 9. prius error 6. in posteriorem positionem. 29. ducatur: et nascentur 174. Item posterior error. 3. in positionem priorem. 32. ductus, educet. 96. Qui numeri producti ambo in unum additi componunt. 270.\n\nEa si per amborum errorum collectam summam, uidelicet. 9. centuris: numerus partitionis. 30. procreabit. quae uera prima summa erat. QUARTAE rules, in which the prior position yields less, the posterior more: this is the same in all senses, which is the third: except that the prior position contributes less, and an error following from it holds the left part of the cross. The posterior position, with its error, holds the right. All other things should be handled in the same way, as in the third rule. We have given an example of this in the theme for you to consider: which anyone can easily explain without a precedent.\n\nPLURIS and minor rules have been dealt with, the observation differing from these same prescriptions due to their being expedited by two false positions: we will add these. And to make it clearer: how much this method of explanation differs from the one already given: we will keep the same theme. Let us imagine the first merchant received 35, 38, 42. Whose sum collected makes 115. Thus, by this conjecture, 15 more than necessary was produced. The position of 35 should be at the left turn of the cross and the error born from it. 15 should be placed at its left foot, and P. [littera, to be added: in the left sinus (pocket). And since conjecture has produced something more just: let us try again. Let us place the first merchant to the left. 31. The second. 34. The third. 38. For what reason did they all receive it? 103. With such a large amount. 100. Let them be divided. In the third position, which is redundant: it erred. Therefore, the third position, 31. at the vertex of the cross to the right: error. 3. at its right foot.\n\nLitera, to be placed in the right sinus for greater significance, should be placed.\n\nThe difference between the positions, discovered through the subtraction of the smaller from the larger, should be noted a little above the middle between the two spaces. This is it. 4. And similarly, the difference between the errors, discovered through the subtraction of one from the other: let a space be placed in the middle between them. This is it. 12. Let us consider the former position. 35. They still hold within themselves an occult excess and something yet unknown, which produced an error exceeding 15. and the second position. 31. an excess yet unknown: they produced an error exceeding 3. From this it is clear that the difference between the positions is 4] quid est inter earum duarum, mediocre est differenzia errorum. 12. Quid est mediocre inter duos erroribus. Et quoniam priora positio praecedat secundam, 4. hoc magis abducit a veritate per 12. quam secunda. At secunda posito, quia minor est priore, 4. hoc magis appropinquat ad veritatem. Nihilominus tamen veritatem excedit in 3. Quare, si magis immutata fuisset, 3. proximius ad veritatem accessisset et minorem aliquantum generaret errorem. Hoc investigandum est: ad quem numerum secunda posito sit minuenda, 31. ut nihil erroris procreet. Hoc manifestum fit, 3. si consideremus tres numeros proportionales nobis notos. Prima est differentia positionum: quae est. 4. Secunda, quae ex ea derivatur: est differentia errorum. 12. Tertius numerus est secundus error. 3. qui ex secunda errante exeunt, qui magis ad verum appropinquat per 12. quam prior error: hoc quod secunda posito minor est priore per 4. longius a veritatem distantem et errorem secundum. 3. propior ad verum accedentem, hoc est inter eandem sobolem: eadem esse debet inter utramque genitricem, id est inter positiones differentiam. 4. quae veritatem intercludens peperit differentiam errorum: et illam secundam positionis differentiam a vero nos excludentem, et adhuc ignotam, quae excessu in secundo errorre edidit. Et permuta. quae proportio est inter errorum differentiam. 12. et differentiam positionum. 4. quae eam peperit: eadem inter secundum errorem, 3. et a vero nos excludentem secundae positionis differentiam: quae eum edidit: et adhuc ignota est: esse debet. Quamobrem regula de tribus notis quartum ignotum proferentibus: rem totam patefaciet. nobis sic coessentibus. Si differentia errorum. 12. venit ex falsarum positiones differentia veritatem intercludente: quae est. 4. de qua falsae positiones differentia veritatem abscondente venit secundus error. 3? Ducantur. 3. in. 4. et surgent 12. quae secta per. 12\n\nTranslation:\nLongius separates truth from error by a small amount, and error by a larger amount. 3. The one approaching truth more closely is the one between the same offspring: it should be the same between both mothers, that is, between position differences. 4. The one that conceals the truth produces a difference in errors: and the second position difference that excludes us from the true one, and the one that is still unknown, which it produced in the second error. And so on. What is the proportion between the difference in errors and the difference in positions. 12. And the difference in positions that conceals the truth produces the second error. 4. From which false position difference the truth is concealed, the second error comes. 3? They are drawn. 3. In. 4. And they rise up 12. which sect follows. 12.  in numero sectionis edunt. 1. tan\u2223tum minui debet secunda positio. 31. ut ueritas in lucem ueniat. Illa enim unitas erat: quae excessum in secu\u0304da po\u2223sitione faciens secundum errorem in. 6. peperit. qua sub\u2223ducta. tollitur omnis error: et. 30. restant. qui numerus ueritatem exprime\u0304s medio inter utram{que} positionem spa\u2223cio notetur.\n{QUOD} SI CVI scire libet: quis erat ille numerus: qui in priore positione excessum fecit: consyderet: quae propor\u2223tio est inter errorum differentiam. 12. et priorem errorem 15. eandem esse debere inter positionum differentiam. 4. et illam prioris positionis differentia\u0304: quae a uero nos ex\u2223cludit: et adhuc ignota est. Et permutatim. quae propor\u2223tio est inter errorum differentiam. 12. et positionum dif\u2223ferentiam. 4. eandem esse debere inter priorem errorem 15. et illam prioris positionis differentiam a uero nos ex\u2223cludentem ct ignotam. quare tractans regulam de tribus notis tertio loco statuat primum errorem. 15. hoc modo. Si differentia errorum, quae est. 12  uenit ex falsarum po\u2223sitionu\u0304 differentia: quae est. 4. de qua fals\u0119 positionis dif\u2223ferentia ueritate\u0304 in priore positione occulta\u0304te nascetur er\u2223ror\nprimus 15? Ducantur. 4. in. 15. et producentur. 60. quae secta per, 12. educent. 5. Is exeessus in priore posi\u2223tione peperit errorem in. 15. quare si subducatur: restat uerus numerus. 30. Sic sublata redundante differentia, quae facit excessum: ueritas sola r\u00e9licta sese offert: siue ex\u2223cessum in secu\u0304da positione, siue in priore inuestigare, at{que} auferre libet. At{que} it a per duas falsas positiones, quarum utra{que} plus iusto conijeit: ueritas eruitur, per obseruatio\u2223nem proportionum: quae sunt inter differentias errorum, et differentias positionum, adiuua\u0304te regula quatuor pro\u2223portionalium: quorum tria sunt nota. \nADEVNDEM modum per differentiarum obser\u2223uationem res expediatur: etiam si utra{que} positio mi\u2223nus {quam} ueru\u0304 afferat. Veluti si ponamus primum mercato\u2223rem habuisse. 27. secundum. 30. tertium. 34. quae sum\u2223mae collectae faciunt. 91. cu\u0304 100 \"sint quaesita. Why is the position. 27. to the left of the cross's knot: error. 9. to the left foot. M. letter in the middle of the sinus be placed. Again trying, let us imagine receiving the first. 29. according. 32. third. 36. which added together make. 97. thus. 3. of. 100.\n\nlacking. Why are all these things standing at their respective places: namely, position to the knot: error to the foot. M. letter in the middle of the sinus. Then, since both collections yielded less result: we will grasp the differences in errors and positions through subtraction. Next, we must consider the first position. 27. still unknown defect in itself producing error. which. 9. lacking. and the second position also having an unknown defect, producing error. they thought. 3. It is clear that there is a difference between the positions. 2. which is in the middle of the two: producing a difference in errors. 6. which is in the middle of the two errors. And therefore, in order to discover the truth: we reason thus. If, 6. the difference in errors arises from position difference. 2\" When truth is concealed by the second position, a second error will be born. By operating according to the rule of the four portionalities, you will find this to be the case. If you want to know how much the first position was lacking, you can find this in the rule of the three notices, in the third question of the first error, in the following way. If the difference between the errors arises from the second difference of position, and the false position conceals the difference in the prior position, the first error will be born. Following the rule, you will find that this was the case. Therefore, the added difference completes the deficiency of the first position.\n\nWhen truth is sought through observation of differences, if each position fails to provide it, that which is lacking for the first or second hypothesis, according to the rule of the three notices, is supplied. When one position offers more, remove what is redundant according to the rule of three knowns, to ensure truth is expressed. As seen in the previous example. Let us now consider how things will be resolved through observation of differences. When the first position offers more than the second, or vice versa. In the given theme, let us first assume the first merchant received three coins, the second received three, six, and ten. Collected together, they make one hundred and nine. Nine are missing from the right side of the first merchant's position, making the position of the letter P significant. Let us try again, assuming the first merchant had two coins, three, and six. Nine are missing from the left side of the third merchant's position, making the position of the letter M significant.  Deinde consyderare oportet prioris positionis ex\u2223cessum edidisse excessum prioris erroris: secund\u0119 aute\u0304 po\u2223sitionis\ndefectu\u0304 peperisse defectu\u0304 erroris secundi. Quocir\u2223ca hi duo errores, quia alter in plus alter in minus peccat: in unu\u0304 sunt addendi: et surge\u0304t. 12. quae erroru\u0304 cumulatio uenit ex differentia positionum, uidelicet. 4. Nam quia prior positio. 9. plura {quam} oportuit attulit: secunda autem positio per. 4. tantum ab ea differens non solu\u0304 excessum illum prioris abstulit: sed etiam plus, {quam} iustum erat: au\u2223ferens, defectum superinduxit: ideo tam ad auferendum prioris positionis excessum, {quam} ad supplendum defectum posterioris, fit ista errorum additio: ut per eam tempera\u2223mentum inter excessum et defectum inueniatur: Nam ita proportionum ratio in utramuis partem, siue excessus, si\u2223ue defectus uim suam porrigere potest. Etenim si in dex\u2223tram ad defectum respicimus The proportion is between the accumulation of errors in the first and second, and the difference in positions, which accumulates and conceals the difference that hides the truth in the second position. This proportion must be the same between the second error and the difference in the second position that conceals the truth. To find out which number is missing in the second position to make it just, consider this:\n\nIf the accumulation of errors, which is 12, arises from the truth being concealed by the difference between positions, which is 4, then a false error of this kind will be born from the second false position. Multiply 3 by 4 and you will get 12.\n\nThese 12 are shown in the 12th section. Add this addition to the second position. 29. qui numerus uno deficitur: supplet id, quod defuit. Et fiunt. Thirty. Who number was the true one of the first merchant.\n\nIf you want to know: how much was to be subtracted from the excess of the prior position to leave a just number: the ratio of proportions should be considered from the left. For just as the sum of both errors is related to the prior error: so the difference of positions should be related to the difference, which conceals the truth in the prior position: this should be the case. And position by position. Just as the sum of both errors is related to the difference of positions: from which arises the error that interposes the truth: this should be the case. And therefore, in making the calculation according to the rule of three notices, the third question should concern the first error: which added more: in the same way, the position prior would have been less, and the second greater, in this manner. Si. 12. The accumulation of errors arises from the difference of positions. 4. \"Despite concealing the truth: from which the false position differs, the first error will be born. 9? They are led. 4. in the ninth. And they increase. 36. which sect follows in the number of partitions, eats 12. In the third, there was an excess: which caused the first position to err. Therefore, with that excess removed, those that were true remain. 30. those were the true numbers of the first merchant. So, when one position yields more, the other less: if you inquire about the number through the rule of the three knowns, the one that caused the excess in the position is found subdued. But if you discover the number that was lacking for the position, it should be added: in order for the truth to appear.\" If anyone wants to know: when does observation of differences bring about a solution; which position, be it one or the other, contributes more or less; what is the subtraction of one position from the other, and what is the error of one from the other? One should adapt to these matters: namely, that the difference from the truth, or the approach to the truth, is discovered through the differences of one position from the other, and the errors from each other. The very difference itself is found through subtraction. From the proportions of these differences, which are intermediate between errors and positions, truth is discovered through the rule of the three lines, whether one looks to the right or to the left. For if one looks to the right, what is the proportion between the difference of errors and the error itself: the one that approaches more closely to the truth, is the same between the difference of positions, which produced the difference of errors, and the number of the position following the error, which still conceals the truth from the error. Et per mutatis, quid est proportio inter errorum differentiam et differentiam positionum generatam suam: idem est inter secundum errorem et ignotum numerum matrem suam. Si autem respesas in sinistram: sicut errorum differentia se habet ad priorem errorem: quid longius a vero recedit, ita positionum differentia ad illam prioris positionis latentem differentiam: quam priorem errorem edidit. Et per mutatis. Sicut errorum differentia se habet ad differentiam positionum: sic priorem errorem ad illam prioris positionis latentem differentiam: quae ipsam edidit, debet habere. Praeterea, nisi haec proportioe ratio in differentiis positionum et errorum esset: nulla quaestio per regulas falsarum positionum solvi quovismodo posset. Succedens autem illa quatuor proportionalium regula, quae quartum numerum ignotum non latet: per tres numeros notos statim eum profert et rem totam explicat.\n\nTranslation: And by substitutions, what is the proportion between the difference of errors and the difference of positions generated by them: it is the same between the second error and the unknown mother of the number. But if you look to the left: just as the difference of errors has to do with the prior error: the farther it strays from the truth, so does the difference of positions have to do with the hidden difference of the prior position: which produced the prior error. And by substitutions. Just as the difference of errors has to do with the difference of positions: so does the prior error have to do with the hidden difference of the prior position: which produced it, it should have. Furthermore, unless this proportionate ratio is in the differences of positions and errors: no question can be solved by any false rule of positions in any way. But following is the rule of the four proportionalities, which does not allow the fourth unknown number to remain hidden: it immediately brings it forth with the three known numbers and explains the whole matter. If you want to know which position, be it of a larger or smaller rule, brings about more or less advantage: the error lies in the second position, and the error in the first position causes a correction of the second. This is due in part to what was said about the difference between the rules, and in part to the first and second rule of Euclid in his second book. As he spoke of lines, these principles are also true for numbers. If there are two numbers, the one that is divided in every part by the other, will be equal to those that are produced from the number divided into any one part of it. For instance, if a number is divided into parts, what is produced from the action of the whole upon itself will be equal to those produced from the action of the same number upon all its parts. I. When an error occurs in the second position: it is as if the differences between the errors and the second error, which are parts of the first error, are brought into the same second position. For instance, in the first example given of rules for the greater and lesser, where either position yields a greater result:\n\nthe prior error is in the second position. 9. is brought: and they are produced. 279. If the difference between the errors is thus brought: 6. alone in 31. is brought: new errors will arise. 186. And if the second error is brought: 3. into the 31st position: they will come into being. 93. Those numbers, which when combined in the second position, make the same number as in the first position. 279. Similarly, when the difference in position is 2 and the second position is a part of the first position. 33. producing 33 errors according to the second error. 3. Let us bring it into the first position. 99. If the number produced is equal to the number producing it: and the second error is brought: 2. into the difference between the positions, and afterwards into the 31st position: whatever arises will be 99. Itaque, since the difference in errors and the ratio of the errors is as follows: what is the relationship between the difference in position and excess in the second position, where we multiply the differences in errors. In the second position, we make 631. And we bring it not only to the total but also to the excess, which conceals the truth, namely.\n\nAnd when we multiply according to the error in the second position, we make 3 x 31. Added to 186, they create 279.\n\nHowever, we silently transferred the first error into the second position. We added 99 above the real number, which is equal to the multiplication of the second error in the first position.\n\nNam multiplication of the differences in errors in the second position's excess is equal to 1.3. In the difference of positions, 2.1.\n\nSince these four are proportional, as stated above. quae ab extremis equaliAre those from the extremes: they are equal to those from the middle, according to the rule of the three knowns. Multiplication with error in the second position. 31. This was done silently: when we first noticed the error in the second position, it makes. 93. They are added above the truth. Therefore, subtraction of the number produced by the second error in the first position, which is also present, removes all excess and leaves the true number hiding among them. But we still do not know what it is. Division made by the error difference. 6. Who is it: it is revealed, namely. 30. For when a number is produced from two multiplications, the division by the other number brought forth is the number in the section. The same rule applies to the subtraction of the product of one from the other, and the error of one from the other, when either position contributes less. \nQVOD si scire desyderas: quando per differentiaru\u0304 obseruationem res expeditur: cur errorum fit ad\u2223ditio: siue prior positio plus, posterior minus affert: siue e conuerso. Ratio illa est: quam cum exempla daremus: attigimus. uidelicet {quod} si contingat priori positioni aliquid superesse, secundae aliquid deesse: differentia positionum non solum excessum prioris positionis aufert: sed etia\u0304 de\u2223fectum in secu\u0304da inducit. Vel si eueniat priorem positio\u2223nem defectum afferre: secundam excessum: tum differen\u2223tia positionum non solum id, quod priori positioni deest: supplet: sed plus iusto aggregans excessum in secu\u0304da po\u2223sitione parit. Quocirca ad inueniendum temperamentum inter excessum et defectum, prior error ex priore positione ueniens et posterior exiens ex posteriore, in unum coniu\u0304\u2223gu\u0304tur: quae errorum cumulatio inter duos errores media, in utramuis partem siue excessus siue defectus comparari potest. Nam si ad defectum respicimus \"Just as an error arises from the difference in position due to a defect or excess, and becomes that error: one born of a defect causes a hidden difference in position that produced the false position. And it changes it. Similarly, an error arises from the difference in position, and the error, which is subject to that difference caused by the defect, should be in that regard. But if we look to the higher part, concerning excess. Just as an error arises from the excess, so the difference in position is to the hidden difference in position that produced the false position, and it changes it. Similarly, an error arises from the difference in position, and the error, which is subject to the excess, should be in that regard. Therefore, the rule of the three knowns easily reveals the unknown difference, whether it is an excess or a defect, which conceals the truth. But if it brings forth excess, it should be subtracted.\" If there is a defect: to be corrected, so that a true number may come out. Furthermore, if you want to know why in rules the greater is before the lesser, or the prior position yields more, the posterior less, or the reverse, the error is always led to the second position, and the second error to the first position; moreover, when the errors of each are produced to each other, they are added, and both are divided by the sum of errors to let truth come out? The entire reasoning of this matter depends, in part, on those things that have been said about ratio and proportion, without which these kinds of questions cannot be explained by false position rules; in part, from the aforementioned rules of Euclid, primarily the third rule given in Book Two. And to make this clearer through examples, I will repeat what was said about the rule of the greater and lesser in the third rule given above: in which the prior position yields more than the subsequent. When the prior error is 6, it is led to the second position. The subsequent error is 174. But the error that comes before is 3. This produces 96 when the numbers are combined. The sum of these is 270. This text appears to be written in an old form of Latin or a Latin-like shorthand. Based on the given requirements, I will attempt to clean and translate the text to modern English as faithfully as possible. However, I cannot be completely certain of the original intent due to the text's age and potential errors in the provided transcription.\n\nHere is the cleaned and translated text:\n\n\"They propose. 30.\nThis multiplication of the first error. 6. is placed in the second position. 29. whatever creates. 174. is the same: as if the first error itself were led to truth. 30. and they would become. 180. besides what, the number of multiplications of the first error in the second position, is lacking to reach the same truth: how much multiplication of the first error in that, which was lacking in truth in the second position, can be added. Therefore, because 1. was lacking in the second position: which would create 6. Therefore, the number produced by the multiplication of the first error in the second position. 174. through 6. differs from the number: which would be produced by the multiplication of the first error in truth. which is 180. which number through 6 exceeds 174. Similarly, the multiplication of the second error. 3. is placed in the first position. 32. is the same: as if the second error itself were led to truth 30. and would be created. 90\"\n\nCleaned and translated text:\n\n\"They propose the following. In the second position, there is a multiplication of the first error. This multiplication is equivalent to if the first error itself were led to truth. The result would be 180, but there are fewer units in the second position than what is required to reach the same truth as the number of multiplications of the first error in the second position. Therefore, since 1 was missing in the second position, which would result in 6, the number produced by the multiplication of the first error in the second position is 174, which is less than the number that would be produced by the multiplication of the first error in truth, which is 180. The number 180 exceeds 174 by 6. Similarly, the multiplication of the second error is in the first position. This multiplication is equivalent to if the second error itself were led to truth. The result would be 30, and 90 would be created.\" \"Although the second error multiplication exceeds the multiplication of the first error in number and surpasses the truth: how many times the second error multiplication produces what is written in the prior position instead of the truth; therefore, there is an excess in the prior position. 2. He who is led into the second error. 3. creates. 6. From this multiplication of the second error, it makes its way back to the prior position, which makes it. 96. Six times more is produced: than they would have produced if the second error were in truth. 30. That one, however, which here exceeds the truth in the prior multiplication, in the second position, was lacking in the prior error. And thus the defect is supplied by the excess. For what the proportion is between the excess of the prior position and the prior error from this, 2. and the defect of the second position and the second error, 1. and the second error from this, 3. is entirely the same.\" \"1. It will be between the prior and the outer one. 6. And the second error. 3. Why are they. 4. the proportional numbers: which are produced from the extremes by multiplication: are equal to those which are from the middle. For they make three twice: what makes six once. Therefore, if both numbers are produced by oblique errors in positions of multiplication, they will always correct each other. And he, who is lacking, will be exhausted by the excess of the other. And he, who remains, will supply the deficiency of the other. This is the reason why things will be reduced to equality. And they will be. 270. The same number, if produced similarly, will also produce 30 if each error is taken separately in truth. Namely, 6 in 30 is created, and 3 in 30 is composed. 90, when added together, make them even 270.\" Since the text is written in old Latin script and contains some errors, it requires translation and correction. Here's the cleaned text:\n\n\"Since the sum of errors in the oblique positions of multiplication is equal to the sum of errors in each position taken separately, and the multiplication of each position error to truth is equal to the sum of errors added together, according to Euclid's first rule: It is necessary that if the sum of errors is cumulated, which is 9, it should divide the truth. The truth itself appears in the number of the section. For any number that is produced by multiplication from two, the same number is divided by the other in the number of the partition.\n\nExercises for false position rules: the more students practice in them: we will add some questions that can be solved by them here.\n\nThree associates had common gold. The second had twice as much as the first, and he had even more gold. The first contributed 44.\n\" Tertius quanto pi\u00f9 ambo reliqui, atque eo amplius aureos. (Tertius received three times as much gold as the others.)\n\nQuanto: quantum ab unoquoque collatum est? (Find out: how much gold was given to each one?)\n\nPer experiments with rules, you will find that the first one received 5. The second, 14. The third, 25.\n\nTres panni telae aureis. (Three pieces of cloth were bought with gold.)\n\nSecunda bis tantum, quanti prima constituit: atque eo pluris aureis. (The second was worth twice as much as the first, and even more gold.)\n\nTertia bis tantum, quanti aliae duae: atque eo pluris aureo uno. (The third was worth twice as much as the other two, and even more gold.)\n\nQuanto: quanti tela quae constituerunt? (Find out: how much gold was spent on all the cloth?)\n\nBy trying with rules, you will find that the first and second cost 24 \u2153, the second 58 \u2154, and the third 167.\n\nObsonator quidam emit in coena domini gallinas. (Someone called Obsonator bought chickens at a feast.)\n\nTres gallinas, quattuor phasianos, quinque nummos. (He bought three chickens, four partridges, and five nummi.)\n\nPerdidit quisque nummos. (Each one spent some nummi.)\n\nNummos tres pluris, quam gallina costitit. (He spent more nummi than the cost of a chicken.)\n\nPhasianus quisque nummos septem pluris, quam perdix. (He spent seven more nummi for a partridge than for a chicken.)\n\nQuanto: quanti quae auis empta est? (Find out: how much was spent on each bird?)\n\nBy trying with rules, you will find that the chicken cost 2 nummos, one chicken 5 nummi, and the partridge 12 nummi. pani rubri et. 7. panni viridis valent aureos. Quanto cubitus quisque utriusque panni constitit? Pone quod precio libet.\nCubitus panni rubri et. 4. panni viridis fuisse empti: duobus coloribus omnium cubitorum precio. Efficiat. Deinde uide si.\nNinety-three and two-thirds. Quanto constitit unaquaeque panni? Sequens regulas duces, gallinam emptam comperies nummo. Unum et tres-quarteres nummis perdicem. Five and two-thirds nummis phasianum.\nMinisteri, a domino numerosum tradita, mandatum erat: ut quemquam tritici modios numerem emere: in mercato profectus, singulos tritici modios denariis licet. quas rationes determining the sum of accepted nummi. 40. He anticipates the need for. A wheat editor indicates the number of individual modii for each. Because the minister refuses to pay the price: because he has discovered that he will be short of nummi. 40. It is asked: what was the number of modiorum? Which number was given to the lord? If the number of modiorum were known: the number of nummi would also be known. Therefore, to find out: how many modii of wheat there were, an investigating number is taken, which is added to. 10. Produce only: as much as is obtained by multiplying by 12, subtracting. 40. This number of modiorum will be revealed. To find out: place these modii as having been 36. Of these, if each consisted of denarii: there would be 360 nummi. The remaining nummi: and they will appear. 400. Given to the minister. Later, let us see: if the vendor indicates this with these quaternions. The vendor demands nummi in 12 parts for each modius. Why do 36 modii consist of nummi? From which, if 40. subducantur: who lacked the price, were restored. 392. thus. 8. are missing: to make it 400. for the total sum given to the minister. And thus the position of modii. 36. reports a defect. Why then let us suppose that modii were missing. 38. if each of them cost denarii: they would amount to. 380. which we add. 40. and what was above and would be. 420. for the total sum given to the minister. Next let us consider: whether the vendor's mark is on. 12. denarii in the modii. 38. will make a square: for thus the coins would be. 456. from which, if we subtract: for those who lacked the price to pay, we would have a surplus. 416. thus are missing. 4. to become. 420. for the total sum given to the minister. Thus both positions and 36 and 38 are false. From these, if you wish to investigate the truth by the regulations given above, you will find the number of modii that the lord wished to buy: as well as the number of coins given to the minister. 440. in which the buyer's bidding with the vendor's mark. To solve this question and similar ones, you can follow this method: Add the number that remained to the sum of the numbers that had existed and those that had not. Subtract the price indicated by the minister of the lessor from the price indicated by the vendor. Divide the remaining sum by the difference between the two prices. The number of modii in the measure of grain will then be shown. This can also be seen in the following question.\n\nThe laborers, summoned by their fathers to work in a vineyard for one day, demanded five denarii each. When called back from work before evening, he only offered three to them. If they were content with three, he took the remaining two denarii for himself. If five had been paid, there would have been none left. It is worth investigating how many laborers there were and how many denarii. To solve such a question quickly, you must add the denarii. Seven of those who had existed were added to the denarii. Thirteen of those who had not appeared. And twenty were needed. qui secundus est numerus. Deinde Deinde should subtract a smaller wage offered than demanded, namely. 3. A second number is to be divided, thus. 20. They will bring forward 2 parts in the number of the division, which are 10. They thought they had dealt with all the workers. But the number of coins that remained: it had to be added to, or subtracted according to the wage demanded and the offered wage: so that the number of sections would show how many wages were asked for. Later, it is easily known: how many coins the father of the family had. For the offered wages were brought into the number of the workers, which would reveal this. Thus, 3 in 10 will produce. 30. If 7 are added to these: they would have surplus.\n\n37. This was the number of the coins that represented the heads of households in the place. Furthermore, if a worker had withdrawn 5 from his wage: 10 workers were necessary. 50. From those, if 13 had been absent: subtract them: there would be a surplus of 37.\n\nYou can also explain further: if you inquire about a number by false positions, the one multiplied by 3, with 7 added, will produce as much as if 5 were in the base. You provided a text written in Latin, which I assume is the original content. I will translate it into modern English and remove unnecessary symbols and formatting. Here's the cleaned text:\n\nGranted, having two AVREOS, whoever that may be, go to the market to determine the number of workers needed to handle the rest. Thirteen. One of them bought cloth cubits, how many gold pieces did he have? 1\nThe merchant bought cloth bolts\nEach one of them declared they would lack coins. 80. I ask, how many cubits of cloth could it hold? And how much space did it have? If through false positions you try to deceive, through these it must be investigated: the number, which when led to eight, adding fifty, produces: how much in ten when subtracted by eighty. To find out that the cloth had sixty-five and a half cubits in length and had coins in it: you can also explain the matter through the given formula.\nTwo merchants bring a ship laden with wool to be sold. One puts seven wool sacks into the ship, the other puts sacks of wool into the ship. 11. He who was alone gave a single sack of wool to the ship's master for sale, paid the price of the ship with it, and if anything remained, he demanded the remainder in payment. When the voyage was completed, the master of the ship gave the man who had carried seven sacks: fifty gold coins. But the man who had carried eleven sacks asked for twenty gold coins. The question is: which sack belonged to which ship, and how many gold coins did the master sell from the sack he received? This can be investigated through positions: if the number in seven is multiplied and fifty is added, it should equal the amount in eleven plus twenty. The master sold 7.5 sacks of gold coins to him, amounting to 102.5 gold coins.\n\nWho bought six pounds in weight from the sacks? When asked about the price of one pound, he replied. How many shillings were six pounds and ten pence? That many shillings. Six pounds and twenty shillings were bought. The question is: what was the price of one pound? To answer this question, the number must be investigated through positions: if the number in six is multiplied and it produces a number greater than ten that is twenty or more when added to the number in ten, the price of one pound will be indicated. 2.5. To solve the enigmas of this genus, it is permissible to follow the compendium up to this point. The smallest number must be subtracted from the middle: and four will remain. This number will be the divisor. Next, the middle number must be subtracted from the largest, and the remainder will remain. By dividing this remainder through the divisor, one number will result. Two and a half will be shown in the number of parts. In the three aforementioned numbers, 6, 10, 20, there are two differences, or two excesses: the smaller one should be cut in this place. After you have the price of one pound, the riddle comes to light. For if you take six pounds for the price of one pound: numbers will be produced. 15. And again, if ten pounds are taken for the price of one pound: coins will arise. 25. Thus is the number. 5. This number is equal to and exceeds 10 and 20 by. at si quos dies cessaret: pro singulis redderet domino nummos. (If each of those days had passed: for each one he would return coins to the lord.) 16. He gave labor to the houses for their repair for so many days: and for so many days he ceased: so that nothing was left of what had been done. I ask how many days he worked and how many he ceased? Considering the agreement carefully, you will find out. 30. Two parts need to be examined: the first, which when brought to 18, produces as much as the second does in 16. That which, due to chance, you will easily discover, if you subtract 30 from the number of days on which he worked, and bring the result to 18, you will then take the remaining part of 30 and divide it by 16 for the number of days of intermission. Afterwards, see which number is greater. If the number produced for the days of intermission is greater: the position of the work days indicates a deficiency. But if the number produced from the work days is greater: the position is in excess. Which of the two will occur: it will be clear by observation and the position must be tested again. Following these regulations given, you will find the number of days on which he worked to be 14 2/17. The remaining ones, however, are up to 30. quibus cessuit: esse. 15 15/17.\nWhen asked, the index of the hourglass indicated some hours: he replied. The third and fourth parts, which it indicated, were the fifth and sixth parts, which it should have indicated. I ask, which hours was it indicating? And which hours should it have indicated? Before you answer, it is necessary to observe the custom of the region where the hourglass is. If it is in Italy: there, the hour calculation progresses throughout the day and night up to 24. To solve this enigma, investigate the two parts of the 24. Of which one-third and one-fourth are one, and one-fifth and one-sixth are another. In this way, you will find that the hours indicated by the hourglass are 9 5/19, but those which should have been indicated are 14 14/19. Since the third part of 9 5/19 is 3 5/57, the fourth part is 2 6/19, and when both are added together, they make 5 437/1083. Similarly, the fifth part of 14 14/19 is 2 18/19, and when both are added together, they make 5 437/1083.\n\nAVREI two hundred were to be divided between two gamblers with equal shares: each should have received. 100. deprived. They poured out gold from their sacks, and a quarrel arose between them: each snatching with fierce hands. Once the quarrel had subsided, they came to an agreement: the one who had taken more was to give up half. The one who had taken less, a third. What the total amount was that had been deposited, and how it had been divided fairly among them, each acknowledged. I inquire, how much did each take? By position, one took eight hundred and seventy-five and 5/7 gold pieces, the other eighteen and 2/7. What is the sum of these? 200.\n\nThe first man made a profit of half the money of the second. The second man made a third from the third man's money. The third man, having finished his game, took away a hundred gold coins from them. I inquire, how much did each bring to the game initially? Trying to find out by position, the first man brought fifty-five and 5/9 gold pieces, the second eleven and 1/9, the third one hundred and thirty-three and 3/9. A man lost the first part of his money, that is, one eleventh part, amounting to 11 1/9. This happened because he had only 44 4/9 left, which was increased by a half of the second part of his money, that is, fifty-five and 5/9. Thus, he carried away one hundred. The second man lost half of his money. The third man made a loss with one third of his money. Thus, he had 88 8/9 left. If the fifth part of the first man's money, which was gained, that is, 11 1/9, is added to this, he would also have one hundred. Two travelers, Cornelius and Antonius, found coins in their journey, one red and one black. In the red one, there were fewer than in the black one, but in both, there were one hundred coins. Cornelius, calculating for himself, said to Antonius, \"If you give me your red coin and ten of your coins, then I will have four times the value of your money.\" Antonius replied. If you give me a black crumb and twenty golden ones, I will be in possession of five times your money. It is asked, how many gold coins were in the red crumb and how many in the black one? And how many did Cornelius have? How many did Antonius have? The principal thing is to make two parts of a hundred. The upper part should exceed the lower part by ten. These are the fifty-five and 45. Again, regarding the dealers who examine the gold coins through positions, imagine that each of them has a number: assigning them also the crumbs as they please. Observe the error in the assignment of the crumbs (served in the same position). Observe which of the two assignments produces a smaller error: keep the other one also in the second position. In this way, you will know which crumb had the red one and which the black one. Afterwards, proceed to the second position and you will grasp the rules given and apprehend Cornelius from his own. 40 5/19.\n\nAntonius. \"36 milites, who were in the station, were asked by a traveler how many were in the garrison: he replied. If the same number of others joined: half of them and a fourth of them would be with you: we would all be together. It is pleasing to know: how many soldiers were in the garrison. By positioning yourself, you will find that there were thirty-six.\n\nA merchant, going to the Triple Nundines, doubled his entire wealth with the first [group of people]. He added twelve pounds of gold. With the second [group of people], he doubled what remained from the first: he added twelve pounds of gold again. With the third [group of people], he doubled what remained from the second: he added gold pounds again. Thus, nothing was left for him.\n\nHe was asked: how much livestock did he bring with him from the beginning? By positioning yourself, you will find that he brought half a pound of gold.\" If we disregard the position, I will explain this: if you begin after the last market day and collect each gain in turn, subtracting expenses, you should go backwards to the first market day. As we demonstrated with a priest and three beggars asking for money from the third book.\n\nThree of them have a sum of gold coins: the first one says to the others, \"If you give me half of your gold, I will have one gold coin with mine.\" 20. The second one says, \"If you give me a third part of your gold, I will have one gold coin with mine.\" 20. The third one also says, \"If you give me a fourth part of your gold, I will have one gold coin with mine.\" 20. I ask, how many gold coins does each of them have? You may assume that the first one has gold coins. 4. The others must have in total. 32. Of these, the first one will give him gold coins. 20. However, if you want to examine the second and third parts separately, other positions are required. Add together all parts in the initial position: they will be. 36. Next, investigate which part of the second is involved: in that, 36. the two parts, of which one makes up the third part of the other. 20. This, which must be done according to other positions and their opposite, 20. should also be examined.\n\nThe second will have only this much. Thus, the first and third will have this. 24. Of which four are the first: as it has been stated: 20. they will be the third. Thus, the second's parts agree with those of the first. However, the third part, through this position, will be. 24. For if the first and second give it: it will be. 4. But it itself has. 20. Thus, they are made. 24. And the excess in the first position will remain. 8. Again, place as desired: for the second and third to have, 24. half of which will make it: the one who is first, 20. And to find the second's parts, add all of these together: they will be. 32. Then, through other positions, investigate the two parts, 32. of which one makes up the third part of the other. 20. And you will find the other to be. 14. Tanntum has the second in the second position, in which the first has. 8. The first and third will have. 18. Of these, the second will make up one third for him. 20. Thus, the first and second have. 22. Of which one fourth is five and a half. That added to ten makes fifteen and a half. Four and a half are missing: to make up. 20. And in this way, the second position makes up for the defect. Therefore, to find the truth between these two false positions, one of which exceeds, the other defects: add both errors, that is. 4 and 4.5. And they will amount to 8.5. This number will be the divisor. Then, the first error is drawn into the second position. 8. And they will produce. 32. Next, the second error. 4.5 is drawn into the first position. 4. And they will make 18. 32. Add that to. 50. When this sum is cut through by eight and a half, the numbers of the partition will come out. 5 15/17. Tanntum had the first. Again, for the second part, investigate the first error. 4. Draw it into the second position. 14. And they will increase. 56. The same second error. 4.5 is drawn into the first position. 12. And they will make 54. 110. ea si secundus exits in eight and a half parts. Twelve sixteen-seventeenths. The second only had that much. such positions are called doubled: those that are presented before the principal ones, and those that should be observed by the eyes, we have set forth here for the reader's better understanding.\n\nIn the Roman MMA, through the false rules of a single question, which often intervene in negotiations, they are easily explained. But those requiring root positions or quadratures demand a higher art: which the Arabs call Algebra. It shows the addition, subtraction, multiplication, and division of radical and cubic quadratures. Through which all things, however difficult they may be, that concern numbers, can be resolved. But we have taken upon ourselves only those things necessary for common life: to explain to the studious.\n\nTHE FOURTH AND WIDE BOOK IS COMPLETED. We believed it necessary to remind our learned readers: just as ancient authors, whether Roman or Greek, can assess the value of their collections of coins and other monies according to our standards: when they take up an old book in their hands, they should recognize the value of the money it contains at once. This was a lesson imparted by BVDAEVS, a man well-educated in our early years, in the book he compiled with great care on the subject of the as and its parts. He admonished his people about the Gallic money, a debt not only to the Gauls but to all nations: for he opened up the ancient treasures, which had been locked away by the barbarian fetters that held them captive. Therefore, those who are familiar with Gallic money will have been sufficiently taught by him: how to evaluate ancient coins and their sums. However, our readers, who are unfamiliar with this, can easily learn the same from an excerpt we have noted down in a certain excellent book. During the reign of Roman Emperor APVD, he took in forty-eight and nine pounds of gold coins, each worth twenty-five denarii. Four ounces of gold were among these, which were worth one denarius each. Later, due to luxury and civil wars, the gold was depleted, and a third part of it turned into powder. Seventy-two of these gold coins remained. They began to be called solidi, as if they were complete, since nothing was missing. The ancients called a solidus an integral solid. Six-denier coins were also named as such, because they contained six ounces. Now, as far as I know, among those peoples, only the Romans had something resembling gold coins in the form of the Four Noble Roses, which the common people called Noble Angels. These ancient Roman gold coins, in weight and purity, were not much or at all inferior to the modern ones. Six of our Noble Angels, which we call Noble English Gold Coins, make up an ounce, and the old solidi, which were called six-deniers, were weighed and marked accordingly. Porro the Romans collected a heap of silver coins, each of which weighed an equal dram: they wanted to enclose them. A denarius among them was called, because it was worth ten asses. As Varro says. A quinarius of silver denarius was worth fifteen asses. But a silver sestertius sestertius was worth two asses and a half, called so as if it were a third semis. Four denarii were worth one sestertius. Therefore, four hundred sestertii were worth a thousand denarii, that is, a libra nummarium. In this way, a sestertius, which was a nummus of two pounds and a half, took a thousand sestertii in the masculine gender. However, the sestertius itself was a nummus of two pounds and a half. For an as was worth twelve ounces, gathering it together. A sextans nummus was worth two asses. A quadrans nummus was worth a quarter of an ass. That is, a triens was worth a third. A semis, however, was worth two quadrantes. This text appears to be in Latin, discussing the conversion of Roman currency to Sterling. Here is the cleaned text:\n\nHuic nummariae rationi, quod ad denarios quinarios et sestertios attinet: uetus pecunia nostra, quam Sterlingam vocamus, omnibus idem est. Quod ueterem Grossum nostrate lingua dicimus, id pondere et indicatura denario Romano aequat. Quod ueterem Semigrossum vocamus, Quinarii. Quod nostratis uulgus ueterem denarium Sterlingum appellat, sestertium nummo aequiparat. Ceterum nos libra Sterlinga vocamus: quod apud Romanos libra dimidia, selibra erat. At quod Romani neutro genere sestertium vocabant, duas argenti libras et semissem complectens, nos quinque libras vocamus: quod libram nostram sint selibram Romanam. ILLUD observandum uidetur: ueteres Sestertia, quod duas libras et semissem caperet, per duos L. et unum S. notabant, ad hoc modum. LLS, siue de sestertijs nummis, siue de alis sestertijs intelligas. Id quod adhuc extant libri ueteres testantur.\n\nSestertia neutri generis. Viginti quinque aurei coronati.\nQuinque librae sterlingae.\nDecem sestertia. Fifty golden crowns.\nFifty pounds sterling.\nTwenty sesterces. Five hundred golden crowns.\nOne hundred pounds sterling.\nThirty sesterces. Seven hundred and fifty golden crowns.\nOne hundred and fifty pounds sterling.\nForty sesterces. One thousand golden crowns.\nTwo hundred pounds sterling.\nFifty sesterces. One thousand two hundred and fifty golden crowns.\nTwo hundred and fifty pounds sterling.\nOne hundred sesterces. One thousand five hundred golden crowns.\nOne thousand sesterces. Two thousand and five hundred golden crowns.\nFive hundred sesterces. Thirteen thousand and five hundred golden crowns.\nThree thousand pounds sterling.\nEight hundred sesterces. Ten thousand golden crowns.\nTwo thousand pounds sterling.\nFive thousand sesterces. Twenty thousand and five hundred golden crowns.\nFive thousand pounds sterling.\nEight thousand sesterces. Ten thousand five hundred golden crowns.\nTwo thousand and five hundred pounds sterling.\nSix thousand sesterces. Thirteen thousand and five hundred golden crowns.\nThree thousand pounds sterling. Decies sestertium = ten million sestertii, meaning the same as mille sestertia and nearly above a million.\nDuodecies sestertii. Triginta millia coronatorum.\nSex millia librarum sterlingarum.\nQuindecies sestertii. Triginta septem millia et quingenti coronati.\nSeptem millia et quingentae librarum sterlingarum.\nVicies sestertii. Quinquaginta millia coronatorum.\nDecem millia librarum sterlingarum.\nTrities sestertii. Septuaginta quinque millia coronatorum.\nQuindecim millia librarum sterlingarum.\nQuadragies sestertii. Centum quinquaginta millia coronatorum.\nViginti millia librarum sterlingarum.\nSexaginta sestertia. Centum quinquaginta millia coronati.\nTriginta millia librarum sterlingarum.\nOctoginta sestertia. Ducenta millia coronatorum.\nQuadraginta millia librarum sterlingarum.\nCenties sestertii. Ducequatquina millia coronatorum.\nQuinquaginta millia librarum sterlingarum.\nDucenties sestertii. Quinteta millions coronators.\nCentum millions librae sterlingas.\nQuadringetes centuries coronae milia.\nDueta millions librae sterlingas.\nQuinquies quindecim millions coronators.\nDueta quinquaginta millions librae sterlingas.\nOctoginta centuries sestertia.\nVicies centena millions corona rum.\nQuadringenta millions librae sterlingas.\nMilles sestertia. Vicies quin quinquaginta millions corona.\nQuinquenta millions librae sterlingas.\nBis millies sestertia. Quinquaginties centena millions coronaators.\nDecies centena millions librae sterlingarum, sive millies millenarum librae.\nTres millies sestertia. Septuagesimas quinquagintas centena millions coronaators.\nQuindecies centena millions librae sterlingas.\nQuater millies sestertia. Centies centena millions coronaators.\nVicies centena millions, hoc est, bis millena millions librae sterlingas.\nSexies millies sestertia. Centies quinquaginties centena millions coronaators. Three hundred thousand, that is, three million three hundred thousand pounds sterling.\nEight hundred thousand sesterters. Two hundred thousand two hundred thousand cornators.\nForty thousand, that is, forty million three hundred thousand pounds sterling.\nTen thousand sesterters. Two hundred fifty thousand five hundred thousand cornators.\nFifty thousand, that is, five hundred thousand million pounds sterling.\nTwenty thousand sesterters. Five hundred thousand five hundred thousand cornators.\nOne hundred thousand, that is, one million three hundred thousand pounds sterling.\nForty thousand sesterters. One million one hundred thousand cornators.\nTwo hundred thirty thousand, that is, two million three hundred thousand pounds sterling.\n\nMina among the Greeks is the same as: what a libra is among the Romans. This mina contained: that is, one hundred denarii Roman, or four sestertii. A drachma and a denarius were of the same estimation. A mina of twelve coronae was valid: that is, two pounds sterling. Drachma was divided into six obolos, three obols into trientes, and quadrantes. A sesquiobolus in Athens was equal to a Roman sestertius.\n\nThe talent was complex, but the Attic one, celebrated by historians, weighed sixty pounds or minas, that is, six hundred coronae or two hundred and fifteen librae sterlingas.\n\nTen talents. Six thousand coronae.\nOne thousand two hundred librae sterlingae.\nTwenty talents. Twelve thousand coronae.\nTwo million two hundred and forty librae sterlingae.\nForty talents. Twenty-four million coronae.\nFour million nine hundred thousand librae sterlingae.\nFifty talents. Thirty million coronae.\nSix million librae sterlingae.\nOne hundred talents. Sixty million coronae.\nTwelve million librae sterlingae.\nTwo hundred talents. One hundred and fifty million coronae.\nTwenty-four million librae sterlingae.\nOne hundred and eighty talents. Two hundred and eighty million coronae.\nThirty-six million librae sterlingae.\nFour hundred talents. Four hundred and forty million coronae. Four hundred and eighty thousand pounds sterling.\nFive hundred talents. Thirteen million corona.\nSix hundred thousand pounds sterling.\nSix hundred talents. Three hundred sixty thousand corona.\nSeventy-two million pounds sterling.\nSeven hundred talents. Quadrigea forty million corona.\nEighty-four million pounds sterling.\nEight hundred and seventy thousand corona.\nNinety-six million pounds sterling.\nNine hundred talents. Fifty-four million corona.\nOne hundred and eight million pounds sterling.\nOne thousand talents. Six hundred million corona.\nTwenty million pounds sterling.\nTwo million talents. Two hundred and thirty million corona.\nTwo hundred and forty million pounds sterling.\nTwo hundred and thirty million corona.\nThree hundred and sixty million pounds sterling.\nFour million talents. Seven hundred and sixty million corona.\nFour hundred and seventy-eight million pounds sterling.\nSix million talents. Trities sexies centenas milliones corones.\nSeptingenta uiginti milliones libros sterlingas.\nOcto milliones talentos. Quadragies octies ce\u0304tenas milliones corones.\nNoningenta sexaginta miliones libros sterlingas.\nDecem milliones talentos. Sexages centenas millones corones.\nMille et ducentas millones libros sterlingas.\nQui\u0304decim milliones talentos. Nonagies centenas millones corones.\nMille et sexcentas millones libros sterlingas.\nViginti milliones talentos. Centies uicies ce\u0304tenas millones corones.\nBis millena et quadringenta millones libros sterlingarum.\nCyathus, duodecima parte de sextarii, pesa dos uncias: drachmas sexdecim.\nAcetabulum, que se llama Oxybaphus: quarta parte de Cotylo, contiene Cyathus y medio. Pesa tres uncias.\nCotyla, que se llama Hemina: contiene la mitad de sextario: Cyathos seis. Pesa una libra.\nSextarius, sexta parte de Congio, contiene dos Cotylas: Cyathos doce. Pesa dos libras.\nCongius contiene Sextarios seis: Heminas doce: Cyathos setenta y dos. Modius, a third part of an Amphora, holds sixteen and a half Sextarii: fifty-two and a half Heminas.\nAmphora, called Quadrantal, holds two Urnas, three Modios, eight Congios, forty-eight Sextarii, and ninety-six Heminas.\nCadus, also called Metreta, holds ten Cogios, sixty Sextarii, one Amphora, and one Quadranta.\nCuleus holds twenty Amphoras, forty Urnas, and sixty Modios.\nMedimnus holds six Modios and two Amphoras.\n\nWe have extracted this from Hactevns' book on weights. Now in our time, each nation almost uniformly uses different standards for coins and weights according to the will of their kings or princes. The names for these coins and weights vary greatly from region to region. However, it appears that in the vast differences in the estimation of coins and weights, many nations agree, in common speech, to call a solid a solidus, and twelve denarii a denarius. A libra: twenty solidi.  Quocirca mer\u2223catorijs supputationibus in hac aetate nostra plurimum conducet: si quis in promptu teneat: quot solidos qu\u0119li\u2223bet denariolorum summa procreet us{que} ad mille. Id, quod ante oculos per gradus quosdam, numerorum notis sig\u2223nandum duximus.\nFINIS APPENDICIS.\nIMPRESS. LONDINI IN AEDIBVS RI\u2223CHARDI PYNSONI. ANNO VER\u2223BI INCARNATI. M. D. XXII. PRIDIE IDVS OCTO\u2223BRIS. CVM PRI\u2223VILEGIO A REGE IN\u2223DVL\u2223TO.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase1"},
{"content": "The pleasant path leading to a virtuous and honest life, no less profitable than delightful. V.L.\n\nPrinted at London by Nicholas Hill, for John Case, dwelling at the sign of the Ball, in Paul's churchyard.\n\nYou who desire to know\nA good ware to take,\nWhereby to riches to grow,\nAnd idleness to forsake,\nThis little book with diligence\nSee that you read and mark,\nThoroughly noting the good sense\nContained in this work,\nBy whose precepts you shall here find,\nRight pleasant for to read,\nWhereof perhaps some youth are blind,\nAnd thereof shall have need,\nDo not therefore despise this book\nBecause it goes in rhyme,\nFor they that look on this book,\nShall find the matter fine.\n\nSeeing gifts are not so much to be valued for the price of them as they are to be esteemed for the good intent of the giver (most gentle reader), my small labors and travail herein bestowed (which I freely give you), I shall humbly desire you to take in good part. Although the matter is not perfectly handled in every detail as my good will and seal would have wished, I have no doubt that you will bear with me kindly, and rather laugh at any faults if there are any, than spitefully criticize others' doings, seeking occasion to disparage them. For the entire effect deals with nothing but good and wholesome counsel interspersed with pleasant mirth and honesty, beneficial to youth no less than necessary.\n\nLicurgus, renowned for his great wisdom and excellence, gave laws to the Lacedaemonians, among which were other laudable and serious statutes. He instituted and ordained a certain kind of exercise, at which the elders who had ruled the commonwealth commonly met, to pass the time with pleasant talk and witty tales, but such as were always either profitable to the commendation and praise of honesty. Or else, to counteract the displeasing and reprimanding of voices, because he perfectly well considered that it was a thing most necessary and required at times for refreshing the weakened powers of the mind, making it more fresh and capable of accomplishing and completing other weightier affairs. I omit other probable authorities which I might now worthily bring herein, to declare that this little work is not to be rejected although after a number of pious precepts, a merry invented matter is placed therein only to quicken spirits and alleviate tediousness. And who is so ignorant, but he well understands that youth commonly takes more delight in reading those things which require it in his age than in grave sentences solemnly pronounced. Yet here he will learn, if he reads it diligently, both how to lead a quiet and godly life in the fear of God, and also to shun the assaults of sin and the dangers of the world. In this brief age, I have written this, not for the prudent, well-strucken in years, whose experience has likely taught them and who have already passed the dangerous rocks and quicksands of this troublesome world. My sincere meaning, gentle reader, if you graciously receive it, I will be more willing to take pains in other matters. Farewell.\n\nIt happened on the eleventh day\nOf the flourishing month, of lovely May,\nWhen Titan began his entry into Taurus,\nAnd the young blood of every creature renewed its strength,\nAnd their powers revived.\nSo that each thing now appeared alive,\nWhich in the stormy winter before\nHad sustained Eolus' Isis blasts,\nAnd the watery snows had torn them,\nBecause of Capricorn's vehement rage.\nWhich cold, cloudy mists, once past their way,\nReceived courage again in his array. And because the sharp cold had passed,\nMavis endeavored first to tune her notes.\nNext, after, the pleasant Nightingale, tempered her voice,\nWhich we her merry melody, every heart, does greatly rejoice,\nThe Thrush, the Blackbird, and the Greenfinch also.\nIn this merry spring time, did she we what they could do,\nAnd then also, the Sun shining very hot,\nCaused the crabbed Cockowe to declare her old note,\nIn this fresh time, (I say) for my recreation,\nInto the woods I walked, to take delight,\nAs well for to hear, the joyful birds sing,\nAs also to behold, how every thing showed itself,\nAgain, alive in its kind,\nThat in the winter, were bare and naked to the rind\nEach tree budded, and his leaves began to spread,\nEach herb had its flowers, in every green mead,\nThe Primrose, the Violet, were then in their prime,\nAnd the sweet-smelling cowslip, flourished at this time,\nThe Hawthorn, the Cherry, and the Damaskine. And each apple tree bloomed, declaring good signs,\nThat if God would grant us his sweet blossoms,\nWe would have a great store of fruit,\nAnd thus going alone, I walked beneath the wood's side.\nA fair aged man, I soon espied,\nAnd directly after me, he came at a soft pace,\nI stayed a little therewith, thinking in this care,\nThis old man seems, a man of gravity,\nAnd therefore I would be very glad, of his company.\nIn this staying a while, at last he overtook me,\nAnd with very grave countenance, saluted me gently,\nAnd with salutation, again I did him greet,\nAs to my duty, I thought it most meet.\nGood father (and if it pleases you), said I,\nTell me, how far is your journey, this way lies,\nAnd of the same lies, anything near,\nI will be very glad, to have your company,\nFor I have, at this time, little to do,\nAnd therefore, I would, learn, some good counsel of you.\nAnd because it is so, I am young and frail,\nAnd not yet instructed, with wholesome counsel. My life to direct, in time for to come,\nGood lessons from you I would learn some.\nWhich might cause me, for you, hereafter to pray,\nWhen I shall follow the same, another day,\nGentle sun (said he), the truth to declare,\nAbout sour miles hence, my journey is to fare,\nAnd because I am aged, and may not well go,\nI take the morning with me, for it behooves me so,\nAnd back again to night, homeward, will I, if I can,\nNine mile a day, is a great journey, for an aged man,\nWherewith, I remembered myself, by and by,\nAnd beheld the Sun shine, so gladly,\nClear was the sky, and lightsome was the air,\nAnd also the way, seemed very fair,\nMy business, indeed (said I), father, lies three miles and more,\nAlong that way, which you must go, and therefore,\nI will go with you softly, and make no great haste,\nFor it requires none, and this forth we past,\nGod thank you young man, said he: that you are so kind,\nSince you will go with me, and not leave me behind. You are young enough to begin. Here is some wisdom I will share. First, make it clear to me where you live and remain, what your state is and your disposition, to which your mind is given in every condition. Also, reveal your name to me, and whether you are single or married I want to know. Before I can show you my good advice or instruct you prudently in any other way, I must know these things. Truly, my dwelling is not far from here, and I was once a worthy gentleman's servant, but he has now passed away. Therefore, I am clearly released from service. Regarding my age, I appear to be young, not yet passing twenty years. Most men call my name Nitne. I am also still single and unmarried. Well then, I perceive you are young enough to begin. And so lead your life thus, and at liberty, without a wife, nor yet committed, to any kind of life, as far as I may or guess can, you were lately as if, a serving man, now whether it is, your mind and will, in the trade of service, to continue still, or leave the same, and by some other means, to seek other ways, your living to sustain, This also to know I earnestly require, or else, I can no way satisfy your desire, In good faith (said I), so more I swear, A servant I have been, about five years, And truly have served to my power, Since I entered into service, the first hour, Wherein, there is so great travel and pain, At most times, and so very little gain, And at other times also, much great idleness, Doing nothing, but sitting, in the fields and streets, Wherein, also there is much great exercise, Almost of every manner and kind of vice, Both pride, drunkenness, and also swearing, By abominable others, God himself tearing. Such quarreling, fighting, and other abominations,\nOf which I could make, to you true relation,\nIf it were not odious, for you to hear,\nAs experience thereof plainly does appear,\nI intend, utterly, the same to refuse,\nAnd some other, more godly state of living to choose,\nWherein I may spend my time, more honestly,\nAnd in the fear of God, live more quietly,\nWithout doubt (he said) thou hast spoken truly,\nFor besides, the aforementioned nothingness, plenty\nWhen a man has served, a great time,\nIf he has done, never so little a crime,\nAway he must go, there is no other remedy,\nThus, is he put, to his shifts, by and by,\nAnd put case, that he continues in service,\nUntil age comes, that he can no more do,\nThen is he cast off, either to beg his bread,\nOr in misery, to live, till time he be dead,\nAnd evermore commonly, it is seen and hard.\nThis to be, of miserable service, the reward.\nAnd to serve any man in the court of renown,\nYou see, how soon, they are up, and down. If you happen to find yourself in a fortunate position with only a few masters, you may grow favor with one, who might provide you with an office or other means of livelihood on some day. This is but a chance, which you may call fortune. They open their mouths for every windfall. If he should later offend the law, you, being his man, would also be at risk of being apprehended. Therefore, I cannot help but highly praise your wit. Now, what other vocation or occupation serves your mind better to earn your living, through trade or merchandise, or by some handy craft? I have no skill in merchandise, nor do I wish to spend my time on it, since it is ruled by chance, and since I am ignorant of it, how can it benefit me? He who intends to live by merchandise needs both good credit and a stock at the beginning, but I lack both. I have none, and because I was once a servant, my credit is gone. Few merchants will trust one of that race, even if it were in the most honest case. Besides, I must swear, swear, and lie if I wish to engage in buying and selling. My wares must now and then be counterfeit if I intend to gain anything through them, which I shall do, thus I shall spin a fair thread, but I would rather despise such unhappy winning that will bring me to hell at the first beginning. You speak well, (said he), and besides all this, to be a merchant, my son, there is another thing. Many merchants In this land, of late, such merchandise has been done and brought behind hand, which could not rise again. The state of merchandise is so bitter, I said, in any other craft or science, I have no understanding. And though I had, yet the world is now in such penury that almost no kind of craftsman can live by it. All kinds of things have become so dear that such has not been seen for many a year. I will tell you, (quoth he), since I was born, I will be eighty years old come tomorrow, was never such misery, necessity, and need. Among each estate, in every place, as is even now at this present day, for though diverse times past, as I say, corn has been as dear here before, yet of all other things, we had plenty and store. But now, the price of all things has leapt such a leap, that neither food, clothing, nor any other thing is good cheap. In whom the fault is, truly I cannot tell. But I would that all things were well. I tell you in earnest, I heartily thank my redeemer, Christ, that my unworthy self is so near the grave. My unworthy self shall have a shorter life, but concerning our first communication, it seems your mind is not occupied with anything. What then, have you good learning, any whit, perhaps to be a priest, you think yourself fit? And surely (said he), if your calling is so well, then you do amiss, never a del. For to truly preach God's word and be a minister, if your calling is to it, you cannot do better. Certainly (said I), my learning is but small, and to such a great matter, it is, for me to meddle with. He should be well learned and of God's calling who occupies the place of preaching. But I am both very unlearned truly, and also far unfit to occupy that place. You speak well (said he), for you might work yourself woe if you would attempt to enter in at the window. Or leap over the shepherd's cot, not called by God,\nSo might you make, for your own sake, a rod,\nWith almighty God (son) is not good, to play,\nFor he may not, be trifled with, by any way,\nYou might thereby, invoke God's vengeance, for your haste,\nAnd eternal damnation, perhaps, at the last,\nTo be a lawyer, then (he said) perhaps, is your desire,\nThereby, you think, to aspire to great riches,\nNot so (I said) for some learning, I lack,\nNor ever, of Sophistry or Logic, had the taste,\nBy which stands the chiefest and principal point,\nAs of that, pleading science, the best jointe,\nAll things considered, I mean not that art.\nNor ever, here, could it agree with my heart,\nCertainly, he said, the law is good, if it is used well,\nBut now, the justice of the right law, is somewhat abused,\nAnd many of their rising, is by extortion.\nWhereby, they claim, to have, of riches, such portion,\nAnd through the same, they almost are grown, to like obliquity,\nAs the Clergy. At the first fall, they were wont to be,\nAnd not without a very urgent cause,\nThey do some without right, so writhe and wrest the laws,\nAnd soon in my judgment, it is plain,\nMuch of their law tends to their own gain.\nWell then, I think, you will go beyond the sea,\nYour youth in learning to occupy so,\nForsooth (quoth I) of all the rest I find\nThat this most pleasing, to my mind,\nIf so it were, my living would extend,\nThere to keep me, two or three years to an end,\nWhereby I might, though it were to my pain,\nSeek countries, and some knowledge obtain,\nBut he that goes thither, without a good purse,\nGoes out of God's blessing into his curse,\nIt is good for such men, to go truly,\nAs intend, the kings' embassadors to be,\nHis grace's weighty affairs, there for to do,\nBut I am not like, to come thereto\nThe truth for to say, and to be plain.\nSince there is so much hazard, and so small gain,\nI am not yet, minded in this care,\nTo seek my living, in so far a place. Therefore, I agree, for if you lack that, you shall fall into great misery, and it little benefits you there to serve any man, for they are all, other than flaws or gentlemen. Further, what will you then apply, to live in the country by your land or annuity, as perhaps you have, tell me, so I may show you my best counsel. I have not much true living left, but what I have lies in the country, whereon I will, if God helps send, quietly live and there end my life. Now I perceive your true intent, in arranging questions, this time have I spent. Therefore, now I will show you, in all that I can, my counsel, how you may best prove an honest man. Say on, let us no longer waste time, for since we began, we have made good progress. My son, you seem very wise, that in this last case, you yourself advise. First, God, and godly things, I will preach to you. And after worldly things, to thee I will teach, in earnest:\nThe principal thing, which thou shouldst ever keep in mind,\nIs to have good respect, continually,\nTherefore, and for what reason, God created thee,\nAnd how, He appointed thee, and in what state,\nTo consider, from thy first creation,\nDirect thy life, in every condition,\nSon, God brought thee into this world,\nFor two good reasons, as it will appear,\nFirst, to give Him thanks, praise, honor, and glory,\nAnd evermore, to praise His eternal majesty,\nBoth for His pleasure, in forming and making thee,\nAnd all things earthly, for thy sake,\nAs well as, for that when thou was lost, by Adam's sin,\nHe would not, of His mere mercy, see thee destroyed,\nBut sent His dear Son, from His high throne,\nInto this wretched world, to redeem us each one,\nWhere He took upon Himself our frail nature,\nIn the blessed virgin's womb, for our aid,\nAnd then, here on earth, us thirty years taught.\nBy His word, and miracles. which he wonderfully wrought,\nAnd at last, on the cross, suffered his bitter passion,\nOnly to redeem us, and to obtain clean remission,\nWhich, his mercy, if it had not been greater,\nWe had remained, damned souls, and no better,\nFor this cause, we are bound to praise him,\nFor from death, sin, and hell, he again, raised us,\nTherefore, him to praise, pray unto, and worship with fear,\nThis, steadfastly, in thy mind, see that thou bear,\nFor it is the chiefest point, of virtue, to laud and know,\nThe creation, of men, and all things, that grow,\nWithout this, other virtues, are nothing regarded,\nIn lacking, this pity, thou shalt be little rewarded,\nFor this, to all virtues, is thentrye full plain,\nSince, by prayer only, each good thing, from God we obtain.\n\nThe other cause, why God brought thee hither,\nWas that thou shouldest, in word, work, and thought,\nAlways endeavor, thyself to thy power,\nHis holy commandments, to obey, every hour. You do sufficiently know,\nIt needs not me to repeat them again,\nFor these two reasons, specifically, my dear son,\nGod brought you into the world,\nThese firm foundations, fixed once in your heart,\nThen justice rightly to embrace, see that you do your part,\nSuffering no man to sustain any injury by you,\nOr to hurt any creature by deed or word, see that you flee,\nAnd so do to other men of every estate and degree,\nAs you would have all men do to you,\nFor this law of Lady Nature every other law does excel,\nWhich if you observe not in every point, well,\nTrust me truly (good son), both the Lord, you shall offend,\nAnd in heaven have no habitation, at your life's end,\nFurthermore, for his honor, his goods, or good name,\nBeware, you hate not by malice or disdain,\nAlso be circumspect that neither bribes, hate or love\nYour heart from equity and justice do remove,\nFor these three things we see, so blind many men's sight. Their judgments and actions are contrary to what is right, but remember, my son, that you will one day die, and if you commit these deeds, you will be punished eternally. The wicked thirst after riches to obtain, and the ungodly covet silver, gold, and other gain. If you also yield to covetousness, know that where covetousness reigns, all kinds of wickedness must remain: impiety, perjury, rapine, and theft, fraud, cunning, and lecherous living, quarreling, treasons, murders, and killings, for lands, treasures, and goods, many men spilling. Finally, no filthier thing exists, or is more detestable, than a man being covetous in any way. For whoever is once given to this vile vice, I may liken him to the blind mole, which neither loves, desires, or in any way knows anything other than this vain world. The wretch does not see how short and how frail the life of man is, and how death assails us. Daily ready, to strike us, with his bow bent,\nWith his deadly, doubtful dart, then is it too late, to repent,\nHe spares not the young, the old, or any degree,\nThe rich, the learned, or the man in authority,\nThe Lawyer, the Landlord, or the lacking poor man, weeping,\nBut without any difference, he strikes every man,\nAnd he is often nearest us, we daily do see,\nWhen we most think him furthest to be,\nBut then (my son), these worldly riches here,\nAnd these vain goods, subject to blind fortune's power,\nDo little esteem, nor much for them care,\nFor these things are not thine, whatever they are,\nWhich either, unstable fortune, her pleasure to fulfill,\nDoes give, grant, and pluck away, at her variable will,\nOr which, when thou diest, will no longer abide,\nBut to seek new masters, away suddenly they do slide\n\nThere be other goods, which thou oughtest to acquire,\nAnd more better riches, thou shouldst desire,\nWhich with thee always will remain And endure, of whom neither fortune nor death has power. Be thou sure to heap up these, night and day, for thyself. In truth, art thou happy and rich in every way. As for the rest, which the common people follow and magnify, if thou hast them as lands, goods, cattle, or money, use them lawfully. No man can forbid thee this, but thou oughtest to use them with justice and moderation. And also when thou mayest, have pity on poverty. Never shut thy ears at the cry of the needy. For he who shows compassion will not refuse to hear the crying of the poor. He himself will cry out and not be heard, I am sure. By relieving the poor, thou layest up in store a treasure in heaven, which neither moth nor rust can corrupt, nor thieves break through and steal. Thus, this heavenly treasure, for worldly trifles, thou shalt have another day. No man is he who lacks clemency, and at other men's misery, takes no pity, or refuses to help his fellow servant here. Since the text is already in modern English and does not contain any meaningless or unreadable content, OCR errors are minimal, and there are no introductions or logistics information to remove, I will simply output the text as is:\n\nSince we are all servants, to one Lord and master,\nBut if it should happen that a poor man be,\nWith patient heart, bear and sustain, the poverty,\nFor the man, who has much, we always well see,\nIs troubled, and tossed, with cares, abundantly,\nAlso he, to whom fortune, has given store,\nThe hot, hasty heat, begins to abate,\nThen after falling, follows, frowning, repentance.\nWith sorrow and shame fastens, bringing great grief,\nTherefore beware of it, and thy mind so moderate,\nThat this foul vice, in thy heart, thou diligently abate,\nPatience is a virtue, of a wonderful strength,\nAnd obtains, the victory, of each thing at length,\nThis godly gift, who soever doth want,\nIn him all goodness and grace, of force, must be scant,\nAnd cruel he must needs be, and also to strife, prone,\nWhich is not the nature, of man, but of beasts alone,\nThe prudent and good, seek chiefly for peace,\nAnd fearing greater mischief, lest. A small spark might increase such a flame,\nThat great peril it would be, again, to quench the same,\nHe who will suffer nothing, nor his way ward off wrath,\nMust flee, the company, of all men we see plain,\nAnd dwell alone, in the woods or mountains high,\nWhere no man may trouble him, nor he any body,\nBut he who will frequent and dwell among men,\nMust learn to suffer disappointments now and then,\nAnd bridle his fury, dissembling his ire,\nAnd in his secret breast, quench the hot fire,\nNor in any way may he, for every light offense,\nViolate the bonds of peace and patience,\nBut as much as he can, forgive other men,\nWho offending in like case, may also be forgiven,\nFurther, beware of gluttony, be ever circumspect,\nWhich with various diseases, you mind and body inflict,\nBesides shortening man's life, it consumes his wealth,\nUnwarily, as it were, him robbing by stealth,\nFor delicacies and dainties are costly. I have known, good son, many who have been utterly ruined\nBy prodigal spending, consuming their patrimony,\nDaily pouring into their bellies, both house, goods, and land,\nUntil poverty pinches them and they are in want,\nExcess and drunkenness are also this vice's brother,\nFor in whom one reigns, the other also dwells,\nFlee this likewise, if you love your welfare,\nOf all others, it is the most detestable snare,\nHe who has once embraced this wicked vice,\nIs clearly defaced by all goodness in him,\nReason then refuses him, and he is left to his will,\nAll sins have free entrance, him then for to shun.\n\nGood son, I have said many years past,\nThat great Alexander, in his drunkenness, commanded in haste,\nHis most dear and familiar friends to be slain,\nBy whose help and good counsel, as it is most plain. He had the whole world conquered, but being thus dead,\nAnd sleep had expelled ebriety from Alexander's head,\nHe then so lamented their deaths in weeping bitterly,\nThat he was ready for very anguish to die presently,\nOh filthy ebriety, the destroyer of the soul,\nOh nourisher of vices and iniquities all,\nWhat thing is it, but you force man's heart to fulfill,\nWhereby he dares to attempt all that is ill,\nQuarreling, strife, cruel fraises, you move,\nNeither regarding discretion, honest friendship, or love,\nThrough the counsel is opened, and secrets revealed,\nThe tongue is not then able to keep the same closed,\nFlee this vice, my son, in all that you may,\nLest it grow, from custom, to nature another day,\nThe tongue also, you must learn to moderate,\nAnd be well aware, what you speak early and late,\nBe ever willing, attend to hear,\nBut speak seldom, as need shall require,\nAbundance of talk is a great sign of folly,\nAnd the busy babbler. He who offends continually,\nHe who seldom speaks and then wisely,\nIs worthy of much praise, and proves to be witty,\nOne principal point, observe in thy communication,\nWhether they be present or absent, our words hurt no man,\nNor speak thou anything except it tend to some purpose,\nLest men laugh thee to scorn for thy babbling,\nRather hold thy peace and be ever silent,\nIt hurts not, neither shall it thee repent,\nIn danger and peril, he remains evermore,\nExperience teaches us, and we see every day,\nSome men's lands and riches to be their casting away,\nIt is no new thing, to see this abuse,\nThat diverse men's wealths have been their confusion,\nAlso, rich men are we the burden of their goods so borne down,\nThat they have no mind to seek after the celestial mansion,\nFor the more, a man desires riches and earthly gain,\nSo much the harder, it is, for him, to attain heaven,\nFor look, where a man's treasure is laid,\nThere is also his heart. It cannot be denied,\nPoverty is profitable to some, disburdening them of countless miseries.\nFurther, pride is to be abhorred no less,\nBeing the source and mistress of strife and debate,\nBy this, the laws are neglected, and the commonwealth is spoiled,\nAnd countless people, this vain pride, has killed,\nWith this pestilence, the famous Rome being once poisoned,\nBy civil war and oppression, was utterly destroyed,\nThis monstrous hellhound, always see thou flee,\nIf thou wilt live eternally with almighty God,\nFor God takes delight in the lowly and meek,\nAnd in humble spirits, gladly has his dwelling,\nHe favors also those who are void of ambition,\nAnd the proud, swelling people, he brings to confusion,\nTherefore, you people, puffed up with pride, what profit is there for you,\nYour pride, high names, and vain styles, forged new,\nWhich death confounds and brings in subjection,\nYour ambitious titles, of such great renown. Some may say, the common people, we will please,\nIn courting of them, therefore to have our praise,\nTell me (I pray), what is the judgment, of the multitude,\nYou shall perceive many times, they mock and delude,\nOr else speak of affection, as some time, naming them wise,\nWhich are perhaps, very fools, rightly to surmise,\nAnd though, the ignorant we mock, God can we not deceive,\nNay, he rather derides us by our leave,\nFor he knows our manners, and our deeds, most secret,\nAnd for them worthy to punish us he will not forget,\nBut many are so blind, that in their own thought,\nThey believe, there is no God, & that there remains nothing\nOf any man, the breath, being once expired,\nTherefore, the present joys, of this life, they have ever desired,\nAnd do daily wish for, deriding the bliss to come,\nHave not they, bestial hearts, under man's shape and form,\nWe spend our life vainly seeking a sinful gain,\nWe ponder not the frailty of our wretched state. We are not daily ready for death, our pride to abate,\nWe are so restless from most to least,\nIn our vocation we never can rest,\nWell if in worldly things blind fortunes governance,\nRuled not without reason at her own will and pleasure,\nAs we see she does, then all things should be well,\nThe laws, and justice, should flourish, and tyranny expel,\nBut since almighty God suffers such things to be done.\nWho if it pleased him, could amend all right away,\nWhy should we grumble, to suffer the same,\nTo repine at God's will, we are greatly to blame,\nWherefore wisdom wills wise men, ever to be patient,\nAnd taking all things, as it comes, to be content,\nBut thou (my son), endeavor thyself, in all that thou can,\nBlind fortune to despise, and the vain praise of man,\nOnly study thou daily, by all manner of ways,\nWith virtuous living, the Lord for to please,\nFor true honor and praise, thou canst not obtain,\nUntil after this life, in heaven thou remain,\nWhich the good and just. Enjoy what follows,\nWho with humility and meekness have led their lives here,\nAlso, I charge you, suppress your anger,\nAnd the hot rage of ire, which causes great busyness,\nWrath works woe and much mischief we see,\nFury makes frays, and then of necessity,\nFollows wide wounds, hurts, and other harm,\nIn which bloody bickering, oft times some men are slain,\nFor the mind of man, being once, incensed with ire,\nIs so blindly oppressed, with that rash, raging fire,\nThat it neither can behold, or judge anything right,\nReason then rules not, and wit has lost its might,\nWhen after once being past, such furious rage,\nShould have been quiet and still, but experience teaches,\nThat the talkative person often repents to his pain,\nFinally, my son, another thing there is,\nWhich I have not yet rehearsed, but above all the rest,\nYour life will be defaced if in youth you do not have the grace,\nThat is the wanton desires of the frail body. With this thy best age, violently assail the lewd lusts, and ensure that you flee from them, bridling them before they blind you. There is nothing so contrary to virtue as the wicked concupiscence of the body. Virtue, striving to ascend to heaven, is only attended by vile lust and vain pleasures, which always look to the earth as a beast to the ground. The living spirits, both of mind and body, are destructive. The devil takes many with his book and snare, sparing no effort. So he may keep them from attaining the bliss of heaven, and after this life, live forever in pain in hell. Therefore, beware carefully of these deceits of Satan, your enemy, and this detestable poison, covered over with honey. Be on your guard against them as much as you can, lest they repent you in vain on another day. When your ripe years of discretion and man's perfect state shall clearly perceive, though it may be too late, your wit, your substance, your members. And yet a little wanton pleasure, to remain,\nThen shalt thou, as many others are wont to say,\nOh youth, and lusty years, how they have vanished away,\nHow evil have I spent you, wretch that I am,\nWhere are you gone, oh unfortunate man,\nIf God, in his goodness, would restore,\nThe joyful days, I would again obtain,\nI would then tread, the plain path of virtue,\nAlthough it were never so narrow a way,\nI would therein walk, and continue night and day,\nThat there is no thing like virtue, I find now true,\nNor to it, to be compared, alas, I often rue,\nWhich ever continues, and always endures,\nGiving to man such worthy praise and honor, as is sure,\nShe increases thy riches, and thy life does preserve,\nYea, after death, she abiding, away will not swerve,\nBut I of all others, think myself most unhappy,\nWho by flattering voluptuousness, was wilfully deceived,\nWhich sliding away, has long since left me. Wrapped in all my mischief and woeful misery,\nFor I, being a young man, the stews did frequent,\nAnd in banqueting, sleep, and play, my time idly spent,\nNothing then would I learn, all study I despised,\nAbhorring in good science to be exercised,\nBut now therefore (it is worth the time) I well see,\nMy self both unlearned and no less in poverty,\nAnd my whole body bruised,\nMy wits altogether dulled, and my senses confused,\nI have hitherto lived, as one who has dreamed,\nHimself to be awake, and yet was deceived,\nSuch things (suns) of some men we are wont to hear,\nThat are far struck in age, and to their grave draw near.\nWhich calling to remembrance, their lusty years past,\nDo now (but too late) bewail their misery at the last,\nThen shutting the stable door, when horses there be none,\nAnd now warning wise, when blind Fortune is gone,\nThen seeking a Surgeon to heal their wounded sore,\nWhen there is no hope of cure, in the same any more,\nMy son, therefore take time. While you may,\nFor it never returns, if it once vanishes away,\nNeither wailing, weeping then will help, or remedy,\nIf the body is once struck, with death's dart deadly.\nThe medicine is profitable, a physician says,\nThat in time is administered, not slackening any day,\nWherefore when your youth first begins to flourish,\nThen you ought to embrace virtue and flee from sin,\nThen you should take the right path of living,\nTo good and honest studies, apply yourself,\nThen use reason and govern your mind by counsel,\nWhile it is pliant, every way to wind,\nWhosoever will be wise, let him be wise while he may,\nFor to be wise a day after the fair, is folly, I say,\nAnd therefore he is worthy, his misery to sustain,\nBewailing the loss, it is never to be recovered again,\nNow concerning this world, understand clearly,\nThat the same is very short and transitory,\nAnd the whole life of man, in which we do run,\nIn comparison to eternity. Of the world to come,\nThe truth, which should be truly declared,\nIs not to be compared to one moment of time,\nAnd touching the mystery of the world, I say,\nHe is happier that is gone, than here far away,\nFor mark I pray thee, how infancy comes out,\nOf his mother's womb naked, without any cloth,\nAnd the first thing he then does, is weeping with tears,\nBecause the misery of this world, as I think he fears,\nAs I might liken it, to some merchant man,\nWhich on some perilous voyage, his way must take,\nAnd fearing both drowning and pirates, and shipwreck,\nTrembling, so dangerous a journey to take,\nYet nature teaches, the infant plainly,\nThat he then enters, into the vale of misery,\nAll other beasts, that nature brings forth,\nTo their dams quickly run with open mouth,\nBut man, as soon as ever he is born,\nIf the mother clothes and feeds him not, he is lost,\nAfter infancy, how long it is,\nOr perception creeps in, in that little breast of his. Childhood follows, neither of them being able to govern themselves, whether willing or unwilling, then what grief, trouble, fear, and pain endure the child, or he attains to adolescence,\n\nIgnorant, frail youth begins to arise, which leaves reason and is commonly ruled by vice,\nHis strength increasing, he puts away fear,\nGood warnings and precepts, he will then no longer hear,\nThen he wears wild, his young blood being warm,\nGiven to anger and lust, which does him much harm,\nGood counsel he refuses, and is then rash in all things,\nTo the evil inclined, the good ever turning,\nNo one dares he doubt, no danger he refrains,\nSo that his frail lust he may obtain by any means.\n\nNo law he then fears, if there is a toy in his mind.\nFew young men in that age can abstain from vice,\nWhom either shame, fastness, fear, or wisdom constrains,\nBy no spot of vice, their youth to stain,\n\nThen comes man's state, grave and sage,\nBy experience and wisdom. He is troubled in this age. Then sound sleep eludes him, he toils with pain,\nTo maintain the living of himself and his house,\nHe gathers together, in this stage of life,\nTo keep his hounds, his family and wife,\nCaring for them all, both to clothe and feed,\nSpending this his best age, with misery in deed,\nAnd then weary rude age, on man swiftly creeps,\nBringing many discommodities, both of mind and body,\nHis strength departs, his fresh color will not stay,\nHis senses shrink away, his sight wanes,\nHis hearing dulls, and his smelling leaves him,\nFurther he is always threatened, with one or other disease,\nNo meat then tastes good to him, all things displease,\nWithout aid of a staff, his legs then fail,\nHis wit is gone, his body shrinks, wearing pale,\nThis every age, has its infirmity, we see,\nWhich of necessity, we are forced to suffer patiently,\nWhich last age, will not leave him, till he has brought. Man passes, of all other troubles to tell,\nWhich man is besieged, while he dwells here,\nWith which man's life is so greatly surrounded,\nThat it is seldom or never, but by some of them consumed,\nNow vehement cold, with lying snow, afflicts us,\nThen harsh burning heat, molests us,\nTrembling, which often causes the ground\nTo widely open, some pleasure to obtain,\nSometimes we are annoyed, by such great wet and rain,\nThat it overflows, whole countries we see plain,\nHunger, thirst, and also poverty,\nWith the lacking, of necessary things to occupy,\nWho can declare, in meter or prose,\nThe great and manifold number of those,\nDiseases and sicknesses, of every kind,\nWhich daily kill our bodies and weaken our mind,\nBy battle and warfare, some men are overwhelmed,\nIn the seas and other waters, many men, are drowned,\nOthers by falling, catch their death,\nOr by crushing, their limbs. Some men are made lame,\nSome are consumed by cruel fire, ashes are their lot,\nMany are choked, and cruel beasts kill some,\nWhat shall I recite, in earth the living man,\nHas no greater enemy than the seed from which he came,\nFrom that seed spring all thieves and robbers,\nAll murderers, perjured and false witness bearers,\nFornicators also, and vile adulterers,\nAnd from mankind descend all wicked doers,\nThis one kills me with his weapon, another with his tongue,\nThe most part work by fraud and deceit,\nO Lord, almost all men now take delight,\nAt other men's harm, such is their spite,\nThe brother, the natural brother, distrusts,\nThe friendship of friends now lies in the dust,\nThe father, his son, and the husband his wife,\nEach suspects the other, here is a goodly life,\nNow son, I have told you, as well as I can,\nThe perils that compass the life of man,\nWherefore thou mayest not use this worldly pleasure,\nExcept thou intendest to refuse eternal life,\nAnd the time is short. that thou here shall remain,\nThe truth whereof, I can well declare plain,\nFor all though that I am, now four score old,\nYet the bedroll of my life, when I do unfold,\nI do now well ponder, and perceive in this case,\nThe same to have been, but a very little space,\nAnd yet to live mine age, of a hundred I know,\nOne shalt thou not find, though thou sought them on a row,\nGood father said I, I think you speak true,\nFor I have seen very few, of thine age,\nNow (quoth he) I have taught thee, sufficiently to know,\nFirst God, thyself, and the world, I believe,\nAnd because quietly, thy life thou wouldst lead,\nIn the fear, and love, of God, as thou saidest,\nI could wish, thou shouldst prudently provide,\nOf some good stock, a wife with thee to abide,\nBy whom, thou shalt, many commodities possess,\nMore than I can, at this time, express,\nFor thy wife, shall leave, both father and mother,\nTo stick to thee only, and to none other,\nHer own kindred, and friends doth she leave,\nDaily, during her life. To you for this, by her, you shall have, the fruit of your own seed, Which shall keep you in remembrance, when time you are dead, Obedient, she will be, and a constant companion always, And a joyful joy also, to prolong your days, Each good thing shall be common, ever between you, Your games, shall be one, your livings to sustain, Further, if age or sickness shall harm you hereafter, She will be a constant and faithful helper, By assisting, ministering, and watching over you, Comforting and relieving you with the best she can do, And then your children, who may flourish afterwards, Will do their endeavor, you gladly to cherish, In whom you shall yet live, when life is gone clean, And your name still live on, by them it shall remain, Further, to them you shall leave, your goods being got, And not unto strange heirs, whom you do not know, Besides this, your wife, with her, will bring Towards the maintenance, of our and her living. Both friends and kindred, by her, will arise. Which may be profitable to you in various ways,\nWhy then, you must have a wife, there is no denying,\nIf you mean living quietly and godly another day,\nGood father (I said), I think you speak well,\nBut of one thing I would gladly hear told,\nWhich way is best, by your consideration,\nHow might I have one, of an honest conversation,\nMary, young man (he said), the matter lies in this,\nIn the little wisdom of him who chooses her,\nFor I would not have you therein beguiled,\nAs I once was, when I was young and wild,\nWhich, if it were not for lack of time and space,\nYou should plainly hear, my sorrow in this case,\nI pray you (I said), take the pains to declare it,\nAnd for lack of time and space, do not spare it,\nNay, not so (he said), for we are now near the farmers' plea\nWith whom I intend, to come a little space,\nTouching, my business, wherefore I came here,\nWhich being once finished, if you will then,\nBear me company, homeward, in the way,\nI will declare. I have yet to say the rest. To this, I answered that I would gladly dispatch his matters quickly if he would. Here ends the first part.\n\nWe saw, the farmer, at his gate standing,\nWho, as soon as he perceived us coming,\nGentlemanly, saluted us, calling him by his name,\nAnd we did likewise greet him, with chunks for the same,\nWith whom this aged man, having walked with us for a while,\nConcerning his errand, held a full conversation,\nThe farmer had us in, where we had good cheer\nAnd tarried the maintenance, for half an hour,\nThen took we our leave of the farmer,\nAnd so straightway departed from him there,\nAnd when we were homeward, a pretty way,\nTo the said old man, I began thus to say,\nNow, Sir, and it please you, in your tale to proceed,\nI am much desirous to hear the end in deed,\nWell said, son (quoth he) then give diligent care,\nWhen I was of the age of twenty-two years,\nI was very lusty and pleasant,\nTo sing, dance. I was skilled in various activities, such as playing ball, running, wrestling, throwing the axe or bar, either with hand or foot, and excelled in all other feats, as none in the town could match. I was fine, fit, neat, proper, and small. I was then, as I say, both beautiful and fair. It seemed no less so to you, he said, since you bear your age well. Well then, let us pass over such things, he said. I could then finally play the harp better than now. By my minstrelsy, fair speech, and sport, all the maidens in the parish were drawn to me. Each loved, lusty Lewes, as they called me. Not one of them refused my company. I was then a parish clerk in the town, helping the priest with mass and singing in the choir. With the living I had, I lived without care, having no wife or child for whom I would spare. There was a neighbor, a very honest man, dwelling within the same parish. A daughter, of mean stature yet well made, possessed the greatest beauty. Her countenance was cheerful, and she had a good favor. Her smile was smooth, and her eyes were wanton. She was the most amiable maiden I had ever seen, with a fine figure and the ability to speak eloquently. Her communication was filled with a very singular grace. She appeared modest, demure, and sad. Despite her demure appearance, she was never too formal, adjusting to the required familiarity in the right moments. Her attire was always white, and seeing her on a holy day was a beautiful sight. Her gait was womanly, with a graceful gesture to and fro. She returned home late from the city, having served there for two or three years. The first time I saw her was on a holy day at noon, returning homeward from the church. Her wanton, beautiful features so captivated me that I was earnestly drawn to love her. She lightly brought me, by whom I was then struck, with such a violent pain, that the holy water bucket, from me it straight I flung, And great haste I made, running swiftly after, Because I would, so gladly overtake her, At last I overtook her, but with much ado, Then she greeted me, and I her also, Her sister went with her, who bade me welcome, Saying, \"whether in such haste, good Lewis, do you run?\" In faith (quoth I) since you inquire, To a neighbor's house, which dwells here by low, The haste that I made, was for your company, And to know what fair maid, this truly is, Mary (quoth she) she is none other, But my own sister, born of father and mother, And sister Grace (quoth she) I pray you, of him be acquainted, For with his melodies, he does us often make merry. I tell you Grace (quoth she) he is an honest man, And on his minstrel's harp, full well he can play. To this Grace answered, very sadly, And I would be acquainted, with him very gladly. And this passed forth, we were nearly home,\nSo then taking leave, I parted them from me,\nAnd this was the first time, of our greeting,\nWhich was to me, an unhappy meeting,\nAs you shall plainly, hereafter perceive.\nBefore the cruelty of cruel Cupid left me,\nBackward I went, where my dinner was prepared,\nAnd still by the way, my heart full for sighing,\nWhen home I was come, to dinner I was seated,\nMy heart was heavy, no meat could I eat,\nAfter dinner, down, on my bed I lay,\nMuch musing with myself, what thing it might be,\nThat so suddenly had struck, my heart with such woe,\nAnd so soon had driven, my mirth and pleasure from me,\nNo way could my wit, by wisdom's device,\nHow this sadness and thought on me should arise,\nExcept it were, by beholding the maid,\nWhose beauty and favor was ever in my head,\nTo have slept in vain I would, but it would not be,\nYet at the last, a short sleep took me,\nIn which sleep, also my thought I did see,\nThe maid. Whose fairness, before it perplexed me,\nI rose up then to the church to hear the servant,\nMy love did not come to evensong as I thought she would,\nTherefore I was sorrowful, my heart was cold,\nAt supper, nothing could I eat, then I thought it best,\nIn a convenient time, to retire to my nest,\nNo rest could I find, my sleep was completely gone,\nMy heart was heavy and cold as a stone,\nThe morning came, when the night had passed,\nThe air was clear, the sun shone bright,\nAbroad I then walked, to hear the birds,\nWhere a friend of mine met me and asked what was wrong,\n\"Truly (I said) I am now exceedingly faint,\"\nYet I do not know the cause that taints my heart,\n\"God bless you (he said) and went his way,\"\nThis in a miserable case, I spent that day,\nAnd many more days until shearing time came,\nHer father shearing sheep, invited me to his feast,\nGlad was Lewes, thinking then I would speak,\nAt leisure, with my love, and my mind. To her breast,\nThen she shall know the woe and smart,\nThe heaviness and sorrow, of my woeful heart,\nThe restless nights and unsettled days,\nThe heavy thoughts that trouble me always,\nI will also then, if I dare,\nThe bottom, of my mind, to her declare,\nThen I softly spoke to myself, God lend me,\nA committed time, and that he will send me,\nTo obtain at her hands, such favor and grace,\nThat my humble request may be heard and take place,\nWhat need for longer process, the shearing day,\nThat I so long looked for, at last came I say,\nTo her father's house I came, as they were at dinner,\nHe had me heartily welcome, and in the best manner,\nTo the table was I seated, down on the bench,\nWhere I might see, full in my eyes, the well-favored whey,\nWho there served the table, as it was the custom,\nAnd surely she waited in most womanly wise,\nWhich she well could do, for as I said before,\nIn the city had she been, good manners to learn.\nLittle meat could I eat. Which was noted well then,\nBy the maid's mother and also by her good man,\n\"Which cheer good Lewes,\" she said, \"tell me, you look very sadly,\nAs one half dismayed. What man, she asked, where has your mirth gone?\nI think you muse on the man in the moon,\nBe merry I pray you, and therewith she kissed me.\nBut my heart was holy, for the maid who served,\nAnd afterward, when dinner was done\nAnd the guests departing, away each one,\nI also having rendered them thanks for my cheer,\nWent homeward, with the company, that were going there,\nBeing both of my purpose, that I came for none\nAnd also for work at ease, both in my heart and heavy head,\nThen, inflamed was my lust, and grew more and more,\nWhich was but a little, kindled before,\nThe malady, which before, might in time, have been healed,\nNow incurable, and that I well felt,\nMy harp, which was wont, so sweetly to sound,\nLay now untouched, for me on the ground,\nMy breast, which before, many souls, did rejoice,\nBegan to toughen. and my voice grew hoarse,\nMy color, which once so vividly appeared,\nWas faded away, and changed its appearance,\nMy legs, which were once nimble to dance,\nWere shrunken clean, by this unhappy chance,\nI was near consumption, all strength was gone,\nSo holy was I altered, that I was scarcely known,\nAt last I considered, the best way to proceed,\nIf I thought, by her help, to be cured in truth,\nWas that she listen, understand my woe and distress,\nAnd if she would, of pity and clemency, relieve my great pain,\nThis way, I thought sure and plain,\nFor physicians do not use, to administer remedy,\nBefore they are informed, of their patients' malady,\nSo it happened, in a morning not long after,\nThat I chanced, to walk, through the common pathway,\nWhere the milk cows of the town fed at daytime,\nAnd all the maidens, in the parish, milked in that place,\nAmongst whom, Grace's sister, was one there,\nAnd therefore, I remembered. To banish all fear,\nAnd make open, and break holy to her,\nThe full care, and effect, of this whole matter,\nBut first, certain words, a farce to prove her,\nI would cast out, to see, how this care, would move her,\nAnd how she would take it, ere I meant to declare,\nUpon the liking, of her answer, not for to spare,\nThus drawing near, I had her good morrow,\nWhat gentle Lewes (she said), God keep you from sorrow,\nHow do you, what wind drives you hither,\nThis morning so early, and I pray you whether,\nAre you thus walking, youre self alone,\nI think surely, you have some pretty one,\nThat causes you daily to this place to come,\nWell wanton, well, though not all, yet I know some,\nLoose Joan (she said), for so was her name,\nIn judging amiss, you are greatly to blame,\nFor if I for love, report, to any in this place,\nIt is truly, to you, or to your sister Grace,\nFor surely, you two, of beauty bear the flower,\nThis judgment, must I give, though I die, within an hour. No, if she [judges not well], for there are many maids who excel us. Yet, for gentleness, I have always sent for, and the honest behavior, which has continually been, in you, I think you are worthy to obtain. As good, and as fair, as any in this parish remains, I would think her well-bestowed, if you had her. I thank you, fair John [quoth I], that it pleases you so kindly to praise me. And if God preserves my life, any while He will, I trust to deserve your gentle kindnesses. And therewith, as it were joking, I stayed a pretty while. What, Lewes [quoth she], do you think I am dismayed, whereon you so study a penny for your thought? In faith [quoth I], if you knew it, yet it would be worth nothing. My thoughts [she said], you are changed in every respect. What, has any maid roused your heart from its place? Tell me [she said], and my best counsel you shall have. With all that I can do, as God is my soul, I say: Oh, (quoth I), my heart is wrapped full of woe, yet have I no faithful friend to show. I shall tell you, Lewes (quoth she), whatever you say. To me, think it sure, under lock and key, for ever hearing it, by me spoken again. Receive the same, for your plain profit. Well, since you will needs know what careful mystery I keep, you shall hear the same, in few words plainly. So it is, that your sister Grace, that swears an oath, has my love and heart in such a case. Neither wandering, nor walking, wherever I go, neither playing nor working, whatever I do, neither waking nor watching, any time or space, but at all times and evermore continually, her amiable countenance restrains my mind daily. No pleasure pleases me, my mirth is amplified, No joys delight me, my liveliness is abated, No music rejoices me, their sounds are unsweet, No pastimes pass on. \"as at this time unmet,\nNo work is well wrought, now under my hands,\nNor I am nothing, as I was, before I entered love's banes,\nSo that I well know I am like to sustain,\nDeath's dart, very shortly, if I do not obtain,\nThe rather, her love who now has the measure,\nMe to slay or revive even at her own pleasure,\nWhich I would she did shortly for the ease of my pain,\nBy the dart of cruel death devouring me clean,\nNow have you heard all I quoth, and more as I say,\nThan ever to any other I told before this day,\nWherein I shall desire you to play an honest part,\nFor the swift quieting of my poor wretched heart,\nMarie (quoth John), now I perceive very well,\nOf your sadness and sorrow, there is no marvel,\nThat such an impostume breeding in your breast,\nWorks you worthy full wayward rest,\nNo wonder it is though you look wan and pale,\nFor love has made you drink a draught of sour ale,\nI took you never so tender, so soon to be caught\" With the lovely links of love which are so quickly formed,\nYou were wont before this time always to say,\nThat they were fools who to love did obey,\nAnd that it was impossible any wise man to be,\nSo earnestly set in love in any degree,\nBut that when he would always, well he might,\nEasily put the same out of his head quite,\nBut now you are caught in the same net,\nWhich in times past you greatly neglected,\nWhen you saw any lover, you laughed him to scorn,\nBut love has now brought you to school to learn,\nAnd surely she said, to judge in my intent,\nYou have deserved love's punishment worthily,\nWell, I (said I), to a man who is fallen in misery and woe,\nGood comfort behooves, and not chiding so,\nMy fault I confess, what need is more,\nI desire your good counsel for curing of my sore,\nWhich way may I best by your good advice,\nAchieve this doubtful and dangerous enterprise,\nWell, Lewis (said John), now that I know,\nWhat woman she is that works you this woe,\nLet me alone. I will first address this matter,\nAt night in bed, I will earnestly attend to her,\nAs you shall perceive here by this time tomorrow,\nOther ease or increase of all your whole sorrow,\nTherefore, in the meantime, be of good cheer,\nAnd I will diligently work in your cause I swear,\nWhich I thanked her for, saying gladly I would,\nHer gentleness consider, if ever I could,\nAnd having once kissed her, I took my leave then,\nThis departing from her, whom shall I entrust,\nThe morrow neared, I came eagerly to the same place,\nTo hear if I were likely to obtain any grace,\nJohn was not then present, I stayed a little while,\nAt last not far from there I saw her coming over a bridge,\nWith her pail in her hand, then I went to meet her,\nAnd gently greeted her, she greeted me in kind,\nEither death and double sorrow (said I, John), do you bring,\nOr life to revive me, who am now dying,\nNeither of them both (said she), but hope have I brought,\nThereby partly to feed and relieve your heavy thoughts. \"Then I said, \"And no more time be wasted, I gladly make haste, for it is night being laid in our bed, I turned myself toward my sister and said, \"Oh, John, I wish I were in your place to declare my whole case to her. Your wish was empty (she said), but listen to what I tell. I asked Grace how she did, and she replied, \"Very well. \"Whatever you do (I said), lie at your ease, I know that others are as wicked as you, causing much misery, mischief, and care for your sake. This grieves me greatly since the world began. God never created a more honest man, and he is as worthy of your love as I can guess. He is about to die, such is his distress. For your love, sister John, Grace spoke to me then, \"In this parish I am known by very few men, and fewer do I know than how it can be that any man fears love of me. Who is it, I pray you, name him to me.\"\" And if I know him, I will tell you, she said,\nNay, Sister Grace (I said), that shall not be,\nBefore first some promise you make to me,\nThat you shall not at the first kill his heart clean,\nBy guile him a new one, or by any other means,\nOf unkindness on your behalf, but you shall, if you can,\nBent him your love, before another man,\nFor I will assure you, if you perfectly knew,\nHis good gentle behavior, both honest and true,\nWhich is so pleasant a parson to sing and to dance,\nAnd is skilled in instruments for your pleasure,\nSo well can shoot, wrestle, and leap so lightly,\nSo handsome a man in every man's sight,\nAnd besides this, more sorrow has sustained,\nFor your sake, and is also so cruelly pained,\nThat death to him were a great deal sweeter,\nThan to live as he thinks it much better,\nAnd if in your default you should suffer him to die,\nFor lack of your love what profit would it bring,\nNay rather, it might name you. A murderer I say, one who would greatly shame you,\nNot an enemy, but a most true lover who loves you heartily,\nWell (said my sister), to love him you shall pardon me,\nFor I will do nothing in that matter truly,\nBut this promise to grant to you I am content,\nAt the first time, no, he shall not have my consent,\nTherefore tell me his name without any delay,\nAnd then you shall hear what I will further say,\nIt is (said I), Lewis, the clear one of the town,\nWho for your sake is tossed up and down in misery,\nWhy sister John then said to me,\nI thought you would not of all others so suddenly,\nHave moved me to this lightness I say,\nBut rather have persuaded me if I had been inclined that way,\nWhy Brace (said I), I mean no dishonest way,\nFor he would have you as his wife very gladly,\nSo much the more she said, it is to be borne,\nBut I tell you now as I told you before,\nI will neither love him nor any other,\nBy other persuasion of sister or brother. You might think, dear sister John,\nIf I became a lover so soon after coming home,\nAnd if he loved me so earnestly as you have told,\nTo have moved himself before this time he would,\nBut he thought of securing his desire,\nWhen you acted as his broker for the first time,\nAnd very late she said, so fall to your rest,\nAnd herewith meddle no more in earnest or jest,\nNot one word more said John, I could get nothing from her,\nFor she gave herself where she slept by and by,\nAnd thus have I (Lewes) for you resolved the matter,\nIt is now your turn to attempt her,\nAnd spare not to speak, if you mean to succeed,\nWho trusts to obtain, must put away pride,\nBut surely (Tewes) since she now knows that she loves,\nShe is not a little proud, I suppose,\nAlas, I said then, I would I were dead,\nThen I would be at ease, both my heart and my mind.\nUnlucky fortune I may call it,\nWhich forces me to love one among all,\nWho neither regards my woeful distress. Neither will comfort me with any words of kindness, John replied. It may be that she will hear, You are much better suited than I to be her messenger. And tomorrow she will come to milk in my place, For I must stay at home to brew and bake bread, And speak to her then you do not need my counsel, For you are wise enough to tell your own tale. And in the meantime, you will be sure, I will do my best to procure your way. Thus we parted with no more words. I went to the church, and she went homeward. The next morning came, which I thought very long. And no wonder, for my pain was so strong. Then to the common tavern I hastily made my way. Where my heart's desire was for milking, I quickened my pace, And at last I came to her. I had a good morning, she said, welcome young man. How do you, I said, my own sweet heart. Your love has caused me much sorrow, So deeply is the beauty of your face engraved in me, Your pleasant tongue and behavior, my own love's grace. The seal. You're beauty and lovely countenance have moved me so,\nI am yours entirely, devoted in every condition,\nTo love you and serve you, with humble submission,\nSubjugating your pleasure, and willing to sustain,\nAs long as life remains within me,\nIn consideration of which I desire to obtain,\nNothing but true love, for true love in return,\nA young man spoke, I am sorry for your pursuit,\nAnd much more sorry that you set your mind on me,\nAs to bear such love and goodwill from you,\nWherein I fear your time you will waste,\nIn hoping for a thing you cannot obtain,\nWhich at length will bring you double pain,\nAnd as for me, I do not yet mean to marry,\nI am young enough, I thank God I can delay,\nAnd also I will, for all I yet know,\nShow you true devotion for two or three years,\nNeither would I have you think I do not esteem you,\nFor truly to judge, I can none otherwise deem you,\nBut for your behavior, qualities, and honesty,\nYou are worthy to have one much better than I. Oh beautiful Grace, if Grace would show any grace,\nI pray you that it may appear in this heavy case,\nAnd not confuse me with such a cruel nay,\nNor be so merciless with your words and utterly slay me,\nAnd do not let pity and mercy be banished,\nFrom a creature so fair by God, formed and fashioned,\nNor deny your name in any time or place,\nBut according to your name, show me some grace,\nSir (said she), what need is there for longer process,\nIt seems to you folly to take such heaviness,\nFor me who am not now disposed to marry,\nNor to whom before this time you revealed your love,\nIn truth (said she) as I do now, I would have clean forsaken it,\nAnd Lewis this folly I could wish you still,\nBy wisdom to be ruled and flee from your will. Finally, I desire you to take it for the best,\nThat I here not your suit, nor grant your request,\nOh Grace (quod I), since it is your pleasure to spy on me,\nI shall abide your mercy to save me or kill me,\nYour hard-hearted heart I pray God once to soften,\nSome compassion to grant me before that I die,\nAnd thus God be with you, my love most unkind,\nFarewell, gentle Lewes (quod she), God change your mind,\nAnd send you to put away this sorrow quietly,\nWhich has brought you in this woeful misery,\nThus away went I then, half in despair,\nMy heart greatly vexed between hope and fear,\nWithin two days after I met with her sister Johan,\nTo whom I showed and made my heavy mourning,\nDeclaring to her all the words that were,\nBetween me and my love Grace her sister,\nWell said Johan, I wish that this next night,\nWith your harp you hold your way to our house right,\nAnd there underneath our chamber window,\nIn singing and playing let her hear what you can do,\nYour melody may cause her stubborn hard heart. To love you perhaps, it may please her so,\nAnd this you can easily do without any ill,\nFor both our parents wish it,\nAnd thus, giving John thanks for his good counsel,\nI hurried home, in haste by the road,\nMy harp to tune, and some ditty to make,\nWhich I might sing and play for her sake,\nThe night at last came, and when the clock had run nine,\nThere I went with my harp, as I thought it was time,\nFor as the summer season required twilight it was then,\nTo her father's house that I came,\nA bed was there, no stirring heard I,\nMy harp out of its case I plucked by and by,\nAnd struck up suddenly a very pretty round,\nWhich my harp then new-strung merrily did sound,\nAnother dance or two, I then also played,\nWhich being once finished, I suddenly stayed,\nAnd this ballad hereafter I began to sing,\nMy harp heard the note, which merely did ring,\nOh my love, Grace,\nYour beautiful face,\nHas pierced so my heart. Your mild countenance,\nWith your tongue so filled,\nIs the cause of all my unrest,\nNot Troilus of Troy,\nBy Cressida's sweet allure,\nWas ever so set on fire,\nNor Pyramus the young,\nBy the love of Thisbe so strong\nOr burned in such hot desire,\nNor Hercules the mighty,\nBy Dianira's beauty,\nWas ever so overcome,\nNor Samson the strong,\nWith love was so wronged,\nBy Delilah the wicked woman,\nNor Dido, that woeful one,\nDid Eneas love so,\nAs I do now heartily love you.\nFor in good faith,\nIt will be my death,\nExcept you extend your mercy,\nAnd when this ballad was fully ended,\nMy comfort was never the more amended\nFor no answer at all would she give me,\nYet without any whit playing a while I stayed,\nAt last her father bade me goodnight,\nSo did her mother and her sister rightly,\nAnd gave me hearty thanks for my pain,\nBut no farewell of Grace could I obtain,\nThen home straightway I went full sadly,\nWhere I languished all that night, in terrible pain. And thus it continued for a month, then one of our neighbors came to me, offering of his own free will his only daughter and heir. She would spend the rest of her life in good land after him, and he asked for five marks by the year until her life ended. The maid was also indifferentally fair, and proved to be a good housewife as later appeared. After expressing my thanks for his good will, I then said to him, \"You should know my mind upon the sight of the maid. She pleases me well, and is content to marry me. Shortly after, we were married and have lived happily together ever since. May God grant you the means to live up to this, for it certainly appears that you have a good memory, which can declare the time of your youth so perfectly. For this long tale, I thank you kindly, good father, but for your love and grace, whose chance it was to have her.\" She took a serving man against her friends' will within less than a year,\nBetween them was nothing but quarreling and strife,\nBrawling and fighting all her long life,\nAnd both of them became beggars at last,\nShe was an evil wife, and he spent as quickly,\nSo that they were compelled within one or two years,\nWandering begging both for to go,\nWhich grieved me much when I knew their misery,\nAnd then I relieved them a little for my old love true,\nTHIS I have told you (my son) for this intent,\nBecause your folly you might prevent hereafter,\nAnd not to love one before her manners you should know,\nBut first know her, then love her, and so it will grow,\nTo avoid good purpose, end, and effect,\nAnd all other vain hasty lovers thou shouldst neglect,\nAnd when you mean to choose a wife,\nMy counsel in this case is that you use,\nFirst her mother's manners learn if you can,\nAnd her father also whether he is an honest man. For children often incline to their parents' conditions, as straight as a line. Further inquire of neighbors dwelling thereabout, of what honesty she is by the voice of the route, or if there is any honest woman dwelling near, send her closely thither, both to see and hear, whether she is fair and of body clean, or diseased or sickly by any means. For no man will buy horse, ox, cow, or calf, but he will first be fully sure they shall be sound and safe. Much more circumspect a man ought to be in choice of his wives, with whom he must live all the days of his life. If she is unclean or infected with any kind of disease, all of your children of nature shall have the same sickness. Also cause the same woman diligently to know, if she is foolish and cannot spin nor sew. For I say each woman that is chaste will exercise herself, and waste no time idly. Idleness is the nurse of vices all. And the same causes the mind in much misery to fall,\nThis idleness causes many wealthy fair cities to decay,\nFleshly lust, also idleness ever follows,\nIf either chaste Penelope or fair Lucretia,\nHad spent their time sitting at home in idleness,\nAnd not given themselves to weaving, and spinning,\nIn their husbands' absence from the beginning,\nOf a thousand wars which daily came to them,\nThey would surely have consented to some one man,\nBut touching the search of her life and modesty,\nThese things yourself may much better perceive,\nFor all people now are so unjust,\nThat few shall you find whom you may trust,\nIt is now the manner of many to deceive and lie,\nFew are there to be found of credit worthy,\nTherefore if you will have your purpose take effect,\nIt behooves you to be careful and circumspect,\nAnd son, herein lies all your marrying and making,\nIf you are not wary in your wife taking,\nConsidering no small time you together shall remain. But even though death shall separate the lives of you two,\nIn all other cases rashness and haste is folly,\nBut in this matter may it bring intolerable mystery,\nTherefore consider carefully your wife's choice first,\nSo that you do not repent later when it is beyond remedy,\nBut if it happens that the woman you marry turns out to be a shrew,\nFirst gently warn her and speak lovingly,\nDo your best to chasten her and teach her mildly,\nAnd often win her favor through fair means,\nEmbrace her with kisses to reconcile her,\nThus, if you can tame her through gentleness,\nWhich if it will not in any way recall her,\nThen you must put sharper remedies in use,\nThrough threats, fear, and scolding to bring her to your control,\nWhich if she does not heed or stand in awe,\nThe last help of all is that blows must follow.\n\nConcerning the preservation of your wife's chastity,\nI will not speak much about that truly. But this I say, and I dare undertake,\nThat oft a wise man an honest woman makes,\nI could here tell you more, but I will not now,\nFor I leave the same holy thing to your discretion,\nYet to learn this one lesson, I would have the good son,\nNever let your fancy or desire after any other to run,\nBut in marriage be to her as faithful and true,\nAs the turtle which never will change her make for a new,\nFor there is no one thing she will take more grievously,\nNor anything else will stir her spirits so vehemently,\nNor of nothing does she desire to be avenged,\nAs in that one point if her husband has offended,\nThen her heart is hotly incensed with ire,\nWith disdain and malice then her heart is on fire,\nIn such a way some seek the way then,\nLikewise (if they may) to deceive their good man,\nBelieve me, sons, seeds of them will keep their honesty,\nIf their husbands live abroad in such a wicked way.\nFurther, your children hereafter see that you bring up. In exercise and good learning, teach them to fear God and obey,\nAnd keep them in obedience as much as you may,\nFor they will always grow bold as necessity requires,\nBut not always so humble as you would desire.\nTherefore, while they are young and tender in years,\nIt is both their help and undoing, as it often appears,\nMuch pain you must take in godly instructing them,\nIf you intend they shall ever prove honest men,\nLoose them all evil company to exclude continually,\nFor just he cannot be who is conversant with the ungodly,\nOne sheep having a perilous pocket,\nOf course must infect all the whole flock,\nYour daughters always with shamefastness adorn,\nFor it is the fairest flower all women can wear,\nLet them never be idle but always doing,\nWith the wheel, the distaff, or with the needle sewing,\nFor the welfare of this sex stands in their honesty,\nWhich when they are idle, is then in most jeopardy,\nAnd at their ripe years do not overlook it. Some husbands should not be bitter (good son) to their servants at any time, nor punish them with rigor for every crime. There is a mean to be observed in correction, I say, by which you may cause them to love and obey you. And although fortune has pointed their master to be, he might likewise have made a servant of you. No man knows what may happen to him in the future, for the whole state of man's life depends on balance. I have known diverse men, both rich and wealthy, who afterward have fallen into such misery that they would have gladly avoided vile beggary by serving to live in great calamity. What we are and have been we know, but what we shall be afterward, we are ignorant of such is our uncertainty. In your first keeping of a house, be not extravagant, neither in food nor apparel, for at your pleasure you may the same always amend. When God's abundance of substance sends it hereafter. But it would be to your shame, if you should neglect,\nThe first honest door you encountered on your take,\nOften call home your neighbors, but most such as are poor,\nTo dine and sup with you, let some of them be sure,\nThe Lord your table shall bless the more,\nAnd for such generosity He will increase your store,\nGo not to law with them, nor be no extortioner,\nFinish their causes if you can, and be no bearer,\nIn no man's matter but in all that you can,\nSet quietness and concord between man and man,\nBut in wrangling matters be in no way a mediator,\nWhich might get you an ill name, and no man the better,\nFurther, if you find a man of approved honesty,\nThat fears God and is given to holy virtues,\nOf whom most men speak well, for his good living,\nThat is no drunkard, quarrelsome, nor delights in fighting,\nBut quiet, sober, and learned in wisdom,\nBeing of good judgment, and also of good experience,\nThis man's friendship seek busily to obtain. For there is no greater gain than a faithful friend,\nBut before you earnestly embrace his friendship,\nLearn how he has treated his other friends in similar cases,\nFor such as his behavior has been to others,\nTrust me (son), he will treat you equally,\nAnd once you have won him as an unf feigned friend,\nThen seek to continue the friendship gently,\nLet no slight, displeasure, the same break or decay,\nBut bear with him rather in all that you can,\nOf all treasures the chiefest that God sends on earth,\nIs a man to have always a sure and steadfast friend,\nFurthermore, if sickness shall befall you or me,\nSeek to minister remedy to it at the same time,\nOr ever the same does augment and increase,\nThe sooner it will mend, and the grief will be the less,\nFor just as fire when first begun\nWith a little water will be quenched soon,\nBut if it is allowed to continue still,\nAnd a while to burn even at its own will,\nThen the flame will be raised in such a great rage. That hole well and conducts can scarcely be surpassed,\nEven so every malady at the first entrance,\nMay be easily cured without great pain,\nThis first point of physics learn thou from me,\nIf the sickness is hot, cold, or moist, you must use a contrary remedy,\nIf overmuch labor and toil be the cause,\nThen by ease and rest from the same, you must pause,\nAnd if the same comes from overmuch ease and rest,\nThen exercise and moderate labor is best,\nIf it be by superfluity of drink or food,\nThen abstinence is the best remedy you can get,\nAnd if need requires a physician, then call,\nOr a surgeon, but good diet is the best healer of all,\nThe surgeon is necessary, for physicians kill,\nThe most part who trust them until,\nFor where by chance some one they save,\nA hundred for him they send to their grave.\n\nALSO my son, this last precept thou must learn,\nWhich diligently to observe I the earnestly warn,\nBe prepared always, and evermore ready. Death comes suddenly to embrace,\nYes, even in his most lust and wealthiest time,\nLet the remembrance of him be still before thine eyes,\nHe assaults men commonly when they think of him least,\nFearfully invading them in their most quiet and rest,\nHe draws ever nearer with his inexorable dart,\nDaily perceiving every age of man to the heart,\nHow often does death strike the young lusty man,\nAnd bereaves him of his best years, we see now and then,\nOh, how great are the trials and pains,\nThat a man in this life sustains with pain,\nHow short is our time and the same also so variable,\nThat nothing in this life can be found stable,\nWith what innumerable ills are we besieged,\nWhich by our frail nature can never be resisted,\nWhat are we but wretches, dust of the air,\nAs brittle as glass seems, it never so fair,\nMost like to a shadow in a sunny day. Which, when the clouds suddenly vanish a way,\nLike a flower which blooms in the morning dew,\nAnd at night is withered and has faded its hue,\nThough we are now alive and lovely in every man's sight,\nFair, amiable, pleasant, full of courage and might,\nPerhaps Phoebus has once run his course,\nDead carcasses we may be and vile meat for the worm,\nWhat profits us then our great sums of money,\nHeaped together by extortion and bribery,\nGold, stones, jewels, or implements most precious,\nLands, houses, or villages, be they never so sumptuous,\nEither worship, honor, or lordly authority,\nRule or dominion or worldly dignity,\nWhich makes many men so proudly look over all,\nAs though they were to the goddesses equal,\nSince death ends all things, and we wretches with misery,\nLike dust and shadow, consume so suddenly,\nSeem all our pride and glory is so soon extinguished. And yet our time so quickly passes, never to be recovered.\nOh life, so fleeting and fragile,\nWhom such a number of dangers and perils do assail.\nOh life, which art so short and uncertain,\nMost like unto smoke, a man can only touch,\nNow this man dies today, and another after him,\nI perhaps, and you tomorrow at the right time,\nSo little, and so little, each man does die,\nEven like a butcher having sheep and beasts many,\nOf which some he slaughters down right now,\nAnd others some to morrow in the morning or at night,\nThe nervous day others go to the block,\nAnd so forth until all the whole flock is consumed,\nThus death daily remembers, but fear not the same,\nFor of itself it is good and worthy of no blame,\nDeath finishes all pains, death ends all care,\nDeath dangers dissolve, and put away fear,\nTo the poor, to the prisoner, and to the comfortless,\nTo the condemned, and to the miserable captive in distress,\nTo the bondman, slave, spoiled, and Lazarus impotent. To those who are suffering and in great torment,\nDeath is embraced and heartily welcome,\nTo such and many others in similar state and condition,\nTo good men, death is unexpected,\nThose who have lived in righteousness,\nAnd who have remained in faith, mercy, and godliness,\nTo them, death is no disadvantage but an advantage,\nAlthough it takes them in their most flourishing age,\nIf thou art good, receive death gladly,\nFor it is a passing out of this valley of misery,\nThen shalt thou return to the earth and again send,\nThy body which but for a time she did lend,\nAnd if thou ponder in thy mind discreetly,\nWhat damage or harm can death do to thee,\nShe spoils thee of thy riches perhaps thou wilt say,\nBut then riches are of no use to thee in any way,\nNor any other thing else thy soul shall desire,\nBut from poverty to riches thou seemest to aspire. For he is the richest of all others, not he who has the most but he who has the fewest things needs to leave children and friends behind. It is a miserable thing to do. Yes, but it would be much more miserable to see them dying while living, and it will not be long before they follow you. When the Lord calls for them accordingly, these worldly riches and pleasures are not yours but lent to you as if for a little time. Naked you came here and naked you must go. Therefore, do not sorrow for terrestrial trifles. This world is like a certain great feast, to which every man is bidden as a guest. For a while, during the Lord's pleasure, we have the fruition of this vain worldly treasure. Upon this condition that we are always ready at his commandment to give place and depart gladly, and to suffer other men who come after us to receive the same fruition. What man would not willingly leave this loathsome life. Which is so wicked and disorderly, with no faith, pity, or justice remaining,\nWhere no peace or quietness exists, and all kinds of sins reign,\nWhere the brother is often the brother's bane,\nWhere the son wishes for his father's short life,\nThe wife her husband, and the husband his wife,\nWhere every man deceives and practices fraud,\nThat this world may worthily be named,\nA den of thieves, being inflamed with pillaging and plundering,\nOf lust and boredom, what shall I mention,\nWhere chastity is banished and virginity defaced,\nAnd the honorable sacrament of Matrimony is disregarded,\nIn this wicked world, there are also so many pearls,\nSo many labors and sicknesses,\nWhere fortune alone rules without reason,\nWhere no man can be assured of himself for any length of time,\nWhat good man would not now leave this world gladly. Which is encompassed by such unbearable mystery,\nWhy it is great folly to fear,\nSince it ends the miseries we endure here,\nAnd consider also that departing from mortality,\nBy death you enter into your region of eternity,\nTherefore, my son, let your life be godly,\nSo you will not stand in any fear to die,\nFirst, in your health, keep yourself in good stead,\nBy will, all your worldly affairs as near as you may,\nSo that in your sickness you are not troubled by them,\nNor your mind then from godliness encumbered,\nAnd every night before you go to your rest,\nTo confess to God your sins with heartfelt repentance is best,\nAnd humbly ask mercy with unfeigned hope,\nPrepare yourself then holy to die,\nAnd so your soul most humbly take to the Lord,\nWho on the cross suffered his passion for your sake,\nThus sudden death will not unexpectedly find you,\nIf you bear well this last lesson in your mind,\nThis rough counsel if you follow in every condition. As I trust you will, according to my expectation,\nWalk well in the right pathway then,\nWhich as I first said leads to the life of an honest man,\nAnd now, my former promise being performed and done,\nWe are nearly come to the place where we met,\nI thank you, good father, said I, for your wholesome counsel,\nI have never heard told a more virtuous tale,\nI beseech Almighty God to grant me the grace,\nTo observe in every point and case,\nAnd the same Lord, of His goodness, reward you generously,\nWho has taken such pains to instruct me virtuously,\nFarewell, my good son, said he, Christ be your guide,\nAnd so departing from him, I headed,\nEND.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase2"},
{"content": "Of the triumph and the verses that Charles Emperor and the mighty reduced king of England, Henry VIII, were greeted with, as they passed through London:\n\nThe great triumph, how one man could discern\n(The lusty fresh devices, the sumptuous rich array,\nThe crafty imagery, so like to life,\nWith bright colors shining, fresher than May)\nThat was in London, on the sixteenth day of June,\nWhen the eagle, persing the sun beams,\nEntered with the lion, dread in all realms.\n\nThe pagent, beautifully wrought and of great value,\nSet with devices and made curiously,\nFilled with personages, all of pleasure,\nIn some virgins, richly attired,\nIn other some children, making sweet harmony,\nAnd some with rich arms, dashing full they were,\nWhich shone and lustred wonders clear.\n\nThe reasons and also proverbs, manyfold,\nVery subtly conveyed, at each place,\nOrnately written in, letters all of gold,\nJustly to write should be too long a space. In various places, as you will understand, there was a child who stood alone. He held a roll in his hand, but what he said, few or none knew. Therefore, many have come to me asking what these children meant. For fierce love and great honor, they had composed lusty verses ornately. Ceasar was to salute, and the high conqueror, Henry the Eighth, our king - the only glory of all earthly kings and of chivalry. The flourished Beloved and feared by great and small. Throughout the whole world, what do the verses mean? And until I know, I have no rest. I saw them so desireously: inquire about it. I thought it best to borrow the same verses and, under correction, make a rough translation. So bold I am, of that most human master, called Lily, to translate his fresh verses into our tongue from their ornate Latin. To each state, learned and unlearned, they should be celebrated. First, find them in Latin:\n\nCarolus Henricus vivat. Defensor utriusque\nHenricus Fidei. Carolus Ecclesiae.\n\nThese verses were written in letters of gold and set up at the cross in the marketplace, and they are translated into English as follows:\n\nGod save noble Charles and King Henry,\nAnd grant to them both: good health, life, and long.\nThe one of holy church, defender right mighty. Most mighty Charles, of the land of Germany,\nThe amiable and sweet-smelling flower\nOf the noble and happy people of Spain:\nOf kings' lineage, you are the highest honor,\nWith your prowess, Charles, like a conqueror,\nYou illustrate the entire world,\nMercifully favoring the people of every state.\nCharles, you have come at the world's request,\nThe only hope in every doubtful chance,\nTo cause wealth, peace, and rest in afflictions: Of Europe, the rich and great kingdoms, cities, and towns rejoice greatly to obey you,\nAnd that you should be their lord and captain.\nGod give you grace, long may you reign,\nThat you may with your shield of high justice,\nThe Christian people fortify and sustain\nAgainst false enemies who always devise\nTo invade us with a much cruel guise,\nMoors, Saracens, Turks, people without pity,\nBy your mighty power, subdue us not.\nWhat joy was it to the people of Mynos?\nWhat time the renowned knight Iason\nHad conquered in Colchis the golden fleece:\nWhat joy also was Scipio's triumph for the city,\nWith the enemy vanquished.\nYou, Caesar, mildly, prince among princes,\nEnter the hospitality of Henry, our prince.\n\nWhat great joy was it to the people of Mynos,\nWhen Iason, the renowned knight,\nHad conquered the golden fleece in Colchis,\nWhat joy also was Rome's, when Scipio triumphed,\nWith the enemy vanquished.\nYou, Caesar, mild prince among princes,\nEnter the hospitality of Henry, our prince. Is it coming with pompant King Henry.\nCarole Christigenu,_ decus et que_ scripta loquuntur,\nA magno ductum Carolus habere genus:\nYou Henry, shining with the praise of virtue,\nDoctrina, ingenio, religione, fide.\nVos. Praetor. Consul. sanctus cum plebe Senatus,\nVectos huc fausto sidere gestit. Ouans.\nCharles the clear lamp / of the Christian nation\nIt is spoken plainly in writing\nOf great Charles / to have offspring\nAnd the thou Henry / our sovereign lord and king\nThy great laude of sweet virtue / so brightly shining\nHigher doctrine / wisdom, faith, and religion\nExcell the fortune / of kings alike.\nWith what joy do you abide / for you two princes?\nThe honorable mayor / with the entire senate\nNo place can the gentle citizens endure /\nSo joyful they / of high and low estate:\nHaving their gaze lifted to heaven elevated:\nPraising God / with all their strength and might\nTo behold / so fair and glorious a sight.\nRome, renowned city, praises the magnanimous Catones,\nCantat Ambalem Punica regna suum. The noble city of Rome commends the worthy Catons and Carthage Annibal. Of Solomon, the glory of David, it was taught: Alexander, first of his people, surpassed all. The same of worthy Arthur shall never falter Among the strong Britons, whose like is not found Through out the world in fierce hardiness.\n\nSo you, Charles, you, Cesar, army-mighty,\nShall cause your fame and honor to blow\nOver the whole world, from Cestus to Occident,\nThat all peoples may know your worthiness.\nFor we shall to the high god bend our knees,\nPraying him to send us the high victory,\nThat peace on earth may reign universally.\n\nCarol, who shines with Scepter and sacred Diadem,\nYou, Henry, bearing joined stems,\nAnother light to the Germans, another clear luxury to the Britons. Miscens Hispano sanguine uterque genus. (Mingling Spanish blood, the whole genus.)\nViuite felices, quot uixit secula Nestor. (Live happily, for how long Nestor lived.)\nViuite Cumanae tempora fatidicae. (Live in the prophetic times of Cumae.)\nO Charles, shining with scepter and diadem,\nAnd Henry likewise, the glory of kings,\nThe one of German lineage, the other clear light of Britain,\nTogether bound by Spanish genealogy,\nGod grant you both to live as long, joyfully,\nAs Nestor and Cumae. God grant my request,\nFor then shall reign among us peace and rest.\nOb quorum adventum toties Britannia\nSupplicavit superis votas precibusque dedit,\nQuas omnis aetas, pueri, iuvenes, senes,\nOptaverunt oculis saepe videre suis.\nVenistis tandem, auspicio Christi, Mariae,\nPacis coniuncti federe perpetuo,\nSalute piis heroes, salute beati,\nExhilarant nostros numina uestra lares.\nO princes, how often the people of Britain\nHave made supplication to God for your coming,\nAll ages prayed with heart glad and willing,\nChildren young, old with devotion,\nDesiring entirely with great affection. Your noble persons, behold and see\nUntil that time contented they cannot be.\nAt last you come, conducted by Christ and Mary,\nTogether with perpetual bond of peace.\nHail most pious princes, full of clemency,\nHail mighty kings, blessed and well at ease.\nI pray the living God: that it may please Him,\nYour great virtues, graces, and goodness,\nInto us and ours, may have a large entrance.\nQuanto amplectetur populus te Caesar amore,\nTestantur varijs gaudia mixta sonis.\nAera, tuba, litui, cantus, citharae, calamis.\nConsonant teresonant organa disparibus.\nVnum te celebrant, te unum sic cuncta salutant.\nO decus, o rerum Gloria, Caesar. Aue.\nWith what joy Charles the people embrace\nTheir right great joys openly testify,\nMixed with sweet sounds of many asect,\nSome sounding trumpets and clarions wonders high,\nSome singing most melodiously,\nSome upon lutes, some upon harps play,\nTo rejoice in all that ever they may.\nSome with pipes, make sweet harmony. Some strike Thorgan Kayes, very sweet and shrill. The sound reverberates up to the sky. All celebrate Charles, both loud and still. All and each one Charles does salute thee and will, saying, \"O Worship: o glory of things human. Hail mighty Charles, emperor of Germany.\" This was all that the children said and meant. That stood alone, before as I have said. Wherefore I pray you, be content with it. Each man it knows, I hold myself well paid. Once, now to you, it cannot be denied. For here may you at length, both read and see. So that you need not, demand more of me. Right honorable mayor and prudent senators, of this noble city, the flower of Christendom, you have well shown what is fitting for high honors. To generosity, nobility, and royal sovereignty. In the house of Fame registered shall it be. For certainly, shortly, it shall be sent thither. And there it shall remain, ever without end. Worthy citizens, contented, you cannot be. Only with Juno: but you will also have. The lady Minerva flourishes in your city, plainly and without further words, with good learning and doctrine. You have a great master for this, the flower of Poetry, whose name is Lily. Finis.\nPrinted by Richard Pynson, printer to the king's noble grace.\nBy royal privilege.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase2"},
{"content": "Each of us was worthy to partake in the cruelty of crucifixion. The king/the queen/their children, two, were ordained by God for me, for by his king, his father, his mother, and the mass were also present. Thou shalt receive of thy husband three male children which shall nurse with thine own milk. And God first ordained Godfrey of Bouillon these provisions for some affairs that they had.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase2"},
{"content": "O mighty Father in heaven, one God and three persons,\nWho made both day and night,\nAnd as it was your will,\nYour only Son you sent to us,\nUntil in a maiden's light,\nThe Jews, who were wild,\nHanged him who was so mild,\nAnd to death he was led,\nWhen he was dead, the truth to say,\nHe rose on the third day,\nThrough his own might,\nThen to hell he went alone,\nAnd took out many souls,\nFrom that hold he freed,\nWith them to heaven he went,\nOn his Father's right hand he set him,\nThat all should know without fail,\nThat he was omnipotent.\nAnd after wisdom he was sent,\nThat all should keep his commands,\nAnd believe in him truly,\nHe is our savior,\nBorn of that blessed flower,\nThat I call Mary,\nWho will judge us without error,\nSome to pain and some to bliss,\nIt dreads the day of full doom.\nThe lady commands,\nThat the gates be undone,\nAnd bring them all before me,\nFor they shall be at ease. They took their pages on horse and all. These two men went into the hall. Ipomydon on his knees set him down. And the fair lady greeted him. I am a man from a strange land, I pray you if it is your will, That I might dwell with you this year, Of your nurture to learn. I have come from far land, For I have heard tell beforehand, Of your nurture and your service, Is held of such great empire, I pray you that I may dwell here, Some of your service to learn. The lady beheld Ipomydon, And seemed a gentle man, She knew none such in all her land, So goodly a man and well-born. She saw also by his bearing, He was a man of great valor. She soon cast in her thoughts, That for no service came he naught, But it was worship to her, In her service him to do. Now Ihesu as thou bought us dear, Give them joy this gesture will here, And hearken to a right tale. Some men love to hear tell, Of valiant knights that were fell, And some of bright ladies, And some miracles that are told, And some of venturous knights old. That our lord did fight\nAs Charles did the noble king,\nWho brought the heathen down,\nThrough the help of almighty God.\nHe won from the heathen hounds\nThe spear and nails of Christ's wounds,\nAnd also the crown of thorns,\nAnd many a rich relic more,\nFrom them he won also,\nAnd killed them even and morn.\nThe Turks and the bold Paynims,\nHe brought down many a fold,\nDurst none stand before him.\nCharles terrified them so,\nThat the captives might curse the day,\nAnd the time that they were born.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase2"},
{"content": "The king our sovereign lord, by his singular wisdom, has providently considered and among other princely virtues profoundly remembered two things specifically most beneficial and necessary for noble and virtuous princes, both for the security conservation and maintenance of their royal estates as well as for the political governance of their realms, dominions, and subjects. That is to say, justice and power. The one most expedient to all subjects in times of peace, and the other most necessary in times of war. And forasmuch as justice without power cannot have due course, and is rather a vice than a virtue, therefore his highness among all other his worldly busyness primarily studies to plant and join these two virtues together throughout his realm, so that the fruits of them which is peace, tranquility, wealth, and prosperity may spread over all to the pleasure of God, his high honor and glory. The king is comforted by the rejoicing of all his subjects. Although, from the beginning of his reign, his grace has singularly minded and tended to justice, he has specifically charged and commanded all and singular his principal officers, judges and counselors, to endeavor themselves with all effect to accomplish this his laudable endeavor, as also Saintes Maries used to keep and behave themselves as true servants to his highness, without leaving it to any other, not sparing to do so for any fear or fear of any man whatsoever. For so it is the king's pleasure, and in their lawful demeanors, the king's highness will show favor and mercy to all such and every one of them, whether they have been or are retained, it being the king's highness' great favor and mercy, and upon hope of amendment, can be agreeable to pardon all such as have been or are retained. The king has been disregarded contrary to his laws for past times. However, his grace explicitly declares that, against similar offenders in the future, he will cause his laws and statutes to be enforced without favor or mercy, as a fearful example to all others. The king commands all his servants and subjects in every shire to aid, help, and assist his justices, ministers, and subjects for the advancement of justice in every case. And just as the king intends to have his servants holy to himself, and no manner of his subjects, by unlawful bearing, maintenance, or otherwise, shall meddle with them. So it is the king's pleasure that his said servants, in such discrete order and just dealing, shall order themselves toward his subjects in every shire, without oppression, unlawful maintenance, or embracery, so that their good demenors may be a laudable precedent and virtuous example to all others. Doing this, it is the king's will that all be it. grace is determined strictly to punish all transgressors and offenders of his laws, especially his grace minds more sharply to look upon his own servants, forasmuch as his grace has chosen them to give example of good governance to all others.\nGod save the king.", "creation_year": 1522, "creation_year_earliest": 1522, "creation_year_latest": 1522, "source_dataset": "EEBO", "source_dataset_detailed": "EEBO_Phase2"}
]