THE ELEMENTS OF ARITHMETIC most methodically delivered. Written in Latin by C. VRSTITIUS professor of the Mathematics in the University of Basill. AND Translated by THOMAS HOOD, Doctor in Physic, and well-willer of them which delight in the Mathematical sciences. Obedire novit, nescit servire Virtus. . ANCHORA. SPIT. LONDON Printed by Richard Field. 1596 TO THE RIGHT WORSHIPFUL Sr CONIER CLIFFORD KNIGHT. health and felicity. ULYSSES (Right worshipful) having spent some time in the Court of Aeolus, where he was friendly entertained, desired at length to be dismissed: his suit was granted, and he sent away in Princely manner, with all the winds tied up in a bottle, to be commanded as he thought best. He was not passed a days journey or two from the shore, but he went to wrack through the misdealing of those, to whose safe conduct he had committed himself, and by the winds was driven back again to the coast of Aeolia. His weather beaten men, and himself being refreshed as time & place then served, he set forward to Aeolus his Court, hoping to find such entertainment as he found before. Not daring suddenly to enter in, he rested himself at the Porter his lodge, where he heard these words: 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. It may be they moved his patience a little, because they touched his former estate, yet having better conceit of others in that place he let them pass, and pressed forward to the presence of the King, sitting not far off with his Nobility about him, to whom he spoke thus in humble manner: 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. And that his request might the better prevail, he added his reason, (For it is not likely though his estate were bad, that he would crave any thing more than reason.) 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. But he was so far off from finding his expected relief, that he was commanded to get him thence with these hot words to his cold comfort: 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. Such as Ulysses his departure was from the Island Aeolia, such was mine from London Northward; such as his entertainment was at his return, such was mine coming thither again. For miscarrying in that voyage besides mine expectation (I would say my desert were it not presumption) I found that of Horace to be true: — Diffugiunt cadis, Cum faece siccatis amici, Far jugum pariter dolosi. Some said they could not, some durst not for fear, and some answered they would not help to repair that loss, which the unkind Northern blast had enforced me to. What should I do, or what other thing could I do in this case, but that which Ulysses had done before? 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. I suffered all, and bore it out, not lying still as a man careless, but knowing mine head and hands (under God) to be my best friends, I set them to their old occupation again, teaching the Mathematical arts, and penning, or translating such books as I thought most convenient for that purpose, of the which this book is some little part. And for so much (Right worshipful) as I intended therein not only your pleasure, but your profit especially among the rest, I am bold to make that intent known in dedicating it unto your Worship, & craving your favourable acceptance thereof, which if it please you to afford, you shall invite me to a farther work, and bind me not to be unthankful for your friendly favour. Your worships to command, THOMAS HOOD. THE ELEMENTS OF ARITHMETIC MOST METHODICAL LIE DELIVERED. CHAP. I. The definition and subject of Arithmetic. Arithmetic is the Art of numbering well. This definition is short & plain, being drawn from the end of Arithmetic, after which sort other Arts also are defined. To number well, is to express and practise the whole force, properties, and use of numbers. It appeareth then that the Subject of Arithmetic (that is to say, the thing whereabout that Art is wholly occupied and whereunto all the rules thereof are to be referred) is number. Number is a Multitude consisting of unities. It is the definition of the seventh book of Euclid. Aristotle in his Categories seemeth to define it thus. A number is a discreet quantity not having position but order. The which definition agreeth with that of Euclid, saving that it distinguisheth an unity from a point by a certain contrary quality which is in them; attributing to a point position or place, to an unity only order & consequence. Whereby it appeareth that a number is the joining together of things following one an other in order not touching one the other much less cleaving together, and that in such sort that you can not determine any bond or common knot in them, whereby they should be knit together and continued. The beginning of number in an Unity. An unity is that, whereby every thing that is, is said to be one. An unity properly is no number, because it is no multitude, for multitudes only are numbered: neither is it a part of a number because that every part of a number ought also to be a number. It answereth in proportion to a moment, and to a point: whereof the one is the beginning of time, the other of magnitude, and yet no part of them: as Aristotle proveth in the fifth and in the beginning of the sixth book of Physics: yet in Arithmetical numeration it is wont to be taken for a number, and that the least that may be. The division of number shall be drawn out of numeration. The Schoolmen divide numbers into Digites, articles & mixed numbers: to this end as I think that they might make a distinction in the natural consequence of numbers: wherein first of all the nine simple figures are to be considered, the which the number 10. succeed as it were a knot knitting them to the mixed numbers following. But for so much as this is no lawful division of numbers, we will draw an other out of the last part of Numeration, which we call Division, the use whereof is exceeding great. In the mean time if the former two numbers metaphorically so termed (which have given occasion to the rest used of the Latins who say Arithmeticum pauciores habere digitos, plures autem articulos homine) if that division I say do displease you, you may peradventure more fitly divide numbers into simple, decades and compound numbers. CHAP. II. Of Notation, and of the first part thereof. Simple, consisting of Notation which hath two parts One more general, wherein the figures are considered, By themseluyes, and so there are Nine signifying figure. One without any signification. Or One with another, where there are to be considered, Degrees, which are Simple, containing single numbers. Multiplied containing Ten. hundreds. and Periods, whereof some are Single Compound both being perfect or imperfect. The other more special, wherein the numbers to be considered are either of One term, as whole numbers More terms, and they be either Of one kind, as parts, otherwise called fractions. Of diverse kinds, as mixed numbers. Numerration. A. Compared. E. Arithmetic is double: Simple and compared, Simple is that which teacheth the practice of simple numbers without comparison. This partition springeth of the diversity of the subject. For numbers are considered either by themselves alone absolutely, or else one with an other in respect of their proportionality, or proportion which they have one to an other. Hereon cometh this foresaid division containing all things belonging to this Art. Simple Aruthmeticke consisteth of two parts, Notation and Numeration. Notation is the way haw to set down numbers in writing, and to express them being written. The Greeks' called it Semeiosis, and it is more fitly called Notation then NUmeration, because it teacheth us how to note, that is, to write every number, and to express it when it is written. There be two parts of it. The first expresseth generally the value of the figures wherewith the numbers are written. The figures wherewith numbers are written are ten. As 1. 2. 3. 4. 5. 6. 7. 8. 9 0. Neither the Greeks' nor the Latins used these figures. For the Rom●●nes used only thesE seven letters, M. D. C. L. X. V. I but the Grecians used all the letters of the Alphabet, being marked with certain pricks. Many therefore think that these figures were invented by the Arabians, the which people in old time practised this Art and the whole science of the Mathematics very diligently, where of beside the word Algorithme, whereby they expressed the Art of numbering, many other words left unto us in this kind of knowledge may be a sufficient testimony. For the Latins in the former rude and barbarous age forsaking the wellsprings of the Grecians, followed after the small rivers of the Arabians. Valla and Cardane think them to be invented by the Indians. Some there be that attribute them to the Phoenicians. They have been retained by the common consent almost of all people, because they be most convenient to number withal, and most fit to receive all kind of practice used in this Art. The figures are considered either by themselves alone, or one with another. By themselves alone, when they are without composition, or as it were without composition, for then the first nine have a certain value. The tenth which hath the form of a circle signifieth nothing standing by itself alone, but being placed with the rest, it maketh to the increase of their value. As, 1. is as much as one, 2. two, 3. three, 4. four, 5. five, 6. six, 7. seven, 8. eight, 9 nine, 0. of it own nature is without value, yet having his place wherein none of the other signifying figures is placed, it increaseth the value of them which follow: for it is never set in the last place. It is called commonly a Cipher. Figures are considered one with an other being joined together, and then the process and order of their places increaseth their value. For so much as every number may be increased infinitely, therefore by reason of their divers increase amounting to ten, hundreds, thousands ten thousands, etc. it is unpossible to write every one of them with a divers figure. For than should the variety of the figures be also infinite. But what is wanting in the figures, that their places through often repeating of them, maketh up. In continuance of which places the figures take unto them a divers signification, & that so much the greater by how much the more in joining the numbers together they come nearer to the left hand. The places of numbers are distinguished by degrees and periods. A degree is a place whereby the value of each number is gathered. A degree is of two sorts: the first is of simple numbers, the other of multiplied numbers, to wit, the second degree is of ten, the third of hundreds, and so forth after the same manner in the rest. Numbers according to the custom of all people proceed from the left hand toward the right, so that the least number occupieth the last place next to the right hand. Wherefore the last figure in value is the first, & every figure in the first place keepeth his first and proper signification. In the other places every one is armed with a double power, to wit, with his own and with an other borrowed for a time. For this consideration is generally to be observed in the places: that, that which followeth is twice as much as that which went before. As for example: the figure 4. by itself alone is four: and so much also it signifieth in the first degree. But being removed into the second, it signifieth forty, as you see here, 40. Again, being set in the third degree it is ten times so much as it was in the second: that is, four hundred, as here, 400. And so forth after the same manner both in the other figures and degrees. A period is that which comprehendeth the degrees, and is either perfect, consisting of three degrees, or imperfect containing less than three. To conclude, that Period which consisteth but of one trinity or ternary of degrees is single, but that which consisteth of more than one, is compound. As 1 2 3 is a perfect single period: 1 2 3 4 5 6 is a perfect compound period. Also 1 2 is an unperfect single period: 1 2 3 4, 1 2 3 4 5 is an unperfect compound period. 1 2 3 4 5 6 7 8. is an unperfect triple period. In numbers the degrees are to the periods, as the commata & cola are the periods in an oration. CHAP. III. Of the other part of Notation. THus much concerning Notation in general: now followeth the other part which expresseth the numbers written more particularly. And here we have to deal with numbers, either of one term, or more. Whole numbers are written with one term. As 20, 24, 360, 1234, 1578, they be also called numbers of one band or rank. How whole numbers of a single period should be set down, may be gathered by tha which hath been said before. But that the compound numbers may be expressed, they must with pricks or small lines be divided into single periods, and be so uttered that at the first prick we name a thousand, at the second a thousand thousand, at the third a thousand thousand thousand, and so forth going forward after the same order, till we have numbered the whole. Numbers be they never so great may easily be expressed, if distinguishing them with periods you utter them by parts, in such sort that both the value of the figures and the power of the degrees, may govern them when they be uttered. So that at the end of the first period you shall name a thousand, at the end of the second a thousand thousand (which some call a million) and to be short, at the end of the third period, you shall name a thousand thousand thousand: observing the same increase in all the periods following, be they never so many. Hereupon it cometh to pass that the least signifying figure of the period following, is greater than any of the period going before. Moreover the ready uttering of numbers maketh a man to write them speedily, & contrariwise the speedy writing of them maketh a man to utter them readily. For example, having distinguished this number 34˙567˙890 with pricks you shall express it thus beginning at the left hand. At the first period you shall say thirty four thousand thousand or else thirty four millions. At the second, five hundred sixty seven thousand. At the third eight hundred and ninety. The number following you shall express thus 2˙016˙542˙009˙873. Two thousand thousand thousand thousand: then sixteen thousand thousand thousand, five hundred forty two thousand thousand, 9 thousand eight hundred seventy three. The Romans' were wont to number by hundred thousands. The which fashion if you list to follow you shall set down a prick next to the second degree of the second period & express the other numbers following, as you did the first. As in this number 77˙89˙320, you shall say thus seventy seven hundred four score and nine thousand three hundred and twenty. After this manner Pliny in his second book 108. Chap. affirmeth according to the saying of Artemidorus that the longitude of the earth inhabited West ward from the pillars of Hercules in India, is eighty five hundred seventy and eight thousand miles. The which number must be set down thus. 8578000. Thus much for the notation of numbers of one term: now follow the numbers of many terms. And they be either of one kind as parts (which we commonly call fractions,) or of diverse kinds as mixed numbers. Parts are set down with two terms divided one from an other with a line: whereof the uppermost numbereth the parts, the nethermost nameth them. Although an unity by itself of it own nature can not be divided, yet if we regard the subject thereof, or the magnitude whereto it is applied, it may be divided. And yet the unities of the number proceeding of this division do not arise of the very unity itself, but of the dividing of the magnitude, that is: they come not of the unity as it is by itself alone, but as it is joined to some other thing. As a shilling can not be divided as it is one, but as it is a piece of money, which customably is divided into testons, groats, three pences, two pences, pence and half pence, etc. Wherefore although parts be called by the name of numbers, and are written with the figures belonging to whole numbers: yet in very deed they are no numbers, if Euclides definition of a number be true. But in the notation of these parts there are two things to be considered, the quantity of the parts (for a whole number may have many parts, both great and final) and the number of them. Hereupon it cometh that when they be written they require two terms or numbers: whereof the uppermost declareth the number of the parts the which are to be found out, whereupon it is called the Number, or Numerator. But the nethermost naming the parts of the whole, expresseth their quantity: wherefore it is called the name, or the Denominator. As 1/2 signifieth one part of the whole divided into two halves, that is; one half. 1/3 one third part 3/4 three fourth parts, or three quarters. 5/6 five sixth parts, that is, five parts of the whole divided into six parts. Hereby it appeareth that the greater the Denominator is, the less is the quantity of the parts. Parts are either principal, or of the second sort. The one have their original of the first division of the whole, the other are parcels of the principal parts. They may be called Simple & Derivative parts: whereof the one ariseth of the first division of whole numbers, the other of breaking the parts into smaller parcels. Their notation and value differeth much. The principal parts are written after the manner before named. But the parcels, or the parts of parts called commonly fractions of fractions, are set down by the first toward the left hand, without any line between them for difference sake, as 5/12 1/3 five twelve parts of one third part, 3/4 7/8 three fourth parts, of seven eight parts: or else otherwise by the preposition 5/6 out of 3/10. Moreover, parts are either proper or improper. They are less than the whole: these are either equal unto, or greater than the whole. True and proper parts in deed are less than the whole, and therefore that which they signify can not be expressed by an unity, whereupon they have the numerator always less than the Denominator. But those parts whose Numerator is equal or bigger than the Denominator, are improper parts, for that they may be expressed by an whole or mixed number: as 3/3 are as much as one whole: 4/2 are two whole, 6/5 are one whole, and 1/5 more than the whole. Wherefore although whole numbers be sometime set down as parts, yet when the Numeration is once finished, they are never written after this manner, but must be reduced to whole numbers: for you cannot fitly say, 2/2 two halves, or 4/3 four third parts, or 8/4 eight quarters: but one whole, one whole with a third part, etc. There remain as yet mixed numbers, which are whole numbers with parts. But their notation may be easily gathered out of the former kinds. In Geometry they are not accounted for numbers: but when the measure of a thing cannot be expressed but by a mixed number, it is called irrational, as though it were not indeed to be expressed by a number. Hereupon cometh the name of Surde numbers. CHAP. FOUR Of Numeration, the other part of simple Arithmetic, and of the first kind thereof. Numeration is double. Simple, or of numbers of one kind which is either of Whole numbers, and is either Prime, as Addition. Subduction. Second as Multiplication. Division. Parts. Mixed, or of numbers of diverse kinds. HItherto we have entreated of Notation, now followeth Numeration: which of two numbers given findeth out the third. As the framing of an argument is in Logic, even so is numeration in this art. For as the Logicians in reasoning do infer the conclusion by the premises, even so the Arithmeticians by numbering do infer a diverse number from the numbers given. The numbers given are either of one and the self same kind, as whole numbers, or parts, or of diverse kinds as mixed numbers. Numeration of whole numbers is when the numbers given are whole numbers. And this Numeration is double. Prime: which numbereth one number with an other only once, and that either by adding or subtracting them: whereupon it is called Addition or Subtraction. As if these numbers 360. and 15. were given either to be added or subtracted, you must either add 360. once to 15. or else subtract them once and not many times as in the second numeration. Wherefore this may well be called single numeration, and the other manifold numeration. Addition is a Prime Numeration, which joining one number to an other, findeth out the total of the numbers given. Touching Addition of simple figures the rule of reason teacheth us how to do it, but the addition of compound figures may be learned by this rule. If, proceeding orderly from the first of the numbers given to those which follow, you set the particular sums (being less than ten) of the figures of every degree added together alone by themselves, each one under his own degree, putting the overplus (that is, for every ten, an unity) to the degree following: you shall find out the total of the numbers given. Addition of single numbers may be wrought by the direction of nature only, wherefore it may be practised in our minds, and brought hither for the speedy dispatching of our work. As 5 and 6 are 11, 7 and 8 are 15, 9 and 9 are 18. But the Addition of compound numbers goeth on by piecemeal from part to part, so that it requireth the help of art. The first precept to be observed in this place, is to number all the figures of every place alone by themselves, as though they were but single figures: and this must be done in all the parts of Numeration following. The second is to number the figures of one and the same degree only together, that you confound not the one with the other. Last of all, you must set down the total made of the parts, in such sort, that if they increase to ten or more, only the figure next to the right hand must be written under that degree, the other which remaineth must be added to the degree following. As for example, add 983. to 402. First, setting down the numbers in such sort one under an other, that the figures of every degree may answer may answer one to an other, I must begin at the first and say, 2 and 3 are 5, the which I set down under that degree: then 0 and 8 are 8, the which I writ under the second place. Then 4 and 9 are 13, which number for that it is in the last place, I set it down whole. So that the total of the numbers given is, 1385. The example standeth thus. 983 402 1385 Item add 89647 unto 78450. The numbers being duly set down one under an other, according to the order of their degrees, and a line drawn underneath them, I begin to number at the right hand saying, 0 & 7 are 7. Then 5 and 4 are 9 Item 4 and 6 are 10. But for so much as the total is written with two figures, I set down the cipher only under that degree, and keep the unity in my mind. Then 8 and 9 are 17, and the unity which I kept maketh 18, where again I set down 8 because it is next to the right hand and keep the unity next to the left hand in my mind. Lastly 7 and 8 are 15, and one which I kept are 16, the which I set down wholly. So the total of the numbers given is, 168097. The example is to be set down in this order. 89647 78450 168097 In this part of numeration and in the rest, there are said to be but only two numbers given, although sometime there be more to be added together, for that the two first numbers only are added together, and then if there be more, the total is added to that which followeth, as in the example following. 90641 4790 2853 98284 678921 54086 560 9522 743089 Here, I say, 2 and 6 are 8, 8 and 1 are 9, taking 8, which is the total made of 2 and 6, for one simple number, and adding it to 1 standing above it, etc. Subduction is a prime Numeration, which taking one number from another, findeth out the Remainder of the numbers given. The use of it is to find out the difference between two numbers: whereupon it is requisite that the numbers given, should be unequal, and that from which the subtraction is made, must be the greater. Subduction of simple numbers may be drawn of the table of our mind. But compound numbers are to be subducted by the the Theorem following. If proceeding orderly from the first of the numbers given, you set the remainder of every figure of one & the same degree, subtracted one from the other (either by itself alone, or by borrowing ten of the degree following) severally each one under his own degree: the remainder of the numbers given may be found. In every part of numeration there must a certain practice go before it: the which must not then be to seek when we should work, but it must be drawn out of the treasure of our minds, and ministered unto us. So that in subduction there is required of us a ready foresight, that we may know what number will remain, when any one of the figures is taken from another by itself alone, or else added unto ten. As 5 from 8, there remain 3, 4 from 9 there remain 5. Or by adding ten to the single figure: as 9 from 17, there remain 8, 6 from 15, there remain 9, etc. In compound and great numbers, it is most convenient to place the greater uppermost, and the less beneath, and then to number by the parts going on toward the left hand. For that way is more easy than the other which proceedeth from the left hand to the right, as in the examples following. 857 345 512 58734 9832 48902 300065 1984 298081 In the first example every thing is plain. In the second I begin to number, 2 from 4 remain 2, 3 from 3 remain 0. Then for so much as 8 can not be taken from 7, the figure of the same degree, I borrow an unity of the degree following, and join it to 7 thus 17, and draw 8 out of 17 there remains 9 to be noted under neath. This may be done, if (as some are wont) I take the distance of the neither number from 10. & add it to the uppermost number, or set down the uppermost number only, if the nethermost be 10. As for example, because 8 differeth 2 from 10, I take 2 and add them to 7 which make 9 to be written under the third degree. But for so much as I borrowed 1 of 8 in the uppermost number following, there remain but 7, I draw therefore 9 from 17 (taking an unity from 5 following) there remain 8. In the last degree, because there is no figure beneath which answereth to that above, I set not down the whole 5 but 4, because I borrowed one of it before. In the third example, that the signifying figures may be subducted out of the ciphers, we must borrow of the numbers following an unity, as we did before. First take 4 from 5 there remains 1, then because you cannot take 8 from 6, you must subtract it from 16, there remains 8. For although the figures following, from whence I borrow the unity, be a cipher, and therefore without signification, yet for so much as it hath a signifying figure before it, I may borrow an unity of it. Then I go on and take 9 from 9, there remaineth nothing. Moreover I take 1, not from 10, because there was an unity borrowed of it before, but from 9, there remain 8. And for so much as there is no figure to be drawn out of that which followeth, I set down 9 Likewise I writ down the last figure less by one than it is, because I presuppose that it wanteth that unity which I took from the first cipher. CHAP. V Of the second kind of Numeration. Hitherto of prime Numeration. Now followeth second Numeration, which numbereth one number with an other, so often as one of the two numbers given requireth, and that either by multiplying or dividing: whereupon it is called Multiplication or Division. Second Numeration is either often Addition, as Multiplication: or often Subtraction, as Division. For one of the numbers given is so often either increased or diminished, as the other, either the multiplier or divisor requireth, according to the number of the unities contained in them. Multiplication is a second Numeration, which joining together the multiplicand, so often as there be unities in the Multiplier, bringeth forth the facit. This is the 15. definition of the 7. Book of Euclid. It maketh no matter which number you make the multiplicand, or which number you make the multiplier, as it appeareth by the 16. prop. of the seventh Book of Euclid, which saith: If two numbers multiplied together, the one into the other, produce any numbers, the numbers produced are equal the one to the other: that is, they make one and the same number. Notwithstanding the Schoolmen set the greatest number uppermost for the multiplicand, and the least nethermost for the multiplier, for that it seemeth most convenient for young beginners. Multiplication of single numbers is to be conceived in our minds by this Theorem. If either of the two numbers given be divided into certain parts, the product coming of the whole numbers multiplied together, is equal to the product made by one of the whole numbers and the parts of the other number so divided. The 1. proposition of the second book, or else, if both the numbers given be divided into certain parts, that which is made of the whole number, is equal to the product made of their parts. That we may multiply easily and readily, we must have in our minds, a table whereby we may know what the pruduct of every simple figure is, being multiplied one by an other: the which thing may be easily done by the rule of whole numbers and their parts. Every child by the direction of nature can tell how many twice four, or four times five, or thrice six do make. But if you happen to stick in greater numbers, a little exercise will make this table very ready. As if you would know how much seven times eight is: divide either of the two numbers given, into as many parts as you list, as 7. into 2. 3. 2. Then multiply 8. by every one of these parts, and add the particular productes together, and you shall have 56. How many are eight times 9 divide 9 into 3. 3. 3. and multiply 8. by those parts, adding the products together, so you shall make 72. The same may be done if you divide both the numbers. The examples must be set down thus. I. 8 2. 3. 2. 7 8 3. 3. 3. 9 16 24 16 56 24 24 24 72 II. 4. 4 8 3. 4 7 5 3 8 5 4 9 16 16 12 12 56 12 20 15 25 72 The Schoolmen frame this Table by this Theorem. Two numbers being given which jointly together are more than ten, if you multiply the difference of each of them from ten one by the other, & then subduct cross-ways one of the differences out of one of the numbers given, the product and the remainder parted into diverse degrees, shall be the product of the numbers given. As you see in these examples. Multiplication of compound figures is drawn out of the Theorem following. If proceeding from the first of the numbers given toward the left hand, you multiply the figures of the multiplier, into every one of the multiplicand, and join together the particular products being less than ten, setting them orderly under their multiplier, and put the overplus to the degree following: the total made of the parts, is the product of the numbers given. For example sake, multiply, 365 by 3. Here the numeration must proceed from part to part. Having therefore set down the numbers, as you did in the former kinds of numeration, multiply the nethermost figure severally into every one of the uppermost thus: three times 5 are 15, I set down 5, and keep 1: thrice 6 are 18, and one which I kept, are 19, I set down 9 and keep one to be added to the degree following: thrice 3 are 9, and one are 10, which I set down wholly. The example standeth thus. 365 3 1095 Now take an example of multiplication to be wrought by the parts of both the numbers given. Multiply 1568 by 54: Item multiply 3508476 by 2509. In both these numbers you shall have so many rows of numbers, as there be figures in the multiplier. Wherefore we must take diligent heed, that we confound not the particular products, and beware that in distinguishing the products, we set the first figure of every one of them under his particular multiplier, as you see here. An abridgement of multiplication. Neither are the shortest ways to be neglected: this therefore is an especial abridgement of multiplication. If one, or both the numbers given, have ciphers in the beginning, then if multiplying only the signifying figures together, you put the ciphers to the total, you shall have the product of the numbers given. The corllary of this abridgement may be this. If the last figure of the multiplier be an unity, and the other ciphers, then setting the ciphers before the multiplicand, you shall have the product of the numbers given. The examples. Division is a numeration, which drawing one number from another, as often as may be, findeth out the quotient of the greater. In division there are three numbers to be considered, the Dividende, the Divisor, & the Quotient, the which must be so placed, that the Dividend may stand above, the Divisor beneath, and the Quotient at the side, or between them both. The use of division is to declare how many times the less is contained in the greater. The Artisicers term division by a Geometrical phrase, calling it Comparison, which is the applying of a measure given to any right line, as here, A— ‑— ‑— ‑— ‑— B C— D the line CD, is compared with the line AB to see how many times it may be contained in it. Or it is the joining of one side to an other side, given to make a rightangled parallelogram. As therefore in multiplication, the multiplying of two sides together maketh a right angled figure, if the lines meet perpendicularly, even so in division, the dividing of the area or platform of the right angled figure given by the length as it were by the divisor findeth out the breadth which is represented by the quotient, as you see here. Multiply the side A C, by the side C D, the product will be the area or platform of the right angled figure A B C D. Likewise divide the platform of the right angled figure E F G H, by the side G H which is the length, you shall find out the other side E G which is the breadth which answereth to the quotient in division. Hereupon it is called the breadth of the comparison or of the compared figure. A quotient is a part of the dividend, having the same denomination with the divisor. This definition is taken out of the 39 prop. of the seventh book of Euclid which saith: If a number measure any number, the number measured shall have a part of the same Denomination with the number which measureth it. And contrariwise, as it is in the 40. prop. If it have a part, the number whereof the part taketh his denomination, shall measure it. For Euclid taketh measuring here for dividing: as for example, because 48 may be divided by 6 into 8 the quotient, I say that 8 being the quotient, is the 6 part of 48, and taketh his denomination of the Divisor. And contrariwise, because 8 is the 6 part of 48, therefore 6 divide 48. Likewise if 6 divide 54, the quotient will be 9, and 9 therefore is the 6 part of 54, and because 9 is the sixth part of 54, therefore 6 divide 54, etc. A part of a number is a less number in respect of a greater, when the less measureth the greater. This is the first definition of the 5 and the third def. of the seventh book. Euclid will not have every number that is less than a greater number, to be a part of the greater, but only that which being taken oftentimes by itself alone, measureth the greater, that is, either maketh it, or taketh it clean away: As two is the third part of 6, five is the half part often. For that 2 being three times taken, either maketh or taketh away 6, and 5 twice taken make 10 or destroyeth ten. This kind of part is called commonly, pars metiens, or mensurans. A measuring part and of the barbarous it is called pars aliquota, an aliquot part, that is, a quotient: but the number which by itself alone being oftentimes taken, maketh not a number without the help of some other part or number, is the parts of the whole, using the word parts in the plural number for distinction sake. As 2 are the parts of five, because two make not five alone, without the help of three, or some other numbers. This kind of part they commonly call, pars constituens, or componens: of the barbarous it is called, pars aliquanta. Division of a simple quotient may be taken out of the table of Multiplication. For by the same numbers that a number is made, by the same it may be divided. The first principle to be observed in division is to consider wittily, and to know readily by what number, and what number every one of the single figures do divide another, as six times 7 are 42, therefore 7 divide 42 by 6, and 6 divide it by 7. Item seven times 9 are 63, therefore 9 divide 63, and the quotient is 7, or 7 divide it, and the quotient is 9 The division of a number that hath many figures, is wrought by the Theorem following. If, beginning at the last figure of the numbers given, you multiply the particular quotient of the divisor contained in the dividend (setting it down aside by itself) into the Divisor, and then subtract the product from the dividend, doing this as often as may be, by setting forward the Divisor toward the right hand, till you come to the first figure of the dividend, the quotient of the parts will be the quotient of the numbers given. Although division commonly be wrought by finding out the quotient by multiplication & subduction: yet the principal work wherein the whole force or virtue of division consisteth, is only the finding out of the quotient, which being once found, the division in mind is supposed to be done. But when we have found out the quotient, we are to think and consider with ourselves into how many parts the divisor cutteth the number which is set over it, and how often it may be drawn out of it. That this therefore may be set before our eyes, and that in removing forward the Divisor, the remainder (wherein afterward the quotient must be sought for) may more manifestly appear, the truth of that which we conceived before, is proved by multiplication and subtraction. To express this by examples, divide 4936 by 4, we must seek how often the Divisor 4 being set under the last figure of the dividend, is contained in the same. I say then that 4 is contained in 4 once: wherefore I set down 1 in the quotient, and take 4 out of 4, there remains nothing. Then set I the Divisor one degree forward toward the right hand, where 4 are contained in 9 twice: wherefore I note 2 in the quotient, and multiply the Divisor by it, the product is 8, which being subducted from 9, there remain 1. The figures out of which the subtraction is made, must strait way be blotted out. Again set forward the divisor: there 4 may be taken from 13 three times, wherefore I set 3 in the quotient, and multiply 4 by it, the product is 12, which being drawn out of 13, there remains 1. To conclude, I find 4 in 16 four times, therefore I set down 4 for the quotient, and multiply 4 by it, which being taken out of 16, there is nothing left. The example. Again divide 1008 by 36. In this example there be many things to be taken heed of. First, because the last figure of the dividend is less than the last of the Divisor, therefore I set not 2 under 1, but one place farther toward the right hand under the o. And first I consider with myself, how often three may be had in ten, I find it to be thrice, and there remaineth one. But for so much as I cannot subduct the Product made by multiplying the quotient into the whole divisor, out of the number standing above it, to wit, 108 out of 100, therefore I take a quotient less than that by one, that is 2, whereby the divisor being multiplied, there arise 72, which being taken out of 100, there remain 28 to be written over the head. Then the divisor being set forward, I see three to be contained in 28 nine times, but because I must have regard also of the figure following, I set but 8 in the quotient, which being multiplied into the divisor, make 288 to be drawn out of the number set over it. The example is thus. It appeareth therefore by this, that the question must be made not of the whole Divisor, unless it be a single number, but only of the last figure of the Divisor, and every particular must be but a single figure, as if you should seek how often 2 were in 21, you can not take it ten times, but 9 times, but 9 times at the most. Moreover the nature of the thing requireth that we find out always such a quotient, as being multiplied by the Divisor, maketh no greater a number then the dividend is. See the examples following. An abridgement of division. If the Divisor end in ciphers, the work may be wrought by the signifying figures alone setting the ciphers in the mean time, under the utmost figures of the dividend, next to the right hand. But if the last figure of the Divisor be an unity, and the rest ciphers, then setting the Divisor as before, and taking the figures that have no cyuphers underneath them for the quotient, the division is dispatched. the figures that remain after the division is done, must be set down as the parts are with the divisor underneath them for their denominator. As for example, divide 165968 by 360. Item 6734 by 100: the example shall stand thus. CHAP. VI Of the double division of numbers. Numbers by division are distinguished 2. manner of ways, First they are either Odd. Or Even: which may be divided either, One way only, and are either, Evenly even. Or Oddely even. Many ways, and are both evenly, and oddly even. Secondly, as they be considered, either By themselves alone, and then they are Prime. or Compound. One with an other, and then also they are Prime. or Compound. NVmbers are divided two manner of ways, first into even and odd numbers. An even number is that which may be divided by 2. An odd number is that which can not be divided by 2. This is the 6 & 7 definition of the seventh book of Euclid, whence this difference of numbers drawn out of their division, is taken, as that also which followeth, whose use in Arithmetic is very great. Even numbers are as these following, 2. 4. 6. 8. 10. 12. 14. and so forth after the same order, always omitting one. Odd numbers are, as 3. 5. 7. 9 11. 13. 15. etc. Of even numbers some may be divided but one way only, other some may be divided many ways. Those which may be divided but one way only, are evenly even, or oddly even. A number evenly even, is that which an even number divideth by an even number. This is the eight definition of the seventh, such are 4. 8. 16. 32. 64. etc. that is to say, all the numbers from 2 upward doubled by 2. As appeareth by the 32 prop. of the 9 book. A number oddly even, is that which an odd number measureth by an even number. As it is in the 9 definition of the seventh book. Euclid calleth it an evenly odd number, as 6. 10. 14. 18. 22. For 3 an odd number divideth 6 by 2 an even number: 5 an odd nunder divideth 10. by 2 an even number. Such are all the numbers whose moiety or half is an odd number, as appeareth by the 33 prop. of the seventh book. The numbers which may be divided many ways, are evenly even and odd: which may be divided both by an even number, and by an odd, into an even number. This third kind of numbers, is let pass of Euclid among the definitions of the seventh book, but yet not neglected in the propositions of the ninth book. The examples thereof are these, 12. 20. 24. 28. 56. 144. As 2 an even number divideth 12 by 6 an even number, and 3 an odd number divideth it by 4 an even number. Item 2 an even number divideth 20 into 10 an even number: and 5 an odd number divideth it into 4 an even number. Such are all those numbers which are neither doubled by two from 2 upward, nor have their half an odd number as it is in the 34. prop. of the 9 book. Again, numbers be distinguished otherways being considered both by themselves, and one with another. Being considered by themselves, they are either prime, which may be divided by an unity only: or compound, which may be divided by some other number. This is the 12 and 13 definition of the seventh book, where Euclid divideth numbers into prime & compound numbers. The matter of this division is taken out of the 34 prop of the seventh book, which saith: That every number is either a prime number, or else divided by a prime number: that is, a compound number, as appeareth by the 33 prop. of the same book. A prime number is that which no other number divideth beside an unity, saving that it measureth itself. It may be called an uncompound number for that it is made of no number: as 2. 3. 5. 7. 11. 13. etc. A compound number is that which some other number maketh, being taken certain times, as 4. 6. 10. 12. Of this sort are first of all, all even numbers, than all those odd numbers which in the eleventh definition of the seventh book are called oddly odd: which may be divided by an odd number into an odd number, as 9 15. 21. 25. 27. etc. Numbers compared one with an other, some are Prime in respect one of an other, which can not commonly be divided by any other number, but by an unity. Other some are compound in respect one of an other, which may be commonly divided by one or more numbers. This difference of numbers compared together, is drawn out of the former, and is set down in the twelfth and eleventh definition of the seventh book. Numbers prime in respect one of another, have no common divisor beside an unity, which measureth all numbers: As 3 and 8, 6 and 7, 5 and 12. Therefore when the numbers given are prime one to an other, there can no less numbers be given in the same proportions. So that prime numbers are the least, and the least are prime numbers, as appeareth in the twenty and twenty four proposition of the seventh book. And therefore, when the numbers given are prime, they are also the terms of the proportion given, as we shall see afterward. Compound numbers in respect one of an other, may commonly be divided by one, or many numbers, as 3 and 9, 9 and 15: for 3 is the common Divisor of them both. Item 12, 18, 24, may be commonly divided not only by 6, but also by 3 and 2. If you would know whether the numbers given, be prime or compound one to an other, you may do it by the Theorem following. Two unequal numbers being given, if in subducting the less from the greater as often as may be, the remainder divideth not that which went before it, until it come to an unity, the numbers given are prime one to an other. 1. prop. 7. As in 4 and 11, 18 and 7, 35 and 12, in subducting the one from the other continually, you shall come to an unity: wherefore I say, that they be prime in respect one of an other. And hereby we may easily conclude, if after the continual subducting of the one from the other, we come not down to an unity, but meet with some one number dividing that which went before: the numbers given are compound in respect one of an other. As in 14 and 6, 30 and 18, 49 and 14. An abridgement of the former work. For so much as division is nothing but an often subduction, if you divide the greater number given by the less, you may do the former work more readily. Yet in dividing the numbers after this manner, you must have no regard to the quotient, but compare the Divisor with the remainder. As I prove that 234 and 17 are prime numbers one to the other, by division thus. In compound numbers take these for example 144 and 27. CHAP. VII. Of the greatest common Divisor, and the least common dividend. NVmbers are divided after the manner before named. Moreover out of the difference of numbers compared together, there ariseth a double invention: one is the finding out of the greatest common divisor, the other of the least common dividend. For so much as compound numbers may be divided often times by many divisors, the drift of the first invention is out of many to choose the greatest number, which may be the common divisor, or as Euclid termeth it the common measure of them all. The numbers therefore given Are either Two only, and no more, as in the first Rule. I. Three or more, & in them we meet with the greatest common divisor, either. At the first. II. Or at the second time. III. The finding out of the greatest common divisor is to be learned by the Theorem following. I. If after the continual subduction of two numbers given, the one from the other, as often as may be (which may more briefly be done by division) there remain some one number which will exactly divide them both: The number remaining shall be their greatest common divisor. As in 76 and 20, their greatest common divisor is 4, as appeareth by this. Item of these compound numbers 63 and 4, their greatest common divisor is 7. II. When there are three numbers given, if the greatest common divisor of the two foremost, being compared with the third, do divide it also, that divisor shall be the greatest common divisor of them all: and so forth in the rest, be they never so many. 3 prop. 7. As 14. 21. 35. 63. Compound numbers. 7 The greatest common divisor. 2. 3. 5. 9 The quotients of the numbers, being prime numbers one to an other. Here 7 being the greatest common divisor of the two for most 14 and 21, compared with the third number 35, divideth it also and in like manner the fourth 63: wherefore it is the greatest common divisor of them all. III. But when the greatest common divisor of the foremost, divideth not those which follow, then comparing the first divisor, found out with the third number, the greatest common divisor of the numbers compared together shall be the common divisor of the numbers given: and so forth in the rest be they never so many 3 prop. 7. As between 18. 12. 9 The greatest common divisor between 18 & 12, is 6, the which number for so much as it cannot divide 9, compare 9 and 6 together by themselves, and then because the greatest common divisor of 6 and 9 is 3, it shall be also the greatest common Divisor of all the numbers given. The example is thus set down. 18 12 9 The numbers given. 6 The numbers compared. 3 The greatest Divisor. See also this example following of four compound numbers in respect one of an other, wherein there is a double comparison made, and the greatest common Divisor divideth itself also. 8 16 28 26 The numbers given. 8 The comparisons. 4 2 The greatest common divisor. The greatest common divisor is found out as is aforesaid. Now followeth the finding out of the least number that may commonly be divided by the numbers given: that is, such a number, than which there cannot be a less, which two or more numbers given may exactly divide. Euclid calleth it the least that may be divided by certain numbers. For as many numbers given may make many compound numbers, even so also they may divide them, out of which numbers to choose the least that may be divided, is greatly available to the readiness and easiness of numbering. The numbers whose least common dividend we seek for, are either Only two alone, & those Prime one to another. I. or Compound. II. Many, wherein we find our the least number that may be divided. At the first. III. Or At the second time. FOUR The finding out of the least number that may be commonly divided by certain numbers given, is learned by the Theorems following. I. If the two numbers given, be prime one to an other, their product is the least number that may be commonly divided by them. As 15 the product of 3 and 5, being prime numbers one to an other, is their least common dividend. Item 63 is the least number commonly divided by 9 and 7 II. But if the two numbers given, be compound one to an other, if the quotient of the one, found out by the greatest common divisor, multiply the other, the product shall be the least number that may be divided by them two. 36 prop. 7. As for example, I divide 9 and 12, being compound one to another by 3, their greatest common divisor, the quotient will be 3 and 4. Then if I multiply either 9 by 4, or 12 by 3 crossways, the product 36 shall be the least number that may be commonly divided by 9 and 12, the numbers given: 9 12 3 3 4 36 15 21 3 5 7 105 III. But when there be more than two numbers given, if the third divideth that which the two former divided, the first numbers found out shall be the least that may commonly be divided by them all. As these three numbers 6. 10. 15. compound one to an other being given, because 30 being the least number to be divided by 6 and 10, may also be divided by 15, which is the third number, therefore it is the least that may be divided by them all. Likewise in 4. 7. 28, because 28 being the least number that may commonly be divided by 4 and 7, it may also be divided by the third number 28, as by itself, it shall be the least number that may be divided by the numbers given. As in these examples. 6 10 15 2 3 5 30 30 4 7 28 28 28 FOUR But if the third number doth not divide that number which the former numbers did divide, then compare the least dividend found out, with the third number: the number which may commonly be divided by the numbers thus compared together, shall be the least that may be divided by them all: and so forth in the rest. 38 prop. 7. As in 4, 8, 12, the least dividend is 24. Item in 6, 16, 28, the least common dividend is, 336, as you see. 4 8 12 4 1 2 8 4 2 3 24 6 16 28 2 3 8 48 4 12 7 336 So, the least number that may be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 30, is 60, whereupon it cometh to pass, that this number hath so greatly pleased the Astronomers in numbering their parts. 1 2 3 4 5 6 10 12 15 30 The number given. 2 The numbers compared. 6 2 3 2 12 60 6 10 1 60 10 6 1 60 12 5 1 60 15 4 1 60 30 2 1 60 The least common dividend. CHAP. VIII. Of the Accidental numeration of parts which commonly they call Reduction. The numeration of parts is Accidental: and is called Reduction, which is of parts unto parts either of Least terms. One denomination. Unto whole numbers, which is either of Proper parts. Improper parts. Or Essential, which is Prime, as Addition. Subduction. Second, as Multiplication. Division. HItherto we have spoken of the numeration of whole numbers: now followeth the numeration of parts, wherein the numbers given are only parts. And it is also double, Accident all and Essential. Accidental numeration is that which by reduction changeth the form and fashion of the parts, changing their value nothing at all. This numeration in one word is commonly called reduction, where the parts are so ordered, that they only take unto them another shape, such an one as is fittest for the work ensuing. Reduction is double: the one is of parts unto parts, the other of parts unto whole numbers. Reduction of parts unto parts, is that which findeth out other parts proportionable to the parts given, by reducing them either to less terms or to terms of one denomination. Reduction of parts to less terms, is that which divideth the terms of the parts given (being compound numbers one to another) by their greatest common divisor: and taketh their quotients in stead of their numbers given. Seeing that parts have their value only of the proportion that is between the upper number and the nethermost, and every proportion is best known when it is set down in least terms: therefore in the parts also it shall not be amiss to reduce them which are written in greater terms (that is, with numbers compound one to an other) or lesser terms, the which thing is done by the common divisor. For the quotients have the same proportion that the numbers divided have, as appeareth by the 35 prop. 7. So that their value is not changed. The proof also of this may be taken out of the 17 prop. 7. which saith, If a number multiply two other numbers, the products shall have the same proportion that the multiplyers have. Contrariwise therefore if a number divide other two numbers, the quotients and the numbers divided shall have one proportion the one to the other. As if 4 multiply 6 and 8, their products 24 and 32 shall have the same proportion which the numbers given have. Again, if you divide these products 24 and 32 by 4, the same numbers to wit, 6 and 8 will return again in the quotient. If the parts given therefore be 6/21 they must not be set down in these terms, but must be reduced strait ways to 2/7. Likewise 4/16 divided by 4, come to 1/4. So 108/252 may be reduced by 36 their greatest common divisor to 3/7. By this kind of Reduction not only the terms of the same parts, but also the terms of diverse parts may be reduced crossways one by an other: that is, the numerator of the one, and the denominator of the other. As 2/3 and 5/8 their terms being reduced cross-ways will be 2/3 5/4. But this rule concerning the Reduction of diverse parts, is not so general as the other of one and the same parts. For it is more special and peculiar, as it were to multiplication, as we shall see hereafter, wherefore we must not think that all such parts given may be reduced after this manner. Reduction of parts to one denomination is that, which (when the parts given are of diverse denomitations) taketh first the least number which may be divided by the denominators, for the common Denominator, and then dividing that number by the denominators of the parts given, setteth down the products made by the quotients, & by the numerators of the parts given for the numerators of the partswhich we seek for. It is a thing of great use to reduce the parts given to one denomination, and to make them proportionable one to an other in the least terms that may be: that is; of parts of diverse kinds, to make parts of one and the same kind. As for example if 3/4 and 5/6 were to be brought to one denomination, I must first seek out the least number that may be divided by 4 and 6, which is 12, than I divide 12 by their denominators 4 and 6, and multiply 3 and 5 the numerators of the parts given by the quotients 3 and 2, the products 9 and 10 shall be the numerators of the parts sought out: and the least number which might be divided by the denominators: to wit, 12, shall be the common denominator to them both. As in this example. See also the examples following. In the first example therefore, the parts of one denomination are 9/12 10/12, and they be likewise the least proportionable to the parts given. For as in 3/4 4 is to 12, so is 3 to 9: and in 5/6 as 6 is to 12, so is 5 to 10, and so forth in the other examples. By this rule of reduction we may know whether the parts being of divers denominations be equal or unequal. For being brought to one denomination, if they have the same numerators they be equal: if not, they be unequal: they that have the greatest numerator are the greatest, and they that have the least numerator are the less. As if it should fall into question what proportion there were between 5/6 and 10/12. Item between 5/7 and 13/21, whether they be equal or unequal, this rule will take away the doubt: and will declare that the former parts are equal, the latter unequal, and 5/7 to be the greater, 13/21 the lesser, as you see here. To this reduction of parts to parts, succeed the reduction of parts to whole numbers, and it is either of proper or improper parts. The first reduction of proper parts taketh the numerators of many parts, having one denomination for the parts given. As 3/5 and 4/5 are reduced to whole numbers when the numerators of the parts given 3 and 4, being as it were proportionable to the parts given, are taken for the parts themselves: so likewise we reduce these parts 1/9 4/9 7/9 to whole numbers, when we take 1, 4, 7, the numerators for the parts themselves. Reduction of improper parts taketh the quotient of the numerator divided by the denominator, for the parts given. This reduction is used when the parts are but one, and their value either equal or greater than the whole, as 3/3 10/5 27/9. If you divide the numerators of every part severally by his denominator, the quotients will be 1, 2, 3. Hitherto also may those parts be referred whose denominators do not exactly divide the numerators, as 36/7 are reduced to 5 1/7 and 29/6 to 4 5/6. CHAP. IX. Of the Essential numeration of parts. WE have spoken of the Accident all numeration of parts. Now followeth the Essential numeration, which numbereth the parts either by increasing, or diminishing them. There be two kinds of Essential numeration. The first giveth the same denominator to the numerator found out, which the parts have that be given: and it is either Addition or Subduction. Addition addeth together the numerators having one denomination, and setteth the common denominator of the parts given under the total. Parts of diverse denominations the one being greater or less than the other, can not be added together without confounding of the denominators, wherefore it is needful to reduce them to one denomination. As if you add 3/5 to 4/5 the total will be 7/5, or by reduction 1 2/5. Item add 3/4 to 5/8, here if you should confound 4 with 8, the total should have his denominator of neither of them both. Wherefore the parts must first be brought to one denomination, and then be added together. As in this example. The Reduction. The Addition. The total. Likewise in adding 1/2 unto 5/6 and 3/11 there arise 106/66, or by reduction, 1 20/33. The example is thus. Reduction. Addition. The parts found out. Subduction is a numeration which subducting the numerators of the parts given, having one denomination, one from the other, taketh the common denominator of the parts given, for the denominator of the remainder. As in subducting 3/8 from 7/8 there remains 4/8 or 1/2. Item subduct 1/3 from 10/13. Here for so much as the parts be of diverse denominations, you must reduce them to 13/39 and 30/39, then subducting the numerators one from another, that is, 13 from 30, there remains 17, under which you must set the common denominator, so that the parts found out are 17/39. As in this example. Thus much of the first kind of the Essential numeration of parts: the second kind giveth to the numerator found out an other denominator than the parts have which are given. And it is either Multiplication or Division. Multiplication is that which multiplying the terms of one kind together (that is the numerators by the numerators, and the denominators by the denominators) taketh the products for the terms of the parts found out. This multiplication is properly the finding out of other parts than are given, which are to one of the parts given, as the other is to an unity: and it is wrought by multiplying both the numerators and the denominators of the parts given one by an other, whereby we may necessarily conclude, that the parts found out are always less than the parts multiplied. As if there were 1/2 to be multiplied by 1/3, multiplying the numerators and the denominators one by the other, the product is 1/6, which is to 1/3, as 1/2 is to 1. For as 1/2 is the half of one, so 1/6 is the half of 1/3, wherefore 1/6 is less than 1/3 or 1/2. This perhaps may be perceived more easily in the numbering of some certain thing. As for example, in a degree there be 60 minutes, therefore 1/2 of a degree are 30 minutes, 1/3 20 minutes, 1/6 10 minutes: as therefore 60 are to 30, so are 20 to 10. Again in a shilling there are twelve pence, therefore 1/2 of a shilling are six pence, 1/3 four pence, 1/6 2 pence: as therefore a shilling is to six pence, so are four pence to two pence. Item if 5/7 be multiplied by 3/4 the product will be 15/28. The abridgement of multiplication. If the opposite terms of the parts given be numbers compound one to an other, then in their stead take their quotients found out by their greatest common divisor: and then work the multiplication as before. As, multiply 3/4 by 5/18. Here for so much as 3 and 18 being opposite terms, are compound one to another, you shall set down in their stead their quotients 1 and 6 found out by their common Divisor, so that the parts thus reduced shall be 1/4 and 5/6. Now multiplying 1 by 5, and 4 by 6, the product will be 5/24. Likewise if 3/7 should be multiplied by 14/15, first you must reduce them to 1/1 and 2/5, which being multiplied one by the other, make 2/5. As in these examples. So if 3/10 5/6 and 1/4 be multiplied together, they give 1/16. For first 3/10 and 5/6 being multiplied together, do make 1/4, which being afterward multiplied with 1/4 make 1/16. Thus: Correlarie gathered of the former abridgement. Of this reduction of numbers this consequence is gathered. If the cross or opposite terms of many parts be equal one to the other, then neglecting them which are equal, the numerator remaining set over the denominator remaining, is the product of them all. As if 4/5 5/6 6/7 7/8, were to be multiplied together, I cast away the middle numbers which are equal, and see 4 the numerator over 8 the denominator, so that the product is 4/8 or by reduction 1/2. The reason hereof is this, because the cross terms being as it were compound one to the other, are by reduction to one denominator brought to unities thus, 4/1 1/1 1/1 1/8. But for so much as an unity in multiplication changeth not a number, therefore the figure 4 multiplied by the three unities in the same rank, maketh but 4, no more doth 8. Moreover it appeareth by this proportion which is in multiplication, that second parts may well be reduced to principal parts by this means, as 1/3 out of 3/4 are 3/12 or 1/4 of the whole, which thing may be easily perceived in the numbering of any thing, as 3/4 of a degree are 45 minutes, whose third part are 15 minutes, which are equal to 1/4 of a degree. So 2/3 of ●/4 parts of a crown are half a crown. For 3/4 are three shillings nine pence, whose 2/3 parts are two shillings six pence. So 5/6 1/2 of 3/8 by multiplication are brought to 15/96 or 5/32. And thus much of multiplication of parts. Division is that which in stead of the parts given divideth whole numbers proportionable unto them. The division of parts is properly the finding out of a quotient which is in proportion to the dividend, as an unity is to the divisor. Whereby it is manifest, that the quotient must needs be greater than the parts which are divided, otherwise than it is in division of whole numbers: for there the quotient is always less than the dividend. This invention cometh of the division of such numbers as are proportionable to the parts given, for so much as the parts themselves can not be divided. First therefore divide these parts of one denomination, to wit, 9/10 by 3/10. Here therefore you shall reduce them (by the former reduction to whole numbers) unto 9 and 3, then shall you divide 9 by 3, the quotient will be 3, which declareth that 3/10 the divisor is contained in 9/10, the dividend three times, and as 3/10 are to one, so are 9/10 to three: the which quotient therefore is bigger than the parts divided. Divided 5/6 by 3/6, that is, see how many times 3/6 are contained in 5/6. First you shall reduce the parts given to whole numbers proportionable unto them, that is, to 5 and 3, then shall you divide 5 by 3, the quotient will be 1 2/3. Contrariwise, 3/6 divided by 5/6 give in the quotient 3/5. For both the denominators being neglected, I set the one numerator over the other as parts, because the one cannot be divided by the other. Furthermore if you divide parts of diverse denominations, as 3/4 by 1/3, you shall first make the parts of one denomination, reducing them to 9/12 and 4/12: then by the former reduction you shall bring them to whole numbers, and divide 9 by 4, the quotient will be 2 1/4. Divide 8/9 by 5/6, reduce the parts to one denomination, that is, to 16/18 and 13/18, then reduce these parts having one denomination to whole numbers 16 and 15, which being divided one by another, make in their quotient 1 1/15. CHAP. X. Of mixed numeration. Mixed numeration is either Accidental, and is reduction of Whole numbers, Or Mixed numbers. Essential, whereof there are two kinds Prime, as Addition. Subduction. Second, as Multiplication. Division. Hitherto of simple numeration. It remaineth now to speak of mixed numeration wherein the numbers given are whole numbers and parts together. And this numeration also as that of the parts is double. First Accident all, which beside the former reduction consisteth of the reduction both of whole and mixed numbers unto parts. Reduction of whole numbers unto parts taketh the product made of the whole numbers given, & the denominator of the parts given, for the numerator of the parts which we seek for. This reduction may be drawn out of the ●●●uction of improper parts, as it were a consequence following it. For as you reduce 4/4 by division unto 1 whole, so contrariwise you shall bring one whole to 4 quarters, if you multiply one whole by 4 which is the denominator of the parts, for they will return to 4/46 Item, as 30/5 by division are reduced to 6 an whole number: so if you multiply 6 by 5 the denominator of the parts, and take the product 30 for the numerator, and 5 for the denominator, there will be again 30/5 which are equal to the whole numbers given. Thus whole numbers as often as the use of numeration requireth, are written as parts, by setting an unity underneath them. As for example these whole numbers, 2, 5, 9, having an unity set underneath them for their denominator shall be as it were parts after this manner, 259/111 Reduction of mixed numbers taketh the product made of the whole numbers given, and the denominator of the parts given being added to the numerator, for the numerator of the parts found out. As you may reduce 4 3/7 into parts, if you multiply 4 by 7, the denominator of the parts given, and add 3 the numerator, to 28 the product, taking the total 31 for the numerator, and setting the denominator given underneath it, thus, 31/7. Likewise reducing 6 3/4, there come forth 27/4, and so forth in the rest. Thus much concerning the accidental numeration of mixed numbers: now remain the special kinds of numeration, whereof the first numbereth the terms of mixed numbers severally by themselves alone: that is, the whole numbers by themselves, and the parts by themselves, and that by adding or subducting them, whereupon it is called addition and subduction. These kinds of numeration differ nothing from the former, but are in proportion all together like unto them. The examples of Addition. I. Add 3 and 2 4/5 the total will be 5 4/5. II. Item add 3/4 to 6 6/7, first the parts added together are 45/28, or by reduction 1 17/28, the total therefore is 7 17/28. III. Add 8 7/9 to 4 1/2, first the parts added together are 23/18, or by reduction 1 5/18, then add, 1 to 4 make 5, which being added to 8, are 13. So that the total is 13 5/18. The examples of Subduction. I. Take from 3 the whole number 5/9, here I reduce an unity of the whole number to 9/9, and there remains 2. Afterward I subduct 5/9 out of 9/9, there remain 4/9. The remainder therefore of the numbers given to be numbered is, 2 4/9. II. Item, take 1/4 from 8 1/5. Here because 1/4 cannot be taken out of 1/5, to wit, the greater from the less: I take 1 from 8, there remain 7, than I bring 1 and 1/5 by the reduction of mixed numbers unto 6/5, from whence (after I have brought the parts to one denomination) I take 1/4, there remain 19/10. So that the remainder of the numbers given is, 7 19/20. III. Take 4 from 10 1/2, the remainder will be 6 1/2. FOUR Take 5 7/12 from 7 5/8. Here first you shall make the parts of one denomination, that is, 15/24 14/24. Then taking 5 from 7 there remain 2, then also subducting the parts having one denomination one from the other, there remains 1/24. The remainder therefore of the parts given is, 2 1/24. The latter kind of mixed numeration numbereth the terms of the mixed numbers given jointly together: that is, the whole numbers with the parts, and that by multiplying or dividing them: whereupon it is called Multiplication or Division. If 3/4 were to be multiplied by 5, setting down the whole numbers after the manner of parts, I multiply 5/1 by 3/4, the product is 15/4 or by reduction 3 3/4. The value of the parts is sought out after this manner. As if you would know what 5/8 of a circle were, I set down the whole parts of a circle were, I set down the whole parts of a circle, that is, 360 degrees after the common order of parts, thus, 360/1: then multiplying the numbers given together, the product is 1800/8, or by reduction unto whole numbers, 225. So many degrees are 5/8 of a circle. So likewise if you would have 1/7 of 10 ●/8, first you shall reduce the last of the two numbers given by the reduction of mixed numbers into 83/8, which being multiplied by 1/7, make 83/56, or by reduction 1 27/56. Likewise if both the numbers given be mixed, as 4 1/3 and 5 3/11, the product made by multiplication will be 754/33, or by the reduction of improper parts, 22 28/33. The examples of Division. If 4 were to be divided by 7/9, you must find out whole numbers proportionable to the numbers given: for the whole 4 being reduced into 9, the parts will be 36/9 and 7/9, whereby I see that the whole numbers 36 and 7, are proportionable unto the numbers given: and they being divided one by the other, make in the quotient 5 1/7. Item if 1/2 be divided by the whole number 3, 3 being reduced unto 6/2, I find the whole numbers 1 and 6 to be proportionable to the numbers given: which being divided the one by the other, make in the quotient 1/6. Item, divide 8 2/3 by 4 3/5: here reduce the terms of both numbers by the reduction of mixed numbers to 26/3 and 23/5, then make them of one denomination thus, 130/15 69/15, afterwards reduce them to whole numbers proportionable unto them, that is, to 130 and 69, which being divided one by the other, the quotient shall be 1 61/69. The end of the first part. THE SECOND PART OF ARITHMETIC. CHAP. XI. Of the kinds of proportionality or reason. E. The second part of Arithmetic consisteth in the comparison. Either of the terms, and is called proportionality or reason, whereof The one is in equal terms: and is called the proportionality or reason of equality. The other is in unequal terms: & is called the proportionality or reason of inequality, and is to be considered Either according to the difference of the terms. Or according to their quotient: and herein we are to note, The kinds: for of the proportionality of inequality some is, Of the greater inequality, & that is Either Prime, as Multiplex. or Simple, which is Muitiplex Superparticular. Or Multiplex Superpartiens. Of the lesser inequality, which is to be divided into as many kinds as the former is divided into. Or The handling of the kinds wherein we are to considertheir Notation. Numeration, which is, Addition. Or Subduction. Or of the proportionalities or reasons. F. HIther to of Simple, now followeth Compare Arithmetic, which consisteth in repressing the nature and the quality of terms compared together. In writing of proportions the numbers whereby they are expressed, are by a Geometrical phrase called terms: by which word are to be understood such quantities as are compared together. Hereupon generally such things as are compared together are called Terms. The comparing of terms together, is called Proportion, which is a certain respect which the terms have one to another according to their quantity. This definition in this place is more general than the 3 def. of the 5 book of Euclid, which faith, that Proportion is a certain respect of two magnitudes of one kind only, according to their quantity. For in that place it was requisite that there should be mention made of magnitudes of one kind only, because that magnitudes are not all of one kind. But here those words may be omitted, because all numbers are of one kind, & every one may be compared together: and moreover the analogy or respect which they have one to another, may be expressed by an other number, which thing cannot be done in all magnitudes, though they be all of one kind itrationall, and surde magnitudes can not be expressed by number. The comparison of terms, is either equal or unequal, whereupon it is called Proportion of equality or unequality. Proportion of equality is one only, and not divided into many kinds. As the proportion of 5 to 5, of 6 to 6, of 12 to 12. For both in these, and in all other equal terms, the respect is all one, neither hath it any more kinds, because that one and the same number cannot be either more or less equal to another. Proportion of inequality is known either by the difference of the terms, or by their quotient. The proportion of difference, is that whereby one term differeth from another. The proportion of the quotient is when one term containeth another. The proportion of difference is known by subduction, the proportion of the quotient is known by division. As in the natural order of the numbers the difference between 1 and 2, 2 and 3, 3 and 4, 4 and 5, 5 and 6, etc. is all one, that is an unity. Again, the difference between 8 and 5 is 3, between 6 and 4 it is 2. But the proportion of the quotient is by an especial phrase simply called proportion. As, the proportion of 2 to 1 is double, of 3 to 2 sesquialter, of 24 to 3 octuple. For the former terms (which commonly are called the antecedents) being divided by the terms following (which are called the consequents) the quotients are 1, 1 1/2, 8: whereby we may gather that there may be now and then the same difference between the terms though as touching the quotient their proportion differ, as in 4 8 and 12, the difference between 4 and 8, and between 8 and 12, is all one, yet the proportion gathered by their quotients is diverse, for in the one it is 2, double, in the other it is but sesquialter 1 1/2. Contrariwise also in terms the proportion of the quotient may be all one, though their difference be diverse: as in 3 and 9, 4 and 12, the quotient is 3 triple in them both: yet the difference is not a like, for in the former it is 6, in the second it is 8. Promportion of inequality is double, for either the greater term is antecedent to the less, or consequent. Hereupon the one is called the proportion of the greater inequality, the other the proportion of the lesser inequality. When two terms are compared together, the one is called the Antecedent, the other the consequent. Therefore, as often as the greater term is the antecedent and goeth before, it is called the greater inequality. As when 6 is compared to 3, and 3 to 2. But if the lesser term be the Antecedent, and we consider what difference there is between that and the greater, it is called the lesser inequality. Both these forenamed proportions are either Prime or second. Prime, is a proportion of one kind only, and is either simple or multiplex. Simple proportion, is when the greater term containeth the less once and somewhat more, as proportion Superparticular and Superpartiens. Superparticular, is a simple proportion wherein the great●●● term containeth the less, and one part more of the less. As when 3 is compared to 2, 6 to 5, 8 to 7. For in these proportions the antecedent containeth the consequent once, and one part beside of the lesser term. As it may be perceived by their quotients when they be divided thus, 1 1/2, 1 1/5, 1 1/7. This kind containeth under it infinite other kinds, whereof the first is sesquialter, the second sesquitertia, the fourth sesquiquarta. Item there is the same proportion between 12 and 8, that is between 15 and 10, as appeareth by their terms reduced to the least denomination thus, 1 1/2. Superpartiens is a simple proportion, wherein the greater term containeth the less, and ●ertaine parts of it. As 5 to 2, 7 to 4, 13 to 8. For the antecedents being divided by the consequents, their quotients declare their proportion on this manner, 1 2/3, 1 3/4, 1 5/8. Likewise the proportion between 15 and 9 is superpartiens, as appeareth by the quotient and the parts reduced to the least denomination. 1 2/3. Multiplex is a prime proportion when the greater term containeth the less certain times exactly. Multiplex is defined after the same manner in the second defin. of the fifth book of Euclid, which saith, that Multiplex is a greater quantity in respect of a less, when the less measureth the greater. As the proportions of 4 to 2, 9 to 3, 16 to 4 are multiplices, that is to say, double, triple, quadruple: for their quotients are 2, 3, 4. Second proportion consisteth of two kinds of proportion, and is either multiplex superparticularis, or multiplex superpartiens. It seemeth that these kinds of proportions are made of the former, by joining two of their names together. For the one is the third kind of proportion joined to the first, and the other is the third joined to the second. Multiplex superparticular is is a compound proportion, wherein one term containeth another certain times, and one part beside. As 5 to 2, 10 to 3, 13 to 4. For the quotients declaring their proportions, are 2 1/29 3 1/3, 3 1/4. Multiplex superpartiens, is when the one term containeth the other, & besides more parts than one. Here we are to understand, that they be the true parts which are not equal to the lesser term, as in 11 to 4, 12 to 5, 22 to 6. For the quotients are 2 3/4, 2 2/3, 3 2/3. CHAP. XII. Of Notation and Numeration of proportions. THus much for the kinds of proportion, now followetg the handling and practice of them: which consisteth in Numeration and Notation. Notation is the writing of the antecedents above, and the consequents beneath. The Antecedents 4 8 6 Or 5 1 The Consequents 3 3 2 Or 6 2 The proportions being written, are expressed by the quotient (the parts if there be any, being reduced to the least terms) which declareth as well their proportion as their denomination. In multiplex proportion it is called double, triple, quadruple, octuple proportion, where the quotients are 2, 3, 4, 8. In proportions of lesser inequality this preposition Sub is generally added in them all. As for example, it is called in Submultiplex, Subduple, Subtriple, Subquadruple, etc. In Superparticular proportion (because the half of the greater number is the proportion) it is called after the manner of the Latins Sesquialtera, Sesquitertia, etc. putting the word Sequi in the beginning, and the denominator of the parts in the ending as when the quotients are 1 1/2, 1 1/3, 1 1/4. In proportions of the lesser inequality we say, Subsesquialtera, Subsesquitertia, etc. Likewise Submultiplex, Superparticular, Submultiplex Superpartiens. For in compound proportions the names are also compound, as in 3 1/1 triple sesquiquarta, 2 4/4 dupla superquadripartiens, etc. And thus much for instruction sake. For I know that these words are very strange to us, and not heard of, nor used among the best writers. For they when they would speak of the proportions of things, were wont to express them by two numbers. As Archimedes in expressing the proportion of the circumference of a circle to the Diameter, said not that it was triple sesquitertia, but as 22 was to 7. Moreover the quotients teach us the way not only how to express the proportions given, but also how to set them down in the least terms. For if you set an unity underneath, or over against the quotients being whole numbers, you shall find out the least terms of the multiplex which you seek for. As in the multiplices, if you set an unity right against 3, 4, 5, the proportion in the least terms will be triple, quadruple, quintuple. Again, if you reduce the quotients being mixed numbers, by the second reduction of mixed numbers unto parts, you shall in like manner find out the least terms in the other kind of proportions: as in Superparticular proportion, if you seek for the least terms of 1 1/2 sesquialtera, 1 1/3 sesquitertia, 1 1/8 sesquioctava, than the quotients reduced after the foresaid manner, the least terms will be these. 3 4 9 2 3 8 The like is in all the other kinds, as if you would find out the least terms in duple superpartiens five six parts, then shall you of 2 5/6 make the antecedent 17, and the consequent 6, and they shall be the least terms in the proportion assigned, and so forth in the rest. Thus much for notation. Numeration is the addition or subtraction of proportions. Addition, is that which taketh the products made by multiplying the like terms together, (that is, by multiplying the antecedents by the antecedents, and the consequents by the consequents) for the terms of the compound portion. The adding of proportions is set down in the fifth def. of the sixth book thus. A Proportion is said to be made of proportions, when the quantities of the proportions multiplied together produce another proportion. By the quantities of the proportions he meaneth the quotients whereof the proportions take their names. As if you add a sesquialtera to an equal proportion, it maketh a sesquialtera. Add a sesquiquarta to a duple superpartiens two third parts, the product will be a sesquitertia: as you see in these examples. Simple proper. Compound proport. Simple. Compound. Anteced. 3 4 12 1 8 40 Conseq. 2 4 8 4 3 12 That is, add 1 1/2 unto 0 1 1/2 1 1/4 unto 2 2/3 3 1/3 That the products made of multiplying the terms together, have a compound proportion appeareth by the 5 prop. of the eight book of Euclid, which saith, that Plain or superficial numbers, are in that proportion one to the other which is composed of their sides. As 3 and 12 being plain numbers, have proportion one to another, according to their sides, as you see. The sides. The plain numbers. 3 4 12 1 3 3 If diverse proportions be to be added together, first add two of them, then add the compound to that which remaineth. As if the proportions given were, 2 6 10 1 5 4 First two of them added together, make 18/5 to which the third proportion being added, maketh a Sextuple. 120/22: 6. Thus every proportion may be tripled or quadrupled: if the terms of the proportion given being set down three or four times be added together after the foresaid manner. So a double proportion being tripled, maketh an octuple. And a sesquialtera being doubled four times, maketh a quintuple sesquidecima, as you see here. Subduction of proportions, is that which taketh the products made of the opposite terms of the proportion given, for the terms of the proportion remaining. Which proportion is least, and which may be subduced one from the other in Multiplices may be easily known by the denomination only. But in the other (the terms being compared together as parts) it may be gathered by that which hath been taught before in the parts. Subduction seemeth to be the consequence of that which was set down in the fifth def of the sixth book. For if the addition of proportions be wrought by multiplication, it followeth then that subduction must be wrought by division. Therefore whether you divide the quotients one by the other, or multiply the opposite terms of the parts given one by the other (for in parts the multiplying of the terms crossways is as much as division) the terms produced will have a proportion left like unto the former. As take a sesquialtera from a double proportion, the remainder will be a sesquitertia. Item, take a sesquiquarta from a double sesquialtera, there remaineth a double, as in these examples. The prop. given. The rem. The propor. given. The remainder. Anteced. 2 3 4 5 5 20 2 or Conseq. 1 2 3 2 4 10 1 2 from 1 1/2 1 1/3 2 1/2 from 1 1/4 2 There is no multiplication in proportions properly as in numbers, unless you take the continuing of the proportions for their multiplication. Neither is there any division required in them beside the finding out of one or more proportionals. Wherefore we will proceed to the rest. CHAP. XIII. Of proportion and the kinds thereof, but especially of Arithmetical proportion. F. The comparison of reasons, is Either of unequal reasons, and is called improportion, whereo● in the 8 d. 5. with the which Arithmetic doth not meddle. Or of equal reasons, and is called proportion, consisting Either in the equality of the differences: and is called Arithmetical proportion, which is Either disiunct, Or continual, and this again is, Either simple, Or multiplex, & is called progression: wherein we are to observe, Either the order of the terms, Continual, as they follow one another. Or Interrupted, when we seek but for certain terms. Or the sum of the terms. Or in the equality of the quotients: and is called Geometrical proportion. G. NOw that we have spoken of comparing the terms together it remaineth to speak of the comparison of reasons, wherein first we have to deal with proportion. Proportion is the equality of reasons. Euclid in the fourth definition of the fifth book, defineth proportion to be a similitude of reasons. Aristotle in the fifth book and 3. Chapter of the Ethics, calleth it an equality of reasons. If we should define it to be the Identity of reasons, we should not vary much from the meaning of Euclid, who in the sixth definition of the fifth book speaketh after the same manner, calling them proportional quantities which have the same reason one to another. And albeit that these definitions do properly agree to Geometrical proportion, yet if you take the word reason in the largest sense, for any kind of respect it shall be general, so that as reason or proportionality is a mutual respect of two terms one to another, so proportion shall be a respect of like terms one to another. Proportion is either Arithmetical or Geometrical. Arithmetical proportion is an equality of differences. As in 8, 7, 6, and in 12, 9, 6, 3. Here are two proportions, wherein you may see an equality of differences, or an equal excess of the antecedents. In the first, the difference is 1, in the latter it is 3. So likewise in 15, 11: 8, 4, as 15 exceed 11 by 4, so 8 exceed 4 by 4. Arithmetical proportion is double, for either it is continued in the terms, or it is severed, whereupon it is called continual, or disiunct proportion. These differences are common both to Arithmetical, and to Geometrical proportion. That proportion is continual wherein not only the extremes are continued together by one mean, but generally every proportion, wherein many and sundry middle terms have a continual respect one to another, so that any of them may be either the antecedent or the consequent. In so much, that not only the proportion between 3, 2, 1 is continual: where 2 is the consequent of the former, and the antecedent of the latter proportion: but this also 5, 7, 9, 11, where the same difference is continued from the first to the last. And any of the middle numbers either 7 or 9, have the same nature in respect of them that go before and follow after, that 2 had in the former example. Proportion is severed in the terms when the first only hath the same respect to the second, that the third hath to the fourth, but the second agreeth not proportionably with the third: that is, when all the terms in order as they follow are not of like proportion one to the other. As 10, 8, 4, 2, are severally proportional because 2 the first difference is not continued between 8 and 4. So that here is a severing not only of the terms, but also of the proportions. Leaving therefore in this place to speak of disiunct proportion, we will handle that which is continual. Continual proportion is either Simple, where the extremes have but one middle term: or manifold, wherein many middle terms go on in order continually: whereupon it is called progression. Progression is a continual enlarging of the terms of the proportion given: wherein there are two extremes given, and many other numbers in the middle between them. In Arithmetical progression we are to find out two things, either the number of the terms as they stand in order, or else their sum. Sometimes the terms be all sought for as they follow in order, and are found out by the continual adding of the difference to the last term in the progression. As if you would know the terms of this progression 1, 5, 9, 13, as they follow in order, you must add 4, which is the difference unto 13, than the fifth term will be 17, to which if you add 4 again, it maketh 21, which is the sixth term, and so forth infinitely. Sometime breaking off their natural order, we seek for some one term in the progression, which may be found out by the Theorem following. The progression increasing, if you take an unity from the name of the term which you seek for, & add the product made of the remainder, and the common difference of the terms, unto the first term, the total will be that which you desire. The progression decreasing, if that product be taken from the first term the remainder is the term sought for. As in this progression 4, 7, 10, where the difference is 3, I desire to know the 12 term beginning at 4. Here 12 is the name of the term which I seek for, from whence I take 1, and multiply 11 which remain by 3 the difference, the product is 33, whereto I add 4 the first term. So the total is 37 the term which I sought for. Likewise the 30 term is 91, and so forth. The terms 4 7 10 37 91 The names or number of the terms. 1 2 3 12 30 Thus much for the finding out of the terms. The finding out of the sum is the half of the product made by multiplying the two extremes added together, and the name of the last term, as it standeth in order. As in this example, 1, 5, 9, 13, 17, 21, both the extremes added together make 22, the name of the last term is 6, for it is the sixth in order. Whereby if you multiply 22, the product will be 132, the half whereof (that is) 66, is the sum sought for. This rule may be proved by the 41 prop. of the first book of Euclid, which saith, If a parallelogram and a triangle have one and the self same base, and be in the same altitude, the parallelogram shall be double to the triangle. The numbers then of the progression represent the triangle: whereof one side is the name of the last term, or (as they commonly term it) the number of the places: the other is the extremes added together. But for so much as the product made of these two sides maketh not a triangle, but a parallelogram, therefore in taking the one half I gather by it how big the platform of the triangle is, which is the sum of the progression. The example of this progression 2, 4, 6, 8, 10, is set down here as you see: 5 1 1 1 1 1 1 1 1 1 1 0 0 The whole parallelogram is 60, wherefore 30 which is the half, is the sum of the progression. 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 12 To set forth the profit and use of this in resolving certain questions, let this be the first example. For so much as a natural day hath 24 hours, I demand how many strokes the clock will strike in a day. In this example we seek for the number of the strokes beginning at one, and adding one till you come to the 24 term. After this manner the clocks in Italy and Norinberge are made. Both the extremes added together make 25, which being multiplied by 24, the name of the last place, make 600, whose half are 300 strokes, which is the sum sought for. Another. There be 30 eggs laid in a rank, every one three foot from another. I must put them in a basket which standeth 13 foot from the first egg: yet so that at every time I go to and fro I must take up but one. The question is, how far a man may go before I have gathered them all. This was wont to be a pastime used amongst young men, when one gathered the eggs and put them in a basket without breaking them, and another went to a certain place and came again, so he which first had done his task, got the game. Here the question is about the sum of the Arithmetical progression. 13 16 19 40 64 100 1 2 3 10 18 30 Both the extremes together are 113, th● product made of this sum, and the name of the last term is 3390, whose half is 1693 feet, which make 339 Geometrical paces. This space he goeth and cometh, wherefore the double of it which was first found out, that is 3390 feet, or 678 paces is the sum which we sought for. CHAP. XIIII. Of Geometrical proportion, and of the golden rule belonging thereunto. G. Geometrical proportion is two fold. disjunct: out of whose propriety a riseth the golden rule: and that is, Simple: simple proportions is Direct: before which there goeth Either no kind of Arithmetical account Chap. 14. Or some Arithmetical account, and that Either simple, as the numeration of Whole numers. Chap. 15. Parts or fractions. Chap. 16. Mixed numbers. Chap. 17. Or, compared as the proportion Arithmetical. Geometrical. Or Reciprocal. Chap. 18. Or Manifold. H. conjunctly or Continual. L. THus much of Arithmetical proportion, Geometrical followeth, which is the equality of reasons. Here this word Reason is not taken generally, but specially, for the equality of the quotients arising of the numbers compared together. As in 2, 6, 4, 12, what reason or proportionality is between 2 and 6, the same is between 4 and 12, that is, subtriple. And if four numbers be directly proportional, they shall be also proportional backward, and crossways in this manner. 2 6 4 12 Direct. 6 2 12 4 Backward. 2 4 6 12 Crossewaies'. Geometrical proportion is double: Disiunct, or Continual. Disiunct is that, that agreeth neither in terms nor proportionality, or reason. As in 6, 3, 16, 8, for here are two double proportions distinquished both in terms and in reason, because that between 3 and 16 there is not the same reason observed, that is between 6 and 3 and between 16 and 8. disjunct proportion consisteth of four terms at the least. It hath this property that the product of the extremes, is equal to the product made of the middle numbers, and contrariwise. 16. p. 6. & 19 p. 7. Among the properties of Geometrical proportion set forth by Euclid, this is most excellent and necessary to be known, because the force and fruitfulness thereof is such, that it may very well be called the head of ciphering, that is, the foundation and offspring of all rules, as we shall see hereafter. It hath this meaning, that the number made by multiplying of the first into the fourth, is equal to the number made by multiplying the second into the third, as in 6, 4, 18, 12, the product made of 6, multiplied by 12 is 72, the which number ariseth by multiplying the middle numbers 4 and 18 one by another. And this thing is proper to continual, and generally to all Geometrical proportion, as we shall see hereafter. This property ministereth unto us the golden Rule, which three proportional numbers being given, dividing the product made by multiplying the third into the second by that which remaineth, findeth out the fourth proportional number. Out of the forenamed property, as out of a most plentiful spring is drawn this Rule, wherein consisteth the chiefest parts of casting accounts. The which some for the excellency of it do call the Rule, others the rule of Proportion, other the rule of three, and all men all most in general, for the excellent use of it, do call it the golden Rule. The drift and end of it, is the finding out of a fourth term, or number, that shall have the same proportion with the three numbers given: the which thing it bringeth to pass by multiplying two terms of the proportions given one by the other, that is, the antecedent of the one, and the consequent of the other, and dividing the product by the other which remaineth: for the quotient will be the fourth number of the proportion given. As in 3, 2, 9, I desire to know the fourth that shall be to 9, as 2 to 3, multiply therefore 2 by 9, and divide 18 the product by 3 which remaineth, the quotient will be 6, the proportional number which was sought for. The cause of this consequence is manifest, because having the product of the middle numbers, it is all one as if you should say, that you had the product made of the first and fourth numbers, and to prove that, or to bring to light the fourth number lying hid in the product, you must divide the product by the first number, and say, that the quotient is the other number unknown which made the dividend: because that every number is divided by those numbers by which it is made, and every product divided by one of the multiplicators bringeth forth the other. As in 9, 6, 12, I desire to find the fourth, multiply therefore 6 by 12, that is the middle number, than I say, the product 72 is the product arising of 9, and the fourth number unknown. The product then being had, and one of the multiplyers known, the other cannot any longer be hid. Wherefore I divide 72 by 9 the quotient will be 8, which is the other multiplier, and also the fourth number in proportion: thus, 9, 6, 12, 8. This rule is used in proportion simple or manifold. Simple proportion is that which consisteth only of four terms or numbers. The form whereof is double, for either the proportion proceedeth directly forward, or else backward, whereupon it is called Direct, or Reciprocal. I call that Direct proportion, wherein as the first term is to the second, so the third is to the fourth, and crossways. Which is Direct proportion, may be easily gathered by this, that by how much the third number is greater or less, by so much the fourth in order-shall be greater or less, whether the numbers be given ordinarily, or crossways, or backwards. And it is handled two manner of ways, for sometimes besides the ordering of the terms there is nothing else required: sometimes it requireth some former kind of numeration. The ordering or disposing of the terms to conclude the proportion attributeth to every one that is given his own place: that is, to the thing which is called into question, it giveth the third place: the first place to the thing of the same kind: the middle place to that which remaineth. The lawful order of the terms is that in distinguishing the reasons, we join the consequent of every reason to his own antecedent of the same kind. As if I should say, seeing jupiter in four months moveth ten degrees in the Zodiac: therefore in six months it moveth 15 degrees. Here the proportion ordinarily shall stand thus, 4, 6, 10, 15. For as 4 months is to 6 months, so is 10 degrees unto 15 degrees. For here also the product made of the extremes is equal to the product made of the middle numbers. But the terms in the Rule of proportions is otherwise ordered, because in things of one kind the question is not so moved, that they should be joined together, but crossways. For we do not move the question customably from the antecedent of the second reason to his consequent thus, if 4 months are 6 months, what are ten degrees? But the question is moved from the consequent of the first reason to the consequent of the second crossways: if 4 months give 10 degrees, than 6 months shall give 15 degrees. Therefore seeing the proportion is moved crossways: and thereby may be handled more conveniently, and may be better conceived by this means of the learner, therefore in the ordering of the terms, we place them which be of like sort cross-ways in sundry places, not jointly together. And hereupon the question being moved somewhat confusedly, we must set that term that hath the question adjoined unto it in the third place, and the term of the same kind, that is the term which betokeneth the same thing, and is called by the same name in the first place: and the other in the midst: then must we follow the Method and direction of the rule, multiplying the third by the second, or contrariwise, the second by the third, and dividing the product by the first. The examples. A traveler is to go 180 miles, whereof he goeth 9 in two days, the question is when he shall end his journey. Here the question being propounded somewhat confusedly, you set in order the terms. And for so much as the question is moved about the 180 miles, in what time they may be gone through, you shall set that number in the third place, the 9 miles being a term of the same sort, shall stand in the first place, the two days in the middle. Then multiply 2 by 180, and divide the product 360 by 9, the quotient will be 40, the fourth number in proportion answering the question. The example is thus: If 9 require 2 days, ergo 180, 40 days. If a pole six foot long standing upright upon the ground casteth a shadow ten foot long, I demand how high that Tower is, which at the same time casteth a shadow 125 foot long. Here I see that the numbers given are proportional. For as the height of the pole being known is unto the shadow, so is the height of the tower unknown unto his shadow. The terms therefore being set duly in order, and handled according unto the rule, the example shall be thus. If 10 give 6: ergo 125 shall give 75. After the same manner in this rule may the parts be ordered and handled, as may be perceived by the examples following: If the sixth part of the Moon increaseth when she is distant from the Sun the twelfth part of a circle: how big shall she be when she is distant from him half a circle. Here the first term shall be 1/12 of a circle, the second 1/6 of the Moon, which in that distance is lightened with the light of the Sun: the third 1/2, the work being wrought there ariseth 1, which signifieth that the Moon is at the full when she is half a circle off from the sun. The terms are thus: 1/12 1/6 ergo 1/2 1. I spend in the fourth part of a month 5/6 of a crown, what do I spend in 7/10 of a month? Here the terms are given in order, wherefore I multiply the third by the second, the facit will be 7/12, which being divided by 1/4 after the manner of parts or fractions, the quotient will be 2 1/3, so that the example shall be thus: 1/4 5/6: ergo 7/10 2 1/3. But here if you desire to know the value of 1/3 of a crown or of any other fraction what so ever, you shall search it out in like manner by the rule. For that they teach commonly, that the number made by multiplying of the numerator, and the parts of the whole being known, must be divided by the denominator, whereupon proceedeth the value of the parts given, it is nothing else but to practise the rule of proportion. For it is all one, as if you should set the denominator of the parts in the first place, the numerator in the third, and the parts of the whole being known in the middlemost place, and so work accordingly. As, put case three parts of a crown be 45 pence, you shall find out the value of 1/3 part thus: If 3 be 45 pence: I shall be 15 pence. If one or more of the terms be mixed, you must use the same method which before was used in whole and broken numbers: saving that both the whole and broken numbers must first be reduced into fractions. As the question being moved, two yards and 1/6 of cloth are bought for five crowns, what shall I pay for 5 3/4 of a yard: you shall reduce therefore the first term unto 13/6, the second unto 5/13 the third unto 23/4, whereof there amount 13 crowns & 7/26, which is the number required. Certain abridgements. In whole numbers it shall not be amiss to have regard unto the abridgement following. If of the proportional numbers given, the first and second, or the first and third be compound one to another, than they being divided by the greatest common divisor: the proportion shall be more briefly concluded by the prime numbers. The reason of this abridgement is this, because the terms being changed after this manner, yet the proportion is not changed by the 15 prop of the fifth book, where it is said, that Like parts of multiplices, and also their multiplices compared together have one and the same proportion. As in 12, 32, 15, because 12 and 32 are compound one unto the other, therefore they being divided by 4, I take their quotients 3 and 8 in stead of the numbers given. Thus, if 3 give 8, therefore 15 shall give 40. Likewise seeing the first and the third that is to say 12 and 15 are also compound the on● unto the other: taking their quotient 4 and 5, being prime numbers found out by their greatest common Divisor, for the numbers given, the proportion may be easily concluded in this manner. If 4 be 32: therefore 5 shall be 40. In the fractions it seemeth a brief kind of working to dispatch the matter by multiplication only: to wit, if you take the number made by multiplying of the denominator of the first term, by the numerator of the second term, and then by the numerator of the third term, for the numerator of the fourth: and the facit of the numerator of the first term, by the denominators of the second, and third for the denominator of the 4. The draft of them is thus: As if 1/4 give 2/3, what shall 5/6 give? Multiply 4 by 2, there ariseth 8, then by 5, the product is 40, which is the numerator of the number which we seek for. Furthermore multiply 1 by 3, and then by 6, the facit will be 18, the denominator of the fourth term. The example is thus: 1/4 2/3 ergo 5/6 40/18, or 2 2/9. CHAP. XV. Of examples of the rule of proportion, requiring some kind of simple numeration before them. IN the former examples, the rule of proportion was simple. But oftentimes it requireth some certain kind of numeration to go before it: the which numeration serveth to the ordering of the terms of the principal proportion, according to the nature and laws of the question propounded, that thereby the proportional number, which we desire to know, might be with more easily found out. And this numeration is drawn out of both the parts of Arithmetic: Sometime out of absolute or simple Arithmetic, and is either simple or mixed. Great is the variety of those questions which may be resolved by Geometrical proportion. Wherefore here wit is to be required, that by the former conditions and nature of the question offered, we may be able to perceive what is meet for the unknitting and conclusion of the proportion, and how also (as it were a certain preparative to the future demonstration) the terms of the proportion lying hid, are to be unfolded, orderred, set down, and to be applied to the rule itself. First therefore in this example addition of whole numbers is required. A Vintner buying at Ansa in Italy 120 hogesheads of wine for 2400 ss. laid out for the carriage of them to Basill 1200 ss. and to the waggeners for lading and unlading them 30 ss. for custom in diverse places he paid 50: his expenses in his journey came to 64. He desireth to gain 800. I demand for what he shall sell an hogshead. In this example Addition ordereth one of the proportional terms: for add the price to his expenses, and to the gain, the total will be 4544. Whereby I perceive that the number of the hogsheads assigned is unto this total sum as one hogshead is to the price which we seek for. Wherefore the example shall be thus. If 120 Hog. be worth 4544 sss.: ergo 1 Hog. gives 37 1●/1● sss. Another. Two travelers going from Basill unto two contrary places, do travail a diverse pace one faster than the other: the one goeth every day five, the other but three miles: I demand how many miles they shall be a sunder at nine days end. Here also Addition helpeth to the ordering of one of the terms. For add their daily journeys 5 and 3, the total will be 8, which is the distance for one day, whereupon the rule concludeth, 1 day. giveth 8 miles.: therefore 9 days 72. miles. Examples of subduction going before. A pipe of water voideth into a cistern containing 250 firkins, every hour 24 firkins, now there run out at another pipe every hour 16 firkins, I demand in what time the Cistern will be filled. Here you must find out by subduction, one of the terms of the proportion to be ordered according unto the rule. For reason telleth me that I must seek out the excess of the filling above the emptying. Subducting therefore 16 firkins from 24, there remains 8 firkins, for so many every hour are left in the cistern. Hereby the proportion may easily be concluded: 8 Remain in 1: ergo 250 31 1/4 A certain man selling an hundred pound of certain kind of merchandise for 15 crowns, found at the length that in an hundred crowns he had lost so much as he paid for every hundred weight: I demand what he paid for an hundred. Here you shall subduct the price of an hundred pound, to wit, 15 crowns, (which number also representeth the loss) out of an hundred crowns, the remainder will be 85, the first term of the proportion. For even as 85 crowns which is the stock diminished, is unto an 100, which is the whole stock: even so 15 the price of an hundred weight which was lost, must be unto the stock whereof it was raised: whereupon the terms shall stand thus: If there remain 85 Crowns. of 100: ergo 15 de 17 11/17. Examples wherein Multiplication goeth before. A Polonian going unto the University of Basill to study, delivereth unto a Merchant 126 golden Ducats (every one being valued at 28 ursatis, to receive again of him for them at Basill silverlings (which commonly are called Thaleri) being worth 18 ursatis. The question is how many silverlings he shall receive. Here you shall reduce by multiplication 126 ducats unto ursatis, accounting to every ducat 28, whereof the whole sum amounteth unto 3528 ursatis. Then look what proportion there is between 18 ursatis, and one silverling, the same there is between the whole number of the ursatis, and between the silverlings which we seek for. Wherefore the terms shall be thus: 18 Vrsatis are worth 1 Thaler.: ergo 3528, 196 This and such like examples may be also resolved by the rule of reciprocal proportion, as shall appear hereafter. He that hath 34 crowns for his yearly wages, what hath he for eight days? Here I see there is a question of proportion moved, but for so much as the first and third terms be of a diverse denomination, I reduce them by multiplication to one kind of denomination. And so generally the terms of diverse kinds are by a former multiplication to be drawn to the least denomination. Therefore I take for a year 365 days, and that the account might be the more exact, I divide the 34 crowns into 510 ursatis (reckoning 15 ursatis to a crown) or if it please you to bring them into a smaller coin, you may resolve them into pence, and then the example shall be thus: 365 510: ergo 8 11 13/73 Sometimes Multiplication goeth before with Addition or Subduction. The examples of the first. He that desireth in six crowns to gain 10 ursatis, how shall he sell a pound of that were which cost 20 ursatis? For so much as the money here set down is of a diverse denomination: first of all I reduce the 6 crowns by multiplication unto ursatis: valuing every crown at 15 ursatis, the facit is 90. Whereunto if I add the gain, the whole sum of the ursatis, to wit, 100 shall be the middle term of the proportion after this manner: If of 90 there arise 100: then of 20 22 2/9 In the least terms thus: 9 10: ergo 20 22 2/9 A Butcher delivering unto his servant 216 crowns, sent him to market willing him to buy as many Oxen, Calves, Wethers, and Hogs of every one an even number, as that money would afford him to buy: I demand how many beasts he bought. First let us appoint the prices: an Ox at 11 crowns, a Calf at two, a Wether at one crown and 5 ursatis, an Hog at 3 crowns and ten ursatis. Here then is there a double multiplication with addition required, to the ordering out of the principal proportion. For here you must resolve the 216 crowns by 15 (for at so many ursatis do Germans commonly value their crowns) into 3240 ursari: and likewise reduce the several prices of every beast unto one denomination: to wit, the prices of the Oxen unto 165, of the Calves unto 30, of the Wethers unto 20, of the Hogs unto 55 ursatis, and then add together the prices of the same denomination, the total 270 shall be the first term of the proportion. For as this number is to one of each kind of beasts, so the whole summo of money is unto the number of the beasts unknown. The example is thus: 270 give 1: ergo 3240, 12 The example of the latter. A thief having by stealth conveyed away certain silver vessels, in his flight goeth every day 128 furlongs. Four days after the owner following the thief on horseback, rideth every day 174 furlongs. I demand when he shall overtake him. For the resolving of this question, two terms of the principal proportion must be set down, the one by multiplication, the other by subduction. For first we must see how far the thief was gone before the owner espying the matter entered his journey. Wherefore I multiply 128 furlongs by 4 days, the facit is 512 furlongs. Secondly, I consider how much the latter outgoeth the former every day. Subducting therefore 128 from 174 the remainder is 46. Then look what proportion the daily excess hath unto a day, the same proportion shall there be between the former flight and all those days wherein the latter overtaketh the former. So that the terms of the proportion shall be thus: 46 give 1: ergo 512 11 4/23. CHAP. XVI. Of examples of the golden Rule, wherein there is required some numeration of broken or mixed numbers. IN the former examples before the proportions could be concluded, there was required a certain numeration of whole numbers: in those which follow, the numeration is of broken or mixed numbers. Example. 1. A certain man selling his house for 75 crowns, findeth he hath gotten the fourth part of the money which he paid for him, I demand what he paid for him? Here add unto the money which he gave (which is supposed to be 4/4) the games, to wit 1/4, the total will be 5/4: for he that of his stock laid out raiseth 1/4, he by 4 gaineth 5. Wherefore the principal proportion shall be thus: 5 75: ergo 4, 60 A Farmer having gotten by one years harvest certain sacks of corn, payeth to his Landlords for his years rend, to one 2/5, to another 1/4, and hath remaining 35 sacks: I demand how many he had at the first. In this and such like examples of broken numbers having divers denominations, out of the least number which may commonly be divided by the denominators of the fractions given, must be drawn whole numbers being unto the number divided as the parts assigned do require. As here the least number which may be divided by the denominators set down which are 5 & 4, is 20: whereof 2/5 are 8, & 1/4 is 5, which being added together, & the common denominator set underneath do make 13/20. The farmer then of his whole heap of corn which is supposed to be 20, paid 13, wherefore 13/20 being subducted out of 20/20, there remain 7/20: whereupon the denominators being omitted, the terms shall thus be set together: 7 are 35: ergo 13 were 65 A certain man selling his horse lost the sixth part and 2/15 of the money which he paid for it. and had remaining 243 crowns: the question is what the house cost him. This example is like to the former. for you must take the least number which may commonly be divided by the denominators of the parts given, 6 and 15, to wit, 30: whose parts being of the same denomination, to wit, 1/6 is 5, and 2/15 are 4. The parts added make 9/30, which being subducted from 20/30 leave 21/30 where upon the proportional numbers according to the rule shall be: 21 are 243: ergo 30 347 1/3 So much was the principal stock wherefore he lost 104 crowns 1/7. Sometimes not only simple, but also mixed numeration helpeth oftentimes to the ordering of proportion. For example. Of three silver pots, the first weigheth 12 half ounces and 2/3, the second 16 1/2, the third 23 1/6. The half ounce is worth 11 ursatis 1/2: what then is the price of the pots? Here of necessity must addition of mixed numbers go before, although the principal proportion also consisteth of mixed numbers. For the weight of all the pots being added together, you shall have half ounces 52 1/3: whereupon the proportion shall be thus framed. If 1 cost 11 1/2, therefore 52 1/3 601 5/6 A Vintner having bought 16 hogsheads of wine for 27 3/4 crowns, the price of the wine falling, could not sell them but for crowns 19 5/6: I demand what he should have lost if there had been 84 hogsheads? Here subduction of mixed numbers must go before. For the price of the thing sold for less than it cost being subducted out of the price which it was bought for, will bring out the second term of the proportion, so that you may easily answer the question. Therefore 19 5/7 being subducted from 27 3/4 there remains 7 11/12, the loss in 16 hogsheads of wine. Wherefore the terms of proportion shall thus be set down: In 16 hogs. there are lost 7 11/12: ergo 84, 41 9/16 Mixed numeration is more common a great deal then any other, because the measures, prices, etc. of things are seldom contained in whole numbers: wherefore the studious may here have plentiful matter to exercise himself in. CHAP. XVII. Of examples of the golden Rule requiring proportion before them. HItherto numeration used in simple and absoulte Arithmetic served to the finding out of the conclusion in Geometrical proportion. Yet oftentimes some part of compared Arithmetic is required thereunto: as for example, proportion either Arithmetical or Geometrical. First then let these be examples wherein Arithmetical proportion must go before. An Espy informeth his Captain of the distance of his enemy's camp after this manner. If we take our journey hence, and go the first day 30, the second 28, the third 26 furlongs, diminishing after that order every day two furlongs, we shall meet with them at fifteen days end. Now the enemy knowing of their coming, made toward them, so that at nine days end they met together. I demand how far the Captain went before he met with the enemy. Here first and foremost we must gather the sum of the proportion Arithmetical (which commonly we call progression) whose terms are 15, and the extremes 30, and 2: the whole sum, then is 240 furlongs, which is the second term serving to conclude the Geometrical proportion on this manner. 15 240: ergo 9 144 So far therefore they traveled, the rest, to wit, 96 furlongs the enemies went. A labourer was hired for this wages, to receive the first day a penny, the second three, the third five, and after the same order his wages should be continued for the days following: at length being dismissed, he received 1500 pence, I demand how many days he served? Here if you gather together all the terms of the Arithmetical progression into one sum, you shall make a way to the Geometrical progression. As for example, if you add together his wages for 30 days, the first term of the progression being an unity, and the difference 2, the whole sum shall be 900 pence: whereupon I may conclude: If 900 require 30: ergo 1500, 50. In the examples following, Geometrical proportion must go before to inform the principal proportion. When a web of cloth containing 40 yards is bought for 50 crowns, for how much shall he sell the yard, that in an hundred crowns, desireth to gain 12? Before you can conclude the question by Geometrical proportion, being as it were the decider of the whole controversy, you must seek out the price of one yard. If 40 yards are bought for 50 crowns: then one yard doth cost 11 1/4 of a crown. This shall be the third term of the principal proportion, in this manner. If of an 100 crowns there arise 112: then of 1 1/4 there arise 1 2/5 A Merchant buying at Venice an hundred yards of silk for a crown a yard, spent in his journey and for the carriage of them 30 crowns. At Lipsia where the yard is more by a quarter then the yard of Venice, he sold every yard for 3 crowns. The question is what he gained? In this example addition of whole numbers, together with proportion goeth before. For his stock laid out, and his charges being added, the whole is 130. Besides, to know how many yards of Lipsia answer the Venice measure, you shall work by proportion on this manner. If 5/4 of Lipsia measure, make a yard of Venice, than 80 yards of Lipsia shall make an hundred Venice yards. So that you may readily infer the principal proportion thus: If 1 is sold for 3: ergo 80 240 Whereupon 130 crowns being the stock and charges subducted out of 240, I see he gained 110. Another merchant bought three hogsheads of honey. The first weighed 349 pound, the second 286 pound, the third 300, out of every hundred are subducted ten pound because of the vessel, and a pound of honey is sold for 5 pence: I demand what he is to pay. For so much as the price of one pound of honey is set down, therefore by proportion seek out the price of an hundred, 1 pound is sold for five pence: ergo 100, for 500 pence, then add the weight of the vessels, the whole is 110, add also the weight of the three hogsheads, the total is 935: whereupon you shall conclude: 110 are sold at 500 pence, ergo 935 at 4250, that is, 170 florentines. Another buying two yards of cloth at three crowns, sold six yards afterwards for 11 crowns. Having got by this means 46 crowns, there ariseth a question how many yards he bought, and how many he sold. Before you can come unto the principal proportion, you must order the terms thereof by a former proportion, and by subduction. First therefore you must consider how much the six yards of cloth which afterwards were sold at an higher price cost at the first, the which may easily be inferred by that which is presupposed in the question: for if two yards cost three crowns, than six yards cost three crowns, than six yards were sold at nine crowns. This price I subduct from 11, and I see that six yards were sold for two crowns more than they cost. So that it is easy to resolve the doubt thus: If 2 come of 6, than 46, 138 A Vintner bought 1. 24 hogsheads of wine on this condition, that subducting out of every two hogsheads because of the lose five gallons, he should pay for every hogshead 26 crowines: Tell me what he is to pay. Proportion going before with multiplication and subduction, will set in order the principal proportion. First I seek how much is subducted out of all the hogsheads because of the drags. Out of two hogsheads are drawn 5 gallons, ergo out of 120 hoggesheds are drawn 310. Then I reduce all the foresaid hogsheads unto gallons, multiplying them by 32, for so many gallons doth every hogshead hold, the facit is 3968. Out of these I subduct according to covenant 310 measures: the remainder is 3658 gallons, whereupon the proportion shall be thus: 32 give 26: ergo 3658 2972 1/8 Of two Carpenters, the one alone would frame an house in 30 days, the other in 40. Now taking another unto them, they finished the work in 15 days. In what time then would the third man have made it? Here must you use proportion with addition, and subduction of fractions. The first would have done the work in 30 days, ergo in 15 days he finished 15/30 or 1/2 of the work, the second would have wrought it in 40 days ergo in 15 days he wrought 15/40 or 3/8. The parts added together are 7/8 of the work: which being subducted out of 8/8, that is, out of the whole work, there remaineth 1/8, the third man's work. In so much that the proportion may thus be concluded: 1/8 is framed in 15: ergo 1/1 120 Put case 14 pound of some ware were bought for 6 crowns, and every hundred were sold for 47 crowns: subduct so many pound out of the whole as will in the sale amount to 80 crowns gain. How many pound than were there sold? To unknit this knot you shall use proportion with subduction of mixed numbers. First I seek what he paid for an hundred pound. If 14 pound be bought for 6 crowns, than 100 pound give 42 6/7. Moreover for so much as an hundred pound were sold for 47 crowns, therefore 42 6/7 being subducted out of 47, the remainder is 4 1/7 the gain of an hundred: whereupon I say, If 4 1/7 come of 100, than 80 of 1931 1/29. Sometimes many and sundry kinds of numeration, as well out of simple as compared Arithmetic, go before the principal proportion, wherein the diligent young practitioner may exercise himself. CHAP. XVIII. Of reciprocal proportion. HItherto of simple direct proportion: now must we come to Reciprocal proportion. Reciprocal proportion is that wherein the terms being set down crossways, as the third is unto the second, so reciprocally the first is to the fourth. A 'mong the vulgar people we call Reciprocation the going backward the same way we begun: which is evidently seen in reciprocal proportion being compared with direct proportion. For that way that direct proportion goeth on forward, the same way reciprocal proportion returneth backward. For in direct proportion the terms are proportional, not only as they stand in order, but also crossways: that is to say, as the first is to the second, so the third is to the fourth, and also as the first is to the third, so the second is to the fourth. But in reciprocal proportion the terms being taken first orderly, are altogether without proportion, and crossways they differ in all respects in that one is of greater inequality, the other always of less: yea rather as the third is to the second, so backwards, the first is to the fourth, in so much that that which in direct proportion was the first term, is the third in reciprocal proportion: and again that which is the third in direct proportion, is the first in reciprocal proportion. For example sake, if twelve hired servants do their work in 8 days, than 24 shall do it in four days. Here are set down four terms of reciprocal proportion, which being placed directly shall stand thus: stion I must use the rule of reciprocal proportion on this manner: If 40 require 7: then 70 how many? Multiply 40 by 7, the facit shall be 280, which being divided by 70, giveth 4 or two ounces, which is the weight of the loaf. Three mills in two days grind 20 bushels of corn, in what time then shall five mills grind so many bushels? Here again by how many more mills there be, in so much less time they shall grind the corn. Wherefore the proportion shall be concluded reciprocally in this manner. 3 grind 2: ergo 5 1 1/5. A certain man borrowing 66 crowns of his friend, and repaying them at seven months end, promiseth to pleasure his credit or in as great a matter. Whereupon the creditor afterward desired to borrow of him 112 crowns, but they doubt how long time he should keep the money, that the lending of his money might be correspondent. For so much as the latter man desired to borrow the greater sum, therefore he is bound to restore it sooner than the first should, whose sum was less: whereupon the reciprocation shall be concluded thus: 66 give 7: ergo 112 4 1/8. So likewise the former example of reducing 126 Hungary ducats (each of them being valued at 28 ursatis) unto Thaleri being worth 18 ursatis, may be handled and resolved after this fashion. 28 126: 18 196. CHAP. XIX. Of manifold proportion, and first of that which is compound by Addition commonly called the rule of Fellowship. Manifold proportion is Either compounded, and is two fold, Prime, wherein there is required Either one kind of numeration only, as Addition. Chap. 19 Multiplication. Chap. 20. Or many kinds of numeration: as that which is compounded by multiplication, and addition together. Chap. 21. Or Second, and is called Alligation. Or continued. Chap. 27. HItherto hath been handled simple proportion, as well direct as reciprocal, being set forth by diverse examples: now followeth manifold proportion. Manifold proportion is that which consisteth of more thenfoure terms. Simple proportion disiunct, consisteth of four terms, but manifold proportion consisteth of more, as when in one and the same proportion there be many consequents to one antecedent, or contrariwise, or else when there be two rows of numbers proportional in two numbers, or some such like manner. Manifold proportion is double: for either it is compounded in the terms, or else continued. The first joineth together the terms given, either by one only kind of numeration, to wit, either by addition or multiplication, or else by more, as by addition and multiplication together. Compound proportionality or proportion is said to be that wherein the terms are united together. But the things so united and joined together, are of many brought into one, and heaped together two manner of ways, either by addition or multiplication alone, or else both ways together. By Addition, as when many as well antecedents as consequents are taken and compared, as one antecedent to one consequent. Although proportions may be compounded many ways by Addition, yet of them all this is the most usual: wherefore this special definition serving most for this purpose is taken in stead of that which is more general. For example sake, if 3 unto 2, be as 9 unto 6, then by addition 3 and 9 that is 12, shall be in like manner unto 2 and 6, that is 8, so that the terms of the proportion may stand as you see, 12 unto 8: are as 3 to 2 and 9 to 6 In this kind of manifold proportion, the rule following is to be used, If in the terms of manifold proportion the consequent of the one proportionality be given added together, and the antecedents severally by themselves, then as the antecedents knit together by addition, are unto the consequents added, so shall the several antecedents be unto their several consequents, and so contrariwise. This is the 12 p. 7, saving that it is reversed and applied some what more fitly to this purpose. For the words of the proposition be these: If there be a multitude of numbers, how many so ever proportional, as one of the antecedents is to one of the consequents, so are all the antecedents unto all the consequents. And the same backward is true also. By which conversion it appeareth how the examples which commonly are referred unto the rule of fellowship, having many proportions coupled by addition, are to be handled and to be resolved by the rule of proportion, in which examples the antecedents of the one proportion are set down severally, to wit, the contributions of the partners, and the consequents are added together, to wit, the common gain or loss. So that the contributions and common stock of the partners, or else generally all those numbers into which the other is to be distributed proportionally must be added together, and the total must be set in the first place in stead of the antecedent of the first proportion, whose consequent must be the foresaid number which is taken to be distributed proportionally. In the third place the terms compounded by addition must be placed severally, & the rule must be repeated as often as there be several numbers, whereupon I may conclude the proportion. As in the examples following. Three men being partners laid their money together, the first 60, the second 100, the third 135 crowns. Now they gained after a while 45 crowns. The question is how much of the gains each of them by duty ought to have. In this and such like examples equity and reason requireth, that that which is gotten by the common stock should also be common, and should be so distributed, that every man's portion should have the same proportion to the whole gain, that the money which he laid down had unto the whole stock: and crossways as the whole stock is unto the whole gain, so then every man's portion laid together, should be unto his part of the gains. Wherefore here this question of manifold proportion is manifest. Now the terms given are these: The first propor. The second propor. I TWO III IIII 60 45 100 135 The consequents of the second proportion are set down but jointly together, to wit, 45, the antecedents are several. Wherefore to conclude the several consequents, I set the whole stock gathered by adding of 60, 100, 135, to wit, 295 in the first place, which is unto the whole gain, as the severally terms in the third place are unto the several terms in the fourth place, which we desire to know. The example is thus: Three partners in a common stock of 500 crowns, lost 124, to the repairing whereof the first was bound to be contributary 20 crowns, the second 43: I demand how much every one put into the common purse. Here we have of the latter proportion three antecedents, two are assigned in express words, but the third is set down somewhat obscurely. But for so much as the total of the three added together is set down, to wit, 124, the third may easily be known by subduction. Add therefore 20 and 43, the total is 63, which being subducted out of 124 there remains 61, which is the third man's portion in the loss. Wherefore as in the former example we gathered every man's gain by every one's contribution, so likewise in this place by the parts of the loss we may gather each man's stock. The example is thus: Cicero in his Oration for Cecinna sayeth: A woman making her will departed this life, she made Cecinna heir of eleven ounces and an half of her goods, and Marcus Fulcinus the free man of her former husband heir of two fixed parts of an ounce. Unto Eutius she gave one sixth part. How much then had every one for his portion. In this example the whole patrimony is divided into eleven ounces, half an ounce, and three sixth parts of an ounce: the which parts are the antecedents of the late proportion. But in that they be of diverse denominations, I resolve them all into the least denomination, to wit, into sixts. If then you allow to an ounce six sixths, and to the pound twelve ounces, there shall be in a pound 72 sixths. Whereof 11 ounces and an: half are worth 69, and two sixths 2, one 1. Whereupon supposing the whole patrimony to be 2500 crowns, the terms shall be thus: Three covenanted between themselves, that of their gains the first should have twice as much as the second. Now they had laid down 24 crowns, and the third 36. But when they had gotten 12 crowns they cannot tell how to divide them lawfully according to their covenant. Here that which the first man laid down being covertly expressed, must be gathered out of the covenant. For in that he was to receive twice so much as the second, it is manifest that that which he laid out was twice as much as that whereof the second was contributory, whether it were money, or whether it were his labour, or whether it were valued in both. Wherefore if for the first you take 48 crowns the double of 24, than they being added together, the whole doubt shall be removed by the rule on this manner. The good man of an house bequeatheth unto four men 584 crowns, to be divided after this rate, that A should receive 1/2, B 2/3, C 3/4, D 5/6, what was every one's portion? Here the proportion of the parts unto the goods which were left is expressed: yet the parts themselves in the goods do not yet appear manifest. The least number therefore that may be divided by the denominators of the fractions, which is 12, will declare the proportion. For 1/2 of 12, be 6, 2/3 are 8, 3/4 are 9, 5/6 are 10. These being added, the example shall be thus: But if he willed to make such a division among them that were to receive the legacy that the first should have 1/2 and 1/5 wanting 20 crowns, the second 1/3 and 30 crowns, the third 3/4 and 10 crowns, the fourth 1/6 wanting 12 crowns: then what should their portions be? In this case that which is more than their portions I take it from the whole substance, that which wanteth I add unto it: as 30 and 50 are 80, which being taken from 584, there remains 504. Hereunto add 20 and 12, or 32, the total is 536. Then I seek out the least number that may be commonly divided by the denominators, to wit, 60, the parts of the same denomination are for the first 30 and 12, for the second 20, for the third 45, for the fourth 10: and then the portion of the first being first added together, and afterward all the rest, the example will be thus: Then that the testator may be satisfied, from the first man's portion I take 20 crowns there remains 172 48/117, unto the second add 30, they make 121 73/117, unto the third add 50, his whole portion will be 256 18/117, lastly from the fourth take 12, there remain unto him 33 95/117, all which parts are answerable unto the 584 crowns, which were to be divided. Three partners laid together three equal portions, but they left them there, some for longer time and some for shorter: the first left his money in the stock two months and an half, the second three months and 2/5, the third four months and 2/3. Put case they gained 12 crowns, I demand what was every man's portion. This example being of mixed numeration, requireth no other work then the former. For the several times being added together, wherein they left their stock for the common traffic, the total will be 317/30: So that the example will be. CHAP. XX. Of multiplied proportion compound by multiplication, which commonly is called the double Rule. THe former examples appertained unto proportion compound by addition: now followeth that which is compound by multiplication, which in stead of two terms of the proportion, useth the facit made of them. This is the right compounding of proportion, which gathereth them together by multiplication as was said before, and appeareth by the fift definition of the sixth book of Euclid, because it doth not only join together the terms themselves, but also maketh the proportion of the compound terms to comprehend the proportion of the simple. Wherefore hitherto are to be referred, and by this rule of proportion are to be handled these examples, wherein two terms by multiplication must be reduced unto one: for there proportion as appeareth by the fifth prop. of the eighth book of Euclid, is gathered of the proportion of the terms multiplied one by the other. Of this sort are all those examples which commonly are referred to the double rule, wherein two terms are set in the same places, that is, the principal proportion hath a certain circumstance adjoined unto it, in so much that here the proportion should be twice concluded, were it not that for those two terms the products made of them are taken as one. The terms propounded by multiplication, are either of the same place, as in direct proportion, or of diverse places, as in reciprocal proportion. The examples of the first are these. Four Students spend in 3 months 19 crowns: how much therefore shall eight Students spend in 9 months. You see here the principal terms have annexed unto them a circumstance of time, & therefore (the manner of ordering the terms in the golden rule being observed) the first and the third are double, on this fashion: 4 19 8 3 9 Wherefore for those two I take the compounds by multiplication. As for 4 and 3 I take 12, for 8 and 9, 72, to wit their products: So that the compound multiplication shall be concluded thus: 12 19 72 114 Or 1 9 Here between the expenses of four students in three months, there is observed the same proportion which is between the expenses of 8 students in 9 months, to wit, the proportion compound of the two, to wit, of 4 unto 8, and 3 unto 9, or to speak more plainly, the expenses are one to another, as thrice four unto nine times eight. The which may easily be proved, if you put in the mean proportional number to whom the terms of the antecedents and consequents may belong. So that if you should set it down that four student in 9 months should make their expense, the example will be thus: Here than if four students in three months dispend every one but one, than so many students in nine months shall spend thrice as much. Wherefore the proportion of the expenses of A unto B, shall be the same that is between three and nine. Likewise if four students in nine months spend every one but one, than 8 students in as many months will spend twice as much. Whereupon there will be the same reason between B and C, as is between 4 and 8. Furthermore if you multiply 3 and 9, the terms of the first reason by one and the same number, namely by 4, the productes, namely 12 and 36, shall keep the same reason by the 17 proposition of the 7 of Euclid: so shall 36 and 72 being made of 4 and 8 multiplied by 9 Whereby it is gathered that as the charges of A is unto B, so is 12 unto 36. And in like manner as B is unto C, so is 36 unto 72. To conclude, seeing there are here three proportional numbers, A, B, C, and as many more in number answerable unto them 12, 36, 72, therefore by the 14 of the 7 (taking the extremes the mean proportionals being withdrawn) the charges of A shall be all one to the charges of C, as 12 is unto 72, whereby the goodness of the working doth appear. another example. Two Printers in four days print 16 forms, ho●e many shall seven Printers print in 14 days? Here the placing of the terms prescribed in the rule of proportion being observed, they shall be set in this manner: Printers. 2 16 7 Days. 4 14 Then, the first terms and the last being multiplied one by another, the proportion is concluded thus: 8 16: therefore 98 196 Eight yards of cloth, 4 yards and 1/4 broad are bought for 11 crowns: therefore how shall 15 yards of cloth be bought 1 yard 3/4 broad? This example of mixed numeration varieth nothing from the former, therefore it is thus dispatched: Three men trading together by ill luck lost 52 crowns. The first put in 110 crowns for 5 months, the second put in 84 crowns, the third 65, I know not for what time. Their traffic being ended, the first found that he had lost 22 crowns, the second found he had lost 18, the third 12. How long therefore was the money of the last two in the common stock? I think that this example (though it seem to belong to the Rule of Fellowship) may well be referred to the rule of proportion compounded by multiplication, yet it is unlike to the other, because of the numbers given, the middlemost only is compounded by multiplication, the other are sought out by the rule of three, proceeding from a term known to those which are unknown. Therefore the first man's 22 crowns being multiplied by his 5 months, the facit is 550. Whereupon I conclude, if his money, that lost 22 crowns, multiplied by the time doth make 550, what shall the loss of 18 crowns? and what shall the loss of 12 crowns make? The terms shall stand thus: 22 550: therefore 18 450 22 550: therefore 12 300 The fourth numbers inferred by the rule of three are compounded of each man's stock multiplied by the time for which it was laid out. Therefore each of them severally divided by his own stock, the time is severally inferred, so that the second man's money was out of his hands five months and 5/14, the third man's money was out four and 8/13 of a month. An abridgement of the former work. If the terms in the first and third place fall out to be equal, than they being taken away, the rest of the terms shall infer the proportion. The reason of this abridgement ariseth out of the 17 proposition of the 7, for the numbers given keep the same proportionality which they would have being multiplied by one and the same number. As: If the gain of 25 crowns in four year be eight crowns, what shall 100 crowns yield in four year? Here the terms are given as you see. Stock 25 8 100 Years 4 4 Forsomuch therefore as 25 hath the same reason to 100 which they would have being multiplied by 4, as it appeareth by the 17 prop. 7, therefore the equal numbers which are in the first and third place, being cast away, I conclude the proportion in simple terms thus: 25 8 therefore 100 32 Likewise the question being thus propounded, the gain of 25 crowns in 4 year, is 8 crowns, therefore what shall 100 yield in 25 years? The terms shall stand thus: 25 8 100 4 25 Here again the alternate terms of the first and third place being omitted, the rest shall conclude the proportion thus: 4 8: therefore 100 200 If the proportion chance to be reciprocal, then are the terms of the first and third place compounded by alternate or cross multiplication, and the question afterward is concluded directly. Some transpose the principal terms setting the first in the third and contrariwise the third in the first place, & then bring the twofold terms into one by multiplication. The example: Eight horses in 12 days eat 9 bushels of oats, in how many days shall 18 horse eat 24 bushes? I easily perceive that there is here a reciprocation, for 18 horses will spend in a great deal lesser time that heap of oats, which 8 horse will consume in 12 days. Therefore by how much the third term is the greater, by so much the fourth term is the lesser. Therefore I must take heed that disposing the terms as I did before, I conclude not directly the reciprocal proportion by multiplying of the first and last terms one by another. The terms in the example are thus: Here therefore I multiply 9 by 18 and make 162 the first term, likewise I multiply 8 by 24 and make 192 the third term, whereby the lawful induction of the proportion is thus dispatched: 162 12: therefore 192 14 2/9 If 100 crowns do give 5 crowns interest every year, in what time will 56 crowns give. 12? In this question for so much as a greater interest is sought for by a lesser stock of money, the reciprocation is manifest, the which interest craveth so much the more time by how much the more the stock is lesser. Therefore the terms being set down as it is convenient, and multiplied crossways, the doubt is answered by the rule of three. CHAP. XXI. Of manifold proportion compounded by multiplication and addition. THus much concerning the examples of proportion compounded by addition or multiplication only: now must we proceed to that proportion which joineth both kinds of numeration together. Therefore proportion compounded by multiplication and addition, is that which first of all multiplieth the manifold terms given one with another, and then addeth their productes together. This kind of proportion by a two fold, yea sometimes by a fourfold composition, bringeth many terms into one. Unto this appertain the examples of the Rule of second fellowship, as they commonly call them, wherein the time is annexed to the stock, or some other circumstance to the principal term. The Theorem following serveth for the working of those examples. If among the terms of the manifold proportion, the consequents of one reason be given added together, but some antecedents be given severally and double: then as the antecedents compounded by multiplication and addition are unto the consequents added together, so shall the several antecedents compounded by multiplication be unto the several consequents. As thus for example. Two men in common traffic got 60 crowns, the first brought 50 crowns for two year, the second 15 for five year: what shall be each man his share. In this question containing a partnership of equal contributions in regard of the diverse times, I see the division of the gain must be made according to the proportion of the time. That this division may be made the consequents of the latter reason are given added together, namely 60, but the antecedents are given not only severally, but also double, as you see here. The first reason. The second reason. I TWO III IIII 50 2 60 15 5 Therefore first of all I compound the double terms by multiplication, so that after a sort I reduce the unequal times to an equality, the products are 100 and 75, which are all one as if you should say, that the first man should have as much gain by 100 crowns in one year, as he should have for 50 crowns in 2 year. In like manner the second man should get by 75 crowns in one year, as much as he should get by 15 crowns in five year. Moreover these two products 100 and 75 set down severally, shall possess the third place of the proportion: but added together they shall stand in the first place: now the rule of three twice repeated, shall deliver the numbers sought for in this manner: 175 60: therefore 100 34 2/7 175 60: therefore 75 25 5/7 I do of set purpose omit the demonstration of this composition of the terms, because it is not much unlike the former. A hundred and sixty footmen and forty horsemen got a booty of 138 crowns to be divided between them so, that as often as the footmen received one, the horsemen should receive three. How much were the footmen, how much were the horsemen to have? Here likewise I place the consequents of the second reason added together in the second place, namely, 138: in the third place I set the products made by multiplying the numbers signifying how often each one should take his share by the number of the soldiers: namely, 160 and 120, these added together and making 280, shall have the first place. The example is thus: 280 138: therefore 160 78 6/7 280 138: therefore 120 59 1/7 Three Butchers hired a meadow together, promising to pay yearly rent for it 30 crowns, the first fed in it 20 Oxen 70 days, the second fed 46 Oxen 56 days, the third fed 32 Oxen 60 days: How much of the rent shall each partner pay? Again therefore I multiply each man's heard by the several times, the products 1400, 2576, 1920, shall possess the third place: and being added together, they shall have the first place, the yearly rent shall be in the midst thus: 5896 30: therefore 1400 7 91/737 5896 30: therefore 2576 13 79/737 5896 30: therefore 1920 9 567/737 Three merchants were partners for a year. The first in the beginning brought 250 crowns, but after 3 months he withdrew 100 The second after 2 months brought 180 crowns, but after 6 months of their partnership were expired, he took away 50. The third after three months brought in 235 crowns, and after five months added 45 crowns. Now having gotten 68 crowns, what is each man's share? These kind of examples being in outward show most intricate, require a little more labour, otherwise they are to be handled by the same art. They will soon be dispatched, if you endeavour but to sever every man's time accordingly as he changed his stock, and make as many multiplications as there were changes. As in this present example: Because the first man left in the partnership 250 crowns for three months, let them be multiplied together, the product will be 750. Then withdrawing 100 crowns, he left but 150 for the nine months remaining. Wherefore these being again multiplied by their time, make 1350. Which numbers being added, the total sum for the first partner is 2100. Likewise the second man his 180 crowns being multiplied by the 4 months wherein he left them in the common traffic, do make 720. Moreover 50 crowns being subducted from 180 (for so many he fetched away from the principal after the sixth month of their partner ship) there remain 130 crowns, which if you multiply by the six months remaining, the product will be 780, which added to the former, maketh 1500. So also multiply the third man his 235 crowns by 9, they make 2115. Item 45 by 7, they make 315, which added to the former, make 2430. These terms being thus compounded, if you proceed as you did before, the proportion will be inferred, which here you see set down, with ordering of the whole example. The stock. month. Numbers compounded first by multiplication. Numbers compounded by addition. 1 250. 3. 750. 2100. 150. 9 1350. II. 180. 4. 720. 1500. 130. 6. 780. III. 235. 9 2115. 2430. 45. 7. 315. 6030. 68 2100. 23 137/201 1500. 16 184/201 2430. 27 81/201 CHAP. XXII. A treatise of Alligation, whereof B. SALIGNACUS was the Author. l. In Aligation we are to consider, The propriety which is this: that in alligation, The measures should be like. The price of the mixed measure should be mean in quantity between the prices of the simple measures. The kinds which are two Prime: which counterchaungeth the differences between the extremes & the mean: & herein we are to consider the Chap. 23. Propriety, which is this, that if the extremes are in number Even: each of them severally must be compared once only with the mean. Odd: then Every extreme of the greatest number must be compared with the mean once only. Every extreme of the least number must be compared diverse times with the mean. Kinds. which are two, The first kind, some part of the numbers sought for being given, inferreth the rest by simple proportion. Chap. 24. The second kind, the total sum made of the numbers sought for being given, inferreth all the particulars by proportion compounded by addition. Chap. 25. Second: which inferreth the mean by proportion compounded by multiplication and addition. Chap. 26. Of the Definition and proprieties of Alligation. Chap. 1. HItherto we have spoken of the first kind of compound proportion: the second followeth, and is called Alligation. Alligation is an art, which (by the means of certain things given) maketh equal the total made of the prices of the measures mingled together, unto the total made of prices of the measures taken severally. There is 100 pound weight of silver worth 17 pound to be mingled with other silver worth 24 pound, so that the total made of the prices of the weights mingled together may be equal to the total made of the prices of the weights taken severally. Here all the prices are given, the art therefore which by the means of these prices given maketh equal the total numbers is Alligation. Item, Of two kinds of corn mingled together suppose there were 10 bushels worth 16 shillings, and 18 bushels worth 12 shillings: four bushels of this mingled corn are to be sold, so that the total sum made of the prices of the 28 bushels mingled together be equal to the total made of the prices of the 10 and 18 bushels severally taken. Here both the number and the price of the simple measures are given, but there is but only a certain part given of the number of the measures mingled together. The art therefore which by the means of the things given maketh equal the totals propounded in the question, is called Alligation. The property of Alligation is this, that the measures in it be alike: but the price of the mingled measure must be in quantity mean between the prices of the measures severally given. A bushel of corn worth 16 shillings is supposed to be mingled with a bushel of corn worth 18 shillings. Here by this property first of all the bushels are like measures. Then the price of a mingled bushel must be in a quantity mean between 16 and 18. that is to say, it must be greater than 16, and less than 18. So that the price of a simple measure is called the Extreme, but the price of a mingled measure is the Mean. CHAP. XXIII. Of Prime Alligation, and the property thereof. ALLigation is two fold. Prime, which doth counterchange the differences of the terms from the mean. By the definition of Alligation generally taken it appeareth, that in all alligation there are certain things given. Therefore in Prime alligation generally let all prices be understood to be given. Moreover to counterchange the differences of the extremes, is nothing else but to attribute the difference of the lesser extreme to the greater extreme, and contrariwise to attribute the difference of the greater extreme to the lesser extreme. Now this counterchaunge is therefore used, that we may thereby make equal the total made of the prices of the measures severally taken unto the total of the prices of the measures mingled together. That this may be the better demonstrated, we must first of all set down two premises. The first is this: If one and the same number do multiply certain numbers severally, and the total made of them being added together, the product made of the total shall be equal to the total made of the products of the parts. As here: let 10 be the total number made of 4 and 6, let the number multiplying them all, be 2, the products made of 4 and 6, be 8 and 12, and the total made of them is 20. Therefore the product made of 10 by 2 shall be 20. The second premisse is this: If three unequal numbers being given, you multiply any one of them by the other two remaining, and augment the self same number by one of them which remain, and diminish it by the other: and then multiply the number augmented by that which was taken away, and the number diminished by that which was added to: the total number made of the latter products shall be equal to the total made of the former products. As here: Let there be three unequal numbers 4, 7, 9, multiply 9 by 4 and 7, and let the products be 36 and 63, then augment and diminish the self same number 9 as is aforesaid, add therefore 7 unto it, and let the total be 16, also from 9 take 4, and let the remainder be 5, then multiply 16 by 4, and 5 by 7, and let the products be 64 and 35: here the total made of 64 and 35, shall be equal to the total made of 36 and 63. These things being thus set down: now of two kinds of wine, let one be worth 14 pence, another worth 11 pence, let the difference be 2 and 1 from the mean, which is 12 pence. The total made of the differences is 3. The mean price is 12, the extremes are 14 and 11. Therefore multiply the mean which is 12, by the total made of the differences, namely by 3, the product is 36, that is to say, three pottles worth 12 pence a piece shall be worth 36 pence, I say therefore that one pottle worth 14 pence, and two pottles worth 11 pence a piece shall be worth 36 pence. That is: I say that the total made of 1 and 2, by 14 and 11, is 36. For here 3 is the total made of 1 and 2 as was aforesaid. Therefore by the first premisse if you multiply 12 by 2 and 1, the total made of the productes 24 and 12 shall be equal to the product made of 12 and 3. But 12 multiplied by 3 maketh 36, therefore the total of the products made of 12 multiplied by 2 and 1 shall be 36. Again these three numbers 11, 12, and 14 are unequal, the differences of the extremes 14 and 11 from 12, which is the mean, are 2 and 1. The greater extreme 14 is 12 more by 2, the lesser extreme 11 is 12 less by 1. Therefore by the second premisse if you multiply 12, augmented by 1, which was taken from it, and 12 diminished by 2, which was added to it: that is to say, if you multiply 14 by 1, and 11 by 2, the total made of their products shall be equal to the total made of the products of 12 multiplied by 1 & 2. But the total of the products made of 12 by 1 and 2 is 36, as it appeared before: therefore the total of the products made of 1 and 2 by 14 and 11, shall be also 36, which was the thing to be demonstrated. Therefore the reason why with use this counterchaunging is as you see. Here this also is to be noted: If each number of the extremes be a number of multitude, the differences may be counterchaunged diversely: and therefore then in one and the same example a diverse alligation may be made. But all Alligation maketh equal the total made of the prices of the mixed measures to the total made of the measures severally taken, as appeareth by the definition of general alligation: wherefore in the alligation intended when each number of the extremes is a number of multitude, the counterchanging of the differences is at our own choice. Hereafter follow two proprieties, and as many kinds of prime alligation. The first propriety is thus: If the extremes be equal in number, each of them is compared with the mean but once only. As here: 14 18 2 24 4 15 20 3 16 1 14 1 12 5 The second propriety is thus: If the extremes be unequal in number, than each of the extremes whose number is the greater, are compared with the mean once only: but touching the lesser number of extremes, if the extreme be but one only, then is that one extreme to be cempared with the mean so often as there is unities in the greater number of the extremes. As here: 11 18 3 14 3 8 7, 3 10 In this example the greater number of the extremes is 2, and therefore the extreme of the lesser number is twice compared with the mean. Where also you shall note that the total made of the manifold differences attributed to one and the same extreme, is taken for one difference: for, for 7 and 3 we take 10. But if the extremes of the lesser number be many, if you compare more than one of them oftentimes with the mean, the number arising of the comparison is without art: but if you compare but one only with the mean, that shall be compared with it so often as the difference of the unequal numbers is being added to an unity, but the rest shall be compared with the mean but once only. As here: 15 4 17 5 17 6 17 7 13 28 8 32 11, 10, 9 30 In this example the greater number of the extremes is 4, the lesser is 2, the difference of these unequal numbers is 2, unto which if you add 1, the total will be 3. Now I compare only one extreme of the lesser number, namely 32, oftentimes with the mean 15, and I compare it three times, but I compare the extreme which remaineth but once only with the mean. And then last of all. I take the total made of the manifold differences attributed to one extreme, for one difference as I did before. Thus much concerning the two proprieties of prime alligation: now follow the two kinds for the understanding whereof we are to note that the numbers sought for are the numbers of the simple measures. CHAP. XXIIII. Concerning the first kind of prime Alligation. PRime alligation of the first kind, is when some part of the numbers sought for being given, we infer the rest by simple proportion. Here we take the difference attributed to that extreme, whose measures are by number given, for the first term of proportion, and for the third we take the number of the measures given. The which kind of working shall be evidently seen in the examples following. My neighbour mingled 12 bushels of fine wheat worth 14 pence, with other corn, namely, wheat, barley, and oats: the bushel of wheat was worth 18 pence, the barley 11 pence, the oats 9 pence, the bushel of corn mingled together was worth 10 pence. The question is, how much wheat? how much barley? how much oats was mingled together? Here a part of the numbers sought for is given namely 12 bushels, therefore first of all I counterchaunge the differences of the extremes as you see: 10 18 Wheat 1 14 Fine wheat 1 11 Barley 1 9 Oats 8, 4, 1 13 Then for the first term of the proportion I take the difference attributed to that extreme whose measures are by number given, namely one bushel worth 14 pence, but for the third term I take the number of the measures given, namely 12 bushels worth 14 pence a bushel. And so conclude the question propounded by two simple proportions. The first proportion is thus: 1 Bushel worth 14 pence requireth 13 bushels worth 9 pence the bushel: therefore 12 bushels at 14 pence the bushel require 156 bushels at nine pence the bushel. The second is thus: 1 bushel worth 14 pence requireth 1 bushel of each other kind: therefore 12 bushels at 14 pence the bushel require 12 bushels of each other kind. Wherefore with 12 bushels at 14 pence the bushel were mingled 156 bushels at 9 pence the bushel, and 12 bushel of each of the other kinds. Suppose there were four kinds of silver: let a pound of the one be worth 20 pound, of the other 16 pound, of the third 14 pound, of the fourth 12 pound. These four kinds of silver are to be mingled together, so that a pound of the mingled silver be worth 15 pound. Now there were taken 33 pound of the silver worth 16 pound, the question is how many pound of each other kind is to be taken? Here a part of the numbers sought for is given, namely 33, therefore the differences being counterchaunged as you see: 15 20 3 16 1 14 1 12 5 For the first term of the proportion, I take the difference added to the extreme whose measures are by number given, namely one pound worth 16 pound, but for the third I take the number of the measures given, namely 33 pound at 16 pound the pound. Then I conclude the question propounded by three simple proportions, the first proportion is thus: 1 pound worth 16 pound, requireth one pound worth 14 pound: therefore 33 pound worth 16 pound the pound, require 33 pound worth 14 pound the pound. The second is thus: 1 pound worth 16 pound, requireth three pound worth 20 pound the pound: therefore 33 pound worth 16 pound the pound require 99 pound worth 20 pound the pound. The third is thus: 1 pound worth 16 pound, requireth five pound worth 12 pound the pound: therefore 33 pound worth 16 pound the pound require 165 pound worth 12 pound the pound. Therefore if there were 33 pound taken of the silver worth 16 pound the pound, there might be taken 33 pound of the silver worth 14 pound the pound, and 99 pound of the silver worth 20 pound the pound, and 165 pound worth 12 pound the pound, I say they may be taken, but it is not needful they should be taken. For seeing that here each number of the extremes is a number of multitude, therefore the alligation may be manifold, according as I noted it before in the definition of prime alligation. The prices of the same kinds of silver being kept, if there had been 6 pound taken of the silver worth 16 pound the pound, and 8 pound of the silver worth 12 pound the pound, that a pound of the mingled silver might be worth 15 pound, you shall say thus: 1 pound worth 16 pound requireth 5 pound worth 12 pound: therefore 6 pound worth 16 pound the pound requireth 30 worth 12 pound the pound. And therefore unto those 8 pound worth 12 pound the pound, there are to be added 22 pound of the same price. The rest is easy by that which hath been said before. CHAP. XXV. Of the second kind of prime Alligation. WE have spoken of the first kind of prime alligation. Prime alligation of the second kind is that, which by the means of the total made of the numbers sought for inferreth all the particulars by the help of proportion compounded by addition. This compound proportion is commonly called the rule of Fellowship. In it for the first term of proportion we take the total made of the differences added together: for the second, we take the total made of the number of the measures, and for the antecedents of the reasons remaining, the counterchannged differences of the extremes from the mean. Mine host hath two sorts of wine, one worth six pence a quart, another worth 12 pence a quart. Of these two he purposeth to draw six quarts at 10 pence: how many quarts therefore shall he draw of each several kind. Here the whole number given made of the numbers of the simple measures is six. Therefore counterchaunging the differences as you see: 10 6 2 12 4 For the first term I take the total made of the differences which is 6. For the second term I take 6, and for the antecedents of the reasons remaining I take 2 and 4, and then I make a compounded proportion in this manner: 6 6 therefore 2 2 6 6 therefore 4 4 Here therefore there must be drawn two quarts at 6 pence the quart, & 4 at 12 pence the quart. Hereby it appeareth that if the total made of the numbers of the simple measures be equal to the total made of the numbers of the differences, the numbers of the differences are the numbers of the simple measures. Hiero King of Syracuse voweth to the gods for the prosperous success of his affairs a crown of gold. Let us suppose that Hiero for the making of this crown gave unto a workman 500 pound of gold: and that the workman made in deed a crown of just weight, but mingled some silver in the crown. The king demandeth of Archimedes how much gold, how much silver was in the mingled crown. For the resolving of this question Archimedes took two lumps of metal of the same weight with the crown, but of the same kind with the gold and silver which was in the crown. I say they must be of the same kind. For let there be two lumps of silver of equal weight one to an other, but of uneqall fineness, the finer lump shall fill a lesser place than the other which is not so fine. The same is to be said of the lump of gold. So that here will be a most manifest error, if the metals be they either gold or silver, be not both of one kind. I say therefore that the lump both of gold and silver was of the same kind with the gold and silver in the crown. Now let there be 3 such bodies chosen: let the one be a lump of gold, the other a lump of silver, the third the crown mingled of gold and silver. And let them be hanged one after another severally in a vessel full of water, and let the weight of the first be 968, of the second 952, of the third 964. Then let the weights be taken for the value of the bodies themselves: the value therefore of the lump of gold shall be 968, of the silver 952, of the crown 964. Now in the mingled crown by supposition there are 500 pound weight: therefore here the total example made of the numbers of the measures shall be 500, so that the differences being counterchaunged as you see: 964 968 12 952 4 For the first term I take the total made of the differences, namely 16, for the second I take the total made of the numbers of the measures sought for, namely 500, for the antecedents of the reasons remaining. I take 12 and 4, and thereof frame a compound proportion in this manner: 16 500: therefore 4 125 16 500: therefore 12 375 Whereby it is inferred that there was in the mingled crown 125 pound of silver, and 375 of gold. Archimedes in vitrvuius is said to have found out the mixture of the gold, grounding his reason on the differences of the water running out of the vessel. But that the art may stand sure, I construe his manner of reason thus: that is to say, that by the known differences of the water running out of the vessel Archimedes attained unto the other differences of the water unknown, and then at length answered the question by prime alligation of the second kind. Otherwise suppose an unequal running out of the water, that is to say, when the gold was put in, say that there ran out 20, when the silver was put in 36, and when the crown was put in 24. These several quantities of water running out are to be taken for the values of the bodies as they were put in, according to the intent of him which gainesayeth mine assertion. Therefore the value of a lump of gold of 500 pound, shall be lesser than the value of a mass of silver of the same weight, which is against reason. Therefore Archimedes by the known differences of the water running out, attained to the differences of the water remaining in this manner. Suppose that the vessel into the which the crowns were put were 488 pints. If therefore the water which ran out when the lump of gold was put in were 20 pints, the rest was 468, the like we may judge of the rest. My host mingled four sorts of wine to the quantity of 300 quarts: A quart of the one wine was worth 12 pence, of another 10 pence, of the third 9 pence, of the fourth 7 pence: He sold a quart of the mingled wine for 11 pence, how many quarts of the first? how many of the second? how many of the third? and how many of the fourth kind did he mingle together? Here the total number given made of the number of the simple measures is 300. Therefore counterchaunging the differences as you see: 11 7 1 9 1 10 1 12 4, 2, 1 7 For the first term I take the total made of the differences, namely 10, for the second I take 300, for the antecedents of the reasons remaining I take 1, 1, 1 and 7, and then I frame a compounded proportion thus: 10 300 therefore 7 210 10 300 therefore 1 30 10 300 therefore 1 30 10 300 therefore 1 30 Whereupon I conclude that the quarts herein mingled, were first 210 at 12 pence a quart, then of each of the rest there were 30 quarts. An Apothecary was to mingle pepper, sugar, cinnamon and ginger to the quantity of fifty ounces. An ounce of pepper was worth 25, sugar 24, cinnamon 22, ginger 18. An ounce of the spice mingled together was worth 23: How many ounces of pepper, how many of sugar, how many of each other kind are to be mingled together? Here first of all the total given being made of the numbers of the simple measures is 500 Therefore the differences being counterchaunged as you see: 23 25 5 24 1 22 1 18 2 For the first term of the proportion I take the total made of the differences, namely 9, for the second I take 500, for the antecedents of the reasons remaining I take 5, 1, 1, and 2, and thereof I frame a compounded proportion in this manner: 9 500 therefore 5 277 7/9 9 500 therefore 1 55 5/9 9 500 therefore 1 55 5/9 9 500 therefore 2 111 1/9 Here therefore may be mingled of pepper 277 ounces and 7/9, of sugar 55 5/9, and so much cinnamon, of ginger 111 and 1/9. I say they may be mingled, but they shall not be mingled. For in this question each number of the extremes is a number of multitude, and therefore a manifold alligation may be made therein, of which manifold alligation I gave a note before in the first kind of prime alligation. CHAP. XXVI. Of second Alligation. HItherto of prime alligation: Second alligation followeth which by proportion compounded by multiplication and addition inferreth the mean. Here both the numbers and prices are given of each several measure, but there is but some part only of the mingled measures given. Therefore here for the first term I take the total made of the numbers of the simple measures, and for the second term, I take the total of the products made by multiplying those numbers by their extremes, and last of all for the third term of the proportion, I take the part given of the number of the mixed measures. Of two kinds of corn there were mingled 10 bushels at 16 pence a bushel, with 18 bushels at 12 pence a bushel: what is the bushel of mingled corn worth? Here first of all I add together the numbers of the simple measures 10 and 18, the total is 28. Then I multiply 10 by 16, and 18 by 12, and make 160 and 216, the total of them is 376. The total therefore of the prices of the simple measures is 376 pence. But alligation maketh the total of the prices of the mixed measures equal to the total of the simple measures as appeareth by the definition thereof. Therefore these 28 mingled bushels shall be worth 376. Therefore I conclude the question thus: 28 mingled bushels are worth 376. therefore one bushel of mingled corn is worth 13 3/7. P. Ramus the most famous Philosopher of our time, calleth this alligation, alligation of the mean sought for: and defineth it to be that, which (two extremes being given) seeketh out the mean by dividing the extremes added together by their number. He addeth these words as if there be two extremes the divisor must be 2, if there be three extremes the divisor must be 3, and so forth. Therefore by this definition if corn at 16 pence and 12 pence a bushel be mingled together to the quantity of 28 bushels as is aforesaid, the price of a bushel of mingled corn shall be 14 pence, for here the number of the extremes is 2, and the extremes 16 and 12 being added make 28, so that if thou divide 28 by 2, the quotient is 14. But let us see how true this is: Here by the rule of prime alligation the counterchaunging of the differences is in this manner: 14 16 2 12 2 By the which counterchange it is insinuated, that when there are taken two bushels of corn at 16 pence, then must there be two also taken of 12 pence the bushel. Therefore when there shall be 10 taken at 16 pence the bushel, then shall there also be 10 taken at 12 pence the bushel. The numbers of the simple measures 10 and 18 being added together make 28. Therefore I reason thus: If when 10 bushels at 16 pence the bushel are mingled with 18 bushels at 12 pence the bushel, the mean is then 14: therefore when 28 bushels of these simple several corns are so mingled, that the price of a mingled bushel is 14 pence, then shall 10 bushels at 16 pence the bushel, be mingled with 18 bushels at 12 pence the bushel. (For it is all one way from Newhause to Heidelberge, and from Heidelberge to Newhause.) But this second assertion is false (for here when 10 bushels at 16 pence the bushel are taken, then also there are taken 10 bushels at 12 pence the bushel as appeared before) therefore the first assertion is also false. Whether with 18, or with 10 bushels at 12 pence a bushel, you mingle 10 bushels at 16 pence a bushel, the extremes added together in both are 28, and their number is 2. Therefore whether you mingle 10 bushels at 16 pence the bushel, with 18, or 10 bushels at 12 pence a bushel, the price of a bushel of mingled corn shall be all one, and so the price of the cheaper wheat shall be the price of the dearer wheat, which is absurd. I always reverenced my master while he was alive as my duty required, and much more do I now embrace the writings of that most holy Martyr being dead. But it is incident to a man to err: and therefore he being a Philosopher of most holy memory, esteemed those men to be mad that thought it an hurtful thing for a common wealth to have their faults amended: therefore following his own decree by my definition, I correct his in this place. For he himself either did correct it before his death, or if he had lived any longer, I doubt not but he would willingly have amended it: if he did correct it, truly it was never my chance as yet to see his correction. Let 6 ounces of cloves at 36, and 8 ounces of cinnamon at 16, be mingled with 4 ounces of pepper at 15: how shall 4 ounces of the mingled spice be sold? Here the numbers of the simple measures are 6, 8, and 4, whereof the total is 18. The products of 36, 16, and 15, made by multiplying them by 6, 8, and 4 are 216, 128, and 60. The total of them is 404. Therefore I will say thus: 18 ounces of mingled spice are worth 404 therefore 4 ounces are worth 89 7/9. Let the prices of the simple spices be as they were before: and let 1/2 of an ounce of cloves, and 1/8 of an ounce of cinnamon be mingled with 1/4 of an ounce of pepper: what shall an ounce of the mingled spice be worth? Here first of all the unlike measures which are given 1/2, 1/3, 1/4 must be made like. For the property of general alligation requireth that the measures should be like. Therefore by reduction of fractions to one denomination, for 1/2, 1/3 and 1/4, I find out 6/12, 4/12, and 3/12. Then that the work may be more easy for the parts found out, by reduction of fractions to whole numbers, I take 6, 4, and 3. The total therefore of the numbers of the measures shall be 13. Then multiply 36, 16, & 15, by the numbers of their measures 6, 4, and 3, you shall produce 216, 64, and 45: whereof the total is 325. Then shall you conclude the question thus: 13 ounces of mingled spice are worth 325, therefore one ounce is worth 25 3/13. CHAP. XXVII. Of Manifold proportion continued in the terms. Manifold proportion compounded in the terms hath been handled hitherto: that remaineth which is continued in the terms. Manifold proportion continued in the terms is, when unto the disiunct terms of the reasons given, other proportional terms do answer, whereof each middlemost term joineth the antecedent reason with the consequent. As in the examples following 2 3 4 5 8 12 15 For the reasons given in the uppermost row of numbers (namely a subsesquialtera, and a subsesquiquarta) are in the nethermost row of numbers so continued and knit one to an other, that of the two middlemost terms there is made one, which according to the reasons given in the first rank of numbers is both the consequent of the former, and the antecedent of the latter. For as 2 is unto 3, so is 8 unto 12: and as 4 is unto 5, so is 12 unto 15. In this kind of proportion therefore there is intended an invention or finding out of the least terms continually proportional with the reasons given how many so ever. This invention is set forth in the 4 prop. 8, where it is demonstrated by what means, how many reasons so ever being given in the least terms, you may find out other least terms continually proportional, keeping the reasons given. Let that problem therefore be turned into two theorems: whereof let the one concern two reasons given, and let the other concern many reasons given in this manner: I If the middlemost terms being prime one to another of the two reasons given, be set crossways, and then the former of the two multiply the terms of the first reason, and the latter the terms of the consequent: the three products shall be the least terms continually proportional as the reasons given are. Therefore we are to note, that in the reasons given the middlemost terms are either prime numbers or compound, if they be compound, they must be reduced to prime numbers. As in 3, 2, 7, 4, the middlemost numbers are prime numbers in respect one of an other. Wherefore being set crosswise one under another, if you multiply 7 by 2 and 3, which are the terms of the former reason: and then multiply 2 by 7 and 4, which are the terms of the reason following, the products 21, 14, 8, shall be the least numbers continually proportional, as the numbers given are, as you see here: For as 3 is unto 2, so is 21 to 14: and as 7 is to 4, so is 14 to 8. Let these two reasons or proportions be given like one to an other 3, 2, 6, 4. Here 2 and 6 are compound numbers in respect one of another, wherefore I reduce them to prime or the least numbers, namely to 1 and 3, and place them crosswise, and multiply 3 into 3 and 2 the antecedents, and likewise 1 into 6 and 4 the consequents, the products are 9, 6, 4, which are the least terms continually proportional, as the reasons given, according as you see them here written. II. Any three reasons being given, if two terms prime in respect one of another, (whereof the one is the last of the least continual proportional terms of the former reasons found out, the other is the antecedent of the third reason) be set crosswise, and then all the numbers found out be multiplied by the first, & the numbers following by the last: the products shall be the least numbers continually proportional as the reasons given are: and so forth if there be more given. For example sake, let there be three reasons given 4 unto 3, 2 unto 1, 5 unto 6, to be continued in the least or prime proportional terms. First therefore I dispatch the former two reasons, and find 8, 6, 3. And because 3 (the last of them that are found out) is a prime number to 5 the antecedent of the reason remaining: therefore they being set crosswise, let 5 multiply 8 and 6, the products are 40 and 30: but let 3 multiply the numbers following 5 and 6, the products are 15, 30, so that we have now four terms in the least numbers continually proportional as the three reasons given were. The example is in this manner: Let these three be given to be continued after the manner aforesaid, 3 2,— 5 4,— 4 3. The same kind of working must be followed, saving that after that 2 reasons are dispatched, in steed of 8 and 4 which are compound numbers in respect one of another, I take 2 and 1, which are prime numbers as you see in this example: In the example following there are four reasons continued together, and after the same manner there may be as many as you please continued together in the least terms. Therefore the questions of proportion wherein any number offereth itself to be parted according unto many reasons in diverse terms so linked together, that the former term hath respect unto that which followeth, are to be answered by the rule of proportion compounded by addition, the least terms being first found out continually proportional as the reasons given are. The example. Two hundred crowns are to be divided on that condition that the first man shall have three times so much as the second: the second shall have four times as much as the third. How much shall every man have▪ Here there are two reasons propounded, the one is triple, the other is quadruple, which are knit together in this manner: 3 1 a c c e 4 1 That is to say, let a be triple to c, but let c be quadruple unto e. These are knit together by two middle terms 1 and 4, which being so linked together, that one and the same term according to the reasons given, may be both consequent to a, and antecedent unto e, I shall easily resolve the doubt. Now this will be done, if I find out the least terms continually proportional as the terms are which are given, namely 3, 1, 4, 1, the which terms found out will be these, 12, 4, 1. For than they being added together, the proportion will be inferred thus: 17 200 therefore 12 141 3/17 17 200 therefore 4 47 1/17 17 200 therefore 1 11 11/17 Let there be 24 crowns so to be parted, that as often as the first hath 3, so often the second hath 4, and as often as he hath two, so often the third hath 3, and as often as the third hath 6, so often the fourth hath 1. What shall each man's portion be? Here the terms of the three reasons 3, 4: 2, 3, 6, 1: are to be reduced (by the last theorem) into four terms continually proportional as the reasons given are, the which four terms are these, 3, 4, 6, 1. The total made of these terms added together, shall be the first term of the proportion, and being severally set down, they shall be the third term. The number which is to be divided shall be the middlemost, and then the proportion shall be concluded thus: 14 24: therefore 3 5 1/7 14 24: therefore 4 6 6/7 14 24: therefore 6 10 2/7 14 24: therefore 1 1 5/7 Four other men were to divide 120 crowns gotten in partnershippe, so that as often as the second had 5, so often the third had 9, and as often as the third had 7, so often the fourth had 11, and as often as the fourth had 9, so often the first had 13, what was each man his portion. This question is somewhat differing in show, but the working is all one. For the matter is dispatched by proportion of addition, the least numbers being found out continually proportional as the reasons given are. As you see here: 340 120 therefore 35 12 6/17 340 120 therefore 63 22 4/17 340 120 therefore 99 34 16/17 340 120 therefore 143 50 8/17 CHAP. XXVIII. Of continual proportion. L conjunct or continual proportion, is either Simple. Or Manifold: and is called progression, wherein we are to consider The order of the terms, Entering into the progression. Or Making a new progression. Or The sum of the terms. HItherto of disiunct proportion: now followeth continual. Continual proportion is that whose middle terms do all supply the place of an antecedent, and a consequent. In continual proportion the first extreme is only an antecedent, and the last is only a consequent: of the other terms, which are in the midst between the extremes any of them in respect of that which went before it is a consequent, in respect of that which followeth it is an antecedent, as this is in three terms 9, 6, 4, in four terms, 8, 12, 18, 27, in five terms 32, 16, 8, 4, 2 etc. Continual proportion is either simple, or manifold, simple continual proportion, is that wherein there is but one only mean between the extremes. This is the 9 d. 5. which affirmeth that proportion consisteth in three terms at the least, as in 9, 6, 4, where the extremes 9 and 4 are linked together by one only term. This is framed in the least terms, if you multiply the terms of the reason given being prime in respect one of another, both by themselves, and one by another, for the products are the terms of the continual proportion, according to the reason given. As for example, of 2, 1, which are the terms of a double reason, you shall make a simple continual proportion if you multiply 2 by itself, and then by 1, and last of all 1 by itself, for the products will be 4, 2, 1. Likewise of 3, 2, the terms of a sesquialter reason, you shall by the same means make 9, 6, 4. The property of it is this: The product made of the extremes, is equal to the product made of the mean, and contrariwise. This is the 17 p. 6. concerning magnitudes, and also the 20 p. 7. where this property is also attributed unto numbers, because in three proportional numbers, the product made of the extremes, is equal unto the square made of the mean. As in 9, 6, 4, the product made of 9 and 4, is 36, which is equal to the square of the mean, namely unto six times six, or which is all one to the product made of the mean multiplied by itself. For it is, as if you should infer the proportion disiunctively in four terms after this manner: 9 give 6: therefore 6 give 4. Wherefore it is to be gathered, that this property is general to all simple proportion whether it consist of three or four terms. Manifold continual proportion, is that wherein many mean terms proceed continually after one and the same reason. Hereupon it is called progression. As 1, 2, 4, 8. Item 16, 24, 36, 54, 81. In progression we have aneye to the finding out of the order of the terms making the progression, or else to the sum of them. The finding out of the terms is two fold: for either the terms given enter into the progression, or else new terms are made by their means. All the terms of a proportion given, can not be extended and continued. In manifold proportion it may be, but in other reasons it can not so well be done. It shall appear therefore by the theorem following, what terms given will admit a progression. If the last term saving one doth divide the product made of the last term, the quotient shall be the proportional term following, Or, If the last term doth divide the product made of the last saving one, the quotient shall be the antecedent proportional. This Theorem is gathered out of the 18 p. 9 It is also deduced as it were a consequence out of the propriety of continual proportion, whereby not only proportion, but any reason given may be continued. As in 3, 9, 27, wherein 9 divideth 729, the product made of the last number 27, the quotient is 81, which is the fourth proportional number. Likewise let 9 which is the second number divide 9 which is the product made of the first number 3, the quotient 1, is the first proportional number. So that the progression shall be in this manner, 1, 3, 9, 27, 81. This continuation of the terms in manifold proportion may be infinitely inferred only by the multiplication of the last term by the denominator of the reason or proportion: Yet in those whose reason is manifold, any term sought for may be found our readily in this manner: If you divide the product made of the last number multiplied in itself, by any of the numbers which went before it: by how many degrees the divisor went before the number multiplied, by so many degrees shall the quotient follow the same number multiplied. For example sake, let this double proportion be given. 2 4 8 16 32 64 128 4 3 2 1 2 3 4 Antecedents Consequents. And let 16 be the number to be multiplied by itself, th' product is 256, divide that by 8 which is the number next before it, the quotient 32 is the number which must follow next after 16. But if you divide the product 256 by 4 which went before 16 in the third place, the quotient will be 64, which must follow in the third place after 16. If you make the division by 2, which went before 16 in the fourth place, the quotient 128 must follow in the fourth place after 16, and so forth in the rest. For it is as if you should set the rule of proportion thus: If 2 give 16: then 16 shall give 128. Out of the foresaid abridgement, another ariseth, whereby we may find out any term of manifold progression, in this manner: If some certain terms of continual progression be given, and the numbers following one another from an unity according to their natural order, do answer the said terms from the second forward, the quotient of the product made by any of the terms whatsoever divided by the first term, shall be the term of the progression more by one, than both the numbers answering to the numbers multiplied, do amount unto being added together. As in this triple proportion, let there be given some terms, and let the numbers be written underneath them in order as you see: 3 9 27 81 243 0 1 2 3 4 Then if you multiply 27 by 81, and divide the product 2187 by 3, which is the first number in the progression, the quotient 729 shall be the sixth term of the progrestion which is more by one, than 2 and 3 (that is to say 5) are. Likewise if you multiply 243 by itself, and divide the product by the first, the quotient shall be 19683, to be placed in the ninth place, and so forth of the rest. The terms aswell of manifold reason as of proportion and progression, may in this manner be continued infinitely. But the nature of numbers doth not bear it so well in the other kinds of reason or proportion. For first of all, if of the reason given the terms be prime numbers one to another, then can not a third, much less a fourth or fifth in whole numbers be adjoined to them in continual proportion: as it appeareth by the 16 p. of the 9 As in the reason of 3 to 5, which is superpartient two fifts, you shall never bring forth a continual proportion, unless you join to it this mixed or surde number 8 1/3, which cannot be expressed by an whole number. Likewise how many proportional numbers soever being given, if the extremes be prime one to another, no such number can be given at the last, as the second is unto the first, by the 17 p. of the 9 As in this subsesquialter 4, 6, 9, because 4 and 9 are prime one to another, therefore there can be no whole number in such proportionality unto 9, as 4 is unto 6. Therefore in such kind of proportions this Theorem following taketh place. If the terms of continual simple proportion, not admitting an usual progression be multiplied by the antecedent of the reason whereof they consist, and the last be multiplied by his consequent: there shall be produced four lest terms continually proportional according to the numbers given: and so forth continually by multiplying the products by the antecedent given, and the last by the consequent given, you shall find the least terms how many so ever continually proportional according to the reasons given: As for example, seeing that this sesquitertia 16, 12, 9, hath not a fourth number in progression. Therefore first of all, I multiply all the numbers given by 4, the antecedent of the reason given between 4 and 3, whereof they were made: the products shall be 64, 48, 36. Likewise I multiply the last number 9 by 3 the consequent given, the product is 27, the which numbers are all continually proportional next unto the numbers given. After this manner many others may be found out. 4 3 The terms of the reason given. 16 12 9 The simple prop. 64 48 36 27 The first progression. 256 192 144 108 81 Behold also this example following of a manifold superparticular reason. 5 2 25 10 4 125 50 20 8 625 250 100 40 16 Thus much concerning the ranging of the terms: now followeth the sum of the progression. The finding out of the sum, is that which (the first term of the progression increasing being subducted from the second and the last) addeth unto the last term given a number, to the which the remainder of the last term hath such proportion as the remainder of the second term hath unto the first. I said (the progression increasing) that ye may understand that the first term is here taken for the least, and the last for the greatest. Now the mastery of this invention is general, being not only of force in manifold proportion, as that rule is which the common sort of Arithmeticians do prescribe, but all progression of what proportionality so ever given: It is taken out of the 33 prop. 9 which affirmeth this, of how many proportional numbers soever following one another. If from the second and the last term there be taken away numbers equal to the first, the remainder of the last term shall be unto all the antecedents going before it, as the remainder of the second term is unto the first term. Therefore these three proportional terms being found out, you shall set the excess or the remainder of the last term in the third place, and the remainder of the second in the first place, and the first or least term of the progression in the middlemost place, and then work by the rule of three, whereby you shall infer a number containing all them which were antecedents to the last, which being therefore added to the last shall contain the sum of them all. As in this manifold progression, 2, 4, 8, 16, 32, if 2 be taken from 4, and 32, there remain 2 and 30. Now as 2 the remainder of the second term is unto the first term 2, so is 30 unto all the antecedents. Therefore for the working of the rule, the terms shall stand thus: 2 give 2: therefore 30 give 30. The reason is in each place alike and equal. Therefore seeing that the total made of the numbers going before the last is 30, it being added to the last, namely unto 32, declareth the sum of the progression to be 62. Likewise in this subtriple progression, 2, 6, 18, 54, let 2 be taken from 6 and 54, the remainders shall be 4 and 52. Now as 4 the remainder of the second term is unto the first term 2, so is 52 the remainder of the last term unto all the antecedents. Wherefore the three terms known shall be thus ordered of the finding out of the fourth term. 4 2: therefore 52 26 And for so much as this fourth number, is equal unto all the antecedents, it being added to the last term 54, the total 80 shall be the sum of the progression. Likewise let there be this subsesquialter progression, 16, 24, 36, 54, 81. The remainders of the second, and last term are 8, and 65, whereupon by these three proportional numbers 8, 16, 65, the fourth number 130 is inferred, which is the total made of all the antecedents except the last, it being therefore added unto the last term 81, declareth the sum of the progression to be 211. Whereby it appeareth that the first, second, and last term of the progression being known the sum cannot lie hid. An Example. A certain man sold his house to be paid for it in wheat after such a manner, that when he that bought it came in at the first door he should give him one grain, at the second two, at the third 4, and so forth proceeding continually by double proportion according to the number of the doors. He that shrunk from his bargain should pay to the other 12 crowns for a penalty. Now being 60 doors, the question is how much wheat he was to pay. A rich man of Basill not ignorant of the huge increase of Geometrical progression, sold his house sometimes upon this condition being among his cups. But the condition of his bargain being brought unto an account, the number of the wheat was found to be unmeasurable, so that all the houses of Basill being turned in garners, were not sufficient to receive it. I thought good to set down the example. Let the 60 term of the progression be compendiously found out. 1 2 4 8 16 32 512 1024 0 1 2 3 4 5 9 10 524288 19 536870912 29 288230376151711744 58 576460752303423488 59 Wherefore the sum of all the grains shall be, 1152921504606846975 The which number is so great, that it soemeth sufficient to match the sand of the Adriatic sea. FINIS. A BRIEF TREATISE ADDED CONCERNING THE RESOlution of the Square and Cube in numbers, whereby the side of them may be found. A Figured number, is a number made by the multiplying of one number by another. The sides of the figured number, are the numbers by whose multiplication it was made. Albeit a number be not a magnitude admitting figure and angles, yet seeing that the multiplying of numbers is like unto the making of right lined plane figures, therefore for that resemblance of a Geometrical thing, the names of figures are attributed unto numbers produced by multiplication. For even as a right line drawn first a long one right line comprehendeth a plane figure, and then a long another, which is the depth, comprehendeth a body: even so the multiplying of one number by another, maketh a plane number (so called because the unities whereof i● consisteth, may be contrived into a plane figure) and that being multiplied by a third number maketh a number, receiving that title because it may be fashioned like a figure. The numbers multiplied one by another represent their sides. A figured number is twofold, a plane and a number. A plane number is that, which is made by multiplying two numbers one by another. 16 d. 7. Multiplication (as is aforesaid) maketh a certain thing like to a right angled parallelogram. Wherefore here are excluded all other numbers, which others call trigonals, pentagonals, etc. made by addition, whereof Euclids Arithmetic maketh no mention. A plane number is either made of two equal numbers as a Square: or of two unequal numbers, as an Oblong. 18 d. 7. A square number is made by a number multiplied by itself, as four times 4 are 16, five times 5 are 25, ten times 10 are 100: for these products may by their several unities be so displayed in a plane surface, that they may represent plane squares, as here you see in the number 16: 1 1 1 1 16 is the platform, 4 is the side. 1 1 1 1 1 1 1 1 1 1 1 1 wherefore by the making of this Square, it is evident that a square may be made of any number given, and that any number may be the side of a square. But as it is requisite that the sides (being lesser than 10) of the squares under an hundred should be gathered by the Table of multiplication: so the sides of greater Squares are to be sought out by art. The squares whose sides are simple numbers, are here set down as you see: The sides 1 2 3 4 5 6 7 8 9 The squares 1 4 9 16 25 36 49 64 81 The Square is known by finding out his side expressed by a whole number. Albeit the finding out of the side of a square, be applied to each number given as to a square, yet the square numbers only have a side to be expressed by a certain number of unities or by rational numbers, the other are to be expressed but in power only. The sides are commonly called roots by a metaphorical phrase. The side of a square is to be found by the Theorem following. If the odd degrees of a square number being marked from the right toward the left hand with points, you subduct from the number given, the particular square of the last period, setting the side thereof alone by itself, 2 then going on if you divide the remainder (if there be any) with the figure going before it, by the double of the side set alone by itself, 3 and multiply the quotient found out (being placed by the side which was first set alone by it self, and also before the doubled number on the right hand) by both the numbers (namely by the doubled number, and the figure set by it) being counted as one divisor, subducting the products from the number given, and then renew this last work of division so many times as there are pricks remaining, the side of the square shallbe found out. This artificial devise is borrowed out of the 4 p. 2. Where by demonstration it is proved, that if a right line be cut into two segments, how soever, the square of the whole line is equal to the squares of the segments, and to the two right angled figures made of the segments: as in the figure here annexed, the two Diagonals k g, and b f, are the squares of the segments a b, and a c. Also the compliments b k and f g, are the right-angled figures made by multiplying the line a b by b c. The self same parts are to be found in any square number. As for example in the number 169, whose side is 13. This side being divided into two pieces 10 and 3, multiply each piece by itself once, namely 10 by 10, and 3 by 3, then multiply one by another, as 10 by 3, & 3 by 10, so shall you have 4 plane numbers, whereof two are Squares as here you see. 10 3 10 3 100 30 30 9 169 Therefore as the Square 169 is made by the adding together of these four plane numbes, so by subducting them severally it is resolved. First therefore I mark each odd place with points because the particular squares are to be found in the odd places. Then for so much as the unity standing under the first point next unto the left hand, and representing the last period, is both a square and the side of a square: that figure therefore being set alone by itself in the quotient, and being subducted from the unity standing over the point, there remaineth nothing. This unity set alone by itself in the quotient shall signify 10, when another figure is set by it representing the side of some other particular square. Whereupon I say, that the greater diagonal kg, is now subducted from the whole square, and the side of it k i or a b (for they are equal one to another) and also the side of one of the compliments is found out. This is the first step to this resolution. Moreover I double the figure found out, because being doubled, it is the side of both the compliments taken jointly together, namely k i and g i. Then setting 2 the doubled number under 6, I divide 6, (which in this place is as much as 60, and representeth both the compliments) by 2, the quotient 10 3, representing the other side remaining of the compliment, namely, i f, or b c, the which number I set in the quotient, and count it for the segment remaining of the right line given. Wherefore because this number 3 is the side of the diagonal remaining, that is to say, of the lesser square b f: therefore being set by the divisor on the right hand, and multiplied both by itself, and by the divisor, it bringeth forth three plane numbers, namely, the square b f, and the two compliments a i and i l, which being subducted from the numbers standing over them, there remaineth nothing. The example is thus: 1̇69̇ (13 123 3 69 which is all one as if you had set down the numbers found out thus: 1̇69̇ 100 The greater diagonal. 60 The two Compliments. 9 The lesser diagonal. 169 Take this for an other example to make this devise more plain. Let the Square given be 1764. This number being marked with two points telleth us that the side thereof is to be written with two figures. First therefore beginning at the point on the left hand, I seek the side of the last period, namely of 17. But for so much as it is no square number, I take 4 the side of the next lesser square, which I set alone by itself in the quotient, and then multiply it by itself, the product is 16, which being subducted from 17 there remaineth 1. Moreover, I double the side found out, the product is 8, I place this doubled number under 6, and by it I divide 16 standing above it, the quotient is 2, which must be set by 4. This quotient 2 must be set before the divisor 8 on the right hand under the point, and then must it be multiplied both into itself and into 8, the product is 164, which being subducted from the figures standing over them, there remaineth nothing, whereby I gather that the number given is a just Square. The example standeth thus: The same manner of working is to be followed in greater square number given, saving that the former part of the work is to be used but once, but the latter part is to be followed so many times as there are points remaining, excepting the last. As in 54756, I say, that the side of the square next unto 5 is 2, therefore 2 being set in the quotient, and multiplied by itself make 4, and taken from 5, the remainder is one. Moreover I double the quotient, the product is 4, which I set under the next figure toward the right hand, and thereby divide 14, the quotient is 3, the which 3 being set both in the quotient, and also before the divisor toward the right hand, I multiply both the numbers by it, the product is 129, this being subducted from 147 standing above it, the remainder is 18. But because there is yet one point remaining with the which I have not meddled, therefore again I double all the whole quotient. For in this case I must take 23 for the one side of one former square, and generally in great numbers when I light upon more particular squares than two, I must esteem them but as two, and take the sides which are first found out but as the sides of one only square. Therefore twice 23 are 46, by this I divide 185, the number to be set in the quotient is 4, the which number also must be set before the divisor on the right hand, then must 464 be multiplied by 4, the product is 1856, this product being subducted from the numbers standing over it, there remaineth nothing. The example standeth thus: See also the example following: 10̇94̇28̇64̇ (3308 Out of this invention this conclusion followeth: The number whose side cannot be expressed by whole numbers, is no square number. Such are all prime numbers, & (the squares themselves excepted) all other compound numbers. For if in them you desire to find out the square side, you shall labour in vain, because they are not squares for to the whole numbers arising in the quotient, there will be some fraction adjoined, whereby it cometh to pass, that the number of the side is not to be expressed by a true number, and commonly it is called a surde number. Yet if you adjoin to the side found out the number remaining, taking his denomination from the double of the side augmented by an unity, you shall find the next side that may be like to the side of a square. As if from 40 you take the nearest square, namely 36, the remainder is 4. Here therefore the side sought for of the square, exceedeth not the side found out by an unity, but either by one, or more parts of some whole number: wherefore I double 6 the side found out, and add an unity to it being doubled, the total is 13, this number I set under 4 the remainder, and say that the side of 40 sought as near as may be, is 6 4/13, the denominator of the fraction being added to the greatest square in the number given, namely unto 36, maketh the next greatest square above it, namely 49, whose side is 7. But this surde side namely 6 4/13, multiplied by itself maketh 39 133/169, which are not just equal unto 40 the number given. The like judgement may be used in the rest which are not squares. Thus much concerning plane numbers, but especially such as are square numbers. The number followeth, which is a number made of three numbers. 17 d. 7. A number is made by a twofold multiplication, even as a body among magnitudes is made by a twofold comprehension: the first is like that which maketh a plane figure, the second is like that which maketh a body. As three times 4 taken five times: for three times 4 are 12, this is a plane number, five times 12 are 60, this is a number. A number is made either of 3 equal numbers as a Cube: or of 3 unequal numbers as an Oblong. 19 d. 7. A cube in numbers is made by multiplying the side thereof by the square of the side: whereby we may gather, that to multiply a number cubically, is to multiply the number made of itself, by himself again as four times 4 four times are 64, which is a cube having relation to a Geometrical thing, partly because their making is like, partly also because the unities of the number set in a certain distance one from another resemble a cube. The cube is known by finding out the side expressed by a whole number. For every Cube in numbers hath such a side as may be expressed by whole numbers, but in magnitudes it is not always so, as in deed in magnitudes there are many things not to be expressed by whole numbers. Now for so much as the side of any Cube under 1000 is a simple figure, it is meet before we undertake to find out the side of any greater number, that we should know what Cube is made of each simple figure, and what is the side of any Cube lesser than 1000, as I have here set them down. 1 2 3 4 5 6 7 8 9 1 4 9 16 25 36 49 64 81 1 8 27 64 125 216 343 512 729 But in searching out of greater Cubes, we must proceed as the Theorem following teacheth us. If you distinguish with points as it were into periods, the Cube given beginning at the first figure on the right hand, and omitting each two figures continually, and first of all subduct the particular Cube of the last period from the number given, setting the side thereof in the quotient: and then set the triple of the quotient under the figure next following the former point on the right hand, and the square of the quotient being tripled beneath it one degree more toward the left hand: and afterward divide the number above written by the triple of the square, setting the quotient by itself, and then multiply the divisor by the quotient found out, and the tripled square by the square of the quotient, and the quotient cubically, subducting the products (so orderly added together, that each figure may answer the numbers whereof it was multiplied) from the number given: and renew this last manner of division so many times as there are points remaining, the side of the Cube shallbe found out. The cube of the greater segment 10. The cube of the lesser segment 3, The greater Parallelipipedons. The lesser parallelipipedons. The cube therefore hath eight particular solids in numbers which are made of the parts of the number given, namely of 10 and 3 in this manner. First let there be four plane numbers made, each part being multiplied by itself, and one by another: 10 3 10 3 100 30 30 9 169 If again you multiply these planes by the same parts, there will arise 8 solides, as you see here: 9 9 30 30 30 30 100 100 3 10 27 90 All these being added together are equal unto 2197, the cube of the whole line. 90 300 90 300 300 1000 The same way therefore that is kept in making the Cube, is also to be followed in resolving the Cube. As for example, I mark the cube given with points in this manner, 2197. Then I subduct the particular cube of the number set under neath the last point: but for so much as that number is no Cube, I take the nearest to it, namely an unity, which also I set in the quotient. This unity in the number given is 1000, but in the quotient it is but 10. The unity subducted from 2, the remainder is 1, which must be written over the number given. So that the greater cube A is to be supposed to be subducted from the number given. This is the first step of this work: I triple the quotient found out (that is to say, I multiply it by 3) this triple representeth the three sides taken jointly together of the three lesser solids marked with C, I place the tripled number under 9 Again I multiply the quotient squarewise, and triple the product which maketh likewise 3. This product resembleth the three squared sides taken jointly together of the three greater solids, marked with D, I place the product one degree lower toward the left hand underneath 1. With it I divide 11, which is written above it, the quotient is 3. This segment or quotient 3, being multiplied by 3 the divisor, maketh 9, which in respect of the place wherein it standeth is 900, and representeth the three greater solids marked with D taken jointly together. Furthermore the same quotient being multiplied squarewise maketh 9, and multiplied afterward by the tripled number standing under 9, it maketh 27, which in respect of the place wherein it standeth is 270, and representeth the three lesser solids marked with C. Last of all, the same quotient multiplied cubically, breedeth the lesser Cube B. These three products therefore being added together, and the total subducted from the numbers standing over it, there remaineth nothing, which importeth that the number given is a Cube. The example is as you see: Take these examples following to make the matter more plane. Let the side be sought out of this Cube, 16387064: contrive it therefore (as it were) into certain periods with points. Then first of all, seek out the side of the cube, which you find in the last number over the point next to the left hand. But for so much as 16 is no cube, take 2 the side of the next cube under it that is to say, of 8, and set it in the quotient, and subduct 8 the cube from 16, there remain 8. This first work is not to be renewed throughout the whole number, but the rules following must be repeated as often as there are points remaining. The first step to find out the side is in this manner. Moreover triple the quotient found out, the product is 6, which are to be placed under 8, namely under the figure following the next prick toward the right hand. Then multiply the quotient by this tripled number (or which is all to one purpose, square the quotient, and then triple the products) it maketh 12, set that number in a lower place one degree nearer to the left hand: and make it the divisor. Divide 83 by 12, observing this rule in choosing your quotient, that it be no greater, then that the numbers afterward produced by multiplication may not exceed the numbers standing over it. So that here you shall take 1 in 8, but five times. Afterward by this number 5, multiply the divisor 12, and by the square of 5 multiply the trebled number 6, and last of all, multiply 5 cubically, so shall you produce three numbers, namely 60, 150, 125, to be described as you see: These numbers added together, and subducted from 8387, the remainder is 762. The second step to find out the side is in this manner: And because there is yet one point remaining, this last manner of division must be wrought again. First therefore I triple the quotient, the product is 75, which must be so placed, that the first figure thereof, namely 5, may stand under 6, the second under the 0. Again multiply the quotient by this tripled number (or which is all one, square the quotient, and triple the product) it maketh 1875, which shall be the divisor, whose first figure namely 5, must be placed under 7, the last figure of the tripled number. Then you see that I may be contained in 7 many times, but I can take it but four times, I set 4 in the quotient, and multiply the divisor by 4, the product is 7500, afterward I square 4, it maketh 16, which I multiply by the tripled number 75, the product is 1200. Last of all, I multiply 4 cubically it maketh 64: these products added all together make 762064, this number being subducted from the Cube, there remaineth nothing, whereby I gather that the number given is a just Cube. The third step to find out the side is in this manner: Behold also the example following: 614̇125̇000̇ (850 Another manner of working: Hitherto the princely high way to find out the side of the Cube hath been declared. But there are other ways also bending thereto, and leaning to the same principles: whereof this is one. Having found out in the Table of the simple cubes, the figure representing the side of the cube contained in the number standing under the first point on the left hand, set it in the quotient, and subduct the particular cube of that figure as you did before: then square, that figure, and triple the square, the product shall be the divisor, whose first figure shall be set under that figure which is on the right hand next of all to the first point aforesaid. 2 See how many times the divisor is contained in the number written over it, multiply the divisor by the quotient, and subduct the product from the dividend. Yet here you must take heed, that you choose not a greater quotient, then that the products made afterward thereby may be subducted from the numbers given. 3 When the subduction is done, triple the first figure which was set in the quotient, and place the last number which was set in the quotient on the right hand of the product. 4 Multiply this whole number by the square of the quotient last found out, and set down the product so, that the first figure thereof toward the right hand may stand under the point next going before on the same hand, and then subduct it from the number given. As in 804̇357̇ the particular Cube, namely, 729, being taken from the number standing under the last period upon the left hand there remain 75357, the side of that particular cube is 9, which I set in the quotient. I square that side, it maketh 81, and triple the square, the product 243 is my divisor, which I set under the number given, so that 3 may stand under 3. With this divisor divide the number standing over it, you shall find 2 to be contained in 7 three times. I set therefore 3 in the quotient, and multiply the divisor by it, the product is 729, which being subducted from 753, the remainder is 24. As you see in this induction. Moreover I I triple 9, the product is 27, by the which on the right hand I set 3 the quotient last found out, the total is 273. This number I multiply by 9 the square of 3 the quotient last found out, the product is 2457, which being subducted from the number standing over it, there remaineth nothing. The induction is on this manner: The self same work may be dispatched another way somewhat differing from the former in this manner. The figure in the quotient being found out by subducting the particular cube, and also the second figure in the quotient being found by division, let the whole quotient be tripled, and let the tripled number be multiplied by the former figure in the quotient. Then let the product be multiplied again by the latter figure found out, and let a cipher be set on the right hand of that product. Last of all, let the cube of the latter figure found out be added to this product, and let the total sum be subducted from the number given. As in 373248. The first induction is in this manner: Moreover I square the side found out, it maketh 49 and triple the square, the product is 147, which shall be the divisor, by this I divide 302, the number written over it, the quotient is 2. Now I triple the whole quotient 72, it maketh 216, and multiply this triple by 7, the former figure in the quotient, the product is 1512. I multiply also this product by 2 the latter figure of the quotient, and set a cipher on the right hand of it, so that it maketh 30240, unto this number I add 8 the cube of the latter figure found out, the total is 30248, which being subducted from the figures above it, there remaineth nothing. The induction is thus: All the points of the number given being examined, if any thing remain, it signifieth that the number given is no Cube: wherefore the true side of it cannot be given exactly in numbers. Yet if it please you to sift out the nearest side that may be, by the first kind of reduction of mixed numbers, you must reduce the number given into a cubical fraction of a great denomination, and afterward seek out the cubical side of that fraction. For example sake, because 120 is no Cube, therefore let it be reduced into sixty cubical parts, after this manner. Multiply 60 cubically in itself, it maketh 216000, by this being taken for the denominator of the fraction, multiply 120 the number given, the product is 25920000, whose cubical side is 295/60 that is, 4 11/12 the nearest to the true side that may be. FINIS.