An introduction for to learn to reckon with the pen, or with the counters according to the true cast of algorithm, in hole numbers or in broken, newly corrected. And certain notable and goodly rules of false positions thereunto added, not before seen in our english tongue, by the which all manner of difficile questions may easily be dissolved and assoiled. Anno. 1546. woodcut of a young man in Tudor dress seated indoors at a table with a pen in his left hand and counters at his right To the reader. THat art and feat (dear reader, whom utility and necessity both do commend, needeth greatly of no other commendation. How profitable and necessary this feat of algorithm is, to all manner of persons, which have reckonings or accounts, other to make, or else to receive, needeth no declaration. Neither is this art only necessary to those, but also in manner to all manner of sciences, and artificyes. For what craft is that but it sometime doth occupy not only one part of this feat, but all the parts. And for bycause that divers rules in this book have not been in times passed, very commodiously expressed, and set forth, and many examples, more than nedid a great sort coheaped together. Therefore pains have been taken both in the better and more clearer declaration and expressing of the said rules: and also in the resecting and cutting of divers superfluous & void things, rather hindrance to the diligent reader, than fortherance. furthermore there is added the rules of false positions, the which how convenient and profitable they be to the ready solution of all hard & misty questions: when ye have red them then judge. Now thenne ye shall understand that in this art there are vii necessary and distinct parts to be known. Numeration, Addition, Subtraction, multiplication, Partition, Progrestion, and Reduction. Of the which vii hereafter shall be singularly entracted of each of them in their chapters. But I advertise you first to begin at the first part, & then successively to the second, and third. etc. learning every part by itself exactly, as they be set forth in this book: for if you leap to the second part, before you have perfectly the first, or to the third, before you have seen the second, you shall never prosper ne profit in this art. Vale. Finis. ¶ The first part is of Numeration. NVmeration is a manner of expressing of numbers by certain figures which are called figures of algorithm, the which be ten, as in this example. i. two. iii. iiii. v. vi. seven. viii. ix. 1 2 3 4 5 6 7 8 9 0 Of the which nine be significative, the tenth called a siphre, signifying nothing of itself, but only set before the other significative fygurs augmenteth their signification. In numeration by this craft ye must evermore begin at the right side of the book, and so towards the life side, as in this example. k i h g f e d c b a 3 2 0 4 6 7 5 2 8 9 This figure 9 under a, standeth in the first place. 8 under b, standeth in the second place, and so forth to the end, so that 3 under k, standeth in the last place. By these ten figures all manner of number possible to be excogitate, may clearly and plainly be expressed: which all be it, that of themself they signify but simple and little number, as ye see afore, yet according to the diversity of the place they stand in, diversly doth their signification amount. Wherefore in numeration ye must note two things, the figure significative, and the place it standeth in for the signification of the figure dependeth upon the number of the place it standeth in: For example, this figure 8 standing alone, or in the first place signifieth but viii but if he stand in the second place, as here 80 he signifieth viii times ten, which is called four score. if he stand in the third place, as here 800 he signifieth viii hundredth. etc. Therefore ye must know perfectly the signification of every place, before ye can perfectly number. wherefore understand ye, that the first place is a place of unitees, so that a figure standing in it, signifieth no more than though he stand alone. The second place is a place of tens. The third is a place of hundrydes'. The fourth place is a place of thousands. The fifth place, a place of ten thousands. The vi place a place of hundredth thousands. The vii place, is of thousand thousands, which is called a million. The viii place, is of ten millions. The ix is a place of hundredth millions. The ten of thousand millions. The xi of ten thousand millions. The xii. of a hundredth thousand millions. The xiii of a thousand thousand million, which is called million upon million. And so forth infinitely, every place ensuenge, signifieth ten times as much as the place going before. This must thou know perfitly what every place giveth and signifieth: for the place giveth denomination, and the figure standing in the same place expresseth how many of the same denomination is to be understand: as in example ye shall more plainly perceive In this sum 3400872619 this figure ● standyth in the four place, now by your rule afore, the four place is a place of thousands, than this figure 2 standing in the same place giveth us to wit, that it is two thousand, Likewise this figure 8 standeth in the vi place, now by your rule afore spoken of, the vi place is of hundred thousands: then this figure 8 situate in the same place receiveth denomination of the place, and representeth to us viii. hundredth thousands. Like wise this figure 1 standeth in the second place and for bycause the second place is a place of tens, therefore this figure 1 standing there is bound to the signification of the place, and so signifieth one ten: if a figure of 4 stood there, it should signify four tens, that is forty, and so forth. Then for a farther declaration of the foresaid sum, and all other like sums. This figure 9 standing in the first place, signifieth but himself, that is ix This figure 1 standing in the second place, by cause the second place is ever a place of tens, signifieth one ten. The figure 6, standing in the third place, because the third place is a place of hundred, doth signify vi hundredth: the figure 2 in the fourth place, signifieth ii thousand: the figure 7 for bycause it standeth in the fift place, and that place is a place of ten thousands, it signifieth seven. times ten thousand, the which is three score thousand and ten: the figure 8 in the vi place signifieth viii hundredth thousand: the sypher 0 that standeth in the seven. place signifieth nothing, but only maketh up a place that the figures significative following may increase there signification. Like judgement is of the sypher standing in the viii place: in the ix. place standeth the figure of 4, and this place is a place of hundredth millions: therefore this figure 4 there signify the iiii, C. millions. In the ten place standeth the figure 3, and this place is a place of thousand millions: therefore it signifieth iii thousand millions. So the hole sum is, three thousand milions four C. millions viii C. thousand, three score thousands xii thousand vi hundredth, and xix Now to exercise yourself in numeration numbered with yourself these sums following, & you shallbe perfect enough. Million. Mil. Mil. Mil. x. M. C. x. M. C. x. one 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 Furthermore thou must note that there be in algorithm three manner of numbres, Diget number, Article, and Composte. The digette number, is all manner of numbres, which are under ten as these. 9 8 7 6 5 4 3 2 1 The article number is, all numbers which are of ten as these. 10 20 30 40 50 60 70 80 90 The compound number is all manner of numbres which are compound or made of the dyget & article together, as follow. 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 39 And so forth of all other. This is sufficient for the knowledge of numbered in algorithm. ¶ The second part called Addition. ADdition is a collection of divers and sundry sums, into one total sum, which containeth as much in him as all the other sums, being before sundry. In addition are two numbers to be considered, the one is, the numbres which must be adjoined together: the other is the numbers which redoundeth of their addition together, which otherwise is called the total sum. Then when ye will add many sums together, first writ them fair the one directly under the other, so that the first figure of the one, be right under the first of the other, & the second under the second: every place correspondent under other: that done draw a line under all these several sums, as is to see in the example following. And when ye will add your numbers together, begin at the first places of your sums & add all the figures that ye see in the first places of all your sums together, and that that cometh of that addition, see whether it be digette number, article, or compost: if it be but dyget, set the dyget beneath the line, directly under the same first places: if it be article put a cipher beneath the line, right under the same first place, & reserve the article to be added to the next places of thy sums & there do likewise, if it be compost, set the dyget under the line right under the same place, & reserve the article in your mind likewise to be added to the next places of thy sums: when the figures standing in the last places of your sums be adjoined together, if any article or articles remain, set them down next to the figure ye set last before under the same line: as by examples shall appear. The first sum. 6 7 8 9 4 The second sum 3 4 5 6 7 The third sum 2 3 4 5 6 The fourth sum 7 8 9 3 4 The fifth sum 6 7 4 2 5 The sixth sum 3 4 3 2 2 Summa totalis 3 0 6 5 9 8 Your figures set after this sort, add all the figures that ye find in the first places of all the sums together, beginning at the nethermost saying, 2 & 5 is 7, and 4 that is 11, and 6 that is 17, and 7 that is 24, ans 4 that 'tis 28. This is the hole sum of the figures added together found in the first places, the which number is composte: wherefore, as is in your rule, ye must set the dyget right under the same place, beneath the line, the which is 8, & keep the articles in your mind, which is 2. Now to the second place, toward the lift hand, say, 2 that I have in mind and 2 is 4, and 2 maketh 6, and 3 is 9, and 5 is 14, and 6 is 20, and 9 is 29, now set the 9 under 2, & keep 2 in mind, and add them to the first figure of the third place, that is 3. Now say 2 and 3 is 5, and 4 is 9, and 9 is 18, and 4 is 22, and 5 is 27, and 8 is 35. Now set 5 under 3, and keep ● in mind. Now to the fourth place, toward the lift hand where 4 standeth, now 3 that ye have in mind and 4 is 7, & 7 is 14, and 8 is 22, and 3 is 25, & 4 is 29, and 7 is 36, set 6 under 4, and keep 3, and add that 3 to the undermost figure of the sixth sum that is 3, and say 3 and 3 is 6, and 6 is 12, and 7 is 19, and 2 is 21, and 3 is 24, and 6 is 30. All the figures of this place added together as ye see, maketh article number wherefore according to your rule set a cipher 0 under that place beneath the line, & the article which is 3 next to the same cipher, & all is finished. And all these sums thus collected together maketh 306598. another example of addition. 1 0 0 6 6 7 8 4 5 6 0 0 0 3 1 9 5 0 5 0 0 5 4 5 1 6 1 8 0 1 2 0 2 6 4 2 1 2 2 0 2 0 5 2 5 7 9 Begin first as ye did before, at the first places, adding them all together, beginning at the nethermost, saying, 1 and 2 is 3, and 1 is 4, and 5 is 9, this is the hole sum of the figures standing in the first place, the which is diget number, and therefore according to the rule, set it right under the same place beneath the line: then proceed to the second place and begin at the neither end saying, 2 and 6 is 8, and 5 is 13, and 4 is 17, these number is compost number, therefore set the diget right under that place beneath the line, which is 7, reserving the article in your mind, and so to the third place, saying, 1 that I have in my mind and 4 that is 5, & 2 is 7, and 1 is 8, and 9 is 17, and 8 is ●5, this number also is compost, wherefore set the diget 5 under that third place, & reserve the article 2 in mind to the next place, then to the next places saying, 2 that I have in mind & 6 is 8, & 1 is 9, & 5 is 14, & 1 is 15, & 7 is 22, this is also compost, therefore set the diget 2 under that four place, & reserve the article 2 to the next places, then to the fifth place saying 2 that I have in mind & 4 is 6, & 3 is 9, & ● is 15, this is also compost number, set the diget 5 under the fifth place, and keep the article in mind, to the vi place saying, 1 the I have in mind & 8 is 9 & 5 is 14, and 6 is 20 this is article number, therefore according to the rule set a syfer under the place beneath the line, & keep the article in mind and cum to the vii place, in the which places for because thou findest nothing but ciphers to the thou mights adjoin thy article reserved, the which was 2, therefore under the same vii place set that same reserved 2, and then come to the viii place and there findest thou nothing but ciphers wherefore under the same place set beneath the line a cipher, according to the rule, then c●●●e to the ix places & say 5 and 6 is 11, & 1 is 12, the which is compost number, therefore set the diget which is 2 under the line & reserve the article in mind, which is 1, now for because there is no more places whereunto ye might add this reserved article therefore according to your rule ye shall set it down next unto the figure that ye did set under the line last, as is in your example. This ii examples were sufficient enough to the readiness of addition, how be it yet that it may be the plainer I will subscribe an other example. 1 4 6 9 9 0 0 0 3 8 2 9 0 2 0 0 0 1 0 9 1 6 0 0 1 0 2 0 0 0 0 5 5 1 0 0 8 0 0 Add the first place together. first there thou findest nothing but cyfers, wherefore set a cipher under the line, and so likewise in the second place. In the third place thou findest 6 and ● which maketh 8, the which for because it is dygette number, set it under that place beneath the line. In the four place is 1 and 9 which maketh 10, and for bycause that this is article number, set a cipher under that place beneath the line, and reserve the article to the next place saying, 1 that I have in my mind and 2 is 3, and 9 is 12, and 9 is 21, and 9 is 30 this is also article number, wherefore set a cipher under that place beneath the line, and reserve the article 3 in mind to the next place. Then come to the vi place: saying, 3 that I have in my mind and 2 is 5, and 6 is 11, this is composte numbered therefore I set the dyget which is 1 right under that place beneath the line, and reserve the article 1 to the ne●te place, saying 1 and 1 is 2, and 1 is 3, and 8 is 11, & 4 is 15, this is also composte, therefore set the dyget 5 under the line, and add the article reserved to the figure in the ne●te place saying, 1 and 3 is 4, and 1 is 5, this is dyget numbered, therefore set it under the line, and all is done. ¶ Certain examples to practise yourself in, touching the exercise of Addition. 1 6 7 6 8 9 0 0 3 6 2 1 9 8 8 0 9 2 0 0 0 0 3 2 1 1 1 1 6 8 4 1 1 9 4 2 1 3 2 6 1 7 5 5 2 6 9 7 9 1 0 0 0 0 0 0 0 1 3 4 5 6 2 8 9 1 0 0 2 0 1 0 1 1 0 0 0 0 0 0 0 3 8 9 2 1 0 0 0 9 2 3 9 7 3 9 0 9 0 9 0 2 0 1 0 0 0 2 6 5 1 2 6 0 0 0 0 2 5 4 3 2 0 7 1 2 1 6 5 4 0 0 0 9 6 2 0 0 1 1 0 0 0 1 1 7 8 9 5 1 0 0 7 2 6 4 0 0 0 0 8 6 0 0 0 0 9 9 8 0 0 0 7 8 0 0 0 0 5 9 0 0 0 0 1 0 0 0 0 3 3 9 6 8 0 0 3 9 0 0 0 1 6 2 0 5 8 1 2 9 0 1 0 0 0 0 0 9 6 1 9 1 0 0 0 0 0 9 0 1 6 4 5 7 3 0 0 0 2 6 8 1 0 6 0 9 8 6 1 0 0 0 9 3 9 1 0 0 0 9 3 9 2 0 0 0 9 1 7 1 0 6 4 6 6 9 ¶ Of the prove of Addition. ¶ For the prove of addition, ye shall make a cross after the fashion that followeth. And then ye shall come first to the addible sums, and pluck out all the 9 that ye find there, and the rest what so ever it be, that will not make 9 set it at the upper side of the cross. Then come to the total sum under the line, and likewise deduck all the 9 that ye can find there and that that remaineth, not able to make 9 set it ●t the undermost part of the cross, and if it be like the remanant of the addyble numbers which standeth in the upper part of the cross, your work was good, if not it was nought, as by example ye shall the better perceive. ¶ An example of the prove. A 1 5 0 6 7 0 B 3 3 0 4 2 8 C 5 8 1 0 9 8 ¶ Now for to make the prove of this numbres, ye shall begin at the first figure that ye have made, in saying, 8 and 0 is 8 and 2 is 10, take away 9 then there resteth 1, than 1 and 7 is 8, and 4 is 12, take away 9 rest 3, than 3 and 6 is 9, than to the two ciphers of nothing that nothing do signify, than 3 and 5 is 8 and 3 is 11, take away 9 rest 2, than 2 and 2 is 4, this 4 it behoveth you to put at the neither end of the cross, than come to the place of C. under the line and say 8, ye shall leave the 9 and the cipher 0 that is nothing worth, and adjoin 1 thereto and make it 9, and leave that, than 8 and 5 is 13, take away 9 rest 4, which 4 ye shall put at the upper end of the cross and than is your prove god, for both the ends be like as ye see in this figure of the cross . And at the two other ends ye shall put two 0 0 in certifying that of them cometh nothing. ¶ An other example. A 7 8 9 1 5 4 3 2 6 li. B 4 9 3 0 0 6 7 1 5 li. C 2 0 9 9 3 4 7 8 4 li. D 4 6 0 6 4 5 5 3 0 li. E 9 3 6 4 5 8 7 7 8 li. F 4 4 5 1 9 3 0 0 1 li. G 3 3 3 4 3 9 3 1 3 4 li. ¶ We shall say semblably 1 and 8 is 9 and always leave them, than 0 that doth nothing, than 4 and 5 been 9, than 6, than we shall return to the tenths, and shall find 0 that doth nothing, than 7 that maketh 9 teste 4, than 3 been 7, than 8 been 9 rest 6, then 1 is 7, then 2 is 9, than 0 that is nothing worth, than 7 and 5 is 9 rest ● then 7 is 9 rest 1, then 7 is 8 and 3 is 9 rest 1. then we come to the place of hundreds, & adjoin the 2 to the 3 that is 5 than 8 is 9 rest 4, and so consequently unto the end. And if peradventure we find this figure 9 because of the brefnes, we shall leave it. And shall find at the end 9, therefore we shall put at the end of the cross 0 in sygnifienge that there is nothing above 9 And so shall we do in the number of G. and we shall find like 9 for the which semblably we shall put 0. And so is the addition good & well made. ¶ The prove As touching of addition in broken numbers, ye shall find that under the title of Reduction hereafter. ¶ Of Subtraction, the third part. SVbtraction is a manner of debating or subducing a less sum out of a greater: or like of like showing what remaineth. In subtraction are two numbers, the first is the number abated, the second, the number abating. ¶ Then when ye will subtrahe any one number out of an other. first ye shall write the number to be abated, and under it directly figure under figure, and place under place writ the abatoure, and beneath these two sums draw a line, then begin your subtraction at the first places, and subduce the figure standing in the first place of the abatour of the first figure standing in the first place of the number to be abated: and the rest that remaineth after the abatement set it right under the same place beneath the line: and so do like wise in the second the third, and all other places. And when ye have all done, the number that shall remain under the line, shall be that, that remaineth after the subduction of the abatour of the number abated. As for example. Lent 8 3 4 5 6 Paid 4 1 1 3 1 rest 4 2 3 2 5 ¶ I lent a man 83456 li. of the which he hath paid me 41111 li. again now I desire to know how much remaineth. Then according to the rule, first I set the lente money, and right under that I set the repaid money, figure under figure, and place under place: as ye see by the example Under both these sums I must draw a line. Begin to subtrahe the under sum out of the upper, saying, 1 out of 6 remaineth 5, this 5 that remaineth according to the rule set under the same place beneath the line: then to the second place pluck 3 out of 5 remaineth 2, set that under the line: then to the third place, pluck 1 out of 4 remaineth 3, set that under the line: then to the fourth place, take 1 out of 3 remaineth 2, set it under the line: then in the fifth place, take 4 out of 8 remaineth 4, set that also under the line, & so thou hast finished: Then thou shalt understand that it which is under the line is the remnant of the money not yet paid. ¶ An other example. 8 7 6 6 0 li. 6 7 5 6 0 li. 2 0 1 0 0 li. ¶ Begin at the first place saying, 0 out of 0 remaineth nothing, set the figure of nothing under the line: then to the second place 6 out of 6, remaineth nothing, set the cipher under the line: then to the third place 5 out of 6 remaineth 1, set 1 under the line: then to the four place 7 out of 7 remaineth nothing, set the figure of nothing under the line, then to the fift place, take 6 out of 8 remaineth 2, set that under the line, and thus thou hast done. Then 20100 remaineth yet to be paid. ¶ Now thou shalt noote, that sometime it chanceth that the figure standing beneath is greater then the figure standing above him in the sum from whom subduction is made. In this case thou shalt in thy mind put ten, to the figure in the upper sum, and then subtrahe the neither figure out of the same, set the remnant under the line, and for the same ten the which thou didst put to the upper figure to make him greater, thou shalt add one to the next figure standing in the neither sum, and then subtrahe that likewise out of the figure above him, if the figure above be bigger than the figure beneath with his addition other else equal, and that remaineth set it under the line, as ye did in the other example. If the figure above be less than the figure beneath, then do to him as ye did to the other before: that is to say add ten to him: and so forth in all other places. Where the neither figure of the abatour is greater then the upper figure from whence it should be abated: as by this example ye shall more clearly perceive. ¶ An example. 5 7 2 9 5 4 9 0 4 8 7 6 5 2 9 7 8 5 3 0 1 9 3 ¶ Begin your subtraction saying, 7 out of 0 that can not be, therefore for bycause that 7 standing in the neither sum is more than the figure standing in the first place of the upper sum, ye must add a ten, then deduc your 7 out of 10 and there remaineth 3, then come to the second place and for the ten that ye borrowed in your mind and avoid it the figure in the first place to make 〈◊〉 sygge enough for the figure under it to 〈◊〉 subduced out of it, for the same ten I say ye shall put to the next figure in the neither place of the neither sum 1, then say 9 and 1 is 10, then subduce this 10 out of the figure of 9 standing above it in the upper sum and that ye can not, therefore do as ye did before in the first place, put 10 to the 9 in your mind saying 10 and 9 is 19, then deduck the 10 beneath out of the 19 above, & there remaineth 9 to be set under the line: then to the figure standing in the third place in the neither sum, put 1 for the ten that ye borrowed in your mind the which ye added to 9 in the second place of the upper sum to make it greater: saying 1 and 2 is 3, subtrahe that 3 out of 4 above it, remaineth 1 to be set under the line. Then to the fourth place, take 5 out of 5, remaineth nothing, set a figure of nothing under the line, and come to the .v. place, take 6 out of 9 remaineth; to be set under the line, so to the vi place, take 7 out of 2 that can not be, therefore put to the same 2 according to thy rule 10, and then it is 12, then subduce 7 out of 12 remaineth 5 to be set under the line: and for the same 10 that thou borrowdest in thy mind to put to the figure of 2 in the upper sum, thou shalt add 1 to the figure standing in the next farther place in the neither sum, coming to the same place which is the seventh place saying 8 and 1 the which I have too set to him is 9, then 9 out of 7 that I can not, wherefore likewise again I must help the same 7 with a ten and then it is 17 out of that now subtrake your 9 and remaineth 8 to be set under the line: now as ye have done before in all other places for the 10 here borrowed & adjoined, then add 1 to the next figure standing in the seventh place of the neither number saying, 4 & 1 is 5, then subduce this 5 out of the 5 above and remaineth nothing, wherefore set a figure of nothing beneath the line, & so ye have done. ¶ How be it ye shall noote that when ye have a cipher to be written in the last place of any sum, ye shall not write it, for in the last place it signifieth nothing of itself, neither doth it augment the signification of any of the other. ¶ Yet one other example will we set and then make an end of Subtraction. 5 0 0 0 0 8 1 0 0 7 1 0 0 4 8 4 0 5 7 4 8 0 0 8 7 5 1 6 0 2 3 5 2 7 0 1 3 ¶ Ye shall begin saying, 7 out of 0 that can not be, for ye can not take 7 out of nothing, wherefore as ye have done always in the example afore, put ten to that cipher, and that maketh 10, then deduc your 7 out of it now, and remaineth 3 to be set under the line: then for this ten that ye add to the figure in the first place of your upper number, set 1 to the figure standing in the second and next place of the neither number saying ● and 1 is 9, then 9 out of the cipher above that can not be, therefore as ye did before make that 0, 10, and then subduce your 9 out of this added 10 remaineth 1 to be set beneath the line: then for this 10 likewise, that ye borrowed in the second place of your upper number, ye shall set one to the next figure standing in the third and next place of the neither sum, saying, 1 and the 0 is one, then take that 1 out of 1 above him, remaineth nothing, set a figure of nothing beneath the line then to the four place take the syfer 0 out of 7 above remaineth 7 still: to be set under the line. So to the .v. place take 8 out of 0 that ye can not, therefore put 10 to the sypher and then subduce it, and remaineth 2 set that under the line: for this ten add 1 to the next figure in the vi place, which is 4, then 4 and 1 is 5, and 5 out of 0 that ye can not, then make 0 10 & take the 5 out of it remaineth 5 to be set under: then for the borrowed ten, likewise set to the next figure in the vii place of the neither number 1, saying, 1 and 7 make 8, & 8 out of 1 that can not be, therefore put ten to that 1 and then 10 and 1 is 11 out of this 11 deduce your 8 remaineth 3 to be set under the line: then for this 10 to the next figure in the viii of the neither some set 1 saying, 5 and 1 is 6 then 6 out of 8 remaineth 2, then to the ix place, take 0 out of 0 remaineth also 0, set that under the line, in the ten place take 4 out of 0 that can not be therefore put 10 to that 0 and subduce your 4, remaineth 6, then to the figure in the next place which is the xi put 1 saying 8 and 1 is 9, then 9 out of 0, that can not be, therefore put ten to it, and then subtrahe your 9 out of 10 remaineth 1, set it under the line: for this borrowed ten put one again to the next figure which is 4, saying 4 and 1 is 5, 5 out of 0 that can not be, therefore likewise again make it 10, and then take 5 out of it, remaineth 5, then again for your borrowed 10 put 1 to the next place: but for because there be no more places and therefore subtrahe it alone out of the figure above, saying, 1 out of 1 remaineth nothing, therefore nothing is to be set under the line, not so much as a 0, for bycause it is in the last place. So then the sum under the line is the remain that remaineth after the subtraction of the lower sum out of the upper sum. ¶ Here after followeth the prove of Subtraction. ¶ The prove of Subtraction. The prove whether ye have subtrahed well or no, ye must add the remain to the number paid, and if they twain added together do make the first sum lent completely then is it well subtracted, if not, it is not well subtracted, as by the last example ye may well perceive: for by the rule of addition, add 3 to 7 thereof cometh 10, set the sypher under the line and reserve the article to the next place forth according to the rule of Addition, and thou shalt see this two sums added together to come to the first lente sum: and this of Subtraction shallbe sufficient. ¶ Of Multiplication. MVltiplication is a manner of increasing or augmenting one sum by another In this feat of multiplication are iii numbers to be noted the multiplied number, the multiplier, & the numbered that redowndeth of the multiplication of the multiplied number by the multypliar, as in example. Multiply this number 4 by 3 & thereof come 12, 4 is the number multiplied: 3 is the number multiplier, 12 the third number that redounded of the multiplication of one of these number by the other then for more experience and ready working in this kind of operation ye shall perfectly know by memory the multiplication of one dyget by an other, the which ye shall have here in this table following, of the which one dygette ye shall look for in the head of the table, and the other in the left side of the table. ¶ Here after followeth the Table. 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 ¶ By this table ye shall sufficiently learn to multiply one dygette by an other. As for example, if ye will multiply 9 by 5, look for the 9, at the heed of the table, and for 5 the multiplier at the left side of the table, then with thy finger descend down from the place where 9 standeth till thou come before the place where the 5 standeth and there in the same angle, thou shalt fing 45, and that cometh of 5 times 9, & so do like wise of other. ¶ There is also a proper rule for the multiplication of one dygette by an other, & it is this, when thou wilt multiply one dyget by an other noote the distance of the greater diget from 10 and by the same distance multiply the less dyget or equal, & that that proceedeth of it deduct out of that article whom the less number doth denominate, and the rest is it that ye seek for, as for example: if ye will multiply 7 by 5, first see the distance between 7 which is the greater number and 5, and that is 3, by this 3 multiply 5, and that is 15 then subduce this 15 out of the article that 5 the less number doth denominate, which is 50, then remaineth 35, that is 5 times 7: so likewise shall ye do if the multyplyar and the multiplied be like. How be it most ready it is to know without book very perfitly the multiplication of every dyget one in an other. Now when ye will multiply any one number the one by the other. first write fair your number to be multiplied, and under it the multiplicator, beneath both these sums, ye shall draw a line. Then shall ye consider whether your multiplier be a dyget or articly, other elles compost number. I● it be dyget number ye shall begin to multiply by the dyget the figure or dygette standing in the fyrs●e place of the number to be multiplied, and that that cometh of it, if it be but a dygette set it unde the line right under the same place and then proceed further to the next place and multiply the figure standing in that place by the same multiplier, and that that redowndeth of it, if it be a dygette ●ette it likewise under the line r●ghte under the same place, and so do likewise in every place following, unto such time as all the figures standing in every place, be multiplied: then that the which shallbe found under the line is the sum coming of the multiplication of this two numbers, the one by the other: as by example ye shall the better perceive. 2 3 1 4 2 4 6 2 8 ¶ If ye will multiply this sum 2314 by this 2● ye shall set your figures after this sort, as ye see them. Begin your multiplication saying 2 times 4 is 8, set that 8 under the line, then come to the next place and say, 2 times 1 is 2, set it under the line, then to the third place, 2 times 3 is 6, set that under the line, so to the fourth place, 2 times 2 is 4, set that under the line also, & then thou hast done: so that this number 4 28 under the line, is it that cometh of the multiplication of this sum 2314 by this number 2. But if it be so that in the multiplication of any figure in the number multyplicable, by the multiplier that it which redoundyth of it be article number, than ye shall set a cipher beneath the line right under the same place where the multiplication is, and reserve the article to be added to the number that proceedeth of the multiplication of the figure in the next place by the aforesaid multiplyar, the which likewise if it amount to an article do likewise as I bid you to do in the first place: but if it be number composte, then shall ye set the dyget under the same place beneath the line, and reserve the article to be added likewise as is before said of article number, as in this example. 8 1 4 1 6 4 2 5 4 0 7 0 8 2 1 0 ¶ if ye will multiply this number 8141642 by this figure 5. Begin at the first place saying 5 times 2 is 10, now for bycause that this number is article, ye shall according to the rule before, set the cipher under the line, and reserve the article 1 to be added to the number that precedeth of the multiplication of the next figure standing in the next place of the sum multiplycable, by the multiplier: so then come to the next place saying, 5 times 4 is 20, to this 20 add 1 for that article that ye reserved, and that maketh 21, therefore because that this is a composte number therefore set the digette under the line beneath the same place, and reserve the article to the next place: then come to the iii place saying 5 times 6 is 30, to this add the article 2 which ye reserved in the place next going before, and then it is 32 set the dygette under the line as ye did before reserving the article to the next place then come to the four place saying, 5 times 1 is 5 to this add the article reserved, which is 3 and that maketh 8, set this digette number under the line, and then come to the v. place saying, 5 times 4 is 20, now for because that this number is article set 0 under that place beneath the live reserving the article 2 to be added unto the next place: then come to the vi place saying, 5 times 1 is 5, to this add the article 2 reserved and then it is 7, set it under the line: them to the vii place, saying, 5 times 8 is 40, now for bycause it is an article number ye shall set a sypher under the line, and reserve the article 4 too the next place, and for as much as there is no more places, ye shall set this 4 under the line next unto the 0 that ye set down last, and then ye have done. ¶ When that your multiplier is composte or article, then shall ye take the first figure of your multiplier, and by him shall ye multiply all the figures of the multiplycable numbers, setting alway that that amounted of it beneath the line as ye did before. And when ye have multiplied the number multiplycable by the first figure of the multyplyar: then multiply it again by the second figure of the multiplier, setting evermore the first figure of the number multiplycate, directly under the figure multiplycatour, in what place so ever it stand: and the number multiplycable is multiplied by all the figures of the multiplicator, then make a strike under them all, adding all the numbers multyplycate together as they stand, and that which proceedeth of that addition is the number multiplycable now multiplied by the hole number multyplycatour, as by this example ye shall plainly perceive. 2 3 4 5 1 2 3 4 9 3 8 0 7 0 3 5 4 6 9 0 2 3 4 5 2 8 9 3 7 3 0 ¶ If ye will multiply this number 2345 by this number 1234, set them first as ye see here 2, under them draw a line: then begin with the first figure of the multiplycatour, which is 4, and by him first according to the rule multiply all the multiplicable number through out, saying 4 times 5 is 20, set the cipher under the line reserving the article 2 to the next place: then to the second place, 4 times 4 is 16, to that put your reserved article 2 and it is 18, set the dygette 8 under the line reserving the article 1: then to the third place, 4 times 3 is 12 and 1 reserved from the place before that is 13, set the dyget 3 under the line, reserving the article 1, then to the four place, 4 times 2 is 8 and 1 reserved is 9, set that dygette 9 under the line, and so haste thou multiplied this number multiplicable by the second figure of multiplicator, Now then according to the rule afore, multiply the multiplicable number by the second figure of the multiplycatour saying, 3 times 5 is 15 set the dyget 5 under the line, according to the rule, which biddeth to set evermore the first figure of the number multiplycate under the place where the figure multiplicator doth stand: as here now thou multipliest the multiplicable by the second figure of the multiplicator, which is 3, than say 3 times 5 is 15 set this diget 5 under the line, and beneath the first number multiplicate right under the figure multiplicator, as thou seest in the example, and reserve the article 1: then to the second place of the multiplycable, 3 times 4 is 12, and 1 that is reserved is 13 set the dyget 3 under the line, as ye see in the example, & reserve the article 1, and so to the iii place 3 times 1 is 9 and 1 reserved is 10, set a sypher under the line & reserve the article 1: so to the four place saying 2 times 3 is 6 and 1 reserved is 7 set it under the line, thus have ye done your multiplication by the second figure of the multiplicator 3. Then take the iii figure of multiplicator which is 2, and multiply also all the numbers multiplicable by him saying 2 times 5 is 10 set the sypher beneath the line right under the place where this figure 2 the multiplicator standeth, as ye see in the example: and reserve the article 1, then to the second place 2 times 4 is 8, and 1 reserved is 9, set that 9 under the line: then to the iii place, 2 times 3 is 6, set that under the line: so to the four place saying 2 times 2 is 4 set that 4 under the line. Now begin to multiply with the fourth and last figure of the multyplycatour, saying 1 times 5 is 5 set the 5 under the line as I warened ye before, and as ye see in the example, then to the second place 1 times 4 is 4, set that 4 under the line, than 1 times 3 is 3, set that 3 under the line, than 1 times 2 is 2 set that 2 under the line & ye have done your multiplication: then must ye add according to your rule afore all this single multiplied number together, and that the which cometh of the addition is the number that cometh of the multiplication of this number 2345 multiplycable by the number 1234, multiplicator. Then come to the first place, and see what is there, & there ye shall find a 0, set it under the line, & so to the second place: there ye shall find 5 and 8 which is 13, set the dygette 3 under the line reserving the article 1 to be added to the next place: then come to the iii place, there is 0, 3 and 3 which is 6 to that add the reserved 1 and that is 7, set that 7 under the line, now to the four place, 5, 9, 0, and 9 maketh 23, set the 3 under the line, reserve the article 2, so to the .v. place 4, 6, and 7, is 17, to that add the reserved 2, which maketh 19, set the 9 under the line, and keep the article 1 in mind then to the vi place 3 and 4 is 7, and 1 reserved is 8 set it under the line, than too the vii place, there find ye but 2, wherefore set it under the line, and then have ye done: so that this sum under the line 2893730 is the hole number multiplicate ¶ another example of multiplication. A 6 4 2 6 0 0 3 B 5 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 8 5 2 0 0 6 0 0 0 0 0 0 0 3 2 1 3 0 0 1 5 C 3 2 2 5 8 5 3 5 0 6 0 0 0 ¶ Your figures set after this sort, A is the multiplicable number. B is the number multiplicator. C is the number multiplicate, which cometh of the addition of all the several numbers together standing between the lines. Begin then your work, taking the first figure of B, the multiplycatour which is 0 & by him multiply all the figures of A, the multiplicable and that that proceedeth of it set under the line as ye see: and so to the second figure of the multiplicator which is also 0 multiply all the figures of A by it like wise, and set that which cometh of it under the line, right under the second place where the multiplicant figure standeth: then to the third figure which also is 0 multiply all the multyplycable number A, and set that which cometh of it right under the third place beneath the line, as ye see plain in your example: for of the multiplication evermore by syphers cometh nothing but ciphers. Now to the four place of B, the multiplicator, there shall ye find the figure 2, multiply then all A the multiplicable number by this figure 2 saying 2 times 3 is 6 set that 6 under the line right under the place where the multiplicator 2 standeth, as it appeareth in your example: then to the second place, 2 times 0 is nothing, set that 0 under the line next the afore said 6, and so to the third place, 2 times 0 is nothing, set the figure of nothing down under the line, and so to the fourth place, 2 times 6 is 12 set the dygette 2 under the line and reserve the article 1 to the next place: then come to the .v. place, 2 times 2 is 4 and 1 that I reserved is 5, set that 5 under the line: now come to the sixth place, saying 2 times 4 is 8 set that 8 under the line: so to the seventh place, 2 times 6 is 12, set the dygette 2 beneath the line, and reserve the article 1 to be set in the next and last place as ye see in the example. Thus have ye multiplied A the multiplycable by four figures of B the multiplycatour, therefore now take 0 th● v. figure of the multiplicator: and by also multiply all the figures of A, t● multiplicable, and thereof shall come a● syphers to be set under the line, as ye se● here in the copy. Then to the sy●the figure of B, the multyplycatoure which is 5, by this 5 also multiply all t● figures of A the multiplitable saying 5 times 3 is 15 set that 5 beneath the ly● right under the sixth place wher● the multiplicator standeth, as is to see in the coopie: and reserve the article to the next place, then come to the second place and say 5 times 0 is nothing, set the 1 which ye reserved in your mind under the line, and so to the third place saying 5 times 0 is nothing, set the 0 under the line: then to the fourth place saying, 5 times 6 is 30, set the cipher 0 under the line, reserving the article 3 unto the next place: then come to the v. place saying, 5 times 2 is 10, and 3 that I reserved is 13 ●●t the dyget 3 under the line, and reserve ●e article 1 to be added to the next place: to the vi place saying 5 times 4 is 20 ●d 1 reserved is 21 set 1 the dygette under ●e line reserving 2 the article to the next ●ace: then to the vii and last place say●●ge 5 times 6 is 30 and 2 that was reser●d is 32 set the dyget under the line and serve 3 the article to be set in the next & ●e place beneath the line as ye may see ●he example, & so is all finished: Then ●er all these particular sums draw ●eke, and add all them together, setting ever that which cometh of the addition, under the line, as is in the example: the which shall amount unto this sum, 3225853506000, & this is it that cometh of the multiplication of the sum A, by the sum B. ¶ certain examples of multiplication in the which ye may exercise yourself to be the more practised in it. A 3 4 5 2 3 6 7 ¶ Too multiply by, B 8 8 9 2 5 3 9 3 1 0 7 1 3 0 3 1 0 3 5 7 1 0 1 1 7 2 6 1 8 3 5 6 9 0 4 7 3 4 1 0 7 1 3 0 3 2 7 6 1 8 9 3 6 2 7 6 1 8 9 2 9 Sum. 3 0 7 0 0 3 0 8 1 8 9 8 1 9 6 4 9 7 0 To mul. 1 3 1 9 4 9 1 0 6 4 9 7 0 Sum. 8 4 4 6 1 0 7 4 3 2 To mul. 3 2 4 2 9 7 2 8 1 4 8 6 4 2 2 2 9 6 Sum. 2 4 0 7 9 6 8 ¶ As for the multiplication by squares is neither worth the writing nor the reading: And where as in other copies is set duplation, triplation, and quadruplation, all that is superfluous, for so much as it contained under the kind of multiplication: and they that are expert in this feet, may right well perceive it. ¶ The proof of multiplication. THe proof of multiplication may be by two means. By the subducing out of all the 9: and the second way is by partition. As concerning the first way: ye shall first make a cross, then behold the multiplicable number, and subdue out of it all the nines, and that that remaineth not able to make 9 set it at the upper end of the cross: then come to the multyplycator, and do likewise in him, and that which remaineth all the 9 being subduced, set it at the under part of the cross: then multiply the figure standing in the upper part of the cross by the figure standing in the neither part of the cross, and out of the same that cometh of it take 9 as oft as ye can: and that that remaineth not able to make 9, set it at the right side of the cross: then come to the total sum multiplicate, and subduce all the 6 out of him likewise, & that which remaineth not able to make 9, set it at the left side of the cross, and if it be like the figure standing at the right side of the cross, then is it well, otherwise it is not well. ¶ An example. A 7 9 6 3 B 1 8 5 2 1 5 9 2 6 3 9 8 1 5 6 3 7 0 4 7 9 6 3 C 1 4 7 4 7 4 7 6 ¶ To know whether the sum C, be the very sum which cometh of the multiplication of A, by B. then first subduce all the 9 that ye find in the multiplycable A, and the rest set it at the upper end of the cross, which ye shall find to be 7: Then to the multiplicator B. do likewise and see what remaineth, and there remaineth also 7, set the also of the neither end of the cross: then multiply this 7 standing in the upper end by 7 standing in the neither end: and thereof cometh 49 when thou hast taken all the 9 out of this 49 there will remain 4, the which thou shalt set at the right side of the cross, Then come to C, the total sum of the multiplication & there likewise take out all the 9 that ye find there: and the rest not sufficient to make 9 set it at the left side of the cross, the which thou shalt find to be 4, and for because that this 4 to be set at the left side is like the figure standing in the right side (for that is 4 also) therefore this multiplication is good and well made: & so likewise in all other examples. ¶ The proof by Partition is to divide the total sum C by the multiplicator B and if the quocient be just A than is it well multiplied other else not. But this way can ye not practise, unto such time as ye have learned the feat of Partition. ¶ Of partition the fourth Kind of algorithm. Partition is a part of algorithm, by the which ye may easily divide any greater sum by a less or equal showing how oftentimes the divisor is contained in the number divisible. ¶ In this feat of partition be four numbers to be noted: the number divisible, the number divisor, the quotient, and the remain if there be any. ¶ Before you come to partition it shall be very needful and necessary for you, right perfitly to know the table of multiplication of dygettes: which is set in the chapter of multiplication: For unless that ye know that perfitly ye shall stick greatly not only in multiplication, but also in this feat of partition, and that exactly had in memory, the rest shall be far easyar. As for example. if ye will know how often times 7 is contained in 68 imagine by & by that this 7 should be contained 8 times: then if ye know without the book perfitly the foresaid table ye shall see that 8 times 7 is but 56 ergo 7 is contained more than 8 times in 68, imagine then and suppose it to be 9 times in 68, then by the table see what 9 times 7 is, and thou shalt see that it is 64 wherefore thou mayst conclude that in 68, 7 is contained 9 times and 4 over. No. quotient. 15077 3 divisor. The prove. ¶ To divide this number 45231 by 3 the 3 is divisor. first ye shall set down your numbers to be divided, and atthe end of that number on the right hand ye shall make a streke, wherein ye shall set your quocient, and then set down your divisor which is 3 under the figure that standeth at the uttermost end at the life hand that is under 4, and than say how many times 3 may I have in 4, once 3 and 3 remaineth over, set 1 within the strike and that 1 that remaineth set over 4 then strike the duysor 3 with a dash of your pen, and set the divisor 3 under the figure 5, then join the article 1 to the dygnet 5, and it is 15, then say how many times 3 may I have in 15, 5 times 3, set that 5 in the strike next to the figure 1 and close up the article 1 and the dyggette 5 with a cipher 0 over either of them, and then strike the divisor 3 with a dash of your pen, and set the divisor 3 under the third figure 2, and see how many times 3 ye may have out of 2 none therefore set down a cipher 0 within the strike next to the figure 5 and strike out your divisor with a dash of your pen, and set the divisor 3 under the fourth figure 3 than join the article 2 to the dyggette 3 and that maketh 23 then see how many times 3 ye may have in 23, 7 times, and 3 remaineth, set that 7 within the strike next to the cipher, and the 2 that remaineth set over the fourth figure 3 & close up the article 2 with a cipher 0, then strike out the divisor and set it under the first figure 1 at the right hand, then join the article 2 to the dygget 1 and it maketh 21, than see how many times 3 ye may have in 21, 7 times and nothing remaineth, than set the 7 within the strike and close the article 2 with a cipher 0 over each of them and strike out the devisor with a dash of your pen, & so the third part of 45231 is 15077. ¶ The second example. No. 6 divisor. quotient. 390 5/0 part. The prove ¶ To divide this number 2345 by ●, the 6 is the divisor, begin your division at the lift hand, as is said in the first example, and set your divisor under the third figure 3, for ye may not have 6 out of 2, and therefore say how many times 6 may ye have in 23, 3 and 5 remaineth, set the 3 within the strike, & the dygget 5 that remnyneth set it over the second figure, and close the article 2 with a cipher 0 over it, and then strike out the divisor with a dash of your pen, & set your divisor again under the third figure 4, and then join the article 5 to the dygget 4, and it is 54, then se how many times 6 ye may have in 54, 9 and nothing remaineth, set the 9 within the strike and close up the article 5 and the dygget 4 with a cipher 0 over either of them, and strike out the divisor with a dash of your pen, and set the divisor under the figure 5 and say how many times 6 may ye have out of 5, no times, therefore set down a cipher 0 with the strike, and let the 5 stand & strike out the divisor with a dash of your pen, and so the 6 part of 2345 is 390, and the 5 that remaineth set at the end of the quotient in this manner. ⅚ & so the quotient is 390 ⅚ ¶ To divide by 2 or 3 figures, or by as many as pleaseth you. ¶ first set down your number to be divided and your divisor under it, beginning at the left side at such a place as ye may take the last figure of your devisor in the last end, and then see how oft ye may have that figure in the figure above it, and that set apart for your quotient, with the which quocient ye shall multiply every figure by itself of your divisor and that that cometh of the multiplication, ye shall abate of the figure right over it, putting out that other figure, and set the rest above it, and so work with every figure by itself throughout the divisor. Then renew your divisor 1 figure forward toward your right hand, as before is rehearsed, and so continue your word following to the first figure of your number to be divided. Then it is to be noted that if it hap that your multiplied number that ye should abate be more than the number over it, then for a general rule, ye shall not take your divisor out of the figure above it except that it may sufficiently yield enough to all the abatementes of the residue, as more plainly shall apere in the example following. No. quotient, 343 7/12 12 divisor The prove. ¶ first set down this number 4123, and divide it be your divisor 12, begin your work at your lift hand, setting the article 1 of your devisor under 4, and the dygget 2 under the third figure 1, and than see how many times the article 1 of your divisor ye many have in the 4 over it, ye would say 4 times 1, but that can not be because there ye may not have the quotient 4 multiplied with the dygget 2 of your divisor, for thereof cometh 8, and then that 8 ye may not take out of 1 over the dygget 2 Therefore say again how many times 2 may ye have in 4, 3 times and 1 remaineth, set the 3 within the strike for the quotient, & the 1 that remaineth set over 4, and strike out the article 2 of your divisor with your pen. Then multiply the quotient 3 with the dygget 2 of your divisor, and thereof cometh 6. Then join the article 1 that remaineth, & the digget 1, and it is 11 thereout take 6 & there remaineth 5, set the 5 over the third figure 1, and close up the article 1 over 4 with a figure over it, and strike out the dygget 2 of your divisor again but one figure forward, as thus: set the article 1 under the third figure 1 is the No. and the diget 2 under the second figure 2 & there see how many times 1 I may have in 5 that remaineth, 4 times, & yet there remaineth 1 which must be set over 5, and strike out the article 1 with your pen. Then multiply the dygget 2 of your divisor with the quotient 4 and it is 8, then join the article 1 that remaineth, and the dygget 2 in No. together & it is 12, then take the 8 out of the 12, and there remaineth 4, set that 4 over the second figure 2 in the No. and close up the article 1 with a cipher 0 over it and strike out the digget 2 of your divisor with your pen. Then renew your divisor again as before is said, & set the article 1 under the second figure No. and thence how many times 1 I have in 4, that remaineth 3 times and 1 remaineth, set that 3 within the strike for the quotient, & the 1 that remaineth set over the 4, and strike out the article 1 of your divisor with your pen. Then multiply the quotient 3 with the digget 2 of your divisor, and it is 6, then join the article 1 that remaineth, and the dygette 5 in No. and it is 13. Then take 6 out of 13 and there remaineth 7, set that 7 over the dygget 3 in the No. and close up the article 1 with a cipher 0 over it, and strike out the dygget 2 of your divisor, and then the 12 part of 4123 is for the quotient 343, and the 7 that remaineth shall be set at the end of your quotient, as thus 7/16 Re. 1 Die, 6 Re. 121 Dy. 200 ¶ ye shall note that in these two examples the quotient standeth in the mids betwixt the two lines, and the number to be divided standeth next above the uppermost line, and the devisor standeth next under that neither line. But than ye must mark that there be two dyvysors, one is called the divisor currant, because it is always removable toward the right hand in the operation, and also it is stricken out, and this divisor standeth alway under the neither line of the quotient. The other divisor is called the divisor permanent, for he is not removed nor blotted as the other is, but standeth alway permanent on the lift hand directly against the number that is to be divided. And just over him there standeth the remain of the whole number which remain can not be divided by the devisor, and therefore it is set over the devisor permanent with a strike betwixt: as ye may see in the first ensample, where 1 is remaining, and 6 is devisor. ¶ For as much as in this ensample we can not take 4 which is the divisor, out of 3, therefore we shall set 4 under 5 & say how many times 4 have ye in 35, ye have 8 times 4, and there resteth 3, ye shall set the 8 betwixt the two lines, and the 3 above 5, then efface the 5 & the 4, than we shall set, 4 under 0 & say, in 30 how many times 4, 7 times, set 7 between the lines at the right side by the 8, and there resteth 2, which we shall set above 0, and efface 0, than set 4 under, and say in 28 how many times 4, 7, & there resteth nothing, set 7 between the lines by the 7, than set 4 under 0 & say how many times 4 in 0, there is none, therefore set 0 between the lines, then shall we say in 9 how many times 4, 2 times, set than 2 between the lines, and resteth 1 which we shall set above 9 and efface 9 than 12 how many times 4, 3 times, set than 3 between the lines by 2, & there resteth nothing. Than in 3 that is the last figure how many times 4, no times therefore at the end of the figure ye shall set the 3 thus ¾ and it is made. 3 2 3 A 3 5 0 8 0 9 2 3 3 C 8 7 7 0 2 3 40 B 4 0 0 0 0 0 0 4 4 4 4 4 ¶ Example when the divisor is an article, it behoveth to do semblably, in saying, in 3 how many times 4, no times and therefore we shall set 4 under 5, and 0 under 0, and say how many times 4 in 35, 8 times, set 8 between the two lines under 5, and there resteth 3 which we shall set over 5. Then set the 3 that standeth over 5 and the 0 together, and that is 30, than say how many times 4 in 30, 7 and always so to the end. And than we shall set 4 under 2, and 0 under 3, and say in 12 in taking the 1 that shall rest of the sum afore and shall be above 9, and the 2 that is after 9, how many times 4, 3 times 4, and then set 3 in the number of C, against 2, & than shall we cease, for there remaineth all only 3 to be parted by 40, now we shall not make 0 under 3 as is afore, but at the end we shall set 3 thus 3/40 ¶ Ensample when the divisor is compost, as in this figure afore, we shall say in 35 that been near A, how many times 4 that are in the number of B, 8 time 4 set that 8 between the two lines in the place of C, and there resteth 3 which we shall set above 5 and efface 35 of A, and 4 of B, then shall we say in multiplyenge the 8 of C, by the second figure of B, that is 2, we shall say 2 times 8 been 16, than abate 16 of the number of A against the same 42, and there be 3 which is over 5 and 0 of the number, that be worth 30 and we shall say, of 30 abate 16 and there resteth 14, of the which 14 we shall set ● over 3, and efface 3, and 4 above 0 and efface 0, than shall we set the divisor somewhat forward, the 4 against 0, that shall be effaced, and 2 against 8, and say in 14 demonstring 1 that shall be above 3 and 4 above 0 how many times 4, 3 times, set the 3 beneath the lines in the number of C. and there resteth 2 which we shall set over 4, and efface 4, than shall we say again in multiplyenge the 3 of C, by the second figure of B, that is 2 we shall say than 2 times 3 been 6 and of 8 that is against it we shall abate 6 and there shall rest 2, which we shall set over 8, and efface 8, and always so unto the end, & when we come to the 2 last fygurs of A, & that we would divide them by 42, we may not, for the first that is but 2 shallbe effaced with 1 that standeth above 5 and because that we may take there nothing we shall set 0 against 2 of A, in the number of C, between the lines, and so it is done, & there shall rest 3 to be divided by 4●, and that 3 shall be set at the end of the partition as thus 3/42 and it is finished. ¶ And it is to be known that as many figures as followeth the first figure in the number of B, shall be multiplied by them of the number of C, then the multiplication that thereof shall come ye shall abate in the number of A, as in this ensample shall more plainly appear. The prove. ¶ In this ensample in the number of B, that is the devisor, be many figures, and therefore we shall say, in 3 of A, how many times 2 of B, 1 time, set that 1 upon C and 1 that remaineth of 3, over 3, and than shall we come to the 4 of B and to 1 of C, and multiply them in saying, 1 times 4 is 4, which 4 we shall abate of the number of A, in taking 1 above 3 and 5 after 3, that shallbe worth 15 and thereof we shall abate 4 and there resteth 11, and for the more shortest way of 5 only abate 4, and set the 1 that remaineth above 5, & there resteth always 11, then shall we come to the 3 of B, and to 1 of C, and make all only the multiplication in saynege 1 times 3 been 3, than of 10 abate 3 in demonstring 1 over 5 and 0 after, and then there resteth 7, which we shall set over 0, then because of the cipher 0 may nothing come, we shall leave it, and go to the next figure and say 1 times 5 that is at the last end of B been 5 but in so much that we may nothing abate of 0 that is against it in the number A we shall borrow of the figure afore that is 3 only one and efface the 8, and set the 7 above the 8, and the 1 that we shall hold shallbe worth 10 to the regard of the number that we be in, than we shall say of 10 abate 5 there resteth 5 which we shall set above 0, then shall we avaunt our partetour consequently under the other fygurs following, that is to say till the last of B, be set under the last of A, and then ye may not advance them any further because ye be come to the ends of both the numbers. ¶ The prove of division or partition is made in this manner: ye shall first make a cross, as ye did before in multiplication and abate the 9 of the partition, and set the rest at the lift end of the cross semblably of the third number that is betwixt the two lines, and set the rest at the right end of the same cross, and if there be no thing rest set 0, Then multiply the two numbers of figure, for they 2 be dyggettes that one by that other, & thereof abate all the 9, if there be nothing in the first number, or if ye may not divide it join it with the same that shall come thereof. And so the rest that may not make 9 set it at the end under the cross. Then shall we come to the first number & semblably do away the 9 thereof, & set the rest above the cross, & if that above & that beneath be like, the partition is good, or if not, it is false. And for to understand it better we make proves by the ensamples aforesaid. prove ¶ For the first we shall take the partetour that is 4 and set it at the lift side of the cross, than shall we abate the 9 of the third number, & there rest 8 which 8 we shall set at the right end of the cross and multiply it by 4 and thereof cometh 32 whereof resteth 5 then adiouste them with the 2 farthings that we might not divide, and they shall make 7, the which 7 we shall set under the cross, than shall we abate the 9 of the first number that been the farthings, and there shall rest 7 which 7 we shall set at the upper end of the cross, and so been the two ends like and it is well made. ¶ Reduction. REduction is a kind of algorithm by the which ye be taught to reduce numbers of less denomination or value to numbers of more denomination or value: or if the case require it, numbers of great denomination to the numbers of less value. Example of the first. 20. li. 63. s. 44. d 10. far. Thus reduce the farthings to pens, & the pens to shillings, and the shillings to pounds: and then this sum is 23 li. 6 s. 10 d̔, and 2 far. so have you reduced the less sum to the more. Example to reduce the more to the less. Take the same example again, and reduce the 20 li. 63 s. 44. d 10. far, all in to farthings, and it will make 22410 farthings. first reducing the pounds to shillings, then to pens, & all that pens to farthings: wherefore it shall be very necessary for you to know what thing your number doth signify, whether weight, money measure, or time: and to be expert in all manner of accounts: it shallbe necessary for you to know all manner of weights, coins, measures, and time. Example in english money 4 farthings make 1 d 12. d maketh a shilling 20 shillings maketh a pound. ¶ In weight, and first of troy weight, every pound hath 12 ounces, and every ounce 20 penny weight, and every penny weight 20 grains. etc. ¶ The haperdepeys' pound hath 16 ounces, an ounce 8 drams, the dram 3 scruples, the scruple 20 grains. ¶ Of measure, the yard hath 3 foot, the foot 12 inches, the inch 3 barley corns of length. ¶ Of time, the year hath 365 days, the day 24 hours, the hour hath 60 minutes, every minute 60 seconds, every second 60 third, every third 60 quarts, every quarts 60 fyftes, every fift 60, syxtes, and so forth infinitely. ¶ To reduce the more sum to the less. ¶ When thou wilt reduce the more to the less, look how many times the less is contained in the more, and by that number multiply the number of the more and that that cometh of the multiplication showeth the more reduced to the less. Example. I would reduce 8 d to farthings look how many times a farthing is contained in a penny, and that is as ye know 4 times, then multiply according to the rule 8 by 4, and that maketh 32 which be 32 farthings, and so 8 d̔, maketh 32 farthings. ¶ An other example. Here is a sum of 28 li. and 6 s. I would have this pounds, which is of more denomination reduced to the shillings, which be of less denomination: then look first how oft a shilling is contained in a pound, and that is 20 times, for 20 s. maketh a li. multiply then the 28 li. by 20, thereof cometh 560, which be all shillings: to this put the other 6 shillings and so all is 566 shillings. ¶ But ye shall note that where there be any sum of mean denominations between the more to be reduced and the less to whom reduction is made: then shall it be easy are to reduce first the more to the mean, and so by the mean to the less. ¶ The example. 43 li. 19 s. 20 d 4 farthings. if ye will reduce all this sums to the farthings: then shall it be better for you to reduce the pounds first to shillings and then being shillings to reduce them to pens, and at the last to farthings: so by your rule 43 pound maketh 860 shillings, to that add the 19 shillings, it maketh 879, then reduce this 879 shillings to pens: look first how many pens are contained in a shilling, and that is 12, multiply 879 by 12 thereof cometh 10548 which be all pens, to this add your 20 pens, & that maketh 10568 d then reduce this pens to farthings, see how many farthings be in a penny that is 4, multiply 10568 by 4 cometh 4●272 to these add the 4 farthings and that maketh 42279 farthings. Thus have ye reduced 43 li. 19 s. 20. d̔, 4 farthings the more by the mean to the less. ¶ To reduce the less to the more. ¶ first mark how many times the more doth contain the less: and by that number divide the less, and the quocient showeth the less reduced to the more. Example. I would have this sum 5600 s. reduced into pounds: for how many times a pound doth contain a shilling that is 20 times, then divide 5600 by 20 the quotient shallbe 280, which be pounds: so that 5600 s. reduced to pounds maketh 280 li. & so likewise in all other rekeninges. ¶ When sums of divers denominations come in addition to be added together, then beginning at the sums of least denomination: add them over together till such time as they make a number of the next denomination, and that that remaineth not able to make any number of greater denomination, set it under the line & proceed to the next sum of greater denomination, to the which add the number of the same denomination reduced out of the sum before of the less denomination, so proceeding to the end. ¶ Ensample. li. s. d far. 1680 10 5 3 8200 29 7 2 1008 3 10 3 ¶ Begin at the least which be farthings: saying 3 and 2 been 5, and 3 been 8 this 8 farthings make 2 pens, therefore take this pens and add them to the next sum which is of the same denomination, saying 2 and 10 be 12 d which is 1 shilling the 7 and 5 be 12 which also maketh a shilling, so among these pens ye have ● shillings to be added to the next order of shillings, saying 2 and 3 be 5 and 9 be 14, put the diggette 4 under the line, and reserve the article 1 to the next place saying 1 and 2 be 3 and 1 be 4, set that 4 under the line also and then is it 44 s. the which reduced to pounds maketh 2 li. and 4 s, remaineth under the title of shillings: then put that 2 li. to the other pounds, and so haste thou done in reduction of the sums of less value to the greatest sum, wh●ch be pounds. And this is sufficiently entreated of reduction. ¶ Here followeth of progression. progression showeth the number when it beginneth at 1 or at 2 in mounting always by one, & one, as doth this number 123456789. Now if ye will know the valour of the numbers first ye must regard two things, that is to wite, if the number proceed continually without leaving any thing betwixt as here 1234567 or if it leave any thing betwixt as here, 13576. Secondly ye must consider if the number be even or odd. And after these two consyderacyous, then by four rules that here followeth ye may know the valour of each whole number. ❧ The first rule is when one number proceedeth in mounting always continually in the beginning, then if it end in an even number, than shall we take the half of that even number, and by it we shall multiply the odd number that cometh of the even number, as ye may see in this ensample following. ¶ Ensample. 12345678 4 9 36 ¶ if ye will know how much this number is worth, than multiply the half of 8 that is 4 and the number that is after 8 is 9, and then thereof cometh 36, and so much is the sum worth, and thus may ye do with all such like questions. ¶ An other example. 1 2 3 4 5 6 7 4 7 28 The prove. ¶ For to multiply this number 7 wherein the greatest and the more half is 4 ye must multiply 7 by 4 and it is 28, and so much in the hole sum, ¶ The third is, if a number proceed not continually, and end in an even number ye shall take the half of the said number that is even, and by him multiply the same that is next coming after the same half, and in thus doing, ye shall have the sum of the same ●umber. ¶ An example. 2 4 6 8 4 5 20 The prove ¶ If ye will know how much this number is worth, than take the half on 8 that is 4, then multyplye by the 4 the number which followeth, that is 5 in saying 4 times 5 is 20, and so much is worth the whole sum. ¶ The fourth is when the said number proceedeth not continually, then if it end ¶ Here followeth the rules, and first the rule of three. MVltiply by the contrary & divide by the semblant or like. This rule may be understand in two manners. first multiply the same that ye will buy by his contrary, that is to wit, by the price, and divide by the semblant that is to wit, by as much as ye have bought: or thus, multiply ihe price by his contrary, that is to wit, by the same that thou wilt buy, and divide it by his semblant, that is that same that ye have bought. And note ye why it is called the rule of three, for with three numbers certain ye may know and find the fourth number uncertain. And it is a rule right no table and necessary in the fayct of merchandise. For to have knowledge of this rule, it behoveth to set some rules different in manner of questions, and firk in measures long. ¶ The rule of hole numbers. ¶ If 9 else of cloth cost 25 crones, how much shall cost 15 by the price. Answer. It behoveth you to set the some, that is to wite, 25 crones. And than ye shall multiply by his contrary, that is to wit, by 15 that been 375, and than divide them by that semblant, that is to w●te, by 9, and thereof cometh 41 crones and an half, and there remaineth 1 crone and and half, the which ye shall make in sꝪ. and there been 54 sꝪ. the which ye shall divide by 9 and thereof cometh 6 sꝪ. Therefore ye may answer that the 15 else shall cost 41 crones and an half and 6 sꝪ. Now if ye will make the prove it behoveth you to form your question thus if 15 else cost. 41 crones and an half and 6 shillings, how much shall cost 9 else by the price. Then it behoveth you first to multiply the 6 sz, by 9 and that been 54 then it behoveth you to make thereof crones, that is 1 crone an an half, & then ye shall multiply the 41 crones and an half by 9, and they been 373 crones and an half, and then set thereto 1 crone and an half, and they be 375 crones which ye shall divide by 15, that been 25, the which 25 been the price of 9 else, and so the rule is good, and thus ye may do of all other semblable. ☞ The second rule of hole numbers with numbers broken semblable. ¶ if 10 else and 2 third parts of cloth cost 35 franc. how much shall cost 14 else by the price. Answer. For to know this rule and other semblable: it behoveth you to reduce the else bought, and them that ye will buy all into thirds because of them that be bought, in saying thus, 3 times 10 been 30, and set thereto 2 thirds, that is than 32 thirds. Then it behoveth you to make division by 32, and than ye shall reduce the 14, else in to 1 third, in saying 3 times 14 been 42. Then 42 shall be the multiplicator. Now set the some, that is to wite 35 franc. the which multiplied by 42 be 1470 the which divided by 32 thereof cometh 45 franc. and an half, and there resteth 14 fran, the which ye shall reduce to shillings, and than divide them by 32 and thereof cometh 8 shillings, and an half, there resteth 8 shillings, and than shall ye make them in pens, and divide them by 32, and thereof cometh 3 pens, therefore ye may answer that the 14 else of cloth shall cost 45 francs and an half 8 shillings and an half and 3 pens. ¶ For to make the prove it behoveth you to make your work by the contrary, for it behoveth you to multiply the some that the 14 else cost by the devisor, and divide it by the multiplycatour. Therefore set the some upon the lift side, and first multiply the 3 d̔, by 32, & when they be multiplied ye shall make of them shillings, and then ye shall multiply the 8 sꝪ. and the half by 32, and then make thereof francs. And then ye shall multiply the 45 francs and the half by 32, and divide them by 42, and so ye shall know if the rule be well made. ¶ The third rule of hole numbers with divers minutes. ¶ if 4 else and 2 thirds of cloth cost 10 crones, how much shall cost 6 else & 2 quarters by the price, For to know this rule, it behoveth you first to reduce the 4 else and 2 thirds thus, 3 times 4 been. 12 And than ye shall adjoin the 2 thirds, & than it is 14. And than the else that ye will buy, ye shall reduce them in to one fourth thus, 4 times 6 been 24. And then set the 2 quarters thereto and than there is 26 quarters. And than ye shall multiply that one by that other, that is to wit the nombrant of the first by the denominant of the second, in saying 4 times 14 been 56. And those 56 shallbe the devisor. Than multiply the numbrant of the second, by the denominant of the first in saying 3 times 26 been 7●, and those 78 shall be the multiplicator. And therefore set 10 crones and multiply them by 78, and divide them by 57 And ye shall find that the 6 else and 2 quarters cost 11 crones and an holfe, 15 shillings & 5 pens. And there resteth 8. The example. Devisor M 59 142 242 78 10 crones 246 2 3 4 ¶ If 4 else ⅔ cost 10 crones, 6 elles 2/4 shall cost 13 crones and an half 15 shillings 5 pens, there resteth 8. For to make the prove it behoveth you to work the contrary, for it behoveth you to multiply the some by the divisor, that is to wite, by 59, and make division by the multiplicator, that is to wite, by 78 and ye shall find 10, otherwise if there be more or less the rules he false. ¶ The fourth rule containing hole numbers to the merchandise that ye have bought and minutes to the same that ye will by. ¶ if 8 else of cloth cost 15 crowns, how much shall cost two quarters by the price. For to know this rule ye must reduce the 8 else into quarters, in saying 4 times 8 been 32 then 32 shall be the devisor, and the 2 quarters shallbe the multiplycator. Now set the 15 crones and multiply them by 2 quarters, and divide by 32 and ye shall find that the 2 quarters cost 0 crones and the half 15 shillings and an half 3 pens. For to make the prove ye must work the contrary, for ye shall multiply the some that the 2 quarters cost, that is to wit, 0 crones, and the half 15 sꝪ. and an half 3 pens by 32 and divide them by 2. ¶ The rule of round measures, that is to wite, measure of corn of wine and oil. ¶ first it behoveth you to presuppose and know the measures of corn. ¶ One muy is worth 12 septiers ¶ One septiers is worth 4 mynotes. ¶ One mynot is worth 3 bushels. ¶ One bushel is worth 4 quarters, ¶ The measures of wine. ¶ One muy of wine holdeth 36 septyers ¶ The septyer holdeth 4 quarters. ¶ The quart holdeth 2 pints. ¶ The pint holdeth 2 choppynes. ¶ The choppyne 2 half septiers. ¶ The half sceptre 2 possions. ¶ The first rule. ¶ If the muy of corn cost 10 francs, how moche is worth the bushel. Answer. For to know this rule ye must know how many bushels been in 1 muy, Therefore multiply the muy by 12, and than by 4, & than by 3, which been 144 bushels, the which shall be the devisor of 10 francs therefore divide 10 by 144. And thereof cometh 1 sꝪ 4, d and an half, resteth 24. d Therefore the bushel costeth 1 sꝪ. 4 pens & an half, resteth 24 d ¶ The second rule. ¶ To the contrary, if the bushel cost 1 sꝪ. how much shall cost a thousand and 4 hundreth Muys by the price. Answer, For to know this rule, it behoveth you to make all the Muys in bushels. And there be 201600 bushels, the which it behoveth you to multiplpe by 2, and there be 403200, and of them ye shall make crones. Therefore divide by 36, and there been 1●20● crones. Therefore ye may answer that if the bushel cost 2 sꝪ. a thousaunde and 4 hundreth muys shall cost eleven thousand and 2 hundredth crones, and thus ye may do of all other semblable. ¶ The third rule. ¶ if the septier of corn be worth 1 francs & the lote of penny tourneys weight 12 ounces, how much aught it to weigh when the sceptre is worth 15 tornoys. Answer. Multiply the first number by the second, that is to wit, 20 by 12, and divide it by 15, & ye shall find that it ought to wegh 16 ounces. And thus ye may do of all other like. ¶ If the muy of wine be worth 12 francs how much aught the pint to be worth Answer. For to know this question, it behoveth you to reduce the 12 muys into septyers, from septiers into quarts, and from quarters into pints, and that been 188 pints. And than ye shall reduce the 12 francs in to sꝪ. that been 240, and that into pens, that been 2880 pens, the which behonethe you to divide by 288 & it cometh to 10 d Therefore if the muye of wine cost 12 francs, the pint is worth 10 d But it is requisite that the taverner have some gains if ye sell 12 d the pint. I demand how moche shall he win upon the muy: Answer. He selleth it 2 d more than it is worth therefore multiply 288 pints by 2 and they be 576, the which ye may divide by 12 and there shall be 48 sꝪ. Therefore may ye answer that he getteth 48 sꝪ. upon the muy. ❧ if the muy cost 10 francs how much is worth the pint. Answer. It behooveth you to do as is above said: and ye shall find that it is worth 8 pens and 1 third. ¶ if the pint cost 6 pens, how much shall cost 12 muys by the price. Answer. It behoveth you to know how many pints been in a muy, that is 288, multyplye 12 muys by 288 that is 3456, pints. And than multiply the pints by 6 that been 207, 6, of whom ye shall make sꝪ by division. and there been 10728 sz, and of shillings ye shall make francs. Therefore ye shall make division by 26, ye shall find 86 francs 8 sꝪ. Therefore ye may answer that the 12 muys shall cost 86 francs 8 sꝪ. ¶ In so much as competently we have tracted of the rule of three in the fayct of measures, it is expedient that we tract thereof in the faycte of weight. ¶ if an hundreth pounds of pepper cost 20 sꝪ. how much shall cost 6 pound by the price: Answer. For to know this question, ye must multiply by the contrary and divide by the semblant, that is to wite, multiply by 6 and divide by 10, and ye shall find that the 6 pounds shall cost 1 francs, and 4 sꝪ. To make the prove ye must multiply by 100 and divide by 6. Now I demand of the 6 pounds cost 1 franc 4 sz, how moche is worth the ounce. For to know this ye shall make the pounds in ounces, the which been 96 ounces, & then make the money in pens, the which been 288 d the which ye shall divide by 96 and thereof cometh 3 pens, therefore the ounce shall cost 3 pens. ¶ If one li of saffron cost 3 francs and an half, how much is worth the ounce. Answer. It behoveth you to know that in a pound been 16 ounces, therefore divide the 3 francs, and the half by 16 and ye shall find that the ounce is worth 4 sꝪ. 4 d and an half, & thus ye may do of other like. ¶ If 4 pound of saffron cost 16 fran. 6 sꝪ 8 d how much shall cost 3 quartrones by the price. For to know this rule, ye shall reduce the 4 li. in thirds and shall say ● times 4 been 12, and 1 third been 14 then ye shall multiply by 4, and shall say 4 times 14 been 56 the devisor, than for the second number we shall say, 3 times 3 been 9 fourths or quarters, the which 9 shall be the multiplier. Now set the 16 francs 6 sꝪ 8 d tourneys, & multiply them by 6 and divide them by 16, and thereof cometh 2 fran. and an half, 2 sꝪ. 6 d therefore ye may answer that the 3 quarters shall cost 2 fran. & an half 2 sꝪ. 6 d For to make the prove ye must work by the contrary in multiplying by the divisor, that is to wit by 56 and make division by 9, and so may ye do of other semblable. ¶ If one pound of tin cost 9 blances, how many hundredth shall I have for a thousand and 4 hundredth francs. It behoveth you to know how much is worth the hundred by 9 blances the pound. And ye shall find that there is 12 francs and an half. Now make division of 1400 frances by 12 frances and an half, ye shall find 112. Therefore ye may say that I shall have 112 pound of tin for 1400 frances, And also as we have made this rule, ye may do in all other merchandises, as in lead, iron, spices pepper, sugar. And as we have done of pounds ye may do of quartrons, ounces, & all other weights, ¶ A rule which is without time Three merchants put their money together for to have gains, the which have bought such merchandise as hath cost 125 francs, whereof the first hath laid 15 francs. The second 64 sꝪ. and the third 36, fꝪ. And they have gotten 54 franc. of clear gains. I demand how shall they divide it, so that each man have gains according to the money that he hath laid down. Answer. In all such rules and questions ye shall multiply each one after the money that he had laid, therefore multiply the gains for the first by 25 and divide by 125, that is the divisor commune. For the second multiply the gains by 64, and divide by 125 the divisor commune. And for the third multiply the gains by 36, and divide 125 the devisor commune. And for to find the devisor commune, ye shall set together the multiplycatours, that is to wite 25, 64 and 36 which is 125 the devisor commune. And so shall ye do in all rules of company Now ye may find & know how much each one hath of gains, and ye may see it by the ensample here present. The first hath 10 fꝪ and half 2 sꝪ. The second hath 27 fꝪ & half 2 sꝪ. and half 5 d & half, resteth 2 d and half, The third hath 15 fꝪ and half 5 sꝪ. resteth 60 pens. 256436 125 multiplicator divisor. And they have yet to be divided among them of rests 62 d and and half. ¶ For to make the prove it behoveth you to divide the rests, and than reduce all togethers, and ye shall find the some divided, for all the rules of company been proved by addition of sums. ¶ The second rule of hole time. ¶ Four merchants lay money together for winning for a certain time, of whom the first hath laid 10 fꝪ. for two year. The second 20 fran. for 3 year. The third 100 francs for one year. And the fourth hath laid 40 franc. for 4 year: and they have gained 454 fran. I demand how much each one ought to have of winning after the money that he hath laid, & after the time that he hath holden his money in gain for company. Answer. For to know this rule and other semblable, ye shall multiply the money that each one hath laid by the time that he hath holden it in company. Example. The first hath laid 10 fran. for 2 year, therefore it behoveth you to multiply 10 by 2 in saying 2 times 10 been 20. For the second 3 times 20 been 60. For the third 1 time 100 is an 100 For the fourth 4 times 40 been 160. & then it behoveth you to find a divisor common, for each hath his multiplycator, that is to wit, the same that he hath laid, and for to find it ye shall set together all the multiplycatours, that is to wit the 20, 60, 100, 160 the which maketh 340, therefore these 340 shall be the divisor commune to all, thenne how much each one ought to have ye may see by the ensample here following 454 sꝪ. The first hath 26 francs and half 4 sꝪ. one penny, resteth 140. d̔, The second hath 80 frances 2 sꝪ 4 d rest 80 pens The third hath 134 frances 1 sꝪ. 5 pens, rest 20 pens. The fourth hath 213 franc. and an half 2 sꝪ and a half 5 d rest. 100 pens. 20, 60, 100, 160, 340 Multiplycatours. Dyvysour. Of rest they have to divide one penny. ¶ The rule of company where as is hole time and parts of time. Three merchants lay money in company for to have gains thereby, of whom the first hath laid 30 frances for two years. The second hath laid 400 fran. for one year & three months. And the third hath laid 60 fran. for three years & two months. And they have gained with this money 44 franc. I demand how they shall divide it to the end, that each one have his right after the money and the time that they have set and holden for to gain. Answer. For this rule & all other semblable, ye shall multiply the time by the money, as we have said above but for as much as there be months ye must set & reduce all the time of each one in months, and also if there were any days ye should set all the time in days. The first hath laid 30 frances for 2 years, in 2 years been 24 months, therefore multiply 30 by 24 there been 720, and these 720 shallbe the multyplycatoure of the first. The second hath laid 40 fran. for 1 year and 3 months, in one year been 12 months, and 3 doth make 15 months, multyplye 40 by 15, they make 600 which is the multiplycatoure of the second. The third hath laid 60 francs for 3 years and 2 months, 3 years been worth 36 months and 2 bene 38 months. Now multiply 60 by 38, and there been 2280, which shall be the multiplicator of the third. Now for to have a divisor comune, ye shall set together all the multiplycatours that is 3600 the divisor commune. They have to divide 44 francs. The first hath 8 francs, and half 6 sꝪ rest ●. The second hath 7 frances. 6 sꝪ. and half. rest 0. The third hath 27 francs and half 7 sꝪ. 4 pens, rest 0. 720, 600, 2280,, 360●. Multiplycatours Dyvysour ¶ A rule of divers silver and divers time. Three marcaunhtes have made company together of whom the first hath laid 10 francs. 4 shillings for 2 months. The second hath laid 15 fran. for one year. And the third hath laid 6 francs 7 sꝪ for 8 months, and they have gotten of this money 24 francs. How they shall divide it after the money and after this time I demand. Answer. For to know this rule and all other semblable it behoveth you to reduce the money of every man in shillings. And all the time in months. And then multiply the money by the time. Ensample. The first hath laid 10 francs that been 200 sꝪ. and 4 been 204 the which ye shall multiply by 2 months, and they shall be 408 the multiplicator of the first. The second hath laid 15 francs for one year, and in 15 francs been 300 sꝪ. and in one year been 12 months, therefore multiply 300 by 12, and there shall be 3600 the multyplyecatour of the second. The third hath laid 6 francs 7 shillings, and in 6 fran. been 120 sꝪ. and 7 been 127 sꝪ. for 8 months, therefore multiply 127 by 8, and they shall be 1016 the multiplicator of the third. And for to have the devisor commune, ye must reduce together all the multiplicators, & that shall be the devisor commune, as ye may see by the example following. They have 24 frances of winning. The first hath 2 francs and half 8 sꝪ. and half 5 pens and half resteth 1360 pens. The second shall have 17 francs, 3 sꝪ. and half d resteth 1952 pens. The third shall have 4 sꝪ. & half, 7 sz, 0 pens, & half resteth 17112 pens. 408, 3600, 1016 5024. Multiplicatours. divisor. And they have to divide 1 d of the rests For to make the prove ye shall reduce together the three sums that they have had. And if there be more or less the rule is evil made. ¶ Here followeth the rule of company of factors with merchants servants. OF this rule of factors ye may make 3 rules in manner of questions that fall among merchants. Example, 8 merchants 5 factors, and 3 servants or varlets have made company together, and have clearly gotten 150 fran. whereof the factors ought to have the half of the merchants, and the servants the third part of the factors, how shall they divide these 150 franc. Answer For all such rules and questions it behoveth you to find a number wherein is an half and a third, and that shall be 6, and these 6 shallbe for the merchant. And the half of 6 been 3, that shall be for the factors, and the third part of the factors is 1 which shall be for the servants. And than ye shall multiply the one by the other, that is to wit, the personages by their number, 6 times 8 been 48, and these 48 shall be the multiplicator of the merchants. And than there been 5 factors, that have 3 and 3 times 5 been 15, and than there been 3 servants that have 1, and 1 times 3 is 3, & therefore the factors shall ye multiply by 15 and the servants by 3. Now for to find the divisor commune ye shall set together all the multyplycatours, that is to wite, 48, 15, 3, which been 66 these 66 shall be the divisor commune. Example they have to divide 150 francs. The merchants have 109 franc. 1 sꝪ. and half 3 d̔, and half, resteth 21 d The factors have 34 fran. 1 sꝪ. and half 3 d and half, resteth 21 d The servants have 6 fran. and half 6 sꝪ. 4 d rest 24 pens. 48, 15, 3 66 Multyplycatours divisor. ¶ They have to divide 1 penny of rests For to make the prove ye shall divide all the rests by the divisor commune. And than ye shall reduce all together, for to have 150 frances. ¶ The rule of factors the which gate the half of the gain and of the principal. ¶ And other rule in manner of a question a merchant hath given 50 franc. to his factor by such covenant that he govern them for 10 years. And at the end of the time, that is to wite, at the end of 10 years. And at the end of 10 years, they shall divide the gain and the principal It happeneth that the factor will go his way at the end of 6 years, and he findeth that he hath gained a thousand francs. I demand how ought the said factor to be paid, and how much aught the said merchant to have: Answer. ye ought to regard how moche he should have gained in those 10 years that he should have holden them in gain as he had promised. Therefore ye may form the question, if 6 have gotten a thousand: how much shall be the gains of 10. Multyplye 1000 by 10 and divide by 6 and ye shall find that he should have gutten 1666 fran, and an half 3 sꝪ. 2 pens. Of the which gains the merchant ought to have the half that been 833 francs. 6 shillings and half and 1 penny. And than take up those 833 fran. 6 shylynges and half 1 penny of 1000 francs that he hath gained, and there remaineth 166 frances, 13 shillings 5 pens for the factor. Now ye may answer that the merchant shall have of the gains 833 francs 6 shillings, and half 18. And the half of the principal, that is to wit of 50, that is 25 and there been 852 francs. 6 sꝪ. and half 1 d And the factor shall have of gain 166 francs. 13 sꝪ. 5 pens. And of the principal 25 that been 191 fran. 13 sꝪ 5 d And thus may ye do of all other semblable. And it is proved by the reduction of the two sums gained. ☞ The third rule of factors with covenants, that the factor shall gain the half of the principal. ¶ An other rule of company of factors & merchants with covenant that the factors shall gain the half of the principal and not of the gain. Example. A merchant giveth unto his factor 400 fran. that he shall govern them for 6 years, & at the end of the time the half of the principal shall be to the factor. It happeneth the factor will go his way at the end of 2 years, & hath gained 200 fran. I demand how ought the factor to be paid. Answer. ye ought to regard how moche he should have gained if he had served all his time, and for to find it ye may work by the rule of three, for ye must multiply by his contrary, that is to wit by 6, and divide by his semblant that is to know, by 2, in saying if 2 have gained 200 frances: how much shall 9 gain, and ye shall find that he should have gotten 900 franc, and he gained but 200 franc. wherefore he ought to make a gain 400 fran. to the merchant: and he ought to have the half of the principal, that been 200 frances, therefore he oweth 200 unto the merchant, and so he hath lost all his time, and 200 fran. of advantage for the merchant ought nothing to lose like as he had accomplished all his time ¶ The third rule of changes for to use deceit or fraud. ¶ Two merchants will change their merchandise, & the one beguiled the other the one hath pepper, and that other cloth. He that hath pepper will sell for 25 franc. the hundredth by change, which is no more worth than 20 fran. in silver contented. I demand for how much aught the other to sell unto him the elle of his cloth, that is worth but 15 sꝪ. to keep himself from loss. Answer. For the rule of three ye may say thus, if 20 frances of content give me 25 fran. at the change how much shall give me 15 of content. It behoveth you to multiply the 25 by 15. which been 373, the which ye shall divide by 20 and thereof cometh 18 sꝪ. 9 d therefore ye may say that he shall sell the elle of cloth for 18 shillings 9 d And thus may ye do of all other. ¶ Two merchants will change their merchandise, of whom that one hath 100 pound of wool, that is no more worth but 15 crones. And he will change with an other in a pyce of cloth that is worth 21 crones, and he will give him the wool for 17 crones. I demand for how moche ought the other to sell the piece of cloth to the end that he be not betromped ● Answer. By the rule of three when 15 are worth 17 demand how moche shall be worth 21 divide by 15 and ye shall find the same that ye require. ¶ Two merchants will change their merchandise, and the one defraud that other that hath pepper, and will sell it 24 fran. the hundreth by change. which is no more worth but 20 frances in money content, and he will have the half in money content. I demand for how much aught the other to sell the elle of his cloth that is no more worth but 15 sꝪ. Answer, ye must take away the money content that the other demandeth, that been 12 franc. for the just price, & of the which he will sell over. Therefore take away, and withdraw 12 of 20 franc. which is the just yrice, and there rest 8 fran. for 8 and 4 been 12, And ye may say by the rule of three, if 8 give me 12, what shall give me 15 sꝪ. which is the just price of the cloth, multiply 12 by 15 and divide by 8, and thereof cometh 22 sꝪ. 6 d And therefore the merchant ought to sell the elle of his cloth after 22 sz, 6 d else he should have loss And thus ye ought to do of all manners of changes and barathes, for if he that hath the pepper, demanded but the third or the fourth or 2 or 3, abate all only the same that he shall demand, and then by the rule, as is said. And note ye well that if he will multiply shillings, ye shall have shillings. And of crones ye shall have crones, and of frances ye shall have frances And in like manner of all other. ¶ Here followeth many rules & questions to have the more knowledge of the science of arismetryke, and the first is of collects and tallyages. Ten men own unto the king of collect and tallyage 244 fran. I demand how shall they divide them to the end that each one pay after the valour of his goods, for it is reason that more be paid by the rich then by the power. For he that is more endowed with goods is more holden unto god and to the prince. Answer. It behoveth to know how much each one is worth in his goods, and in his possessions. The first is worth 100 francs The second is worth 400 franc. The third is worth 154 franc. The fourth is wourth 1000 franc. The fift is wourth 1150 franc. The sixth is worth 40 franc. The seventh is worth 440 franc. The eight is worth 80 franc. The ninth is worth 600 franc. The tenth is worth 360 francs. Now it behoveth you to find the multyplycator and the divisor. The multyplycator shallbe each one by himself, and so for the first it behoveth you to multiply by 100, for the second by 400. for the third by 154, and so must ye do of the other: And for to find the divisor, ye shall set together all the multiplicatours, as 100, 400, 154 etc. and all that together shall be the divisor commune, which is 4464 Therefore multiply the collect, that is to wit, 244 for each one his valour, and divide by 4464, or by the half that is 2232, and then ye shall write how moche each one ought to pay. Example. The first should pay 5 franc. 9 shillings 3 d and half rest 1464. The second should pay 12 franc. 17 shillings 3 pens, re●●eth 1●92. The third should pay 8 frances 8 sꝪ. 4 pens resteth 660. The fourth should pay 54 frances 15 shillings 2 pens, resteth 1248. The fift should pay 62 frances 17 sꝪ. 2 pens resteth 96 The sixth should pay 2 frances 3 sꝪ. and half 2 pens and half, resteth 1032. The seventh should pay 24 frances 1 shilling 0 pens, resteth 192 The eight should pay 4 frances 7 sꝪ. 5 pens resteth 2064. The ninth should pay 23 frances 15 sꝪ. 10 pens and half, resteth 2088 The tenth should pay 27 frances 6 sꝪ. 7 pens resteth 624 ¶ And they have to divide 2 pens and half of rests. Then when ye have all divided and write the some and the rests, ye shall set together all the rests, and divide them by the devisor commune, or by the half. And if there be more or less. the rule is not well made, for the remainant of all aught to be divided by the divisor commune. And the prove of this rule is reduction. And mark well this rule for it is right good unto the country where all the goods be praised by all the towns and castles, as it is in many places of Daulphyne, and of Provence ¶ The rule of the milns. ¶ One man hath three milns of whom one grindeth each day 5 septyers of corn and the other grindeth 7 and the third 8. There cometh a merchant that will have gronden one hundredth septyers of corn, I demand how ought the mylner to divide the corn to the milns to the end that each one have assoon done as an other. Answer. For to know this question and rule. ye must find the divisor and the multiplicator, the multyp. shall be each one by himself, and the devisor shallbe the three multiplycatours set together that been 20. Therefore if ye will know how much corn ought to be laid upon the first miln, ye must multiply the 100 septiers of corn by 5 & divide by 20, which shall be 25 septiers, that shallbe laid upon the first miln. And for the second ye shall multiply 100 by 7 and divide by 20, and there shall be 35 septiers, the which ye shall put upon the second miln, and for the third ye shall multiply 100 by 8 and divide by 20 & there shall be 40 septiers, which ye shall put upon the third miln. And thus may ye do of all other semblable. It may be made otherwise, set together the sums that the three milns grind that is 20, and by the rule of three ye shall say, if 20 give me an 100, how moche shall give me 5 or 7 or 8. And it is proved by addition. Example. 100 The first shall have 25 septiers. The second 35 septiers. The third 40 septiers, 7, 5, 8, 20 Multiplicatours. Dyvysour. The rule and question of a shepherd or pastor. Four men have 300 sheep or moutons, of whom the first hath an 100 sheep/ the second 40, the third 150 and the fourth 10 And they give unto a shepherd for to keep these sheep 25 fꝪ. for a year. I demand how ought the one to pay of the 25 fran. after the sheep that he hath. And how long time ought each one to have him at commence or meat. Answer. For to know this rule and all other semblable, it behoveth you to find the multyplycator and the devisor, the multyplycator of the first shall be 100, of the second 40, of the third 150, and of the fourth 10, & than set together all these sums the which been 300 the divisor commune. Or ye may make it by the rule of three in saying, if 300 give me 25, how moche shall give me 100 or 40 or 150 or 10, & always divide by 300 and thus of all other rules. ¶ Ensample of the first. And for to know in how many hours this vessel shall void, ye shall set together the three numbers, that is to wit, 1, 2, 5 which been 8 and that 8 is the divisor therefore divide 60 by 8, and ye shall find that in 7 hours and an half it shallbe empty And thus may ye do of all other semblable ¶ The rule and question of zarasins for to cast them within the see. THere is a gall upon the see wherein be thirty merchants, that is to wit 15 christian men, and 15 saracens, there falleth great tempest where upon it behoveth them to cast all the merchandise in to the see, and yet for all that they be not in surety from perishing, for the gall is feeble and week, so that by ordinance made by the patron, it is necessary that there be cast into the see the half of the thirty machauntes, but the saracens will not be cast in, nor also the Christians: then by an appointment made, they shall set them down upon a row. & then count them unto 9 and he that should fall upon the 9 to be cast into the see, how would ye set them that none of the chrystyens should be cast into the see. Answer. ye shall ordain them after these metres following. Post four quique da post duos unum colloca Tres numerabis, postea unum collocabis unum dic panther, & duo consequenter, Duos post ponas &, iii. siml hic apponas Semel dic ann bis. post ii unum terminabꝭ Primi christiani, sunt saracenique secundi. That is to wit, 4 christiens 5 saracens 2 christyens 1 sarazyn, 3 christyens 1 sarazin ● christian 2 sarasyns, 2 christiens 3 saracens, 1 chrystyens 2 saracens, 2 christiens 1 saracen. Or for to know it more shortly ye may work by this verse following, by the number of the vovels. Populeam virgan matrem regina tenebat ¶ The rule and question of a testament. A Man hath made his testament, the which hath left his wife great, and hath ordained in his testament that if she brought forth a son, he should have two parts of his goods, that is to wit, of 1200 crones, and his wife the other part, and if she brought forth a daughter, than the mother should have two parts, and the daughter the other part. It happeneth when the man is deed, the wife bringeth forth a son and a daughter. I demand how shall they divide the 1200 crones. Answer. ye shall set 1 for the daughter, and 3 for the mother, for the mother ought to have two parts against the daughter, and set 4 for the son for he ought to have two parts against the mother. Therefore ye shall multiply the 1200 crones by 4 for the son, by 2 for the mother, and by 1 for the daughter. And for to find the divisor ye shall set together 1, 2, and 4, which been 7, therefore divide by 7. Example. 4 The son shall have 685 crones & an half, 7 sꝪ. 8 d & half, resteth a half d̔ 2 The mother shall have 342 crones, & and half 12 shillings and half, 4 d̔ Resteth 2 pens. 2 The daughter shall have 171 crones. 15 sꝪ. 5 pens Resteth 1 d 7 Multiplicators devisor. They have to divide an half penny. ¶ The rule and question for to build. And first for the place. A Man hath a ground that is in length 100 yards, and in breathe 70 yards, where as he will edify and buylde● houses, of length 5 yards, and 4 breathe I demand how many houses shall he have upon that ground. Answer. ye shall multiply the length by the breath in saying 70 times 100 been 7000, an be●he house must have 5 yards of length, and 4 of breed/ multyplye that one by the other, and they make 20, which 20 shall be the divisor common, therefore divide 7000 by 20, & ye shall find that there shall be 350 houses Note well this rule. ¶ The rule & question of the walls. A Man will make a wall 32 foot in length, and 2 of thickness, and the height 25 foot, and each foot shall cost the making 2 sꝪ. I demand how much shall cost the making of all the wall. Answer. For to know this rule, ye shall multiply the length by the thickness in saying 2 times 32 been 64/ & than ye shall multiply it by height in saying 25 times 64 been 1600, and than multiply by the price, that is to wit by 2 shillings the which been 3200 shillings, whereof ye shall make francs, therefore divide them by 20 and they been 160 francs. And so much shall cost the making of the wall. ¶ The rule and question of the covering. ¶ If ye will have a house covered with tyelles, ye must know how many tyelles behoveth you to have unto the length of a line, and how many to the bred. Example. If the house had need of 54 for the length, and 34 for the breath, I demand how many should be requysy●e unto all the house. Answer. Multyplye the length by the breathe in saying 34 times 54 been 1836 tyelles, and so many must ye have to cover the house. The rule & question of a graden. ¶ A lover did entre into a garden for to gather apples for his lady, and unto the said garden been three gates, and in each gate is a porter, and when he shall ysue after that he hath gathered the apples, he must give the half of his apples & one, to the first porter, and when he is at the second porter, he must give unto him the half and one/ and to the third porter the half and one, & when he is forth he hath no more but one apple to give unto his lady paramour. I demand how many apples had he gathered. Answer. He had one, apple when he was forth, set to it one, and than it is 2, and then double the 2 and it is 4, therefore he had 4 at the third porter. Then to this 4 set 1 & that is 5, and then double them and that is 10 therefore he hath 10 apples at the second porter, to this 10 set 1 and it is 11, double them, and that be 22 apples. Therefore ye may say that he had gathered 22 apples. ¶ The rule and question of a ladder or stair. ¶ I have seen a stair that had 100 step, pes, in the first step was 1 douffe in the second step 2, in the third 3, in the fourth 4, and so unto 100, I was demanded how many douffes were in all the stair. I answered 5050. probation I will give you certain of all number that do proceed naturally, that is to wit, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, And infinitely as ye will, for all number natural is ended in number even or in number not even, if it be ended in number even, than by the half thereof multiply the number not oven, that encloseth it Example. 1, 2, 3, 4, will ye know what all amounteth unto in saying 2 times 5 been 10, for 2 is the half of 4, and 5 is the number not even that encloseth 4, And if the number end in number not even. As by ensample, 1, 2, 3, 4, 5, will ye know what all amounteth unto. Multyply 5 by his greater half, that is 3, saying 3 time 5 been 15, And thus shall ye always do in what number so ever it be even or not even etc. ¶ The rule and question of two men. IF two men go by one way, and that they go in to any far place, and proceed in such wife, that the one proceed each day certain number of my les, that is to say 4 and 6 more or less. And that other man goeth increasing the first day one mile, the second day two, the third three, and so increasing after progression. Be ye all certain that in some day the one overtaketh the other. It is demanded in what day, and how many miles they shall go. Answer. Double the number of his mills that goeth each day an equal number of miles And of the number double take away one unity, and the remanant shall show you what day they shall meet either other ¶ Example. We shall set it that the one goeth a day 6 miles, double that and it is 12, and fro that 12 withdraw one unity, as it is said in the rule, and there remaineth 11, that is the number of the day that they shall meet together. And for to know the number of the miles that they have gone. Multyplye 11 by 6 in saying 6 times 11 been 66 miles that they have gone. Thus ye may know it by the rule of progression continued, 11 is a number not even, be it therefore multiplied by the greater half that is to wit by 6 in saying 11 times 6 or 6 times 11 been 66. And also one only number amount by progression, & by multiplication, whereby it appeareth that upon the elleventh day they meet each other, & have gone 66 miles. ¶ The rule and question of the women that bore apples to the market. Three women bare apples well & honestly trimmed to the market, of whom the one bare 50 the other 30, and the third 10, their housboundes were brethren and gave commandment to them that they should make as good market one as an other, that is, that they sell all after one price, & that the one bring as much money home as the other. I demand how that may be done. Answer. It is possible. For first there cometh a merchant to her that hath 50 apples: and saith to her how many for one penny, and she answered 7 and so she maketh 7 d of her 50 apples and hath remaining one apple. The other sold after the same price. And she that had 30 apples sold hers for 4 d. and she had remaining 3 apples. The other that had 10 apels sold hers for 4 d and she had remaing 2 apples. And then came there another merchant that gave 3 d for an apple. And so each one bare home 10 d̔ as ye see in this ensample. And thus may ye do of all other semblable. ❧ The rule and question of the bag A Merchant hath a bag that weigheth 19 ounces of three metals, whereof 7 ounces been of gold 8 of silver, and 4 of copre. And he will take thereout 5 ounces. I demand how moche of gold, how much of silver, and how moche of copre is in these 5 ounces. Answer, ye shall multiply the 5 for to know the gold by 7, for the silver by ● and for the copper by 4. And for to find the devisor, ye shall set all the multiplycators together, that been 19, therefore divide by 19 The answer is in this ensample 5 ounces. 7 Of gold 1 ounce an half 8 pens 5 grain. Resteth 1 pens. 8 Of silver 2 ounces, 2 pens and half, 1 half grain. Resteth 2 pens. 4 Of copper 1 ounce 1 penny 6 grains. Resteth 6. ¶ Now set the remeynaunt together and divide it by the divisor common, that is 19 And it is 1 half grain. ¶ The rule and question of the bell. IN a church is made a bell, and there in is put 30 pound of gold 50 li. of silver, 100 of tin, and 102 of copper. When the bell is made there remaineth 40 pound in one piece, that they will sell I demand how much is there of gold how much of silver, how much of tin, and how moche of copre. Answer. ye shall do as above is said of the bag for ye shall multiply 40 each one by himself, and divide by 282. ¶ Example. 30 Of gold 4 pound 4 ounces 4 pens, 1 grain. Resteth 6 50 Of silver 7 pound 1 ounce 11 pens, 9 grains and an half. Resteth 57 100 Of tin 14 pound 2 ounces & half 10 pens, and half 7 grains. Rest 114 pens. 102 Of copre 14 pound 7 ounces 11 pens and an half 5 grains and half. Resteth 105. Multyplycators, 282 divisor common And all divided they have of rests 1 grain. ¶ This rule is proved by reduction, set to the same that remaineth/ and divide by the divisor common, & thereof cometh 1 grain. ¶ The rule and question to change gold into silver. A Merchant hath 100 trancz in gold and he goth unto a changer & saith, I have 100 francs in pieces of gold I would have the money thereof in small pyces, that is to wit, of 2 pens, of 3 pens of 4 pens, of 5 pens, of 6 pens, of 8 pens, 4 of 10 pens, & I would have as many pyecz of one as of an other. I demand how many pieces of every money ought the changer to give him. Answer, ye must set together all these numbers 2, 3, 4, 5, 6, 8, and 10, that been 2 the devisor commune and then ye must make of the francs pens that is 24000 pens, which ye shall divide by 32 and there been 750 pieces of each money, and thus ye may do of all other semblable. ¶ The rule and question of cloth of divers colours I have a piece of cloth whereof the third part is white, the fourth part black, and 8 else of grey. I demand how much hath it of length. Answer. Set 12, for in 12 ye shall find one third and one fourth, the third and the fourth of 12 is 7, and there remaineth 5/ therefore form the rule of three, if 5, be comen of 12, of how moche shall come 58 multiply 12, by 3 that it 96, & divide by 5, and thereof cometh 19 else and 1 fift, therefore ye may answer that the piece of cloth hath of length 19 else and one fift. ¶ The rule and question of spiceries. ¶ A bourgesse said unto h●s servant hold these 13 frances, and go and buy the pepper that costeth 15 sꝪ. the pound, and sugar that costeth 18 sꝪ. the pound, & of fine spices that costeth 9 sꝪ. the pound, and ginger that costeth 13 sꝪ. the pound, and cloves that costeth 10 sꝪ. the pound, and bring me as many pounds of one as of another. I demand how many pounds ought the apotycarye to give him for 13 francs. Answer. ye shall set all the prices together 15, 18, 9, 13 and 10, that been 65 which shall be the divisor, & then ye shall make the francs in shillings, that is 260 shillings. And than ye shall divide by 65, & thereof cometh 4 pound/ therefore ye may answer, that he ought to give him 4 pounds of all these spiceries. ¶ The rule and question of the eggs. A young maiden bayreth eggs to the market for to sell and her meeteth a young man that would play with her in so mich that he overthroweth & breaketh the eggs every one, & will not pay for them. The maid doth him to be called afore the judge. The judge condemneth him to pay for the eggs, but the judge knoweth not how many eggs there were. An that he demandeth of the maid, she answereth that she is but young, and can not well count, but she and her mother had ordained and disposed them by 2 and 2 & there remained 1 egg. Than by 3 and 3 & there remained 1/ than by 4 and 4 and there remained 1/ than by 5 & 5, & there remained 1/ than by 6 and 6 & there remained 1, and at the last by 7 and 7 and there remained none/ I demand how many eggs there were. Answer. 721. And for to prove it, multiply the numbers one by another in saying 2 times 3 been 6, 4 times 6 been 24, 5 times 24 been 120, 6 times 120 been 720, and set thereto 1 that remained always & than they been 721 that which ye shall divide by 7/ & there remaineth nothing/ and so she had 721 eggs. And fater this ensample may the judge judge the young man to pay. ¶ The rule and question of money forgotten with a chaungeour. AN advocate hath given to a chaun, geour money, & hath forgotten how much. For to know how moche and for to have all his money, he findeth subtlety that ensueth, he sayeth to one of his sons, of whom he hath many go unto such a chaungeour & bring me a france and the tenth part of the money that I delivered him, and so was it done And an other time he said unto another son, go unto the chaungeour & bring me 2 frances and the tenth part of the remainant, and so he said unto all, but unto the last he said unto the chaungeour, and bring me all the remainant of the money, and so was it done, and as much brought the one as the other. I demand how moche money he had, how many sons, & how much money each one of them brought. Answer For this three questions pose the number that they all brought, that is to wit, the tenth been 10, and of 10 take one and there do remain 9, therefore ye may say that he had 9 sons, and each one brought 9 sꝪ. And for to know how moche he had given to the chaungeour, ye must multypyl 9 by himself, and it is 81. Therefore he had delivered 81 frances to the chaungeour. For to make the prove lay 81 and take up for the first son 1 and the tenth part of the remainant, and in like manner ye must do of all other. ¶ The rule and question, of time. etc. ¶ A man saith if I had as much more of time as I have, and the half, the third and the fourth of my time that I have set to. I should have of years 50, I demand what age he hath. Answer. Say 12, for in 12 ye find an half, a third, and a fourth And then set there once as much, & that been 24, than set thereto 1 half, 1 third, & ● fourth of 12, and they been 37, and then form thy question. if 37 be comen of 12 of how much shall come 50. Multyply 12 by 50 and divide by 37, and ye shall find that he hath 16 years 78 days and a half 10 hours resteth 2. ¶ The rule and question for to divide distributions. IN a church bell 12 cannons 9 priests and 6 clerks they have to divide a distribution of 400 frnces, werof the canons have 3, the priests 2 and the clerks ●, I demand how moche shall have the canons, how moche the priests and how much the clerks. Answer. Multyplye one number by an other in saying 3 times 12 been 36 that is the multyplycatour for the canons, 2 times 6 been 18, the multyplycatour for the priests, 1 time 6 been 6, the multyplycatour for the clerks. How moche each one ought to have ye may see in the ensample by the devisor. Set together all the multyplycatours & they been 60, the devisor commune. 36 240 frances 38 120 frances 6 40 frances Multiplycatours devisor 60 ¶ The rule and question of the speyre. A Speyr is the half and the third part within the water, and 9 foot with out. I demand how much of length hath the spear. Answer. Set 6, for in 6 is found a half and a third the half and the third of 6 been 5, and there remaineth 1, form the rule of three if 1 be comen of 6, of how many shall come 9 multiply 6 by 9, and they been 54, divide them by 1 and they been 54, therefore ye may answer that the spear hath 54 foot of length, the half is 27, & the third is 18, and there been 45 foot within the water, and 9 without that is 54. And so may ye do of all other semblable, as of a tour. ¶ The rule and question of two men that went that one against that other. TWo men begin to go and take their journey that one against that other upon one day and in one hour. For that one that goeth fro Paris, to London and goeth every day 7 miles, that other goeth from Lion to Paris, and goeth each day 9 miles, and from Lion unto Paris been 80 miles. I demand how long time shall it be or they meet. Answer Set together the miles that they go in one day, the is to wit, 7 and 9 been 16, form now the rule of 16 come of 1 day, of how moche shall come 80 that they have to go, multiply 80 by 1 & it is 80 they which he may divide by 16 & thereof cometh 5, therefore in 5 days they meet. The prove is, for he the from Paris to Lion goeth in 5 days goeth 35 miles/ & that other 45 the which been 80 miles. ¶ The rule and question of a cat. THere is a cat at the foot of a tree the length of 300 foot, this cat goeth up ward each day 17 foot, and descendeth the night 12 foot. I demand in how long time shall she be at the top, Answer. Take up and abate the night of the day, that is 12 of 17 and there remaineth 5, therefore the cat mounteth each day 5 foot/ divide now 300 by 5 and thereof cometh 60 days then she shall be at the top. And thus ye may do of all other semblable. For of this rule ye may make 4 questions, as it appeareth in the practise thereof. ¶ The rule and question of 20 scholars. IF ●0 scholars own unto their host 5 d tourneys, how ought they to pay, so that each one pay his duty & give the money of his purse. How moche shall each one pay. Answer. Each one shall pay 1 penny Paris, and the host shall rethurne unto him again 1 penny tournoys and so each one shall pay the 4 part of a tournoys. ¶ The rule and question of pilgrims. ¶ Twenty pilgrims, that is to wit, men, women, and little children, have spended in drink 20 pens, whereof the men pay 3, pens, the women 2 pens, and the little children half pens. I demand how many men, & how many women, and how many children be there, for to pay this 20 pens, so that there be 20 persons. Answer. There shall be one man. 5 women, and 14 children. ¶ The rule & question of a chanter. ¶ A chanter hath each day of rent fro the court of the prince 12 sꝪ. the which is paid by knights, damoiselles, and squires of whom the knights pay 2 sꝪ the damosels 6 pens, and the squires 3 pens. I demand how many knights how many damosels, & how many esquires aught there to be, to pay this 12 sꝪ. so that there be 12 persons. Answer. There must be 5 knights, 1 damosel, and 6 esquires. ¶ The rule and question for to divine, ¶ If ye will cause your fellow to believe that ye shall divine how many pieces of silver he hath in his right hand say unto him that he put as many pieces in that one hand as in that other. And than that he take five from the life hand to the right hand and than that he put forth of the right hand into the life hand as many pieces as he hath remaining in the lift hand. And there shall remain 10 in the right hand. ¶ The rule and question of three saints. A Holy hermit is entered within a church wherein there been three saints: that is to wit/ saint Peter, saint Paul, and saint Francoys/ this hemite cometh first to saint Peter and saith to him in a manner of his orayson, I pray the that it pleas the to double me the great blances that I have in my purse, and I shall give the 6, and so was it done. Than came he te saint Paul & said to him, please it the to double me the great blances that I have in my purse and I shall give the 6 & so was it done. Then came he to saint Francoys & said, if it wold● please to double me the great blances that I have in my purse I shall give the 6, and so was it done, and nothing had he remaining. I demand how many great blances had he in his purse. answer. He had 5 and 1 fourth. And for to know it double them and they been 10 and an half, and then ye must give 6 to saint Peier, and there remaineth 4 & an half, double them. & they been 9 And then giveth he 6 to saint Paul, & then there remaineth 3 double them and there been 6 and that 6 giveth he to saint Francoys, and so he hat nothing remaining. ¶ Here follow divers other proper rules and questions. A Lord hireth a servant, the which he should give every year ●0 nobles and a gown, and the same servant dwelleth 7 months with him, and then they vary in so much that his lord gave him licence to go his way. And saith, go thy ways out of my house and take thy gown with thee, and then I am nothing in thy debt. Now I demand what was the gown worth, will ye know that, then mark how many months 7 less than a year, that is 5 months less. And had the servant tarried so long yet by his master than should he have had the gown & 10 nobles. Therefore say thus 5 months giveth 10 nobles, what giveth 7. Make it after the rule of three/ & it cometh 14 nobles. ¶ Of three fellows or young men. ¶ Three fellows play together the one to win the others money. For the one had more money than the other. And the first casteth, that the one of them three loseth just so much money as the other two had. Than casteth the second and loseth also as much as the other two had Then casteth the third and loseth also just as much as the other two had. And than was the money just divided, & had each like much. Now I demand how moche had each or they began to play, & how moche money that each had when they played. Will ye know that, then mark how many persons died play, & add 1 to them, as here add 1 to 3 maketh 4. So many nobles had the first. Now double 4 cometh 8, & subtra 1 from 8 rest 7: so many nobles had the second. Then double 7 cometh 14, thereof subtra 3 rest 13, so many nobles had the third. another question. A Man buyeth 46 pound of saffron for 30 pound, what shall cost 63 pounds of saffron. will ye know that, then multiply the 30 pounds with the 63 pounds of saffron, cometh 1890. Now divide them with 46 cometh 41 pounds and 4/49 part of a pound to pay for the 63 pounds of saffron. Now will ye know how many shillings that 4/46 part of a li. is. than multiply 4 by 20, for 20 sꝪ maketh a li. cometh 80 sꝪ. divide them with 46 cometh 1 sz, and 4/64 part of a sꝪ. Now will ye know how many pens that 34/64 part of a shilling is, them multiply 34 with 12 12 pens maketh a sz, cometh 408. divide them with 46 cometh 8 pens and 40/46 part of a penny. Now will ye know how many farthings that ●●/4● part of a penny is, then multypplye 40 with 4, for 4 farthings maketh a penny, comet 160 farthings. Now divide them with 49, cometh 3 farthings and 22/46 part of a farthing, Thus done ye shall find that 63 li. of, saffron cost 41 li. 1 sꝪ. 8 farthings and 22/46 part of a farthing. ¶ Item a 165 pounds of alum cost 2 pounds 5 shillings 6 pens 9 fa●thynges: what shall cost 22 pounds of alum. If ye will soil this question, than make of your pounds shillings & add thereto the odd 5 shillings: cometh 45 sꝪ. Then make of the 45 sꝪ. penes, and add 6 pens, cometh 546 pens, than make of your pens farthings, and add thereto the 9 odd farthings, cometh 2193 farthings. Now multiply the farthings with 22 cometh 48246 farthings. Now divide them with 165 cometh 592 and 66/165 part of a farthing, for so many farthings shall cost 22 li. of alum. Now will ye know how many pens that the forewritten farthings make, then divide them with 4, for 4 farthings make a penny. Then will ye know how many shillings that they make, then divide the pens with 12, for 12 pens maketh a sꝪ. Thus done ye shall find that 22 pound of alum cost ● sꝪ. 3. d 1 farthing, and it is done. ¶ An other question. ¶ A merchant hath bought a bag of pepper, I say not how heavy, but when he giveth for a pound of pepper 12 pens, then remaineth him yet 37 d And when that he giveth for a pound of pepper 15 pens, than he lacketh 44 pens to pay for the pepper. Now I demand how heavy the bag of pepper was, and how much money that the merchant had. For to know this & such other like question, ye shall take and subtra 12 from 15 and there resteth 3, which 3 shallbe your devisor. Then shall ye add 44 and 37 together/ and that maketh 81. Then must ye divide 81 with 2, & thereof cometh 27, so many pound weigheth the bag of pepper. Now will ye know how moche money the merchant had, then must ye multiply 12 with 27. and add 37 thereto, or multiply 15 with 27 and subtra 44, cometh 361, so many pens had the merchant. another question. A Dronkart drinketh a barrel of bear in the space of 14 days, and when his wife drinketh with him than they drink it out within 10 days. Now I demand in what space that his wife should drink that barrel of bear alone. For to soil this question & such other like, ye shall first subtra the lee●t drinker from the more that is 10 from 14 and there remaineth 4, & that is your divisor. Now say 4 giveth 10 what giveth 14 Make it after the golden rule, and ye shall find that she should drink it in 35 days. ¶ Here endeth the introduction of awgrym for the pen. ¶ Here beginneth the introduction for to learn to reckon with the counters, with divers rules belonging to the same. diagram of counters C. thousand, X. thousand. Thousand hundreth Ten. One. FOr as much as there been many persons that been unlearned, and can not write, yet nevertheless the craft or science of awgrym and reckoning is needful for them to know, wherefore I shall hereafter declare & write of this science in the best & shortest wise that may be possible, how that ye shall order yourself in reckoning and to cast counter. ¶ first ye shall understand that in the craft of awgrym be 9 letters or figures that men may lay and write all manner of sums withal. Therefore first of all a man must know in this craft or science for to lay 9 counters in the places of that 9 syfers, for they must lay evermore still for a remembrance, so that ye may remember your place by them. And ye must lay them the one right above the other, that is to say, in the first place every counter standeth for one, and the nethermost counter is the first place, in the second place every counter standeth for 10 In the third place for a 100 In the fourth place for a thousand. In the fift place for 10 thousand. In the sixth place for 100 thousand. In the seventh place for a million. In the eight place for 10 millions. In the ninth place for a hundreth millions. In the tenth place for a thousand millions, and so forth infinitely. And note well that every counter that is laid between the lygnes, betokeneth evermore 5 times more then the counter that lieth in the place next unde him/ that is to say the first counter dying alone above the first place betokeneth 5 the counter dying alone between the second and the third liar and place, standeth fore 50, above the third place 5 honderth, above the fourth 5 thousand above the fift place 50 thousand, above the sixth place 500 thousand, above the seven 5 millions, above the eight 50 millions, above the 9 place 500 millions, above the tenth place five thousand millions. But if ye will the more surer know your places it is necessary for you to mark every place with a mark, as to lay a counter or some other thing which shall ever lay still and in no wise be removed/ but ye must take heed if ye lay counters for the mark of your places, that ye lay them not to nigh the counter that ye must work with all, lest that ye take the one for the other, but lay them as ye see them marked in the ensamples following. And when ye have laid marks and know the order of your places, ye may add, and subtra, multyply and divide what numbers ye list, that is to say, to cast and to abate at your pleasure. ¶ Item when there lie 2 counters between two liars, take him up and lay 1 beside the next liar above them. And when there lie 5 counters beside any liar take them up and lay 1 in the next space above them. ¶ Of addition. addition is none other thing but to set together 2 or 3 numbers and to make of them a total somme, as in the ensample following. ¶ There is a man which owe 20 li. 18 pound, 100 pound, 50 pound, and 69 pound. Now if ye will know how. much that all these sums maketh together Then for the first somme ye must lay two countries beside the seconnde liar, for the two stand for 20/ that is for the first some. Now for the second some lay 1 counter beside the second liar, for that is 10/ and lay 1 counter betwixt the nethermost & the second liar, for that 1 standeth for 5/ and then lay 3 counters beside the nethermost liar/ and they all together make 18. Now for the third some ye shall lay 1 counter beside the third liar/ for that is an 100 For the fourth somme lay 1 counter beside the third and the second liar, that is 50. Now for the fift somme lay 1 counter betwixt the third and the second, and 1 beside the second liar, and 1 bewene the second and the nedermost, and 4 beside the ne, thermost liar, and that maketh together 69/ and in so doing ye shall find that all the forewryten sums make together 247 as ye shall see in the figure following And evermore for a general rule remember your places, for every counter that lieth beside the first liar standeth but for 1, in the second place every counter standeth fof 10, in the third place for 100 as is afore rehearsed. diagram of counters ¶ Will ye prove whither ye have added well or not, than subtra all your sums one after an other. And in like wise as ye do with this ensample so ye shall do with all other of addition, ❧ Of subraction. SVbtraction is, if ye will withdraw any sum from an other sum, ye must know two numbers, that is to wit, the number that ye will withdraw, and the number where fro ye will withdraw. An ensample. There is a man that oweth you 9756 pounds, and there upon he hath paid you 5989 pounds, Now if ye will know what there resteth then set down your sum that he ought you, & thereof withdrw the sum that he hath paid you, and that the remaineth is the some that he doth yet owe you, as ye more plainly may see it in the ensamples hereafter following. diagram of counters And when ye have set your death/ that is to say 975● pounndes under this manner as afore showed. Then if ye will know the rest, them take thereof that ye have paid as 5989 pounds. Now for to do this/ ye shall first take up the counter that lieth between the fourth & fift liar for that is 5000. Then take up one of the counters which lieth beside the fourth liar, & that is a thousand/ & ye should take away but 900, therefore ye must lay down 1 counter again beside the third liar, that is a hundreth. Then take up one of the counters that lieth beside the third liar, which is a hundreth/ and ye should take up but 80, therefore ye must lay 2 counters beside the second liar/ that is 20 and 80, that ye have take up maketh 100, then take up one of the counters that lieth beside the second liar/ that is 10/ and ye should have take away but 9, therefore ye must lay one counter beside the nethermost liar, that is 1 and the 9 that ye have subtrahed or take up, maketh 10, and there remaineth 3767 pound det, and stand thus. diagram of counters Will ye prove whither ye have subtrahed well or not, then add thereto that ye have paid, and if the some come then so great as it was afore, then is your subtraction true, else not. Multiplication. MVltiplycation is nothing e●les but to multiply one number by an other, as thus, to know what is 6 times 9 or 6 times 12 and such like. And in multiplication ye must consider two numbers, that is to wit, the number that ye will multiply, and the number whereby ye will multiply, and ye must work in multiplication after this manner: first ye shall lay down the lesser number, which is 6, and this 6 is the number that shall be multiplied, and the 9 is the number that ye shall mutiply withal And ye shall lay the number that shall be multiplied, at the right side of your liars & when ye work your multiplication, ye shall lay them at the lift side, as in this ensample here after following shall more plainly appear. diagram of counters ¶ first ye must lay down the lesser number, which is 4, as in this ensample, as ye see them laid here on the right hand of the liars. And when that ye have thus done, ye must take up one counter and lay 9 for it on the other side of the marks, that is to wit/ at the lift side. And after that take up another counter/ and lay also 9 for it/ and so forth for every counter that ye take up ye must lay 9 for it at the other side. And when that ye have so wrought your work, it will come just to 36, as ye see the counters before laid on the lift hand of the lygnes. ¶ And if ye will multiplplye by greater numbers as thus/ to know what is 24 times 14. first lay 14 on the right hand of your liars or markers, as this ensample following showeth. place next above your finger/ and reke● every eachone of them for 2 which maketh just 6, then 10 and 6 maketh 16/ as the figure before showed/ when ye have so done, then take away your finger, and for every one of the 3 counters that lieth in the first place on the right side lay 16 on the life side/ and than take them of the right side away/ and ye shall see that the number shall just come to 128, as the ensample before showed. And this wise ye must reckon all counters that lieth in the spaces if the multiplication shallbe truly made. ¶ An other ensample ¶ For to know how many groats be in 4563 nobles. first ye shall set down the less number, that is the number that ye shall multiylye, as this figure following plainly here showeth, diagram of counters ¶ Now for to make of these nobles groats, ye must multiply them with 20, for 20 groats maketh a noble. Now for to multiply this number/ evermore ye must set down the number that ye will multiply at the right side at your marks and set your finger against the mark that ye begin at, for your finger shall be a remembrance to you for that place where your finger standeth is the first place and damneth all the places underneath him. ¶ Now for to make groats of ●hese 4563 nobles. first ye shall set your finger against the fourth liar, & take up one of the four counters that lieth against the said fourth liar, & lay two counters beside the nerte liar above that, where your finger standeth, for that is the second place from your finger, and the two counters so laid standeth for 20 that is one noble, and like as ye have done with this one counter, so shall ye do with the other 3 following. Then take up the counter that lieth between the third and the fourth liar and lay two counters in the next space above that, and that is also 20 or else ye may take it up and lay one counter beside the second liar, for the place where your finger standeth and that is also 20. Tken take up the counter that lieth betwixt the second and the third liar, and lay 2 in the next space above that/ then take up the counter that lieth beside the second liar, and lay two counters beside the next liar above that same. Then set your finger against the first liar, and take up one of the 3 counters and lay 2 counters for it beside the next liar above that, & as ye have done with that, so must ye do with the othertwo, and then ye shall find that 4563 nobles maketh 91290 groats, and standeth thus as the ensample hereafter showeth. And as ye have done with these forewyten ensample of multyplcation, so shall ye do with all other of multiplication. diagram of counters ¶ Will ye know or prove whither ye have multiplied well or not, then divide the groats, that is 91260 by 20, & if the some come to stand as it was afore, than ye have multiplied well. And thus alway ye may make your prove upon all manner of multyplycatours. ❧ Of division. DYuysyon is to divide a some through an other somme: and in this division must be knowen two numbers, that is the nober that ye will divide, and the nober whereby ye will divide it, as to know how many times ye may have a small number out of a great as by ensample, if ye will divide 336, by 14 as in this ensample hereafter. first lay 336 on the right hand of your liars, & then set your finger at the highest place where any counter lieth, for as I showed you before, the damneth all the other places beneath, so the then there as your finger is, is the first place. And then look ye may take 14 from that place, which ye can not do, for every counter standeth but for one, because your finger is there, therefore ye must remove your finger to the next place beneath where the other 3 counters lie/ and then look if ye may take 14 from that place which ye may do right well/ for these 3 counters at your finger standeth but for, 3 and the other 3 counters above standeth for 30, and then see how many times 14 ye may have out of 33 and so many counters ye must lay on the other side just against your finger/ that is to say, ye may have 28 out of 33, that is two times 14 out of 33/ and therefore ye must take up 28 and lay 2 counters on the diagram of counters other side against your finger, & than ye can have 14 no more, than ye must remove your finger to the next place beneath, & then reckon that place at your finger to be the first place, as ye did before: & then look how many times ye may have 14 from that place, which ye can not, for that counter at your finger standeth but for 1, & the other in the space above standeth but for 10 which is in all but 11, therefore ye must remove your finger to the next place beneath/ and than ye shall see that that number is 56 out of the which ye may well take out 4 times 14 which maketh just 56 therefore ye must take up 56 and lay 4 counters on the other side against your finger/ and than take away your finger, and ye shall see that that number that ye have laid on the lift side of your marks cometh just to 24 as the ensample before showed, and then ye have your question soiled; for if ye divide 336 by 14 it cometh just to 24 for 24 times 14 maketh just 336, as I have showed you before in the rule of multiplication. And likewise as ye have divided this number, ye may do with all other numbers. And if ye will prove whither ye have well divided or not, take the number that cometh of your division, and multiply it with the small number that is your devisor, and add that remaineth thereto if there be any, and than it will come just to the great number that was the number to be divided. And likewise if ye will prove whither ye have truly multiplied or no, take the great number that cometh of your multiplication, and divide him by the number that is to be multiplied, and it will come just to the third number that was your multiplier, diagram of counters ¶ if ye desire to know how many groats been in 79992 pens. first ye shall set down your pens as ye see in this figure afore, and ye shall divide them with 4 for 4 d̔ maketh a groat. Now to the operation thereof, when that ye have set down your pens, as the figure afore showed/ than set your finger at the highest counter. and see if ye may have 4 from that place/ which of a surety ye can not, for there lieth but one, and it standeth but for one, by cause you finger standeth there therefore ye shall remove your finger and set it against the fift liar, and see if ye may take away 4 the which ye may do, for there lieth 7. Now your finger standeth against the fift liar, therefore ye shall take up the counter that lieth in the next space above your finger, for that counter is 5, and ye should take up but 4 therefore ye shall take it up and lie it beside the fift liar of the right side, and lay 1 on the life side also beside the fift liar, then see if ye can have 4 any more from the place that which ye can not, therefore remove your finger and set it against the fourth liar, and then see how many times 4, ye may have out of 39 ye may have 9 times 4, and for this 9 times 4 ye shall lay 1 counter in the space between the fourth and fift liars, and 4 beside the fourth liar, & that make 9 And as ye have done with these so shall ye do with all other following, & when ye have finished your work ye shall find the 79992 pens make 19908 gretes as ye may see plainly here in this figure. diagram of counters ¶ The prove. ¶ If ye will prove whither ye have divided well or not, then multiply the groats with 4, for 4 pens maketh a groat, and if the some come to stand as it died afore then ye have divided well. ¶ Iten when ye divide any some with d if there remaineth any thing it is pens And if ye divide by shillings, if there remaineth any thing it is shillings. And as ye have done with these forewryten examples so may ye do with all other. ¶ The golden rule. ¶ Regula aurea is called the golden rule for like as gold passeth all other metal so this rule passeth all other rules in awgrym. And to the operation of this rule must always be noted three things or three numbers, of the which two of them must be like of names and of kind, that is to wit, the first and the third number, & always ye shall multiply the second number with the third, & that that cometh of the multiplication is the number to be to divided by the first number that is general divisor/ and the quotient of the divisor, showeth a number of solution of name and kind of the middle numbers, as in these ensamples following shall appear. ¶ If a man buyeth 40 eggs for 20 pens how many for 12 pens, if ye will soil this question, ye must mutyplye the second and the third number together/ and the product or the some that cometh of that multiplication, ye shall divide by the first number, like as here is showed in this ensample, when ye buy 40 eggs for 20 pens. what shall one pay for 12 eggs ye shall multiply 20 with 12, cometh 240, the which ye shall divide with 40 cometh 6 pens, and so moche shall ye pay for the 12 eggs. And thus ye may do with all other such questions. ❧ An other qustyon. ¶ Iten a 100 apples cost 12 pens, what shall one pay for 87, ye shall multiply 12, with 87, cometh ●044, the which ye shall divide with a 100 cometh 10 pens 44/10 part of a d for the 87 apples. Will ye know how many farthings that the 44 hundreth part of a penny is wdrth, then multiply 44 with as many farthings as a penny is worth, that is 4 cometh 176 the which ye shall divide with 100 cometh 1 farthing and 67/100 part of a farthing. ❧ Item 165 pound of wax cost 2 li. 5. sꝪ. 6. d 9 mites. what shall cost 22 l. For to soil this question and such other liker, first ye must make of pounds sz, & add thereto the odd 5 sꝪ. the which stand in this question, and the come together to 45 sꝪ. then make of the sꝪ. pens, and add thereto the odd 6 pens that standeth in the question cometh 546 d than make of the pens mites, and 24 mites is a penny, and thereto add the odd 9 mites, that standeth in the question cometh together to 23113 mites, and that is the total some of all the li. sꝪ. d, & mites together Now make it after the rule and say 165 pound of wax cost 13113 mites, what shall cost ●2 pound, first multiply the myldelmost and the last together, that is ye shall multiply the mites with the last, that is with 22, and it shall come to 388486, divide them with 165 and it shall come to 1748 mites and 6/165 part of a mite, so many mites shall the 22 pound of wax cost. Now will ye know how many pens that the forewryten mites make: then divide them with 24, for 24 mites maketh 2 penny Then will ye know how many sꝪ. that the pens make, then divide them with 12, for 12 d maketh a sꝪ. And thus doing ye shall find that the 22 li. of wax shall cost 6 sꝪ. 20 mites & 66/165 patt of 1 mite and it is done. ¶ Item when there standeth 1 in the first place, As 1 goose cost 3 d what shall cost 28. ye shall multiply the myddels●e with the last in saying 3 times 28 is 84 so many pens cost 28 geese, and it is finished. ¶ Item in the contrary as when that 1 cometh in the latter end, as here in this ensample 20 capons cost 23 pens/ what shall cost 1 capon. For to soil this question ye shall divide the middlemost with the first that is 23 with 20 cometh 1 d and 1/20 part of a penny, that is 0 mites & ⅘ part of 1 mite for 1 capon, ☞ Item 17 else and ½ cost 14 nobles and 1/ part of a noble, what shall cost 32 else and 2/● part. For to soil this question and such like/ first ye must break the first and last broken together crosswise in saying 1 times 4 is 4, set the 4 by 17 else Then say 3 times 2 is 6/ set the 6 by the 22 else. Then multiply both the numbers of your fractyons together/ in saying 4 times 2 is 8 the which ye shall set under 4 and 6/ then it standeth thus: as in the rule of three/ if 17 else & 4/8 part of an elle cost 14 nobles & ⅓ part of a noble/ what shall cost 32 else and ●/● part of an elle. first mulyply the first hole number with the nethermost of his broken that is 17 with 8 cometh 136 & thereto ye shall add that 4 that standeth above 8 comeeh 140, the which ye shall multiply with the 3 that standeth under the second broken cometh 430 that is your devisor, then multiply 14 with the 3 that standeth under 1 and add that 1 thereto cometh 43 that is your multiplicator, then multiply 32 with the nethermost figure of the broken, that is with 8 and add thereto the same 6 that standeth above the same 8 cometh 262. Now set it in the rule of three in saying 420 else cost 43 nobles, what shall cost 262, multiply the second with the third and then divide that that cometh of that multiplication with the first, and ye shall find that the 32 else of cloth cost 26 nobbles 16 d 34 mites and 150/420 part of a mite. ¶ Item when that there standeth at the beginning a hole number with a broken and in the second and third place no broken, as here. 36 else and 1/1 cost ● li. what shall cost 16 else. For to soil this question, ye must multiply the first hole number with the undermost figure of his broken, that is 36 with 2, and add the 1 thereto that standeth aboveth 2 coming 73, and that is your devisor, then multiply 8 li. also with the undermost figure of the broken, that is to with with 2, and it cometh to 16, then multiply 16 with 16 cometh to 259 the which ye shall divide with 37, and it will show you that there shall be to pay for the foresaid cloth 3 pound 10 shillings 1 penny 17 mites, and 2●/73 part of a mite. ¶ Item when there standeth in the first nor in the second no broken number, but in the latter end a hole number with a broken, as here 1● ounces of grain cost 14 sꝪ. what for 9 ounces of grain and one third. For to know this ye shall multiply the first 14 with 3 that is your devisor, then multiply 9 with 3 and add that 1 thereto the standeth above 3 cometh 28, than set it thus 4● giveth 14 what giveth 18, make if forth after the rule and ye shall find that there is to pay 9 shillings 4 pens for the 9 ounces of grain and one third part. ¶ Item when that ye find neither at the beginning nor at the latter end no broken number, but in the mids a hole with a broken as here. A man bought 48 sheep for 64 crones and ●/4 what shall cost 18 sheep, For to soil this question ye must multiply 48 with 4 cometh 192 that is your devisor, then multiply 64 with 4, and add thereto the 2 that standdeth above 4, cometh 258, the which ye shall multiply with 18 cometh 494● the which ye shall dyvind with 192. And thus ye shall find that ye must pay for the 16 sheep 24 crones 4 stuners & 36 mites ¶ Item when ye find no broken at the beginning but in the second and third one hole with a broken. As 7 else for 6 pound ¼ what shall cost 16 else & ●/● For to soil this question ye must multiply the two undermost borken numbers together, in saying 3 times 4 is 12, the which ye shall multiply with 7 cometh 84 that is the divisor. Then multiply each with his broken cometh 25 and 46 the which ye shall multiply one with the other, cometh 1225, the which ye shall deu●●e with 84, and the solution shallbe to pay 14 li 12 sꝪ. 18 d ¶ when that ye find at the beginning and the mids a hole with a broken, and at the latter end standing a hole without a broken as 9 else & ●/4 for 5 pound ⅝ what shall cost 15 else multiply 9 with 4 and add thereto 3 cometh 59, the which ye shall multiply with 8 cometh 312, that is your devisor then multiply 5 with 8 and add thereto the 5 that standeth above 8 cometh 45 the which ye shall multiply with 4 that standeth under the first broken, cometh 180. Now set it in the rule of three in saying 312 gives 180 what giveth 15, make i● after the rule and it cometh 8 li. 13 sꝪ. 22 mites and 1●/59 of a mite. ☞ The rule of company. ¶ There be 3 merchants or companions the which lay together their money in merchandise, and each to win after his inlayenge, whereof he first laid in 170 crones. The second 60 crones The third 40 crones and therewith they have won 50 crones beside all uncost Now I demand how moche that each shall have after his inlaying. Now for to soil this question and all such other rules of company, ye must make of their money that they have laid in a total somme cometh 250/ now say 250 giveth 50 what giveth 150 make it after the rule of theridamas and it cometh to the first man 30 crones winning. Now for know what the second hath won, ye shall say 250 giveth 50 what giveth 60 make it after the rule, and ye shall find that the second hath won 12 crones, will ye know what the third hath, then say 250 giveth 50 what giveth 40 make it after the rule, and ye shall find that the third hath won 8 crones. And thus shall ye do with all other rules of company. The rule of company with time. ¶ Three fellows doth merchandise together, whereof the first layeth in 50 crones for 4 months. The second 80 crones for 2 months. The third 100 crones for 5 months, & withal this money they have won 6 crones beside all uncost paid. Now I demand what each hath won with his money, for to know this ye must mulyplye each man's money with his time, that is for the first 50 with 4 cometh 200, set the as though he hath laid in so much, for the second multyply 80 with 2 cometh 160 set it also as though he had laid in so moche, for the third multyplye 10● with 5 cometh 500, set that also as though he had laid in so moche. Now add the 3 numbers togythere and then make it after the rule of company and then shall ye find what each hath won with his money. The rule of batering. ¶ Two merchant men will change their wares together and the one hath a fine black cloth the which is 43 else long, and he will give the elle no less than 18 pens. The other merchant hath pepper, and he will sell the pound no less than 13 d Now I demand how many pound of pepper the first merchant shall have for his 43 else of cloth. For to soil this question ye shall say 13 giveth 43 what giveth 18, make it after the rule of 3 and ye shall find that the first shall have for his cloth 59 pound of pepper 8 ounces 12 engelsche and 4/13 part of an englysche ☞ Of a watte. ¶ A wat runneth in the field and overronneth in one minute/ there be 60 in an hour/ 12 rods of ground. And a grahounde being her enemy/ followeth her and overronneth in one minute 15 rods of ground. But or the grahonde began to run the hare had run 200 rods of land. Now is to be demanded in how many minutes and how many rods of land was the hare taken. For to soil this question and such like, ye shall subtra the less running out of the more that is 12 out of 15, and there remaineth 3 and therewith ye shall divide the space that the hare hath run afore or the grahounde began to run/ that is 200 rods, And in so doing ye shall find that the grahounde overtook the hare in the 66 minutes and ⅔ parts of a minute that is one hour and 6 mynues and ⅔ of a minute, will ye know how many rods that the grahonde did run or that he took the hare/ then multiply 66 and 2/● with 15 cometh 3000/●3 the which ye shall divide with 3 cometh 1000 hole so many rods did the hound run or that he took the hare. ¶ The rule of two fellows. ¶ Two fellows went together out of a town, and the one goeth every day 12 mile, and the other goeth the first day but 1 mile, and the second day 2 miles the third day 3 miles/ and so forth ever day one mile more. Now I demand in how many days/ and how many mile went he or that he overtook his fellow. For to soil this question, ye shall double the miles of him that went every day like moche, that is 12 and 2 times 12 is 24, thereof ye shall subra the one mile that the other goeth the first day, and there resteth 23, upon the same day was the first man overtaken of his companion? will ye know in how many mile, then multiply 23 with 12 cometh 276 for so many miles went he or he overtook him. ☞ A man hath a golden crown of 34 stivers, and a phylyppus' gulden of 15 stivers, and a ducat of 28 stivers, and with this money he goeth to the changer and will have for it negenmannekens crowns of 9 mites, & of 3 mites and of 2 mites and half mites. Now I demand how moche that he shall receive of each for the foresaid gold and receive of each like moche. For to soil this question and such like then make of all the great money that he will change mites, for that is the lest coin that he will have/ and cometh 9072 mites, then look how many mites that all the small pens be worth that he will have, that is 25. Now divide the great sum of the mites, that is to wit, 9072 with 25 and ye shall find that he must have of each 362 and 1●/25 and it is done, ¶ Of four carpenters. ¶ Four carpentes will make a house whereof the first taketh upon him to make it himself alone in a year. The second will make it in two years, The third will make it in 3 years. And the fourth in 4 years/ Now I demand if all these 4 wrought upon that house in what space would they 4 make that house. will ye know that, then say the first would make it in one year that were 12 times in 12 year The second in 2 year, that were 6 times in 12 year. The third in three year that were 4 times in 12 year. And the fourth in four year that were three times in 12 year. Now some them all together that is 12, 6, 4, 3, cometh 25, therewith divide 12 cometh 12/25 prat of a year Now if ye will know how many days that it is, then multiply 12 wthy 365 for so many days be in a year/ & that that cometh of that multiplication divide it by 25 cometh 175 days and 5/25 part of a day. ¶ The rule of false positions, by the which all manner of difficult and hard questions may easily be dissolved & first of one self position. NOw shall ye know how by false positions or conjectures one or two ye shall find out the very truth of that the which ye seek fore, and first ye shall understand how to find the truth of a question proposed by one conjecture or position. When that any question is put forth unto you, too be assoiled, of the which one part is known and the other unknown. Answer to that question by and by, with yourself at all adventure, and then consider with yourself whether ye have made right answer or no, if not, look what proportion is between your conjecture and that that followeth of your conjecture and the same proportion is between the thing known and that the partaynyth unto the self thing being yet unknown. As by example ye shall more plainly perceive, ¶ A certain way faring man coming by the way, found so many crowns. that the second, the third, & the fourth part of them added together made 50, I demand what sum he found. To make answer to this question by one position, imagine some sum that hath these parts in it, that is to say: a second a third, and a fourth part, and be it 12, whose second part or half is 6, third part 4, the fourth part 3, which all added together 643, make 13. but the sum that he found, the second, third, and the fourth of it made 50. wherefore 12 is not the some he found, therefore this position is false/ & yet by this false shall ye come to the light of truth, by the help of the rule of three. For look what proportion is between the second, the third, and the fourth part of 12 added together, the which maketh 13, and 12 whose parts therebe, the same proportion is between 50 which is the second, the third, and the fourth part added together of the number unknown, & the same unknown number itself. Then say thus with thyself: if 13 which containeth the foresaid parts in them, come of 12 of whom come 50, then set them thus 13, 12, 50, then by the rule of three multiply 50 by 12, and thereof cometh 600 divide the same by the first number 13 and in the quotient thou shalt find 46 2/13 the which was the sum of the corwnes the which the man found: of the which sum the half part is 23 1/13 the third part is 15 1/1, the fourth part 11 7/1● the which parts added together make just 50. Thus thou seest how that by one false position or conjecture with the help of the rule of three, this question is soon dissolved. ¶ An other question. ¶ find me a number in the which 5 is ⅔ that is to say, two third parts of him Answer. imagine any number ye list 3 that hath thirds in it, as be it 6, then look what is the third part of 6, that is 2. then two of this third parts of 6 maketh 4 wherefore this position is false, yet by this false position with the help of the rule of three, thou shalt find out the truth, after this manner. If 4 be the ⅔ part of 6 to whom is 5 ⅔ parts search by the rule of three, thou shalt find it 5 ⅓ ¶ An other question. ¶ What number is that in the which after that the third, the fourth, & the fift part be deducted out of it, there shall yet remain 24? Answer. imagine any number that hath a third, a fourth, & a fift in him. As for example: say it is 60, then subtrahe out of him his third, his fourth and his fift part: and thou shalt find remain but 13. Lo how much thou hast missed thou shouldest have found such a number, in the which after the foresaid parts were subtracted should remain 24 and here remaineth but 13, yet prove by the rule of three, and thou shalt find the true numb. If 13 remain after the substraccyon of the aforesaid parts is 60 what number is that out of the which after like substraccyon of his third, fourth, & fifth, part shall remain 24 prove by the rule & thou shalt find it 110 ●/1● whose third part is 36 12/13 the fourth 27 10/1● the fift 22 ⅔ which all added together make 86 20/●● the which deducted out of 110 10/13 shall remain 24. These and diverse other questions before rehearsed by the same craft one false position may soon be assoiled. Now will I show you how to dissolve all manner of questions how difficulty so ever they be by two false positions, For by one false position ye shall 〈◊〉 answer to all manner of questions, but two false posions, what ever question it be/ it may soon have solution. How to answer by two false positions. INnumerable questions do chance in numbres, the which though they can not be dissolved by one position or conjecture yet shall it not miss but be assoiled by two positions: in the which manner ye must diligently note how far above the truth or under both positions do fall: For by the observation of ii conjectures how near they be to the truth, & the difference of the errors which ensue of the positions, the verity cometh to light which may be done iii ways: one way by the rule of both more or both less. Another way by the rule of the one more and the other less ¶ when both positions be more than the verity or both less, then subduce the less error out of the more error, & that that remaineth shallbe the divisor: then multiply the first error by the second position, & the latter error by the first position, and then this ii numbers being multiplied, deduct the less out of the more, and that that remayneh divide it by the foresaid divisor, and the quocient shall show the verity. Example ¶ Three merchants divided a 100 crowns so that the second should have 3 crowns more than the first, and the third 4 more than the second: I demand now how many crowns each of them received? Answer. first make saint Androwes cross, as ye see hereafter, then conjecture what ye list, as for example, Say the first had 33, and then must the second have 36, and the third merchant 40, which sum gathered together maketh 109 but ye had but 100 to divide, wherefore ye have missed, and your position redowndyth to more than the very sum by 9, which came of your first position 33 wherefore set the first position 33 at the upper end of the cross on the left side of the cross, and the error which hath ensued of that at the foot of the cross on the same side, as ye see in the example. And for because that this conjecture came to more than the truth, therefore set this letter M. in the space between the upper end of the cross and the neither. And for as much as in this first conjecture ye have erred thus much, conject again and spupose that the first merchant had 31 than must the second have 34, the third 38, all these collect make 103, so that now ye have erred again, your position being to much/ so that your error is 3, and for because that this second position is more than the verity as the first was, set the position 32 at the upper end of the cross on the right side, & the error 3 the foot of the cross on the same side, & put this letter M. between the space to sygnyfey more. Both these pasytion then be more than the verity, wherefore according to the rule first subduce the less error 3 at the foot of the right side of the cross from the greater error at the foot of the left side of the same cross, remaineth ● to be set in the space between both the feet as ye see: which shallbe the divisor. Then according to the rule, multiply the first position which is 33 by the error of the second position which is 3, and thereof cometh 99, than the second position 31 by the error 9 of the first position, and thereof cometh 279, then deduct the less sum 99 out of this more sum 279, remaineth 180, divide this sum by the difference of the errors which is 6, and the quotient shallbe 30 which is the true position: For the first man having 30, the second must have 33, and third 37, which all set together, make just 100 Thus wonderful craftily by these ii false positions the true & just position is brought to light. ☞ The example. ☞ An example when both positions come to less than the verity. WHen both positions come to less than is the verity, the which is all one matter with the other, as ye shall perceive by the same example again. As suppose ye had coviected that the first had received 27, then most the second receive 30, and the third 34, which added together make 91 which is less by ● then the sum 100, the which should be divided among them. Set then this first false position 27 at the upper end of the cross on the left side, and the error ensuing of that at the foot of the same cross on the same side. And for because the this position come to less than the verity therefore set this letter L. for less, in the space between the position and the error, as ye see in the example following. Then conject again and suppose that the first merchant had received 29 then by that rekenning the second should receive 3● the third 36, which all set together make 97, so that yet this position cometh not to the verity 100, but lacketh 3 of it, wherefore set this position 29 at the uppar end of the cross at the right side, and this error 3 at the foot of the cross, and in the space between the position and the error set this letter L, for less. Now for so much as both these positions be less than the verity, work as ye did before, according to the rule, subduce the less error 3 out of the greater error 9, remaineth 6 for the divisor to be set in the space between the two errors. Then multiply the first position 27 by the second error 2 cometh 81, than the second conjecture 29 by the first error 9 cometh 261 then deduct 81 out of 261 remaineth 180 which divided by 6 the divisor afore said, the quotient shallbe 30 which is the just and very conjecture, the which ye should have conjected. Thus ye have had sufficient example of this first rule of both more, and both less. ¶ Here after folweth the example of both less. ¶ Here followeth the rule of one more, and the other less. WHen the one position amounteth to more than the verity, and the other less than the verity, then add the errors together and that added number shallbe the divisor. Then multiply the first position by the second error, and the second posyon by the first error, and that that cometh of both these multyplycatours add them together also, then divide this added number by the added errors the divisor aforesaid, & the quotient showeth the true position. ¶ The example, ¶ we will take the first case again, and suppose that the first merchants have received 32 crowns, then must the second receive 35, & the third 39, which all added together make 106. wherefore that position is false and to much buy set the position 32 at the upper end of the cross/ and the error 6 at the neither end of the cross in the space between, ye shall set this letter M, for more. And for by case that this position hath exceedeth the verity, coviecte again less, and suppose that the first hath received 29, then must the second receive 32, the third 36 all added together maketh 97, which is less than the verity by 3, wherefore set this false position 29 at the uppar end of the right side of the cross, and the error 3 at the neither foot of the cross, in the space between set this letter L, for less. Of this two false positions the one is more than the truth the other is less, wherefore according to the rule add both the errors 6 and 3 together, that maketh 9 for the divisor: then multiply the first position 32 by the second error 3, which maketh ●6. and the second position 19 by the first error 6, which maketh 174, and that that en●ueth of both these multiplications add it together, and it maketh 270 divide this added number by the added errors, which was 9, and the quotient shallbe 30 which is the true position as ye may prove. ¶ The example. ❧ Thus may ye dissolve all other manner of questions, which have been set before in this book, without great pain or study, ¶ Finis. ¶ Imprinted at London in Aldersgate street by John Herford.