The Wellspring of Sciences, which teacheth the perfect work and practise of Arithmetic both in whole numbers & Fractions, with such easy and compendious instruction into the said art, as hath not heretofore been by any set out nor laboured.  
Beautified with most necessary Rules and Questions, not only profitable for Merchants, but also for all Artificers, as in the Table doth plainly appear.  
Now newly printed and corrected.  
Set forth by Humphrey Baker Citizen of London.  
IMPRINTED AT LONdon by Henry Denham, for james Rowbothum.  
¶ Anno Domini. 1564.   

TO THE Right Worshipful Master john Fitzwilliams' Governor of the most famous society of Merchants Adventurers into Flaunders. And to the right worshipful the Consuls, Assistants, and commonalty of the same society, Humphrey Baker wisheth health with continual increase of commodity by their worthy travel.  
ALTHOUGH heretofore (right worshipful) divers and sundry well skilled men in the feat, and science of numbering, have not spared their labour, not only in beautifying, but also in augmenting the same with such increase of wit, as hath exceedingly contented the students therein, and made them riper in the exact knowledge and readiness of it: Yet because an art can not have to much setting forth, neither can be to much commended, I (among, and after so many pregnant wits) am bold to put forth the fruit of my blunt understanding and travail, in the like science, not correcting or reproving any man's former edition, but making an attempt of my gross, and mean study herein, to the intent that this my enterprise may (by other men's doings) be either gently amended, or friendly received. And because I desire specially that it may be gratefully accepted, I have chosen you (worshipful) to be as defenders of this little lucubration, to whom I know it shall be welcome, not so much for my sake, as for the scien●●●oue itself, wherein it is well known you have ripe judgement, and good understanding, as by your daily exercises it may appear. And although some foolish heads of fancy overthwart, think this arithmetical art peculiar only to a few using the trade of merchandise, as though it were not so necessary for other men: Yet I am sure, the wisest and best learned perceiving the wonderful art, the deep devices, and the cunning conclusions that are comprised in it: affirm it (and that of right) to be the best whetstone, or sharpening of the wit of every man, that ever was invented, and think it most necessary to be taught unto children, without the which, nothing either private or common can be well ordered. And true it is, that whosoever is ignorant of this science of numbering, he sleepeth in all the rest, and lacketh that promptitude of wit, which other have in casting and accounting of great sums. Yea, I say he that hath the exact knowledge of it, hath a special gift of GOD, and carrieth about him a note and token of a good wit: and I doubt whether ever any was counted or esteemed simple or foolish, that had this science in his head.  
Now if some curious brain would ask me this question: Sir you commend much your Art of numbering, can you tell who was the first inventor of it, or what this word ARITHMETICA doth signify, I would answer him that it is uncertain who invented it first, but well I know, it is one of the seven liberal sciences: which if I say God gave unto man to adorn his life with all, I say as it is, and as we aught to believe, of whom we acknowledge to have received them and not of the Poetes or heathen Gods, as some schoolmen will say. Moreover, I know that Abraham taught it first unto the Egyptians, and no man will deny but that he received all his knowledge of God, for he worshipped no strange Gods neither received any gifts or sciences of them. And although many years after, Pythagoras declared it to the Greeks, yet was not he the first inventor of it, but god. Plato also commended it as most necessary, willing it to be taught to all youth before all other things. And thanks be to God, it appeareth that his precept hath been very well followed here in England, for I find not in any nation (I speak not flattering my countrymen but commending them) more ready knowledge in this science, than there is in the most part of our youth here in England, and namely in London, which I seeing it is in them, it is (out of question) in their masters & tutors, which by their great care over them, have trained them up unto the practice thereof, to whom the greatest part of the praise redoundeth. And now right worshipful, when I consider how little profit my simple travail shall bring, being but as it were an assay of further attempt, I am compelled to crave at your hands only to accept my good will with the work: for will and ability are somewhat unequal in me, but if ability were correspondent to will, I would gladly pleasure you in greater things than this is, which is but only a shadow in comparison of other men's substantial labour & travail in this art: yet I trust it is not altogether so unpollished, but it hath passed under the file of some workmanship, whereby some men may take occasion of profit & furtherance herein, which is my chief desire, although I know many are able to do better: for England is not so destitute of learned heads, but my doings may soon be followed, yea rather amended. But yet forasmuch as I would they should know, that know not, I thought it not good to hide that which hath been opened unto me. And so casting with myself the sum of all, I desire you eftsoons, right worshipful (to whom I thought it most meet to present the fruit of my study) thankfully to receive it, as your wont wisdoms doth all things well and virtuously purposed.  
Thus far you well, desiring God to maintain you in your estate, to prospero and further all your good purposes, all your traffics and voyages, increase you in virtue, and keep you in good health.  
AMEN.   


The definition of number.  
NUMBER IS AS much to say, as a multitude compound of many unities, as two is compound of two unities, three is compound of three unities, four of four unities, five of five unities ten of ten, fourteen of fourteen, fifteen of fifteen, twenty of twenty unities, etc.  
And therefore an unity is no number, but the beginning and original of number, as if you do multiply or divide an unity by itself, it is resolved into itself without any increase. But it is in number otherwise, for there can be no number, how great soever it be, but that it may continually be increased by adding evermore one unity unto the same.  

¶ Numeration. The first Chapter.  
NVmeration is the art whereby to express and declare the value of any sum or number proposed, and is of two kinds, the one gathereth the value of a sum proposed, and the other expresseth any sum conceived by due figures & places, for the value is one thing, & the figures are another thing: & that cometh partly by the diversity of figures, but chief of the places wherein they be orderly set. And first mark, that there are but ten figures or characters which are used in Arithmetic, whereof nine of them are called signifying figures, & the tenth is called a Cipher, which is made like an 0, & of itself signifieth nothing, but it being joined with any of the other figures, increaseth their valu, & these be they. 1 2 3 4 5 6 7 8 9 one two three four five six seven eight nine.  
Also you shall understand that every one of these figures hath two values: One is alway certain and hath his signification of his own form, and the other is uncertain which he taketh of his place.  
A place is called the seat or room that a figure standeth in, 
A place.  and how many figures soever are written in one sum or number, so many places hath the whole value thereof. And that is called the first place (which next is toward the right hand) of any sum, and so reckoning by order toward the left hand, so that that place is last which is next the left hand. And contrariwise, when you express the value of the figures in any sum, you must begin at the left hand, and so reckon toward the right hand.  
Every of these nine figures, (which are called signifying figures) hath his own simple value when he is found alone, or in the first place of any sum. In the second place toward the left hand, he betokeneth his own value ten times. As 70. is, seven times ten: that is to say seventy. 80. is eight times ten: that is to say eighty. In the third place every figure betokeneth his own value a hundredth times. As 700. in that third place betokeneth a hundredth times 7. that is to say, seven hundred: In the fourth place every figure betokeneth his own value a thousand times. As 7000. is seven thousand, and 8000. is eight thousand. These four first places must be had perfectly in mind, yea and that by heart, for by the knowledge of them you may express all kind of numbers how great so ever they be.  
In the fift place every figure betokeneth his own value ten thousand times. As 70000. is ten times seven thousand, that is to say seventy thousand: In the sixth place every figure standeth for his own value, a hundredth thousand times. As 700000. is seven hundredth thousand. The seventh place M. M. times, or a million: as 7000000. is seven M. M. or seven millions. And the eight place ten M. M. times, or ten millions, so that every place toward the left hand, exceedeth the former ten times. But now for the easy reading, and ready expressing orderly of any sum proposed, you shall practise this manner, following. And for example I propone this number 765432658. in the which are ix places. In the first place is .8. & betokeneth but eight, in the second place is 5. and betokeneth ten times five that is fifty, in the third place is 6. and betokeneth a hundredth times six, that is vj. C. In the fourth place is 2. and that is two M. And 3. in the fift place is ten M. times 3. that is xxx. M. So .4. in the first place is C. thousand times 4. four, that is four. C.M. then .5. in the seventh place is a M. M. times 5. that is five M.M. or rather five millions. And 6. in the eight place is six times ten millions, that is lx. millions. And last of all vij in the ix. place, is seven. C. millions. Now followeth the practice. First put a prick over the fourth figure, and so over the seventh, and likewise over the tenth. And also over the 13.16. or .19. if you had so many, and so still leaving two figures between every two pricks and these rooms from one prick to an other are called ternaries, 
Ternaries.  than you must pronounce every three figures from one prick to an other as though they were written alone from the rest. And at th'end of their value, add so many times a thousand, as your number hath pricks (that is to say, if there be but 1. prick, it is but 1. M. if 2. pricks a M.M. or else a million, if 3. pricks M.M.M. or a M. million, & so consequentli of all other figures following) Then come likewise to the next iij. figures, and sound them as if they were a part from the rest, and add to their value so many times thousands as there are pricks between them & the first place of your whole number. And so do by the next iij. figures following & of all the rest likewise as in example. 451234678567. The first prick over 8. in the fourth place, which is the place of a M. the second prick is over 4. in the seventh place, which is the place of a M.M. or one million, the third prick is over the tenth place which is the place of a M.M.M. or of a M. million, as in the former example. Then for the expressing of this number by the value of every figure according to the place wherein they stand, you shall first begin at the last prick over 1. and take it and the other two figures 5. and 4. which do follow him, and value them alone and they are four Clj. MMM. or else CCCCli. M. millions. Then take the other three figures from 1. to the next prick, and value them as if they were a part from the other, and they are .234. which are CCxxxiiij. millions, or 234. MM. Then come to the third prick over .8. and take the other two figures behind it, and reckon them likewise as if they were alone, and they are six Clxxviij. M. And last of all come to the other three figures which remain, that is .567. and they are five Clxvij. Thus the whole sum of these figures, is four Clj. M. two Cxxxiiij. millions, six Clxxviij. M. five Clxvij, as before.  

Three kinds of Number.  Note also that whole number is divided into three kinds, that is to say, diget number, article and mixed or compound number. The diget number, 
Diget.  is all manner of numbers under ten, which are these nine figures, 1 2 3 4 5 6 7 8 9 of the which I have spoken before. 
Article.  The Article number is any kind which beginneth with a Cipher as this 0. and they may ever be divided just by 10. without any remain, as these, 10. 20. 30. 40. 50. 100 & all other such like. The mixed or compound number, 
Mixed or compound.  containeth divers and many articles, or at the lest one article, and a diget, as .11. 12. 16. 19 22. 38. 108. 1007. and so forth. And as any article number may be made a compound, by putting thereto a diget, even so likewise every compound number, may be made an Article number by adding thereunto a 0.  
¶ And here followeth a brief rehearsal of the order and Denominatours of the places. And this shallbe sufficient for Numeration.  
The order of the places. Tenth. Ninth. Eight. Seventh. sixth. fifth. Fourth. third. second. first.   4 3 2 1 0 1 8 3 4 5   M. of Millions. C. of Millions. X. of Millions. Millions. C. of thousands. X. thousands. thousands. hundreds. tenths. Unities. The Denominatours of the places.   

Addition in whole number. Chap. 2.   example of numbers to be added, with a horizontal line below the bottom number 
431 334 223  
Now begin always at the first places toward your right hand, and put together the three first figures of these three sums, and look what cometh of them, writ that under them beneath the line, as in saying 431 334 223   8 3.4. and 1. being put together do make 8. write 8. under three as thus.  
And then go to the second figures and do likewise: 431 334 223  88 as in saying. 2.3. & 3. maketh 8. write 8. under 2. as here you see.  
And likewise do with the figures that be in the third place, in saying 2. 3. and 4. 431 334 223 988 are 9 put nine under them, & so will your whole sum appear thus: whereby you may perceive that those three sums being added together do make 988. li. And this is the art of addition according to his simplicity, if the sum of any place do not exceed a diget number. But in case the sum of ani one place cannot be expressed by one figure, but by two, you shall put the first of those figures under the line, and keep the other in your mind, for to add it unto the first figure of the next place. And if the same next place cannot be avalued but by two figures, you must in like manner put the first of those figures under the line, and reserve the second for the other place next after, and thus must you do from one place to another until you have come to the last place, where in case you do find that the sum be of two figures, you must set them both down because it is the end of that work, as in this example.  
 734682456  450932345   13467891    4672123 1203754815  
Where the first figures are 3.1.5.6. which added together maketh 15. & for that, that 15. is of two figures, I do put the first figure 5. under the line, and keep the second figure (which is 1.) in my mind, the which. I must add with the next figures of the second place, that is to say with 2.9.4. and 5. the which together make 21. I writ 1. under the line for the second figure of that addition, that is to say after 5. and I keep 2. to be added unto the third place the which with the other figures 1.8.3.4. do make 18. therefore I put 8. next after 1. in the third place under the line, and keep 1. to be added unto the figures of the fourth place, which is with 2.7.2.2. the which with the 1. that I keep do make 14. I set down 4. for the forth figure (under the line) that is to say, after 8. and I keep 1. to be added unto the figures of the fift place, the which is 7.6.3.8. with the 1. that I keep maketh 25. I put 5. in the fift place under the line next after 4. and keep 2. in mind to be added with the figures of the sixth place, that is with 6.4.9.6. and that 2. which I keep, maketh 27. I writ down 7. under the line in the sixth place, and I keep two which I add with the figures in the seventh place, and they make 13. I put down 3. under the line in the seventh place, and add 1. unto the figures in the eight place and they are 10. I do put 0. under the line in the eight place, and then I add 1. unto the ninth place, that is to say with 4.7. and they make 12. the which 12. I writ at length under the line because it is the end of this addition, and this is to be done of all such like. And for the easier understanding of that which we have spoken of addition, you may examine these two other examples following, in the which the first hath these numbers. 3570.2763.579.28. which being added together, do make this number 6940, and in the second example doth result this number 51683. by adding together of these numbers, 47630, 3756, 272, 25, as here under written.  
The numbers to be added. 3570, 47630  2763 3756  579 272  28 25 The line put between.   The sum of this addition. 6940 51683   

Of Substraction in whole number. The. 3. Chapter.  
SVbstraction teacheth how you shall abate one lesser number from a greater, & what there doth remain after that you shall have abated the same, I speak not of the abating of one equal number, from an other equal unto it, for the facility thereof requireth no rule.  
In Substraction are found three numbers, the one is that, from the which the substraction is made. The second is the number that is to be substracted and the third is the number which remaineth after the substraction is ended. As when I would subtract. 25. from 40. The 40. is the number from the which the substraction is made. 25. is the number to be substracted, & 15. is the number which remaineth after you have done the substraction, here followeth the practice. You shall put the lesser number under the greater in such sort that every figure of the one number may answer unto every figure of the other orderly, and then draw a right line under those two numbers as you did in Addition. Then must you begin at the right hand and take the first figure of the undermost number & abate that from the first figure of the uppermost number, & that which remaineth you must set underneath the line right under the figure which you have substracted. Then afterward take likewise the second figure of the nethermost number, and abate that also from the second figure of the higher number. The third from the third, and so forth of all the rest till you come to the end, putting always the remain of every figure under the line in his order, example. I will subtract. 2345. from 9876. after that I have put them down according to the manner aforesaid. 9876 2345 7531  
I take first 5. from 6. and there resteth 1. the which I set under the line right against 5. Secondly I abate 4. from 7. and there resteth 3. the which I set in the second place under the line, next after 1. thirdly, I abate 3. from 8. and there resteth 5. The which I put under the line in the third place, finally I do abate 2. from 9 and there resteth 7. the which I put under the line in the fourth and last place, and thus is this Substraction ended, by the which there resteth. 7531.  
But when two figures of one likeness do chance to meet, so that the one must be abated from the other, as if I should abate 7. from 7. there remaineth nothing, and then must I set a cipher 0. under the line. But when the figure which is to be abated, doth exceed the figure which is over him, so that it cannot be taken out of the same figure. Then must you abate the neither figure from 10. And that which doth remain you shall add unto the same figure which is uppermost. And the sum which cometh thereof shall you set under the line. But whensoever you do borrow any such 10. of the over number: you must add 1. unto the next neither figure following which is to be abated. And there is nothing else to be done in substraction. example, I will subtract 93576. from 4037479. after that I have placed my two numbers, 4037479. 93576. 3943903. as I aught to do, I do first abate. 6. from 9 and there resteth 3. then I put the 3. under the line right against 6. And secondly I abate 7. from 7. And there resteth nothing. I do put a cipher 0. under the line right against 7. in the second place. Then I come to the third place where I find 5. which I cannot bate from the figure over him, which is but 4. therefore I do abate it from 10. as before I taught and there resteth 5. the which I do add with the 4. which is over him, and that maketh 9 I put 9 in the third place under the line for the third figure. Fourthly, for the 10. which I borrowed I add one unto the next neither figure which is three, and they make 4. the which I do abate from the over figure 7. and there resteth 3. I put 3. under the line for the fourth figure. And then I come to the fift place where I do find 9 which I cannot abate from the figure over him which is but 3. but I abate 9 from 10. and there resteth 1. the which I do add with 3. and they make 4. I put 4. under the line for the fift figure. And if it were not, for that I did last borrow 10. the substraction should have been ended. But for because that I must (for every such ten that I borrow) always add 1. unto the next neither figure following, I must therefore proceed unto the substraction. And for that that there is no other figure following in the neither number, it shall suffice to have kept the unity, and to abate it from the next over figure. But I find there 0. & cannot abate 1. from 0. therefore I abate it from 10. and there resteth 9 which I do put under the line in the sixth place, finally for the ten which I borrowed, I keep 1. in mind. The which I do abate from 4. and there remaineth 3. the which I do put under the line in the seventh place after 9 And the operation is thus ended.  
another example.  
576084026 485675437 90408589  
But if there were many numbers to be substracted from one number alone, then must you first add those numbers together according unto the doctrine of the Chapter going before, and then to make your substraction as above said. As if I would abate these three sums 123.234.456. from. 98925. first I do add the three sums into one, & they are. 813. The which I do abate from 98925. and there resteth. 98112.   

Of Multiplication. Chapter. 4.  
IN multiplication there are iij. numbers to be noted, that is to say the numbered which is to be multiplied, the which we will call the multiplicand: and the number by the which we multiply, we call the multiplier, or multiplicator. And the third number is that which cometh of the multiplication of the one by the other, which is called the product. As when I would know how much mounteth 10. multiplied by 9 that is to say how much are ten times nine. I find that they are worth. 90. then 10. is the multiplicand, 9 is the multiplier.  
And 90. is called the product. Then for to multiply, is none other thing, but to find a number which containeth the multiplicand so many times, as the multiplier containeth unities, As 10. multiplied by 9 do make 90. as before said. And 90. containeth 10. so many times, as 9 containeth unities, that is to say nine times.  
In multiplication, it forceth not much which of the two numbers be the multiplicand, nor which be the multiplier. For ten multiplied by 9 maketh as many as 9 multiplied by 10. yet nevertheless it shall be more commodious that the lesser number be always the multiplier.  
And for that, that the multiplication of figures the one by the other, is the most chief & necessariest kind whereby to know how to work in the multiplication of compound numbers, and that every man hath not at the finger's end: I will therefore give you here certain easy ways of multiplication of diget numbers. When you would multiply two simple figures, or digets the one by the other, abate each of those dyget numbers from 10. Then multiply the two remains the one by the other. And if the sum do exceed 10. writ only the first figure, & keep the other to be added to the next operation, which is thus. Add your two simple figures together: And of that which resulteth of the addition, take only the first figure, unto the which you must add the unity which you keep before. And that shall be the second figure of the sum which you do seek. Example, I would multipli 7. by 6. I take 7. from 10. and there resteth 3. likewise I abate 6. from 10. and there resteth 4. then I say thus 3. times 4. make 12. I writ 2. for my first figure, & I keep 1. in my mind, them I add 6. with 7. & they are 13. of the which I cast away the second figure 1. and I take only the first figure 3. unto the which I add the unity which I kept, & they make 4. which I writ in the second place, after 2. And thus I find 42. which is the valour of 7. multiplied by 6.  
Otherwise, and all cometh to one effect, set down your two diget numbers the one right over the other, & right against every of them toward the right hand writ his own distance from. 10: Then multiple the two differences together, the figure which cometh thereof, shall you set down under both the differences. But if there be two figures, set down but the first, and keep the other in your mind afterwards abate (from one of the two diget numbers) the difference of the other diget number that is to say, crosswise. And unto the remain add the figure which you kept, and that shallbe the second number, and thus you shall have your multiplication. Example of the like figures that is to say of 7. multiplied by 6. the distance of 7. unto 10. is 3. And the distance of 6. from 10. is 4. I set them down croswayes as you see: And then I say three times four are 12. I set down two and keep one in my mind, than I abate 4. from 7. or else three from 6. it forceth not from which of them: and there resteth always 3. unto the which I add the unity which I kept in my mind, and they are four, which shallbe the second figure of the multiplication. And thus I find that 7. multiplied by 6. maketh 42. as in the other operation. This practice hath no place where the 12. diget numbers (do not exceed 10.) by adding them together, and then is multiplication easy enough without any rule.  
Another way to know the multiplication of simple numbers, is by this table following: the use whereof is thus.  
first you shall understand that the numbers from 1. and so downwards to 9 set in the left part or hanging margin of this table do betook the multiplyers of all simple numbers. And the elements or figures being put highest in every square room drawing toward your right hand right against every of the multiplyers, do signify the multiplicands, unto the multiplyers of the hanging margin. And the lower or inferior numbers in every square room, do betoken the product of that multiplication, which is made in multiplying the upper number over it, with the figure in the hanging margin, answering directly unto the said square: as by example, 
The Table of multiplication by all the Diget numbers.  1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 2 2 3 4 5 6 7 8 9 4 6 8 10 12 14 16 18 3 3 4 5 6 7 8 9 9 12 15 18 21 24 27 4 4 5 6 7 8 9 16 20 24 28 32 36 5 5 6 7 8 9 25 30 35 40 45 6 6 7 8 9 36 42 48 54 7 7 8 9 49 56 63 8 8 9 64 72 9 9 81  
First because 1. doth not multiply, I set in the upper margin the figures from 1. to 9 both in the higher and also in the inferior rows, for 1. in the hanging margin, multiplied by 1. the upper number in the first square bringeth but 1. so likewise 2. being the higher number in the second square of the upper margin multiplied by 1. in the hanging margin, bringeth two for the lower number in the second square of the upper margin, for one times 1. maketh but 1. and 1. times. 2. maketh 2. then 1. times 3. maketh 3. and 1. times 4. maketh 4. and so continuing toward the right hand until you come to the figure of 9 which is 1. times 9 maketh 9 Then after multiple two of the hanging margin by 2. the upper number of the square next toward the right hand and that maketh four which is the product of 2. multiplied by 2. which 4. is set under the 2. for 2. times. 2. are. 4. and 2. times. 3. maketh 6. then 2. times 4. maketh 8. and two times 5. maketh 10. and so continuing unto 2. times 9 which maketh 18. The like is to be done with the third row, and so likewise of all the residue.  
example, I would know what is the product of 9 multiplied by 8. I seek in the hanging margin the multiplier 8. and amongst the squares directly against eight drawing toward the right hand, I seek the multiplicand 9 in the higher row, and I find the product right under 9 to be 72. Then 72. is the number which cometh of the multiplication of 9 by 8. and so is to be understand of all the rest of the table, which table must be of all men learned by heart, or as they say without book, which being learned you shall the better attain to the rest of multiplication.  
To come now unto the practice of multiplication, when you would multiply two numbers the one by the other, you must set them down after the same manner as you did in addition, and in substraction. That is to say, the first figure of the multiplier under the first figure of the multiplicand, the second under the second, and the third under the third, if there be so many, and then draw a right line under them, as in the other operations going before. After you shall multiply all the figures of the multiplicand by the multiplier, and set down the figures (coming of any such multiplication) under the line every one in order.  
example, I would multiply 123. by 3. that is to say, I would know how much amounteth three times one hundredth, twenty and three. The two numbers being placed in such order as is before said, you must begin towards the right hand: and say thus 3. times 123   3 369 3. are 9 writ down 9 under the line, right against 3. for the first figure: secondly by the same 3. you must multiply the second figure 2. & they make 6. put down 6. after the 9 under the line: Thirdly by the same 3. you shall multiply the last figure 1. and they are but three, set down 3. after 6. for the third & last figure. And thus is the work ended: whereby you shall find, that 123. being multiplied by 3. maketh 369.  
But when that of the multiplication of one figure by an other. The sum which cometh thereof shallbe of two figures, as it happeneth most often, then shall you writ down the first figure, and keep the other figure to be added unto the multiplication of the next figure.  
example, six men have gained (every one of them) 345. crowns I would know how many crowns they had in all.  345    6 2070 first I multiply 6. times 5. are 30. I writ 0. under the line, & keep 3. to be added to the next multiplication: Secondly I say 6. times 4. are 24. unto the which I add 3 which I reserved. And they make 27. I writ 7. in the second place under the line, and I keep 2. to be added to the next multiplication, thirdly I say six times 3. are 18. unto the which I add the 2. which I keep. And they make 20. the which I writ all down for because that is the last work. And so I find the 345. being multiplied by 6. do make 2070. When the multiplier is of many figures you must multiply all the whole multiplicand by every one of those figures, and writ the productes every one under his own figure.  
Example. I would know how many days are passed from the nativity of jesus Christ until the year 1560. full complete. I have to multiply 1560. by 365. which are the days of one whole year. The leap years not being reckoned, which have every one of them 366. days.  
first by the figure 5. I 1560  365 7800 multiply all the higher figures, saying thus five times 0. maketh 0. I writ 0. under the line for the first figure, and because I keep nothing for the next place, I proceed & say 5. times 6. are 30. I set 0. under the line for the second figure, & I keep 3. to be added to the next multiplication, thirdly I say 5. times 5. are 25. The which with 3. that I keep are 28. I set down 8. and keep 2. to be added with the next multiplication. Then coming unto the fourth and last figure, I say 5. times 1. are 5. the which with 2. that I reserved are 7: I put 7. for the last figure of this first operation by the figure 5. with the which figure we have no more to do. And therefore I cancel the same 5. with a little strike thorough it, to signify that we have finished with that figure. And forasmuch that in multiplication there is always as many simple operations, as the multiplier containeth figures. There resteth yet 2. operatonis to be made. I come then unto the second operation, which is by the figure 6. by the which I must again multiply all the figures of the multiplicand as I did by 5 & the first figure (which shallbe produced) you must put one rank more lower than the figures of the operation even now made by 5. not right under the first figure of the multiplier 5. but under 6. that is to say: one place more forward than the fift toward the left hand: & one rank more lower than the first operation: and you shall put afterward every of the other figures which cometh of the same multiplication in their order: thirdly you must make the multiplication by the third figure & that which shall come thereof you must set in his rank, as here under you se. And now we need make no further discourse hereof, because that he which can do the first multiplication by 5. may as easily do all the others. It shall therefore suffice to set here under the examples.  
 1560   365  7800 9360    1560    365   7800  9360 4680    
Now, if you will know how much the operations thus placed do amount unto, which in value are but one number: you must add those three numbers together, but not after the same manner as we have done in the chapter of addition, the first figure of the first rank with the first figure of the second rank, & of the third: but you must add them in the same sort as you shall find them situated or placed: that is to say, the first figure of the first rank alone by itself: the second of that rank with the first of the second rank. The third of the first rank with the second figure of the second rank and the first of the third rank: and so of all other as hereafter doth appear.  
And thus the 1560   1560    365   7800  9360  4680   569400 years do contain five hundredth sixty & nine thousand four hundredth days not counting herein the days of the leap years, which are here in number 390. for then the whole sum of the days should be 569790.  
Another example.  
   34560     2456   207360  172800  138240   69120    84879360  
The sum of multiplication, when you would multiply any number by 10. you must only add one cipher unto all the number. As 345. multiplied by 10. maketh 3450. if you will multiply by 100 Add unto the whole number two ciphers. 00. if by 1000 add 000. And to be brief. when the last figure of the multiplier is 1. and all the rest be ciphers, add so many ciphers to your multiplicande, as there shallbe found in your multiplier. But if in multiplying, the last figure were not 1. but that there were only certain ciphers in the beginning: & that the other were signifying figures, and likewise those of the multiplicand, then shall you put those ciphers apart, and multiply the signifying figures of the one by the signifying figures of the other. Then add unto the product of that multiplication, all the ciphers which you did before put a part. As if I would multiply. 46000. by 3500. I put a part the three ciphers of the first, and the two ciphers of the second numbers. And then I multiply 46. by 35. & thereof cometh 1610: unto the which I add the 00000. and then the whole product will be. 161000000.  
  46        35       230      138       161000000   

Of Division the fift Chap.  
Division or partition is, to seek how many times one number doth contain an other for in this operation are first required two numbers for the finding out of the third. The first number is called the dividend or number which is to be divided & that must be the greater number, the other number is called the divisor, & that is the lesser. And the third number which we seek is called the quotient. As if I would divide 36. by 9 the dividend shall be .36. and the divisor is 9 And for because that 9 is contained in 36. four times, that is to say that 4. times 9 do make 36. The quotient shall be 4. as in marking how many times 9 is contained in .36.  

¶ The practice.  
Write down first the devidende in the higher number, and the divisor underneath, in such sort, that the first figure of the divisor toward the left hand be under the first of the dividend and every figure of the same divisor under his like, that is to say, the first under the first, the second under the second, the third under the third, and so consequently of the other, if there be any more, which is contrary to the other three kinds before specified, but you must consider if all the lower figures of the divisor, may be taken out of the higher figures of the dividend, by the order of substraction. The which if you can not do, then must you set the first figure of the Divisor (toward the left hand) under the second figure of the dividend, and so consequently the rest, if any be to be set down every one of them under his like as before is said. And then draw a line between the dividend and the divisor. And at the end of them an other crooked line, behind the which toward the right hand, shall be set your quotient. As by this example following where the divisor is but of one figure.  
If you would divide 860. by 4. you must set down 4. under the 8. with a line between them as here under you may see.  
The dividend 860 Divisor. 4  
And then you must seek how many times the divisor is contained in the higher number, or dividend answering to him, as in this our example I must seek how many times 4 is contained in 8. in the which I find 2. times, than I writ down 2. apart behind the crooked line, as you see, which shallbe the first figure of the quotient to come, secondly by this figure (being thus 860 4  (2 8   put apart) I must multiple the divisor: and under the same multiplication. I must set that number which cometh of the same multiplication as 2. times 4. do make 8. which 8. I do set under the 4. which is the divisor. Thirdly, I do subtract the product of the said multiplication (of the quotient by the divisor) from the higher number correspondant to the same, as if I abate 8. from 8. there remaineth nothing, and then I cansell or strike out that which is done as you see. In these three operations is comprehended the art of division. The which are to be observed from point to point for there is no diversity in the finishing of the same which is thus.  
I must remove my divisor one place nearer toward my right hand: as in proceeding with our 2 260 (21 4 example I remove my divisor 4. which was under 8. and I set it under 6. then I seek how many times 4. is contained in 6. where I find but one time them I set 1. behind the crooked line behind 2. afterward by this last & new figure 1. I multiply the divisor 4. & that maketh but 4. (for an unity which is but 1. increaseth nothing) I abate 4. from the higher figure 6. and there resteth 2. the which 2. I set over the 6. & I cancel the 6. for so must you do when there resteth any thing after you have made the substraction. Thirdly for that there yet remaineth another figure in the dividend, I remove again the divisor, and I set it under the cipher 0. Then I seek how many times 4. is in the higher number which is 20. where I 2 860 (215 4 20 find 5. times, I put 5. behind the crooked line for the third and last figure of the quotient. Then by the same 5. I multiply the divisor 4. and that maketh 20. the which I abate from the higher number, and there resteth nothing. And so is this division ended: & I have found yt. 860. being divided by 4. bringeth for the quotient 215. that is to say, that 4. is contained in 860. two hundredth & fifteen times. This is the most easiest working that is in division, but that which followeth, appertaineth to the whole and perfect understanding of the same. When the first figure of your divisor toward your left hand is greater than the first of the dividend, you must not place the first figure of your divisor right underneath the first of the dividend, but under the second figure of the same dividend, nearer to your right hand, as before is said. When the divisor is of many figures, and that you have to seek how many times it is contained in the higher number (for the more easier working) you must not seek to abate the divisor all at one time, but you must see and mark how many times the first figure of the same toward the left hand is contained in the higher number answering to the said number, & then to work after the same manner as is before taught.  
Example, I have 316215. crowns to be divided among 45. men for to make my division I must not put the first figure of the divisor which is 4. under the first of the dividend, which is 3. because that 4. is greater number than 3. And further, I cannot take 4. out of three, wherefore I must set the 4. under the second figure of the higher number which is 1. and the figure 5. of the divisor next right under the 6. as you may see.  
316215 45 I must first seek, how many times 45. is contained in 316. which is but part of the dividend, wherefore for the more easy working I need but to seek how many times 4. is contained in 31. & because I may have it seven times I put 7. behind the crooked line, as is aforesaid, then by 7. I multiply all the divisor 45. and they are 315: the which I set under the same divisor, the first figure under the first: And the other in order toward the left hand. Then I subtract. 315. from the higher number 316. and of this first working there remaineth but 1. the which I set over 1 316 215 45 (7 315 the 6. and I cancel the. 315. & the other figures 3, 1, 6, and also the divisor: and then it will stand thus.  
And when I come to remove the divisor, and that I must seek how many times it is contained in the higher number, if I see that I cannot find it there, that is to say that if the higher number be lesser than the divisor, as it is in this example, then must I put a cipher in the quotient behind the crooked line, & if there remain any figures in the dividende which are not finished, I must remove the divisor again nearer toward my right hand by one place, for to find a new figure in the quotient. As in this our example, for after that I have removed the divisor, 1 52 6215 45 (70 I seek how many times. 45. is contained in. 12. and because I cannot have 45. in 12. I put a 0. behind the crooked line after 7. then without multiplying or abating, I remove again the divisor nearer toward my right hand, and I seek how many times 4. (which is the first figure of the divisor) is in the higher 1 316215 45 (703 135 numbered, that is to say, in 12. whereas I find it 3. times: I put 3. behind the crooked line, for the third figure of the quotient: then by 3. I multiply the divisor. 45. and thereof cometh. 135.  

A note.  But here is to be noted, that if it happen that the figure being last found which is put in the quotient, do produce or bring forth a greater numbered (in multiplying all the divisor by the same) then that which is over the said divisor: you must then make the same figure of your quotient (which you do put down) lesser by one: and after that you have canceled the first multiplication, you must make a new. And the same must be so done as often times: as (in decreasing the same) it produceth a lesser number, or at the lest, a numbered equal to that which is over it. As in the last work: for because that the divisor, being multiplied by 3. bringeth forth. 135. which amounteth more then. 121. the same product must be canceled. And likewise the figure. 3. which I did put in the quotient, must be changed into a figure of 2. Then by the said 2. I must multiply the divisor. 45. and thereof cometh 90, the which I abate from. 121. and there remaineth. 31. And then will the somme stand thus.  
13 316215 45 2 135 (703 90  
And here is also to be noted that the some which remaineth must be always lesser than the divisor. 
A note.  Then finally I remove the divisor to the 2. last figures toward the right hand, and I seek how many times 4. is in 31. And for because I find it 7. times, I put 7. in the quotient: by the which I multiply the divisor, and thereof cometh 315. the which I abate from the higher numbered of the dividend, and there remaineth nothing as here you may see.  
13 516215 45 (7027 515  
But in case that after the division is ended, there do remain any thing in the dividend, as most often times there doth: I must them set that remain apart behind the crooked line after the entire quotient, and the divisor right under the same remain, with a line between them both, as in this division following, where there remaineth 3. in the last work of the same. And we shall see what the same doth signify, when we shall treat of fractions or broken numbers.  
 
In sum, all the whole practice of division may be kept in remembrance by three letters, that is to say: S. M. A. which three letters do signify to seek, to multiply, to abate.  
First, I must seek how many times the divisor is contained in the higher numbered: then, by the quotient (which I find) I must multiply the divisor: finally, I must abate the product of that multiplication, from the higher numbered to the same correspondent, that is to say: out of the dividend, answering to the divisor.  
And further, besides this kind of working in division. The which is regular and common: I will here put an other manner of working very easy. The which shall serve for such divisions as are difficyll to be wrought. That is to wit, when the number to be divided is very great, and the divisor great also, and it shall serve again for to avoid error in supputation, and for the placing of lesser figures in the quotient: and consequently it shall save much labour unto them which as yet have much studied in this art. The practice whereof is thus, as followeth.  
I have to divide 7894658. by 643. In the first place, you shall understand, that although the first figure of the divisor toward your left hand, may be found many times in the higher numbered as 10. times 12. times or more: yet is it so, that you must never put but one figure only at a time in your quotient.  
And thus you shall at no time put any noumbre in your quotient which exceedeth the figure of. 9 that is to say any noumbre being greater than 9 for to come then unto our practice, write down your divisor one time: and behind it toward your right hand, draw a line down straight, and right against the same divisor behind the line put this figure 1. Then double your said divisor, and right against the same (being doubled) put behind the line the figure of 2. After, add unto the same numbered (which you doubled) your said divisor and right against the same product, behind the line put the figure of. 3. And unto this third product, you must add again your divisor: and right against the same product behind the line set the figure. 4. And this must you do, until you come to the figure of. 9: in such sort that every of the productes do surmount so much his former noumbre, as all the divisor doth amount unto: placing at the right side of every product behind the line, the numbered which signifieth how much he is in order. That is to say, right against the fift product, you must put. 5. right against the sixth product, you must put. 6: And so likewise of all the other.  
Example of the divisor proponed, 643. first, I write down 643. and  643 1 1286 2 1929 3 2572 4 3215 5 3858 6 4501 7 5144 8 5787 9 right against the same behind the line I put .1. secondly, I double 643. and they make 1286 & right against him behind the line I put .2. thirdly, unto that same 1286. I add the divisor 643. and they are 1929. and right against the same I set 3. fourthly, unto the said 1929. I add the divisor 643. and they are 2572. and right against the same I put .4. And thus must you do always by increasing so much every product, as the divisor doth amount unto, until you have so done nine times, as you see in this present table.  
This being done, you must set down your divisor under the divident after the same manner as is before declared: that is to say, 643. under the three first figures of the dividend toward your right hand, which are .789. Then must you seek how many times .643. are contained in .789. And for to know the same, I look in my foresaid table if I may there find the same numbers, 789. the which is not there: Therefore I must take a lesser number the nearest to it in quantity that I can find in the table, the which is .643. which numbered hath against it on the right hand of the line this diget. 1. Then I take the said 1. and I put it behind the crooked line, for the first figure of the quotient.  
Then I do abate .643. from 789. and there remaineth .146. the which I put over the .789. and I cancel the 789. and thus is the first operation ended. Then I set forward the devisor one figure nearer to my right hand, and I seek a new quotient as I sought this, where I find the higher number over my divisor to be 1464. The which I do seek in the table, and because I can not find it there, I take a lesser number, the nighest to it that I can find, and that is 1286: which numbered hath against it this digette .2. I put .2. for the second figure of the quotient behind the line, and I do abate 1286. from .1464. and there remaineth .178. Thirdly, I remove forward the divisor, as before, and I find the higher number to be .1786. and that the next lesser noumbre to it in my table, is again .1286. I put therefore once again .2. in the quotient for the third figure: and I abate .1286. from, 1786. and there remaineth .500. Fourthly, I set forward the divisor, and the higher number over it, is 5005. and the next lesser number to it in my table is .4501. right against the which numbered is .7. I put my .7 in the quotient, for the fourth figure. And after that I have abated .4501. from .5005. there remaineth .504. Finally, I remove forward my divisor unto the last place: and I find the higher number to be .5048. And the next lesser noumbre to it in my table, is. 4501. I set .7. again in the quotient, for the fift and last figure. Then I take .4501. from .5048. and there remaineth .547. which must be put at the end of the whole quotient with the divisor under it, and a line between them in this manner following.  
(1227 547/043   

¶ The sum of division.  
WHen you would divide any numbered by .10. you must take away the last figure next towards your right hand & the rest shallbe the quotient. As if you would divide .46845. by .10. take away the .5. and then .4684. shall be the quotient, and the .5, shallbe the number that doth remain. Likewise when you would divide any numbered by 100 take away the two last figures towards your right hand, and if you would divide by .1000. take away three figures, if by .10000. take away four figures. And so of all other, when the first figure of the divisor toward the left hand shallbe only 1. and the rest of the same divisor being but ciphers.   

¶ Here followeth the proofs of addition, substraction, multiplication, and division.  

¶ The proof of Addition.  
WHen you would prove whether your addition be well made, consider the figures of the number which be added, every one in his simple value: not having any regard to the place where he standeth, but to reckon him as though he were alone by himself and then reckon them all, one after an other, casting away from them the number of 9 as oft as you may.  
And after your discourse made, keep in mind the same figure which remaineth after the nines be taken away, or set the same in a void place at the upper end of a line. For if your addition be well made, the like figure will remain, after that you have taken away all the nines, out of the total sum of the 24567 2 5329  431  30377 2 same addition, as of ten as you mayther find any: as in this addition which here you see. There remaineth .2. for echepart.   

¶ The proof of substraction.  
Add the numbered which you do subtract with that numbered which remaineth after the substraction, is made: and if the total somme of that addition, be like unto the number from the which the substraction was made you have done well, otherwise 5463 3584 1879 5463 not: as in this example doth appear, where you see the number which is to be substracted is, 3584. and the numbered which doth remain, is .1879. the which two sums being added together, do make .5463. which is like to the higher number, out of the which the substraction was made, as before is said.   

¶ The proof of multiplication.  
THe proof of multiplication is made by the help of division, for if you divide the number produced of the multiplication, by the multiplier: you shall find the higher number, which is the multiplicand.   

¶ The proof of division.  
TO know if your division be well made: you must multiply all the quotient by your divisor, and if any thing remained after your division was made. Thesame shall you add unto the product which cometh of the multiplication: and you shall find the like number unto your dividend if you have well divided: otherwise not.     

¶ Of progression the vi. Chapter.  
PRogression arithmetical, 
Progression arithmetical.  is a brief & speedy assembling or adding together of divers figures or numbers, every one surmounting the other continually by equal difference: as 1.2.3.4.5. etc. here the difference, from the first to the second is but of 1. and so do all the other, every one exceed another by 1. still to th'end. Like ways. Here .2.4.6.8 etc. do proceed by the difference of 2. also 3.6.9. 12. etc. do every one differ from other by 3. and so may these noumbres continued. Infinitely after this order, in adding unto the third number, the quantity wherein the second doth differ from the first: Like ways adding the same difference unto the fourth number, also to the fift, and so unto all the other. As .1.4. the difference of the second to the first is 3. add 3. unto 4: and they are 7. for the third number: Then add 3. unto 7: and they make 10. for the fourth noumbre, and so of all other.  
Then if you will add quickly the numbered of any progession, you shall do thus, first tell how many noumbres there are, and write their some down by itself, as in this example, 2. 5. 8. 11. 14. where the noumbres are 5. as you may see, therefore you must set down 5. in a place alone, as I have done here in the margin. Then shall you add the first number and the last together, which in this example are .14. and 2. and they make .16. take half thereof which is .8. and multiply it by the 5. which I nooted in the margente for the noumbre of the places, and the some which amounteth of that multiplication, is the just somme of all those figures added together, as in this example: 8. multiplied by .5. do make .40. and that is the some of all the figures. another example of parcels that are even, as thus .1.2.3.4.5.6. in this example you must likeways note down the number of the places, as before is taught, and than add together the last number and the first. And the some, which cometh of that addition, shall you multiply by half the number of the places, which before are noted, and that, which resulteth of the same multiplication, is the wholesomme of all those figures, as in this former example, where the number of the places is .6. I note the .6. a part, and then I add .6. and .1. together, which are the last and first numbers, and they make .7. the which I multiply by .3. which is half the number of places, and they make 21 and so moche amounteth all those figures, added together.  
Progression Geometrical is, 
Progession, geometrical.  when the second number containeth the first in any proportion: 2.3. or .4. times and so forth. And in like proportion shall the third number contain the second, and the fourth, the third, and the fift the fourth. etc. As .2.4.8.16.32, 64: here the proportion is double.  
Likewaies .3.9.27.81.243. are in triple proportion.  
And .2.8.32.128.512. are in proportion quadruple.  
That is to say, in the first example, where the proportion is double, every number containeth the other .2. times. In the second example of triple proportion, the numbers exceed each other three times. And in the third example, the numbers exceed each other four times, and thus you see that progression Arithmeticalle, differeth from Progression Geometricalle for that, that in the Arithmeticalle.  
The excess is only in quantity, but in the Geometricalle, the excess is in proportion.  
Now if you will easily find the some of any such numbers, you shall do thus, consider by what number they be multiplied, whether by .2.3.4. 5: or any other, and by the same number, you must multiply the last somme in the progression. And from the product of the same multiplication, you shall abate the first number of the progression. And that which remaineth of the said multiplication, you shall divide by .1. less than was the number, by the which I did multiply. And the quocient shall show you the sum of all the numbers in any Progression. As in this example .5.15.45.135. 405. which are in triple proportion: now must you multiply .405. by .3. and they are .1215. from the which you shall abate the first number of the progression, which is .5: and there resteth 1210. the which you shall divide by the number less by .1. then by the which you did multiply, that is to say, by .2: and you shall find in the quocient 605: which is the total somme of the numbers of that progression. Like wise .4. 16.64 256.1024. which are in proportion quadruple: therefore multiply 1024. by .4. and thereof cometh 4096 from the which abate the first number .4. and there resteth .4092: The which you must divide by .3. and you shall find in your quotient .1364. which is the total somme of that progression, and this shallbe sufficient for progression.   

¶ The vii chapter treateth of the Rule of .3. called the golden Rule.  
THe rule of three is the chiefest the most profitable, and the most excellent rule of all the rules of Arithmetic. For all other rules have need of it, and it passeth all the other, for the which cause it is said, that the Philosophers did name it the golden rule. And after others opinion and judgement, it is called the rule of proportions of numbers. But now in these days, by us it is called the rule of three, because it requireth three numbers in his operation. Of the which three numbers, the two first are set in a certain proportion. And in such proportion as their be established, this rule serveth to find out unto the third number, the fourth number to him proportioned, in such sort as the second is proportioned unto the first. Not for that, that the four numbers, nor yet the three, are or be proporcionall, or set in one proportion, but such proportion, as is from the first to the second, aught to be from the third unto the fourth, that is to say, if the second number do contain the first two times or more, so many times shall the fourth number contain the third. And note well that the first number, and the third in every rule of three, ought and must be always semblable, and of one condition. And the second number, and the fourth must likewise be of one sembleaunce and nature.  
And are dissemblaunte, and contrary to the other two numbers: that is to say to the first, and the third. And if you do multiply the first by the fourth. And the second number by the third. The two multiplications will be equal. Likewise if you divide the one sembleaunte by the other, that is to say, the third number by the first. And likewise the one dissembleaunte by the other: that is to say, the fourth number by the second (which are dissembleaunte to the other two numbers) your two quocientes will be equal.  

Regul.  The style of this rule is thus, you must set down your three numbers in a certain order, as by example here under shall appear. And then multiply the third number, by the second. And the product thereof you must divide by the first number, or otherwise, divide the first number by the second. And the quotient thereof shallbe divisor unto the third number, that is to say, the third number shall be divided by the quotient of the foresaid division, that is of the first number divided by the second. Or otherwise divide the second number by the first. And that which cometh into your quotient, you shall multiply it by the third number. And thus shall you have the fourth number, which you seek for.  

¶ Example.  
IF .8. be worth .12. what are 14. worth after the rate, or else if .8. require .12. for his proporcionall, what will .14. demand? The which three numbers may conveniently be set in such order, as hereafter doth appear.  
If .8. 12. 14. multiply the third number .14. by the second, which is .12. And thereof cometh (for the whole product of this multiplication) 168. the which (as the rule teacheth) you must divide by the first number, that is to say by .8. and thereof cometh .21. And so moche are the 14. worth. This is the way, which is most used.  
Otherwise divide .8. by .12. which you can not do, for they are 8/12. wherefore abrevie 8/12 and they are 1/9 for your quotient, then divide the third number .14. by the said ⅔, and you shall have .21. as before. Or else divide the second number .12. by the first number .8. thereof cometh .1.1/2. the which 1 ½ you shall multiply by .14. and thereof will come .21. as is above said, and thus must you do of all other. And although, that the numbers of this rule may be found in three differences, for sometimes they are whole numbers and broken together, sometimes broken and broken together, and sometimes all whole numbers, if they be whole numbers, you must do none otherwise, than you did in the last example. But in case thai be broken numbers, or broken and whole numbers together, the manner and way to do them, receiveth a certain variation, and difficulty, according to the variety of the numbers, the which operation easily to do, and unvariably, this rule teacheth.  
The three numbers being set down, according unto the order of the whole numbers aforesaid, without any broken number, let .1. be put always underneath every whole number, with a line between them fraction wise, as thus 8/1. and that .1. is denominatour to every such whole number. When you have whole number and broken, they must be reduced and added with their broken number, and if there be broken number without any whole number, the same broken must remain in their estate.   

¶ A Rule.  
This being done, you shall multiply the denominatour of the first number, by the numerator of the second, and the product thereof again by the numerator of the third number And so shall you have the dividend, or number which must be divided, then multiply the numerator of the first number, by the denominator of the second, and the product thereof by the denominator of the third number, and that which cometh of this multiplication shallbe your divisor. Then divide the number, which is to be divided, by the divisor, and you shall find the fourth number that you seek. Of the which manner and fashions of the rule of .3. are divers kinds, whereof the first is of 3. whole numbers, as was the last example, and here followeth the second  
If .15. pounds do buy me two clotheses, how many clotheses will .300. pounds buy me of the same price, that the two clotheses did cost, set down your three numbers thus.  
Lib. Clotheses. Lib.  
15. 2. 300. 1    2 600    600 155 (40    1   
And than as you see, multiply the third number, which is .300. li. by .2. which is the second number, and thereof cometh 600. the which .600. you must divide by the first number .15. and you shall found in your quotient 40. which is .40. clotheses, and so many clotheses shall I buy for .300. li. as appeareth by practice here above written. And here you must mark that the first number & the third in this question be of one denomination, and likewise the second & the fourth which you have found are of one semblance: but in case that the first number and the third in any question: be not of like denomination, you must in working bring them into one, as in this example following. If .12. nobles do gain me 6. nobles, how many nobles will .48. pounds gain me: Here you see that the denomination of the first number is nobles, and the denomination of the third, is pounds, wherefore, before you do proceed to work by the rule of three, you must first turn the pounds into nobles in multiplying .48. pounds by three nobles and they make .144. nobles, for that there is in every pound of money .3. nobles, or otherwise if you will, you may bring the first number being .12 nobles, into pounds, by dividing them by .3. and thus shall your first and third numbers, be brought into one denomination. Then shall you set down your .3. numbers in order thus.  
If .12. nobles do gain me .6. nobles, what shall .144. nobles gain? the which. 144. are the nobles which are in .48. li. Then multiple the third number .144. by the second number 6. and thereof cometh 864. the which you must divide by .12. nobles, and thereof cometh .72. nobles.  
But here it may perchance make some men muse, to see all the three numbers in this rule of three, to be of one denomination, which can not otherwise be done, if you reduce the third number, to the denomination of the first. But if you will reduce the first number, to the denomination of the third, that is to say the .12. nobles into pounds, then shall the first and the third numbers only agreed in one denomination, and the fourth number which you seek, shallbe of the same denomination as is the second, as in the former example. If .12. nobles do yield me .6. nobles, what will .48. pounds yield me: first you shall divide 12. nobles by three to bring them in pounds, and they shall be like to the third number, which is also pounds, then will they stand thus.  
Pounds. Nobles. Pounds.   4. 6. 48.     6.  Nobles.   288. 288 (72    44   
There is yet a more exact way to work in this rule of three, which is thus. You must mark if the third and first numbers in the rule of three, may be both divided by one like divisor: the which after you have divided them, you shall writ down each of the quocientes orderly, in the said rule of .3. every one of them in his own place, as though those were two of the numbers of your question, and not changing the middle number, that is to say the second, as thus, if .50. Crowns do buy me .44. yards of cloth, how many yards shall I have for 120: here you may see that the third and the first numbers, may be divided by .10. which in the third number is .12. times, and in the first .5. times. Wherefore you shall put .12. for the third number in the rule of three, in stead of 120: and 5. for the first number in stead of .50. and let .44. remain still in the midst for the second number, after this sort as followeth, and then work by the rule as before.  
Crowns. yards. Crowns. 5. 44. 12.  12.   88 3  44 528 (105 ⅕  528 355  
Multiply .44. by .12. and thereof cometh .528. divide the same .528. by 5. and you shall find in your quotient 105. ⅗. and even so many yards should you have found, if you had wrought the rule of three, by the first numbers proposed. There is yet certain other varieties, in working by the rule of three, but for that they require the knowledge of fractions, and because they are not so easy as this first way, which is common, therefore content yourselves with this same, until you have tasted the fractions, the which by gods help I intend to set forth in second part of this book, incontinently after that I have first taught you the backer rule of three.  
The backer rule of three is so called: because it requireth a contrary working to that, which doth the rule of three direct, whereof we have now treated. For in the direct rule of three the greater the third number is, so much the greater will the fourth be But here in this backer rule it is contrariwise, for the greater the third number is, so much lesser will the fourth be. Then, where as in the rule of .3. direct, the third number is multiplied by the second, and the product thereof divided by the first. Here you must multiply the second number by the first, and divide the product of the same by the third, and the number which cometh in the quociente, answereth to the question. For such practice cometh often times in use: In such sort, that if you work the same by the rule of three direct (not having a regard unto the Proposition of the question) you should then commit an evident and open error.   

¶ Example.  
If 15. shillings worth of wine will serve for the ordinary of 46. men, when the Ton of wine is worth 12. pounds: for how many men will the same 15. shillings suffice when the ton of wine is worth but eight pounds? It is certain, that the lower the price is that the ton of wine doth cost, and so many more persons will the said 15. shillings in wine suffice. Therefore set down your numbers thus, if 12. pounds suffice 46. men, how many will 8. pounds suffice, you must multiply 46. by 12. and thereof cometh 552. the which 552. you shall divide by 8, and thereof cometh 69. and unto 69. men will the said 15. shillings worth in wine suffice, when the ton of wine is worth but eight pounds, as hereafter doth appear by practice.  
Lib. Men. Lib.   12. 46. 8.    12.  7   92.  552 (69.  46.  88   552.     
Likewise, a messenger maketh a journey in 24 days, when the day is but 12. hours long: how many days shall he be upon the same journey when the day is 16. hours in length? Here you must perceive, that the more hours are in a day, the fewer days will the messenger be in going his journey. Therefore write down your numbers thus as here you may see.  
Hours. Days. Hours.   12. 24. 16. 4   12.  12   48.  288   24  166   288.  1 (18  
And then multiply 24. days by 12. hours, and thereof cometh 288: divide the same 288. by the third number 16. and you shall find 18. the which is 18. days, and in so many days will the messenger make his journey when the day is 16. hours long.  
Likewise, when the bushel of wheat doth cost 3. shillings, the penny loaf of bread weigheth 4. lib.  
I demand what the same penny loaf shall way, when the bushel of wheat is worth but two shillings: here is to be considered that the better cheap the wheat is, the heavier shall the penny loaf way, and therefore writ down your 3. numbers thus.  
Shill. Lib. Shill.   3. 4. 2.    3.  12   12.  2 (6.  
Then multiply 4. lib. which is the second number, by the first number 3. and they make 12. the which 12. you shall divide by the third number 2. and thereof cometh 6. li. & so much must the penny loaf of bread way, when the bushel of wheat is worth but two shillings as may appear.  
And now, according to my former promise, shall follow the second part of Arithmetic, which teacheth the working by Fractions.    
¶ Here endeth the first part of Arithmetic.   

The second part of Arithmetic, which treateth of Fractions or broken numbers.  

¶ The first Chapter treateth of Fractions, or broken numbers, and the difference thereof.  
BRoken number is as much as a part or many parts of 1. whereof there are two numbers with a line between them both: that is to say, the one which is above the line is called the numerator. And the other underneath the line is called the denominator, as by example, three quarters, which must be set down thus, ¾: whereof 3. which is the higher number above the line is called the numerator, and 4. which is under the line is called the denominator. And it is always convenient that the numerator be less in number, than the denominator. For if the numerator, and the denominator were equal in value: then should they represent a whole number thus, as 1/1, 2/2, 3/3, which are whole numbers: by reason that the numerators of these, and all such like, may be divided by their denominators, and their quotientes will always be but 1. But in case that the numerator do exceed his denominator, than it is more than one whole: as 20/18, is more than a whole number by 2/18, other definition doth not hereunto appertain. Furthermore it is to be understand that the midst of all broken numbers is the just half of 1. whole, as 6/12, 7/14, 8/16, 9/18, and other like, are the halves of one whole number, whereof doth grow, and come forth 2. progressions natural: the one progreding by augmenting, or increasing, as these.  
½ ⅔ ¾ ⅘ ⅚ 6/7 ⅞ 8/9 9/ 10. etc.  
And they do proceed infinitely and will never reach to make a whole number thus 1/1. And the other progression, doth progrede by diminishing or decreasing, as thus.  
½ ⅓ ¼ ⅕ ⅙ 1/7 ⅛ 1/9 1/10. etc.  
And these do proceed infinitely, and shall never come to make a 0. which signifieth nothing, but shall ever retain some certain number whatsoever, whereby it doth appear that broken numbers are infinite.   

¶ The second Chapter treateth of the reducing or bringing together, of 2. numbers, or many broken dissembling, unto one broken sembling.  
REduction, is as much as to bring together, or to put in semblance 2. or many numbers dissembling one from the other, in reducing them unto a common denominator. For because the diversity and difference of the broken numbers, do come of the denominators part, or of divers denominators, and for the understanding hereof, there is a general rule whose operation is thus. Multiply the Denominators the one by the other, and so you shall have a new denominator common to all, the which denominator divide by the particular denominators, and multiply every quotient by his numerator and so you shall have new numerators, for the numbers which you would reduce, as appeareth by this example following.  

¶ Reduction in common denomination.  
IF you will reduce ⅔ and ⅘ together, you must first multiply the two denominators the one by the other, that is to say 3. by 5. maketh 15. which is your common denominator, that set under the cross, then divide 15. by the denominator 3. & you shall have 5. which multiply by the numerator 2. and you shall find 10. set that over the ⅔ and they are 10/11, for the ⅔. Afterwards divide 15. by the denominator 5. and thereof cometh 3. the which multiply by the numerator 4. and you shall find 12. which set over the head of the ⅘ and they make 12/15 for the ⅘: as appeareth more plainer in the margin.  
2. If you will reduce ½, ⅔, ¾, ⅚, together, you must multiply all the denominators the one by the other, that is to say, 2. by 3. maketh 6, than 6. by 4. and mounteth 24. Last of all 24. by 6. and thereof cometh 144. for the common denominator. Then, for the first divide 144. by the denominator 2. and thereof cometh 72. the which multiply by the numerator 1. and it is still 72. set that over the ½ and it is 72/144, for the ½: Then divide 144. by the second denominator 3. & thereof cometh 48: the which multiply by the second numerator 2. and they are 96. which set over the ⅔ and they make 96/144, for the ⅔: Then divide 144. by the third denominator 4. & thereof cometh 36. the which multiply by the third numerator 3. & they make 108. which set over the ¾ and they are 108/144 for the ¾.  
Finally divide 144. by the last denominator 6, & thereof cometh 24: The which multiply by the last numerator 5. & thereof cometh 120. Which set over the ⅚ and they are 120/144, for the ⅚, as appeareth here by practice.   

¶ The example.  
  

¶ Reduction of broken numbers of broken.  
IF you will reduce the broken of broken together, as thus, the ⅔ of 7/4 of ⅘, you must multiply the numerators the one by the other to make one broken number of the three broken numbers, that is to say 2. by 1. maketh 2. and then 2. by 4. maketh 8. which is your numerator. Then  8  ⅔ ¼ ⅘  60.  multiply the Denominators the one by the other, that is to say 3. by 4. maketh 12. and then 12. by 5. maketh 60. for your denominator, set 8. over 60. with a line between them, and they be 1/60 which being abbrevied are 2/15 and so much are the ⅔ of ⅙ of ⅘ as appeareth in the margin.   

¶ Another example of the same reduction and of the second reduction.  
IF you will reduce ⅔ of, ¼, of ⅘, the ¾, of 5/7: And the ½, of the ½, of the ⅔ of ⅓. First it behoveth you of every party of the broken numbers to make of each of them one broken, as by the third reduction is taught: that is to say, in multiplying the numerators by numerators & denominators by denominators: first, for the first part which is ⅔ of ¼ of ⅘, you must as is said before, multiply 2. by 1. and then by 4. & you shall have 8. for the numerator likewise multiply 3. by 4. and the product by 5. and you shall have 60. for the denominator, so they make, 1/60 which being abrevied are 2/15 for the first part, that is to say, for the ⅔ of ¼ of ⅘, secondly for the ¾ of 5/7 multiply likewise the numerator 3. by 5. maketh 15. for the numerator, & multiply 4. by 7. maketh 28. for the denominator, and then they be 15/29 for the second part that is to say for the ¾ of 5/7. thirdly for the ½ of ½ of ⅔ of ⅓ multiply the numerators the one by the other, that is to say, 1. by 1. and then by 2. and last by 1. and all maketh but 2. for the numerator, likewise multiply 2. by 2. maketh 4. and 4. by 3. maketh 12. and then 12. by 3. maketh 36. for denominator, and they are 2/36, which being abrevied maketh 1/18, for the third part, the is to say for ½ of the ½ of ⅔ of ⅓. Last of all take the 2/15 the 15/28 and the 1/18 & reduce them according to the order of the second reduction, and you shall find 1008/7560 for the 2/15. And 4050/7500 for the 15/28. And 420/7560 for the 1/18: and thus are broken numbers of broken, reduced as appeareth by practice.  
8 15 2    ⅔¼⅘ ¾ 5/7 ½½⅔⅓    1008 4050 420 60 28 36 2/15 15/28 1/18     760    

¶ Reduction of broken numbers, and the parts of broken together.  
IF you will reduce ⅓ and the ½ of ⅓ together to bring them into one broken number, you must first set down the ⅓ and ½ as appeareth in the margot with a cross between them, & then multiply the two denominators the one by the other, that is to say, 2. by 3. maketh 6. set that under the cross, then multiply the first Numerator, one by the last denominator two, and that maketh 2. unto the which add the last numerator one, and they be three, which set above your cross, so you shall find that the ⅓ and the ½ of ⅓ do make ⅓ which being abbrevied doth make ⅓, which is as much as the ⅓ and the ½ of ⅓. Likewise if you will reduce the ⅔ and the ¼ of ⅓, you must do as before, set down the ⅔ and ¼ with a cross between them, multiply the two denominators the one by the other, that is to say, 3. by 4. maketh 12. which set under the cross as you see in the margin and then multiply the first numerator 2. by the last denominator 4. and thereof cometh 8. whereunto add the last numerator 1. and that maketh 9 which set over the Cross, so shall you find that the ⅖ and the ¼ of ⅓ are worth 9/12, which abbrevied do make ¼, as appeareth by example in the margin.   

Reduction of whole numbers and broken together into a Fraction.  
IF you will reduce whole number with broken, you must bring the whole number into broken, as by this example may appear: reduce 17. ⅝ into a broken number, first you must multiply the whole number 17. by the denominator of the broken, which is eight in saying eight times 17. do make 136. unto the which you must add the numerator of ⅝ which is 5. and all amounteth to 141. which set over 8. with a line between them, & they will be 141/8 so much is 17. ⅛ worth in a Fraction as appeareth here by practice.  
17.  141.   8.  5.   136. 17. 8. maketh 141./ 8. 5.     141.      
In case you have whole number and broken to be reduced, with broken you must bring the whole number into his broken, in multiplying it by the denominator of the broken number going therewith, and add thereunto the numerator of the said broken number, as in the last example, and then reduce that broken number with the other broken, as here appeareth by this example. Reduce 10. ⅔ & 4/7 together, first bring 10 ⅔ all into thirds, as by the sixth reduction, and you shall find 32/3, then reduce the 32/3 and 4/7 together, by the first reduction, and you shall find 224/21 for the 32/3: and 12/21 for 4/7, as appeareth here by practice.  
 
Also in case you have in both parts of your Reduction, as well whole number as broken, you must always put the whole into their broken (as by the sixth reduction) of either part.   

¶ Example.  
If you will reduce 12. ¼ with 14. ⅔ to bring them into one denomination, first bring the 12. ¼ all into fourthes, and you shall find 49/4: then likewise reduce 14. ⅔ all into thirds, and you shall have 44/3, for the 14. ⅔, then reduce 49/4 and 44/3 together, by the order of the first Reduction, and you shall find 147/12 for the 49/4. And 170/12 for the 14. ⅔ as here by practice doth plainly appear.  
   

¶ The third Chapter treateth of abbreviation of one great broken number into a lesser broken.  
Abbreviation is as much as to set down, or to writ a broken number by figures of less signification, & not diminishing the value thereof. The which to do, there is a rule whose operation is thus, divide the numerator and likewise the denominator, by one whole number the greatest that you may find in the same broken number, & of the quotient of that numerator, make it the numerator, and likewise of that of the denominator make it your denominator, as by example.  
1. If you will abreviat 54/81, you shall understand that the greatest whole number that you may take, by the which you may divide the numerator & denominator is 27, which is the half of the numerator, & that is a whole number, for you cannot take a whole number out of the denominator, 81. but that there will be either more or less than a whole number, therefore if you divide 54. 54./ 81.   1   27   27 (2    ⅔ 2   81   27 (3  by 27. you shall find 2. for the numerator, likewise if you divide 81. by 27. you shall find 3. for the denominator, then put 2. over the 3. with a line between them, and you shall found ⅔ and thus by this rule the 54/81 are abrevied unto ⅔, as appeareth in the margin, and so is to be understand all other.  
¶ The form & manner how to find out the greater number, by the which you may wholly divide the numerator & denominator (to th'end that you may abreviat them) is thus.  
first, divide the denominator by his numerator, and if any number do remain, let your divisor be divided by the same number, and so you must continued until you have so divided that there may nothing remain, then is it to be understand, that your last divisor (whereat you did end, and that 0. did remain after your last division) is the greatest number, by the which you must abreviat, as you did in the last example, but in case that your last divisor be 1. it is a token that the same number can not be abrevied. Example, of 54/11 divide 81. (which is the denominator) by 54. which is his numerator, and there resteth 27. then divide 54. by 27. and there remaineth nothing, wherefore your last divisor 27. is the number, by the which you must abreviat 54/81 as in the last example is specified.  

¶ Another style of abbreviation.  
2. Mediate the numerator, and also the denominator of your fraction in case the numbers be even, that is to say, take always the half of the numerator, and likewise of the denominator, and of the mediation or half of the numerator, make it your numerator, also of ½ the denominator, make your denominator, and so continued as often as you may in taking always the ½ of the numerator, and semblably of the denominator, or else see if you may abbreviate the numbers which do remain, by 3. by 4. by 5.6.7.8.9. or by 10: for you must abbreviate them as often as you can by any of the said numbers, and it is to be noted, that with whatsoever number of these, you do abbreviate the Numerator of your Fraction, by the same you must abbreviate likewise the Denominator, so continuing until they can no more be abbrevied. And it is to be understand that if the Numerator and the Denominator be even numbers, as you may know when the first figure is an even number, or a 0, may you perceive if both the Numerator and the Denominator may be abbrevied by 10. by 8. by 4. or by 2. although that some times they may be abbrevied by three. And if they be odd numbers, then must you consider if they may be abbrevied by 9 by 7. by 5. or by 3: but when the first number, as well of the Numerator, as of the Denominator are even numbers, then may you well know that such numbers may be abbrevied by 2. as is aforesaid. And if you add the figures of the Numerator together, in such manner as you do in making the proof by nine in whole numbers: that is, if you find 9 it appeareth that you may abbrevye that number by 9 And likewise by 3. and sometimes by 6. if you find 6. it may be abbrevied by 6. and always by 3. if you find 3. it is a sign that you may abbreviate by 3. And by whatsoever number that you do abbreviate the numerator, by the same must you abbreviate likewise the denominator, and if the first figures of the same number be .5. or 0. you may abbreviate them by 5. but if the first figures be both 0. they may be abrevied by 10. in cutting away the two Ciphers thus, as ⅔ 0/0 which maketh ⅔, as sometimes by 100 thus, as ½ 00/00 in cutting away the four ciphers after this sort, ½ 00/00 and then the 100/200 do make ½, and after this manner have I set here divers examples, although that all numbers cannot be abrevied by this rule, that is to say, all those which may be well abrevyed by the first rule aforesaid.  
abbreviated.  
by 10. 3840/7080 by 9 1890/4725 by 8. 384/768 by 7. 210/525 by 6. 48/69 by 5. 30/75 by 4. 8/16 by 3. 6/15 by 2. 2/4   2/6   ½     
3. Furthermore you shall understand that sometimes it happeneth, that all the figures of the numerator are equal unto them of the denominator, which when it so happeneth, you may then take one of them of the numerator, and also one of them of the denominator, and it shall be abrevyed as 555/888, being abbreviated after this manner cometh to ⅝. And yet it happeneth sometimes, that two, or many figures of the numerator are proportioned unto two, or many figures of their denominators and the other figures of the same number do behold the one the other in this proportion? Then may you take two or many figures, as well of the numerator as of the denominator, and by this manner the same number shall be abbrevied, as 4747/5959 which being abbrevied by this rule, do come to 47/59.  
4. Also it happeneth sometimes that you would abbreviate one number unto the semblance or likeness of another. And for to know if the same may be abbrevied, and also by what number it may be abbrevied, you must divide the numerator of the one number by the numerator of the other, and likewise the denominator of the one by the denominator of the other, for in case that after every division there do remain 0. and that the two quotients be equal, then is one of them the number by the which the said fraction must be abbrevied, as by example of 115/207. I would know if they may be abbrevied unto 5/9, and for to do this, you must divide 115. by 5. and you must divide 207. by 9 and there will come into both the quotients 23. by the which it appeareth that this number may be abbrevied by 23.  
5/9. 115/207 115 ●   55 (23 207    99 (23.    

¶ The 4. Chapter treateth of the assembling of two or many broken numbers together, as by example.  
FOr to add broken numbers together, there is a general rule, which is thus, if the numbers be unlike the one to the other you must reduce than into a common denomination, which after you have reduced them, you must then add both the numerators together, & set the product of the said addition over the cross, & divide the same by the common denominator as by this example following.  
1. If you will add ⅔ with ¾, you must first reduce the two fractions both into one denomination, according to the introduction of the first reduction, that is to say, in multiplying the denominator of the first fraction which is 3, by the denominator of the other fraction which is four, and they make 12. for your common denominator, the which 12. set under the cross, them multiply the first numerator 2. by the last denominator 4. and thereof cometh 8. which set over the ⅔, and then multiply the last numerator 3. by the first denominator 3. and thereof cometh 9 which you must set over the ¾, then add the numerator 8. with the numerator 9 & they make 17. which set over the cross, and then your fraction will be 17/12 which is the addition of the ⅔ with ¾. And because your numerator 17. is greater than his denominator 12. therefore you must divide 17. by 12. and thereof will come 1. and 5. remaining, which 5. are worth 5/12, and so much are the ⅔ added with ¾ as doth appear.  

¶ Addition in broken numbers.  
2. Also if you will add ½, ⅔, ¾, ⅘, together, you must first add the ½ and ⅔ together, according to the doctrine of the last rule, and you shall find 7/6: then add ¾ and ⅘ together by the said last chapter, and they make 31/20. Then finally add the 7/6 (which came of the ½ and ⅔ added together) with 13/20, & you shall find by the foresaid addition that they amount unto 326/120. Wherefore divide 326. by 120. & thereof cometh 2. and 86. remaineth, which is 86/120 of one whole, & they being abrevied do make 43/60: & thus the ½, ⅔, ¾, ⅘, added together do amount to 2.43/00, as here under doth appear.  
  

¶ Addition of broken numbers of broken.  
3. Furthermore, if you will add the broken numbers of broken together, as to add the ⅔ of ¾ of ⅘ with the ⅚ of ½ of ⅝: first you must reduce the numbers according to the order of the forth reduction, in multiplying the numerator of the first 3. fractions, the one by the other, and of the product make your numerator, & likewise you must multiply the denominators of the foresaid three fractions, the one by the other and of the product make your denominator, and you shall find 24/60 for the first three broken numbers, which being abbrevied do make ⅖, them reduce the other 3. fractions, by the said fourth reduction, in multiplying the numerators by numerators, & denominators, by denominators, as you did by the first 3. broken numbers, & you shall find 25/96 then must you add that ⅖ which came of the first 3. broken numbers, & 25/96 which are of the last 3. fractions, both together, by the instruction of the first addition & you shall found 317/480 which cannot be abbrevied, but is the product of the addition: so much are the ⅔ of ¾ of ⅘ added with the ⅚ of ½ of ⅝ as hereafter by practice doth evidently appear.  
  

¶ Addition of broken number with the parts of broken together.  
4. Likewise if you will add the ⅓, and the ½ of ⅓ with the ⅘ and ¼ of ⅕, you must reduce the ⅔ ½ by the fift reduction and thereof cometh ⅚ for the ⅔ & ½, of one of the said thirds, them reduce the ⅘ and ¼ by the said fift reduction, and thereof cometh 17/20.  
Last of all add the ⅚ and 17/20 together according to the first rule of addition, and you shall find 202/120 which being divided bringeth 1. & 82/120 part remaining, which abrevied maketh 41/60 and thus you do perceive that the ⅔ & ½ added with the ⅘ and ¼ do amount unto 1.41/60 as hereafter by practice doth plainly appear.  
 
  

¶ Addition of whole number and broken together.  
5. Also if you will add 12. ⅘ with 20. ⅚, you may (if you will) add 12. & 20. together, and they make 32. & then add the two broken numbers together, that is to say ⅘ and ⅚ by the order of the first addition & they make 49/30: therefore divide 49. by 30. and thereof cometh 1. and 19/30 parts remain, which 1. you must add unto the 32. & the whole addition will be 33. 19/30, or otherwise, you may reduce 12. ⅘ into the likeness of a Fraction by the sixth reduction and they will be 64/5, & likewise by the same reduction, reduce 20. ⅚ and they be 125/6, then add 84/5 with the 125/6, by the first addition and you shall find 1009/30. Therefore divide 1009. by 30. and thereof cometh 33.19/30 as before, & as by practice of the same both the ways, doth here under appear.  
   

¶ The fift Chapter treateth of Substraction in broken numbers.  
IF you will subtract ⅔ from ¾ you must first reduce both the fractions into a common denomination by the first reduction, and you shall find 8/12 for the ⅔, and 9/12 for the ¾. Therefore abate the numerator 8. from the numerator 9 & there remaineth 1/12 as may appear here by practice.  
 
2. But if you have a broken number to be substracted from a whole number, you must borrow one of the whole number, & resolve it into a fraction of like denomination as is that fraction, which you would abate from the same whole number, & then abate the said fraction therefrom, & you shall find what doth remain, as by this example. If you abate ⅘ from 8. you must borrow one of the said 8. and resolve it into fifths like unto the fraction, because it is 4. fifts, and that 1. will be 5. fifths thus 5/5, therefore abate ⅘ from 5/5 & there will remain ⅕, and subtract the 1. which you borrowed from 8. and there doth remain 7. and also the ⅕. Thus the ⅘ being substracted from 8. doth leave 7. ⅓ as by practice doth plainly appear.  
 
3. If you will subtract broken number from whole number and broken being together: thus, as if you would subtract ¾ from 6. ⅚, you may by the first substraction, abate ¾ from ⅚, and there will remain 1/12, and the 6. doth still remain whole, because the ¾ may be abated from the ⅚, thus ¾ being abated from 6 ⅚ leaveth 6.1/12 as appeareth by practice.  
 
Likewise if you will abate ⅔, from 14. ⅖, you must first reduce 14. ⅖ all into fifths by the 6. reduction, and they be 72/5, then reduce ⅔ into a common denomination, by the first reduction, and you shall find 10/15 for the ⅔: and 216/15 for the 72/5: them subtract the numerator 10. of the first fraction, from the numerator 216. of the second fraction, & there remaineth 206/15. Therefore divide 206. by 15. and thereof cometh 13.11/15, and so much remain of this substraction, as may appear.  
 
4. If you will subtract whole number and broken from whole & broken, as thus, if you will subtract 9 ¼ from 20. ½ you must reduce 9 ¼ into fourths, & likewise the 20. ½ into halves by the sixth reduction: & you shall found 37/4 for the 9 ¼ And 41/2. for the 20. ½. Then reduce 37/4. and 41/2 into one denomination, according unto the first reduction and you shall find 74/8 for the 37/4, and 164/8 for the 41/2 them abate the numerator of 74/8 which is 74. from the numerator of 164/8 and there remaineth 90/8 then divide 90. by 8. and thereof cometh 11. ¼ which is the remain of this substraction.  
 

¶ Substraction of broken numbers of broken.  
5. If you will subtract, the 2/1 of ⅔ of ⅗ from the ⅚ of ¾ of ⅞ you must first bring the ½ of ⅔ of ⅗ into one fraction by the 3. reduction, & the 3/6 of ¾ of ⅞ likewise into one fraction by the same reduction, & you shall find 6/30 for the first 3. broken numbers, which being abbrevied do make ⅕: & for the other 3. broken numbers, you shall find 105/192: which being likewise abbrevied do make 35/64, than you shall subtract ⅕ from 35/64 by the instruction of the first substraction, in reducing both the fractions into a common denomination, as before is done, & you shall found remaining 111/320 as may appear by example.  
   

¶ The sixth Chapter is of multiplication in broken numbers.  
FIrst for to multiply in broken number, there is a rule which is thus, multiply the numerator of the one fraction by the numerator of the other. And then divide the fraction if you may, or else abbreviate it, and you have done, but if there be whole number & broken together, you must reduce the whole numbers into broken, & add thereunto the numerator of his broken, and then multiply as is before said, as also hereafter by examples shall more plainly appear.  
1. If you will multiply ⅔ by ¾, you must multiply the numerator 2. by the numerator 3. & thereof cometh 6. for the numerator: Likewise multiply the denominators the one by the other, that is to say 3. by 4. & thereof cometh 12. fpr the denominator, so that this multiplication cometh to 6/12, which being abbreviated do make ½ and so much amounteth the multiplication of the ⅔ by ¾ as by practice.  
 6  ⅔  ¾  12.   
2. Likewise if you will multiply a broken number by whole number, or whole number by broken, which is all one, as ⅘ by 18, or else 18. by ⅘, you must set 1. under 18. thus 8/1: and then multiply 18. by the numerator 4. and thereof cometh 72. the which divide by the denominator 5. & thereof cometh 14. ⅖ for the whole multiplication, or otherwise abate from 18. his ⅕ part which is 3. ⅗, and there remaineth 14. ⅖ as above.  
 72  22  ⅘  18/1 72   5  55 (14 2/5  
 18  3   1●/1  ⅕ 18 18   5  5 (3. 3 ⅗     14 ⅖  
3. Also if you will multiply a whole number, by whole number and broken, or else whole number & broken by a whole number, which is all one. As by example, if you multiply 15. by 16. ¾ or else 16. ¾ by 15. First reduce 16 ¾ all into fourthes, in multiplying 16. by the denominator of ¾ which is 4. and thereof cometh 64. whereunto add the numerator 3. and it maketh 67/4 which multiply by 15/1 according unto the instruction of the last example, & you shall find the product of this multiplication to be 251. ¼ as by practice doth here appear.  
     67  67  1005  15 16 ¾ 1●/●  ●7/4 335    4  67      1005  
2 1  1005  444 (251. ¼  
4. And if you will multiply a broken number, by whole number and broken, or else whole number & broken by a broken. As by example, if you will multiply ¼ by 18. ⅔, or else 18. ⅔ by ¼, which is all one: you must reduce the whole number into his broken by the sixth reduction. And you shall find 56/3, which you shall multiply by the ¼, after the doctrine of the first multiplication, that is to say: in multiplying the Numerator 56. by the Numerator of ¼, which is 1. And it is still 56. because 1. doth neither multiply nor divide. And likewise you must multiply the Denominator 3. by the Denominator 4. and it maketh 12. then divide 56. by 12. and thereof cometh 4. ⅔. And so much amounteth the multiplication of the 18. ⅔ multiplied by ¼ as by example.  
 56  56  18 18 ⅔ ¼  ●6/3 56    12  12 (4 2/3  
5. If you will multiply whole number and broken, with whole and broken, you must first put either whole number into his broken, according to the instruction of the sixth reduction, and then multiply the one numerator by the other, and of the product make your numerator. And likewise multiply the denominators the one by the other, and thereof make the denominator, then divide the numerator by the denominator, and the quotient shall be the increase of this multiplication. Example, If you would multiply 12. ⅘ by 6. ¾: first by the sixth reduction the 12. ⅘ will make 64/3, and the 6. ¾ will make 27/4, then multiply the numerator 64. by the numerator 27. and thereof cometh 1728. for the numerator. And then you must multiply the denominator 25. by the denominator 4. and they do make 20. then divide 1728. by 20. and thereof cometh 86. ⅖ for the whole multiplication, as by example.  
  1728  64  64  27 27 12 ⅘  6 ¾ 448   20  128     1728   1     1728    200 (86.    2    
6. If you will multiply one broken number by many broken numbers, thus: As to multiply ⅔ by 5/7 & by 4/9: you must multiply the numerators of all the fractions, the one by the other, & of the product make the numerator that is to say: 2. by 5. and they be 10. then 10. by 4. and they be 40. for the Numerator. Likewise you must multiply the denominators the one by the other, that is to say 3. by 7. maketh 21. then 21. by 9 maketh 189. for the denominator: then set 40. over the 189 with a line between them, and they make 40/189. And so much amounteth the whole multiplication of the ⅔ multiplied by 5/7 and 4/9 as by example following. And thus is to be understand of all such like.  
   2 3     5 7   40  10 21  ⅔ 5/7 4/9 4 9 40/189  189  40 189    

¶ The 7. Chapter treateth of Division in broken numbers.  
NOte that in Division of broken numbers, you must set your divisor down first, next unto the left hand, and the dividend or number which is to be divided always toward the right hand. And then multiply cross, that is to say the numerator of your divisor by the denominator of the dividend, and the product shallbe the denominator, which afterward shall be your Divisor. And likewise you must multiply the Denominator of your first number, that is to say of your Divisor: by the Numerator of the Dividende, which afterward shall be the Dividend, and that must be set over the Cross, and the Denominator under the Cross, then shall you divide the Numerator by the Denominator if it may be divided, if not, you must abbreviate them, as hereafter by examples shall more plainly appear.  
1. If you will divide ¾ by ⅖, you must set the Divisor (which is 2/3) next to the left hand, and the dividend ¾ toward your right hand, with a cross between them: as may appear by this example in the margin . Then you shall multiply the numerator of the ⅔, which is 2. by the denominator of the ¾ which is 4. and and thereof cometh 8. which shallbe your new divisor: set that 8. under the cross, as the denominator, then multiply the numerator of the dividend, that is to say of the ¾ which is 3. by the denominator of the divisor, that is to wit of the ⅔ which is 3. set that over the cross, and it is 9 for the numerator, which shallbe now the dividend, or number to be divided. Then finally you shall divide 9 by 8. and thereof cometh into the quotient 1. ⅛, and so oftentimes is ⅔ contained in ¾. as doth appear before in the margin. But in case you would divide ⅔ by ¾, you must likewise set your divisor ¾ next to your left hand, as is before said. And then proceed, as is above declared, & you shall find that ⅔ divided by ¾ bringeth into the quotient 8/9, which cannot be divided nor abbrevied, wherefore it appeareth that ⅔ divided by ¾ bringeth but 8/9 of one unity into the quotient as doth appear.  
 
2. Likewise if you will divide a broken number by a whole number or else a whole number by a broken, as to divide ¾ by 13. you shall put 1. under 13. and it will be 13/1 which is your divisor, set that toward your left hand, and then multiply 13. by 4. according to the first division, & thereof cometh 52. for the denominator, set that under the cross & multiply 3. by 1. which is 3. for the numerator, that set over the cross, and it is 3/52 as appeareth in the margin. But if you will divide 13. by ¾ them set the ¾ next your left hand and put one under 13. as in the last example, & it is 13/1 set that toward your right hand thus, as appeareth in the margin, and then work according to the doctrine of the first division, & you shall find that 13. being divided by ¾ bringeth into the quotient 52/3, then divide 52. by 3. and thereof cometh 17. ⅓, and so oftentimes is ¾ contained in 13. as doth appear. 21 21  52  33 (17. ●/●  
3. And if you will divide whole number by whole number and broken, or else whole number and broken by whole number, as to divide 20. by 5. ⅚, you shall reduce 5. ⅚ into his broken by the sixth reduction, & it maketh 35/6 for your divisor, than put 1. under 20. And it will be 20/1, then shall you multiply 35. by 1. and 20. by 6. as is taught in the other divisions, and you shall find 120/35: then divide 120. by 35. and you shall find in your quotient 3. 3/7 & so many times is 5. ⅚ contained in 20. as in the margin doth appear. 1 35 35 120 (3. ●/7 35  
But if you will divide 5. 9/6 by 20. you must divide 35. by 120, which you can not, wherefore you shall abbreviate 35/126, and thereof cometh. 7/24.  
4. If you will divide a broken number, by whole number and broken, or else a whole number and broken, by a broken number. As to divide ¾ by 13. ⅔, you must reduce 13. ⅔, into his broken, by the sixth reduction And they be 41/3 for your divisor, then multiply 41. by 4. & they make 164. for your denominator, likewise multiply 3. by 3. and they make 9 for the numerator, and then will your sum be 9/164. But if you will divide 13. ⅔ by ¾ than you must divide 164. by 9 and you shall find 18. 2/ 9  
5. If you will divide whole number and broken, by whole number & broken, as to divide 7. ¾ by 13. ⅔ you must reduce the whole numbers into their broken, by the doctrine of the sixth reduction, & you shall find 31/4 for the 7. ¾, & 41/3 for the 13. ⅔. Then set down 41/3 toward the left hand because it is your divisor, and the 31/4 toward the right hand, and multiply 41. by 4. for your denominator, and thereof cometh 164. Likewise multiply 3. by 3. for your Numerator, and it amounteth to 93. the which division will be thus 93/164 as before doth appear.  
But if you will divide 13. ⅔ by 7. ¾ you must contrariwise to the other example, divide 164. by 93. and you shall find in the quotient 1. 71/ 93.  
6. The broken numbers of broken, must be divided in such manner as broken numbers are, & there is no difference, saving only that of many broken numbers you must make but two broken numbers, that is to say the divisor, and the dividend, or number that is to be divided, example. If you will divide the ¾ of ⅗ of ½, by the 2/2 of 4/7. For the first, the ¾ of ⅗ of ½ are 49/40 by the third reduction: and the ⅔ of 4/7 are by the same Reduction 8/21, then have you 8/21 for your divisor, & 9/40 for your number to be divided, then multiply 8. by 40. which maketh 320. set that under the cross and multiply 9 by 21: & thereof cometh 189. which set over the cross for the numerator, and they make 189/320 for this division as doth appear.  
But if you would divide 8/21 by 9/40. you must work contrary to the last example, that is to say, you must divide 320. by 189. And thereof cometh in the quotient 1. 131/ 189.   

¶ The eight Chapter treateth of duplation, triplation, and quadruplation of all broken numbers.  
IF you will double any broken number, you shall divide the same by ½: likewise if you will triple any fraction you must divide it by 1/3. And for to quaduple any broken number, you shall divide it by ¼, and so is to be understand of all other.  

Example of Duplation.  
IF you will double ⅜ you shall divide ⅜ by ½, and thereof cometh 6/8 , which being abbrevied are ¾: as by example. Or otherwise, in case the denominator of any fraction be an even number, you may take half the said denominator, without any other operation, and the numerator to abide still the numerator, unto the said half of the denominator of the Fraction, as by the other example before rehearsed: that is to say of ⅜, take ½ of 8. which is 4. and that is the denominator, and 3. remaineth still numerator to 4. and it maketh ¾ and so of all other. But in case the denominator be an odd number, that is to say, not even, them you may multiply the numerator by 2. or else double the numerator, which is all one thing, and that fraction shall be doubled. Example, if you will double ⅗ you must only multiply the numerator 3. by 2. & they be 6. which maketh that fraction to be 6/5, the which 6. being divided by 5. bringeth 1. ⅕ and so much is the double of ⅗.   

Example of Triplation.  
If you will triple ⅗ you must divide ⅗ by ⅕ and thereof cometh 9/5 which being divided bringeth 1 ⅘, or otherwise, because the denominator is an odd number you may multiply the numerator 3. by 3, and thereof cometh 9 which maketh 9/3 as before.   

Example of quadruplation.  
If you will quadruple ⅘, you shall divide ⅘ by ¼ and thereof cometh 16/5 which 16. being divided by 5. bringeth 3 ⅕, or otherwise, because the denominator of the fraction is an odd number, you shall multiply the numerator of the ⅘ that is to say 4. by 4. and thereof cometh 16. the which divide by 5. and you shall find 3. ⅕ as before, and this sufficeth for duplation, triplation and quadruplation.    

¶ The 9 Chapter treateth of the proves of broken numbers. And first of Reduction.  
IF you do abbreviate the broken numbers which be reduced, you shall return them into their first estate: as by example, if you reduce ⅖ with ⅘ you shall find 10/15 and 12/15, then abbreviate 10/15 and you shall find ⅔, abbreviate likewise 12/15 and thereof cometh ⅘ as before.  

The proof of Abbreviation.  
IF you do multiply that number which you have abbrevied by that or those numbers, by the which you have abbrevied them, you shall return them again into their first estate. Example, if you will abbreviate 32/48 by 16. in taking the 1/16 part both of the numerator, and also of the denominator, you shall find ⅔, the proof is thus, you must multiply both the numerator & denominator of ⅔ by 16. that is to say, three by 16. maketh 48. for the denominator, & 2. by 16. maketh 32. for the numerator, them set the numerator 32. over the denominator 48 and they be 32/48 as before.   

The proof of Addition.  
If you do subtract one of the numbers, or many of them (which you have added) from the total sum, there shall remain the other, or others. Example: if you do add ⅓ with ¼ you shall find 7/12. The proof is, if you subtract ⅓ from 7/12 you shall find remaining the other number which is ¼, or else if you do subtract ¼ from 7/12 there will remain the other number, which is ⅓.   

The proof of Substraction.  
If you do add that number which remaineth, with the number which you did subtract, you shall find the total sum, out of the which you made the abatement: or otherwise, if you add the two lesser numbers together, you shall find the greater. Example: if you do abate or substract ¼ from ⅓ there will remain 1/12. The proof is thus: you must add 1/12 & ¼ together, and you shall find ⅓, which is the greatest number.   

The proof of Multiplication.  
If you divide the product of the whole multiplication, by the multiplicator, you shall find in your quotient, the multiplicand or number by the which you have multiplied: or else if you divide the total sum which is come of the multiplication, by the multiplicand: you shall find in the quotient the multiplicator. Example, if you multiply ⅔ by ⅘, the product of this multiplication will be 8/15. The proof is thus: you shall divide 8/15 by the multiplicator ⅘, and thereof cometh ⅔. Or else divide 8/15 by ⅔ & you shall find the ⅘ which is the multiplicator.   

The proof of Division.  
If you do multiply the quotient by the divisor, you shall find the number which you did divide, that is to say, your dividend. Example: if you divide ⅔ by ¾, your quotient will be 8/9 the proof is thus, you must multiply 8/9 by ¾, and thereof cometh 24/36 which being abbreviated are ⅔ which is your dividend, & by this manner all whole numbers have their proofs as well as broken numbers.    

¶ The tenth Chapter treateth of certain questions done by broken numbers. And first by Reduction.  
Find two numbers, whereof the 2/7 of the one number may be equal unto the ⅜ of the other. Answer: you shall reduce 2/7 & ⅜ crosswise, and you shall find 16. over the 2/7 and 21. over the ⅜, which are the two numbers that you seek: for the ⅜ of 16. are 6. and so are the 2/7 of 21. likewise 6. wherefore you may perceive that the ⅜ of 16. which are 6. are equal unto the 2/7 of 21. which is also 6.  
2. Find two numbers, whereof the ⅔ of the one may be double to the ¼ of the other. Answer: double ¼ & you shall have 2/4, which being abbreviated is ½: them reduce ⅔ & ½ crosswise, & you shall find 4. over the ⅔ & three over the ½ which are the two numbers that you seek. For the ⅔ of 3. which is 2. is double unto the ¼ of 4. which is but 1.  
3. Find two numbers whereof the ⅓ and the ¼ of the one, may be equal unto the ¼ & ⅕ of the other. Answer: Add the ⅓ and ¼ together, and they make 7/12 then add ¼ and ⅕ together, & they are 9/20, then reduce 7/17 & 9/20 crosswise, & you shall have 140. over the 7/12 & 108. over the 9/20, which are the two numbers that you seek. For 63. which are the 7/12 of 108. are also the 9/20 of 140.  
4. Find two numbers, whereof the ½ the ⅓ and the ¼ of the one of them, may be equal unto the ⅕ the ⅙ and 1/7 of the other number. Answer: first you must add ½, ⅓ and ¼ together, & they make 13/12: then add ⅕, ⅙ and 1/7 together, & they make 107/210. Then reduce 13/12 and 107/210 crosswise, as by the first question of reduction, and you shall find 2730. over the 13/12 and 1284. over the 107/210, which are the two numbers that you seek: for 1391 which is the ½ the ⅓ and ¼ of 1284. is like to the 11/56 & 1/7 of 2730, which is also 1391.  
5. Find three numbers, whereof the ⅖ of the first, the 3/7 of the second, & the 4/9 of the third, may be equal the one to the other. Answer: set down the 23/57 and 4/9, and then multiply the Denominator of the ⅖ that is to say 5. by the Numerators of the other two Fractions, that is to say, by the Numerator of 3/7, and by the Numerator of 4/9, which is 3. and 4. And thereof cometh 60. for your first number, then shall you multiply the Denominator of the 3/7 which is 7. by the Numerators of ⅕ and 4/9, that to to say by 2. and 4. and thereof cometh 56. for the second number. Then multiply the Denominator of 4/9, that is 9 by the Numerators of ⅖ and 3/7 that is by 2. and by 3. and thereof cometh 54. for the third number.  
And thus the ⅖ of 60. which is 24. is likewise the 3/7 of 56. which is the second number: and the 4/9 of 54. which is the third number.  
6. Find three numbers, of which the first and the second may be in such proportion as ½ and ⅓, and the second and third in such proportion as ¼ and ⅕. Answer: reduce ½ and ⅓ crosswise, and you shall have 3. over the ½ and 2. over the ⅓, then reduce ¼ and ⅕ in like manner, and you shall find 5. over the ¼ and 4. over the ⅕. Then say by the Rule of three, if 5. do give me 4. what shall two give me, which is the second proportional, multiply the second number 4. by the third number two, and thereof cometh eight, the which divide by the first number 5. and thereof cometh 1. ⅗ for the third proportional, and you shall find that 3.2.1. ⅗ are the three numbers proportional that I demand, or else 15.10. & 8. in whole numbers.  
1. What number is that, unto the which if you do add 13. the whole amounteth to 31. Answer: rebate 13. from 31. and there will remain 18. which is the number that you seek.  
2. What number is that, unto the which if you add ⅖ the addition will be ⅚. Answer:: abate ⅖ from ⅚, and there will remain 13/30, which is the number that you desire.  
3. What number is that, whereunto if you add 7. ⅔, the whole addition will be 12. ¼. Answer: abate 7. ⅔ from 12. ¼. & the remain will be 4.7/12 which is the number that you desire to know.  
4. What number is that, whereunto if you add the ¾ of itself, that is to say, of the number that you seek, the whole addition may be ⅚.  
Answer: Here followeth a general rule for all such like questions. first, of 3. which is the numerator of ¾ make still the numerator and likewise of 3. and 4. together, which is both the numerator, and the denominator of the ¾ make your denominator, so you shall find 3/7, then take the 3/7 of ⅚ which is 15/42 or 5/14, and subtract them from ⅚, and there will remain 10/21 which is the number that you seek.  
5. What number is that, unto the which if you add his own ⅔ that is to say ⅔ of itself, the whole addition shall be 20. Answer: do as in the last question: of the numerator of ⅔, that is to say, of 2. make still your numerator. And likewise of the numerator. 2. and the denominator 3. of the ⅔, make of them both, your denominator, and you shall find ⅖, them take the ⅖ of 20. which are 8. And abate them from 20. and there will remain 12. which is the number that you desire and so is to be done of all such like reasons.  
1. What number is that, from the which if you do abate 17. the rest may be 19 Answer: add 17. and 19 together and you shall find 36. which is the number that you seek.  
2. What number is that, from the which if you abate ⅗ the rest may be ⅛. Answer: add ⅗ and ⅛ together: and you shall find 29/40 which is the number that you demand.  
3. What number is that, from the which if you deduct 13. ½ the rest may be 5.5/7. Answer: add 13. ½ and 5.5/7 together, and thereof cometh 19.3/14, which is the number that you seek.  
4. What number is that, from the which if you substract his ⅖ the rest may be. 12. Answer: & a rule for such like reasons, that is to say from the denominator of ⅖ which is 5. abate 2. which is his numerator, and there resteth 3. for the denominator, and thus of ⅖ you have now made ⅔, then take the ⅔ of 12. which are 8: and add them unto 12. and thereof cometh 20. for the number which you desire.  
5. What number is that, from the which if you do abate his ¾, the rest may be 8/9. Answer: from the Denominator of ¾ which is 4. substract his Numerator 3. and there resteth 1. Thus of ¾ you have made 3/1. Then multiply 3/1 by 8/9, and thereof cometh 2 ⅔, the which add unto 8/9, and you shall have 3.5/9, which is the number that you seek.  
6. What number is that, from the which if you abate his ⅘, the rest may be 12. ⅔. Answer: Do as you did in the last question, and you shall find that the ⅘ will be 4/1. And therefore multiply 12. ⅔ by 4/1, and thereof cometh 50. ⅔, the which add unto 12. ⅔, and you shall find 93. ⅓, for the number that you demand. And thus of all such like questions.  
1. What number is that, which being multiplied by 13. the whole. Multiplication shall mount to 221. Answer: divide 221. by 13. and thereof cometh 17. which is the number that you seek.  
2. What number is that which being multipled by 15. the whole multiplication will amount to ¾. Answer: divide ¼ by 15/1 and thereof cometh 1/20 which is the number, that you seek.  
3. What number is that, which being multiplied by 21. the whole multiplication will be 16 ⅘. Answer: divide 16. ⅘ by 21/1, and you shall find ⅘ which is the number that you demand.  
4. What number is that, which being multiplied by ¾ the multiplication will amount to 18. Answer: divide 18/1 by ¾, and thereof cometh 24. which is the number that you desire to know.  
5. What number is that, which if it be multiplied by ⅔ the whole multiplication will be ¼. Answer: divide ¼ by ⅔ and the quotient will be ⅜ which is the number that you require to know.  
6. What number is that, which being multiplied by ⅝, the product of the multiplication will be 16. ⅔. Answer: divide 16.2/3 by ⅝ and thereof cometh 26. ⅔ which is the number that you seek.  

¶ Here ensueth other necessary Questions, which are wrought by Multiplication in broken numbers.  
I Demand how much the ⅝ of 20. shillings are worth, or what are the ⅝ of 20. shillings. Answer: you must multiply ⅝ by 20/1 and the product will be 100/8, therefore divide 100 by 8. and thereof cometh 12. ½, which is to say, 12. s. 6.d. and so much are the ⅝ of 20. shillings worth.  
2. I demand what the ¾ of ⅚ of a pound of money are worth, that is to say of 20. s. Auns. multiply ¾ by ⅚. And thereof cometh ⅝. Then take the ⅝ of 20. shil. as in the last question going before, and you shall find 12. s 6. pence, and so much are the ¾ of ⅚ of 20. s. worth.  
3. I demand what the ⅔ of 8. d. ½ are worth. Answer: multiply 8. ½ by ⅔ or else ⅔ by 8. ½, which is all one, and you shall find 34/6. Then divide 34. by 6. and your quotient will be five pence ⅔, and so much are the ⅔ of 8. pence ½ worth.  
4. What are the ¾ of 14. pence ⅗. Answer: multiply 14 ⅗ by ¾, and thereof cometh 219/20. Therefore divide 219. by 20. and your quotient will be ten pence 19/20, and so much are the ¾ of 14. ⅗.  
5. How many quarters or fourth parts are contained in 7. ⅔. Auns. multiply 7. ⅔ by 4/1 (because one whole containeth 4. quarters) and thereof cometh 30. ⅔, and so many quarters are in the 7. ⅔, that is to say 30. quarters, and ⅔ of a quarter.  
6. How many thirds are in ¾ and ½, that is to say in 3. quarters, and ½ of one quarter, which are ⅞ by the fift reduction. Answer: multiply ⅞ by 3/1 (for because that in 1 whole are contained 3. thirds) and thereof will come ⅔ and ⅝ of a third, and so many thirds are in ¾ and ½ or in ⅞, which is all one.   

¶ Questions done by division in broken number.  
1. What number is that, which being divided by 17. the quotient will be 13. Answer: multiply 17. by 13. And thereof cometh 221. which is the number that you seek.  
2. What number is that, which being divided by ¾ the quotient will be 21. Answer, multiply 21/1 by ¾ & thereof cometh 63/4. Then divide 63. by 4. and thereof cometh 15. ¾, which is the number that you seek.  
3. What number is that, which being divided by ⅛, the quotient willbe ⅔. Answer: multiply ⅔ by ⅓ & thereof cometh 2/24 which being abbreviated are 1/12 for the number which you require.  
4. What number is that, which being divided by ⅘ the quotient will be 16. ⅔? Answer: multiply 16. ⅔, by ⅘, and thereof cometh 200/15. Therefore divide 200. by 15. and thereof cometh 13. ⅓ which is the number that you desire to find.  
5. What number is that, which being divided by 13 ⅓, the quotient will be 20. Answer: multiply 20/1 by 13 ⅓, and thereof cometh 800/3, then divide 800. by 3. and thereof cometh 266. ⅔, for the number, which you seek.  
6. What number is that, which if it be divided, by 12. ½ the quotient will be ⅞ Answer: multiply by 12 ½ and thereof cometh 175/16, then divide 175 by 16. and thereof cometh 10.15/16, for the number which you desire.   

¶ Other necessary questions done by Division in broken number.  
I Demand what part 30. is of 70. Answer: divide 30. by 70. which you can not, for they are 30/70, but abbreviate them & they are 3/7. Thus 30. are the 3/7 of 70.  
2. I demand what part 10. is of 16 ⅔. Answer: divide 10/1 by 16. ⅔, and thereof cometh 30/50 which being abbreviated are ⅗. And thus 10. is found to be ⅗ of 16 ⅔.  
3. Moore, what part is 25. of ⅝. Auns. divide ⅝ by 25/1, and thereof cometh 5/200, which being abbreviated is, 1/40. And thus ⅝ is but the 1/40 of 25.  
4. Moore, ⅚ what part are they of ⅞. Answer: divide ⅚ by ⅞, and you shall find 40/42 which abbreviated are 20/21.  
5. Moore, ⅘ what part are they of 13. 1/ 3. Answer: divide ⅘ by 13. ⅓ and you shall find 12/200, which being all breviated are 3/80. And thus 4/1 are the 5/50 of 13. ⅓.  
6. More 12. ½ what part are they of 30. Answer, divide 12. ½ by 30/1, & you shall found 25/60 which being abbreviated are 5/12 and thus 12. ½, are the 5/12 of 30.  
7. Moore, 16 ⅔ what part are they of of 57 1/7. Answer, divide 16. ⅔ by 57 1/7 and thereof cometh 350/1200 which being abbreviated are 7/24, and thus 16. ⅔ are the 7/24 of 57.1/7.  
8. Moore, ¾ and ⅔ of ¼, or 3. quarters & ⅔ of one quarter, what part are they of 1. Answer, reduce 5/4 and the ⅔ of ¼ into one broken by the first reduction, and you shall find 11/12. And thus the ¾ and ⅔ of ¾ are the 11/12 of one whole.  
9 Moore, of what number are 9 the 2/3. Answer, divide 9 by ⅔, & thereof cometh 13. ½, which is the number whereof 9 are the ⅔.  
10. Moore of what number are ⅖ the ¾. Answer, divide ⅖ by ¾, and thereof cometh 8/15 which is the number whereof ⅔ are the ¾ of the same number.  
11. Moore, of what number are 5. ¾ the 3/7. Answer: divide 5. ¾ by 3/7, and you shall find 13.5/12 which is the number whereof 5. ¾ are the 3/7.  
12. Moore, 9 ⅔ what part are they of 33. ½. Answer: divide 9 ⅔ by 33. ½. And thereof cometh 58/201: and thus 9 ⅔ are the 58/201 of 33. ½ as appeareth.     

The third part treateth of certain brief rules, called rules of practice, with divers necessary questions profitable for Merchants.  

The first Chapter.  
SOme there be, which do call these rules of practice brief rules, for that by them many questions may be done with quicker expedition, than by the rule of three. There be others which call them the small multiplication, for because that the product, is always less in quantity, than the number which is to be multiplied. This practice cometh not in use, but only among small kinds of numbers, which have over them, other numbers that are greater. And this being well considered, is no other thing but to convert lesser and particular kinds of number, into greater, the which may be done by the means of division, in taking the half, the third, the fourth, the fift, or such other parts of the sum, which is to be multiplied, as the multiplier is part of his greater kind, and that which cometh thereof is worth as much (not in quantity, but in his own form) as if you did multiply simply the two sums, the one by the other: And for the better understanding of such conversions, you must have respect to one of these two considerations. The first is, when one would demand this question. At 6.d. the yard of Cotton, what are 18. yards worth by the price? It is manifest that they are worth 18. pieces of 6. pence the piece, or 18. half shillings, which must be turned into shillings, in taking the half of 18. s. & they make 9. s. Or otherwise you must consider, that at 1. s. the yard, the 18. yards are worth 18. s. wherefore at 6.d. they shall be but half so much, for 6.d. is but the ½ of 1. s. Therefore you must take the ½ of 18. and they make 9. s. which are worth as much as 108.d. that is to say, as 18. times 6. pence.  
2. First, if you will multiply any number after this manner by pence whereof the number of the same pennies, do not extend unto 12. and thereof to bring shillings into the product: you must know the certain parts of 12. which are these: that is to say, 6.4.3. 2. and 1. For 6. is the ½ of 12. and 4. is the ⅓ of 12: 3. is the ¼: 2. is the ⅙: and 1, is the 1/12. Then for 6.d. which is the half of 1. shilling, you must take the ½ of all the number which is to be multiplied. And that which cometh thereof, shall be shillings, if there do remain 1. it is 6. pence.  
For four pence you must take the ½ of all the number that is to be multiplied: and if any unities do remain, they shall be thirds of a shilling, every one being in value 4. pence.  
For 3. pence you must take the ¼ of all the sum: if any unities do remain, they shall be fourths of a shilling, every one being worth three pence.  
For 2. pence you must take the ⅙ of all the sum, and if any unities do remain, they shall be sixth parts of a shilling, being every one of them worth two pence.  
For d. take the 1/12 of the whole sum, if any unities remain, they are 12. parts of a shilling, each of them being in value 1.d. as by these examples following doth plainly appear.  

j  At 6. Pence the yard. What 59 yards. 29. shill. 6. Pence. 
ij.  At 4. Pence. What 82. 27. shill. 4. Pence. 
iij.  At 3. Pence. What 927. 24. shill. 3. Pence. 
iiij.  At 2. Pence. What 346. 57 shil. 8. Pence. 
v.  At 1. Pence. What 343. 28. shil. 7. Pence.  
Here you may see in the first example that 59 yards, at 6. pence the yard is worth .29. shil. 6.d. in taking the ½ of 59 And in the second example, the 82. yards at 4. pennies the yard, is worth, 27. s. 4.d. in taking the ⅓ of 82.  
Likewise, in the third example, 97. yards, at three pence the yard, bringeth 24. shil. 3. pence, in taking the ¼ of 97. Also in the fourth example, 346. yards, at 2. pence the yard, maketh 57 shillings eight pence in taking the ⅙ of 346. And finally in the fift example .343. yards, at 1.d. the yard, amount to 28. shil. 7.d. in taking the 1/12 of 343. And so is to be done of all such like, when the number of the pence, is any of the certain parts of 12.  
But if the number of the pence be not a certain part of 12. you must reduce them into some certain parts of 12. and after the foresaid manner you shall make two or three productes as need shall require, and add them together into one sum as 5.d. may be reduced into 4. & 1. or else into 3. & 2: wherefore if you will work by 4. & by 1: you must for 4.d. take first the ⅓. of the number, that is to be multiplied, and for 1.d. take the 2/12, or rather for 1.d. ye may take the ¼ of the product which did come of the 4.d. because that 1.d. is the ¼ of 4.d. But if you will work by 3, and 2, you shall take for 3.d. the ¼. of the number which is to be multiplied: and likewise for 2.d. the ⅙ of the same number, adding together both the productes. The total sum of those two numbers shall be the solution to the question. And in like manner is to be done of all other. As by these forms following may appear.  

j  At 5. d. the yard. What 49. yards? 16. shil. 4. d. 4. shil. 1. 20. shil. 5. d. 
ij.  At 7. d. What 5●? 18. shil. 0 13. shil. 6 31. shil. 6. d. 
iij.  At 8. d. What 40? 13. shil. 4 13. shil. 4 26. shil. 8. d. 
iiij.  At 9 d. What 73? 36. shil. 6 18. shil. 3 54. shil. 9 d. 
v.  At 10. d. What 32? 16. shil. 0 10. shil. 8 26. shil. 8. d. 
vj.  At 11. d. What 27? 9 shil. 0 9 shil. 0 6. shil. 9 24. shil. 9 d.  
Here in this same first example where it is demanded (at 5. pence the yard) how much are nine and forty yards worth? first for four pence, I take the ⅓ of 49. s. and thereof cometh 16. s. 4.d. them for 1.d. I take the ¼ of the same product, that is to say, of 16. s. 4. d. and that bringeth .4. shil. 1.d. these two sums added together, do make 20. s. 5.d. And so much are the 49. yards worth at 5.d. the yard.  
For 7.d. take the ⅓ and the ¼ of the whole sum which is to be multiplied, and add them together, that is to say, for 4.d. the ⅓ and for 3.d. the ¼: because 4.d. is the ⅓ of 12.d. and 3.d. is the ¼ as in the second example before doth appear: Where the question is thus, at 7.d. the yard what are 54. yards worth? first for 4.d. I take the ⅓ of 54: and they make 18. s. Likewise for 3.d. I take the ¼ of 54. and they are 13. s. 6.d. Then I add 18. s. and 13. s. 6. d. together, so both amount to 31. s. 6.d and so much are the 54. yards worth at 5.d. the yard.  
Otherwise for 7.d. take first the ½, of the whole sum for 6.d. Then for 1.d. take the ⅙ of the same product, and add them together, so shall you have the like sum as before.  
For eight pence you must first take ⅓ of the whole sum for 4. pence, and another ⅓ for other 4.d. and add than together as in the example doth evidently appear. Where the question is thus, at 8.d. the yard, what are 40 yards worth? first for 4.d. I take the ⅓ of 40. which is 13. s. 4.d. Again, I take another ⅓ for the other 4 pence which is also 13. shillings & 4. pence. These two sums being added together, do make 26. shillings 8. pence, and so much are the 40. yard's worth at 8. pence the yard as in the third example abovesaid doth appear.  
Otherways, for eight pence you may take first the ½ of the whole sum for 6.d. Then for 2.d. you shall take the ⅓ of the product, which did come of the said ½, and add them together, so shall you have likewise the solution to the question. As in the same third example of 40. yards, I take first the ½ of 40. for 6.d. and thereof cometh 20. shil. then for 2.d. I take ⅓ of the said product, that is to say of 20. s. which bringeth 6. s. 8.d. these two sums (20. s and 6. s. 8.d.) I add together & they make 26. s. 8.d. as before.  
For 9.d. you must take the ½ & the ¼ of the whole sum, and add them together: or else for 6.d. take first ½ of the whole sum, then for 3.d. take the ½ of the same product, because 3.d. is the half of 6.d. And 6.d. added with 3.d. bringeth 9.d. as by the fourth example, where it is demanded after this sort: at 9.d. the yard, what are 73. yards worth. First for 6.d. I take the ½ of 73. and thereof cometh 36. s. 6.d. then for 3.d. I take ½ of the same 36. shil. 6.d. which is 18. s. 3.d. these two sums do I add together, & they make 54. shil. 9.d. as in the said fourth example is evident.  
For 10.d. take first the ½, then the ⅓ of the whole sum, & add them together  
For 11.d. take first ½ for 4. pence, secondly, another ⅓ for other 4.d. and thirdly ¼ for 3.d. of all the whole sum: and add them together.  
Or else for 11.d. take first the ½ than the ⅓ of the whole sum, and finally the ¼ of the last product, adding them together.  
3. Likewise by the same reason, when you will multiply (by shillings) any number that is under xx. s. you shall have in the product pounds, if you know the certain parts of 20: which are these: 10.5.4.2. &. 1. For 10. is the ½ of 20. 5 is the ¼ part: 4 is the ⅕: 2. is the 1/10: and 1. is the 1/20.  
Then for 10. s. which is the ½ of a pound: you must take the ½ of the number, which is to be multiplied, and you shall have pounds in the product. If there do remain 1, it shallbe worth ten shillings.  
For 5. shillings you must take the ¼ of the number which is to be multiplied, & if there do remain any unities, they shall be four parts of a pound, every one being in value 5. s.  
For 4. s. you must take the ⅕ of the number which is to be multiplied. And if there do remain any unities, they shall be fift parts of a pound every one being worth four shillings.  
At 10. shillings the Piece. What 75. Pieces? 37. li. 10. shil. At 5. shil. What 89. 22. li. 5. shil. At 4. shil. What 93. 18. li. 12. shil.  
For 2. shillings you must take the 1/10 of the number that is to be multiplied. Wherefore, if you will take the 1/10 of any number: you must separate the last figure of the same number which is nearest your right hand, from all the other figures. For all the other figures which do remain toward your left hand, from the same figure, which is separated, shall be the said 1/10 of a pound: and that separated figure, toward your right hand shall be so many pieces of 2. shillings the piece: the which figure must be doubled, to make thereof shillings, as by these examples appeareth.  
At 2. shil. What 9/8. 9 li 16. shil. At 2. shil. What 40/3. 40. li. 6. shil.  
Hereupon dependeth another exact way for to multiply by shillings (if the number of shillings be even) which is thus: you shall take ½ the number of the same shillings, and convert them into pieces of 2. shillings. Then by the number of this half, you must first multiply the last figure toward your right hand, of the number which is to be multiplied: And if there be any tens in the same product, those must ye reserve in your mind: But if (with the same or else without the same) you do find any diget number, the same diget number shall you double, & put it in the place of shillings: Then must you proceed to the multiplication of the other figures, adding unto the product the tens which you before reserved: and thereof shall come pounds.  
Now, for your better understanding of this which hath been said and by the way of example, I will propone unto you this question.  
At 8. shillings the gross, what are 97. gross worth after the rate?  
first in this example I take half the number of Shillings, as before is taught, that is to say of eight shillings, which is four shillings: this 4. shil. I put apart, behind a crooked line right against 97. towards the left hand, as here you may see and as here after appeareth by divers examples.  
At 8. shil. the Gross. 4) What 9/1 38. li. 16. shil. At 6. shil. 3) What 9/9 29. li. 14. shil. At 12. shil. 9) What 34/5 207. li. 0 shil. At 14. shil. 7) What 21/0 127. li. 0. shil.  
Now in the first example, where it is demanded, at 8. s. the gross, what are 97 gross? First the ½ of 8. s. which is 4. s. being set apart behind the crooked line, as before is said: them I multiply the 97 by 4. saying first, 4. times 7. is 28. I double the diget number 8. and that maketh 16, the which 16, I do put under the line, in the place of shillings & I keep the tens in my mind, which here are 2. For 20. are two times ten: Then secondly, I multiply 9, by the said 4, and thereof cometh 36: whereunto I add the 2, tens, which before I did reserve, and they make 38. Therefore I put 38, under the line in the place of pounds, and the whole sum will be 38. li. 16. s. Thus much are the 97. gross worth, at eight shillings the gross: the like is to be done of all other. As of 12. shillings in multiplying by 6. Likewise of 6. shillings if you multiply by 3, also of 14. if you multiply by 7. And so of all even numbers after the same manner.  
For 1. Shilling you must take the ½ of the 1/10 part of any number that is to be multiplied. At 1. shil. What 35/0 17. li. 10. shil. And if any thing do remain, they are shil. Thus by this manner shil. are converted into pounds: for it is even like, as if you did divide them by 20. s. as by this example in the margin doth appear. Where it is demanded at 1. s. the yard, the piece, or any other thing, what are 350. worth?  
First I separate the last figure of 350. next to my right hand, which is the 0 with a line between it and the figure 5. Then I make a line under the 35/0, and I take the ½ of 35, after this manner: saying the ½ of 3. is 1. and 1. remaineth, which remain signifieth 10. in that second place. Then I put 1. under the line against 3, & I proceed to the rest, saying: the half of 15, is 7. (which 15. came of the 1. that remained, and of the 5. in the first place) I put 7. under the line right against 5, and they make 17. li. The 1, which did last remain, is 10. s. Therefore I put 10. s apart under the line, and the whole sum is 17. li. 10. s. so much are 350. worth at 1. s. the piece.  
But when the number of shillings is not some certain part of 20. shil. you must then convert the same number of shillings, into the certain parts of 20. and make two or three products, as need shall require, the which must be added together after this manner following.  
For 3. shillings you must first take for 2. shil. the 1/10 of the number that is to be multiplied, then for 1. shilling you must take the ½ of the product which did come of the same 1/10 part: and add those two sums together, as appeareth by this example following.  
At 3. s. the piece of any thing, what shall 684 pieces cost me after the rate. first, for 2 shillings I take the 1/10 of 684, which is 68: At 3. shil. What 68/4? 68 li. 0. sh 34. 4. 102. li. 12. sh in separating the last figure 4, which I must double, and they be 8. I set eight shillings apart from the place of pounds, and then I have 68 pounds 8. s. for the 1/10 part, that is to say, for the 2. s. secondly, for 1. shil. I take the ½ of the product, that is to say: of 68 li. 8. s. which is 34. li. 4. s. and I put the same under the 68 li. 8. shil. Then finally, I add those two sums together, that is to say, 68 li. 8. s. and 34. li. 4. shil, so they make 102. li. 12. s. and so much are the 684. pieces worth at 3. shillings the piece, as may appear in the margin.  
For 6. shil. take 3/10 of the number which is to be multiplied: that is to say, first 1/10, then double the product of the same 1/10 and add them together. Or otherwise for 4. s. take first the ⅕ of the number that is to be multiplied, them take the ½ of the product which is for two. s. and add them together.  
Or else take for 5. shil. the ¼ of the whole sum, then for 1. shil. the ⅕ of the product and add them together.  
Likewise for 7. shil. take first for 5. shil. the ¼ then for ●. shillings take the 1/10 of the number which is to be multiplied, and add them together.  
For eight shillings take the ⅖ at two sundry times, that is to say, first ⅕ for 4. shil. and then as much more for other 4. shil. and add them together.  
For 9 shil. take first the ¼ and likewise the ⅕ of the number that is to be multiplied, and add them together.  
For 11. shil. take first ½ for 10. s. Then for 1. shil. take the 1/10 of the product, & add them together.  
For 12. shil. take first the ½ for 10. shil then for 2. s. take the ⅕ part of the product, and add them together.  
For 13. shil. take the ¼ than the ⅕, & again another ⅕ of the number which is to be multiplied. And add the productes together, that is to say: first for 5. shil. take the ¼, then for 4. shil. take the ⅕. And again, another ⅕ for the other 4. shil. and assemble the three productes, the like is to be done in all others, when the price of the thing which is valued, is only of shillings. And as by these examples following doth plainly appear.  
At 6. shil. Was 67. 13. .8 6. .14 20. li. 2. shil. At 7. shil. What 347. 86. .15 34. .14 121. li. 9 shil. At 8. shil. What 540. 108. .0 108. .0 216. li. 0. shil. At 9 shil. What 230. 57 .10 46. .00 103. li. 10. shil. At 11. shil. What 159. 79. .10 7. .19 87. li 9 shil. At 12. shil. What 349. 174. .10 34. .18 209. li 8. shil. At 13. shil. What 267. 66. .15 53. .8 53. .8 173. li. 11. shil.  
4. Likewise in multiplying by pence you shall have (at the first instant) pounds in the product, in case you know the certain parts of the 1/10 of a pound or of 24. pence, which are these 12, 8, 6, 4, 3, and 2. For 12, is the ½ of 24: 8. is the ⅓: 6 is the ¼: 4 is the ⅙: 3 is the ⅛: and 2. the 1/12: but for 12.d. which is 1. shil. we have before made mention thereof.  
For 8.d. you must take the ⅓ of the 1/10 and the rest which are the pieces of 8.d. must be doubled to make of them pieces of 4.d. And of the same number being doubled, you must take the ⅓ which will be shillings, & if there do yet remain any thing, they are thirds of a shilling being in value 4. pence the piece.  
For 6.d. take the ¼ of the 1/10, and of that which remaineth you must take the ½ which shall be shillings, if there do yet remain 1, it shall be in value 6. pence.  
For 4.d. you must take the ⅙ of the 1/10 and of that which resteth, take the 1/● to make thereof shillings, if any thing do yet remain, they are thirds of a shilling, being in value 4.d. the piece.  
For 3.d. take the ⅛ of the 1/10, and of that which remaineth, take the ¼, to make of them shillings: if any thing do yet remain, they are fourths of a shilling, every one of them being worth 3.d.  
For 2.d. take the 1/12 of the 1/10: and of that which resteth take the ⅙ the which are shillings, if there do still remain any thing, they shall be sixth parts of a shilling, every one being in value 2.d  
For 1. d. it is not possible with ease, to bring of pence, pounds (into the product) upon the total sum: But first you must bring them into shillings by the order of the second rule of this chapter, and then afterward you shall convert them into pounds, if need so require. As by this example following may appear.  
At 8. d.   What 59/5.    19 li. 17. shil. 4. d. At 6. d.   What 67/8.    16. li. 19 shil. At 4. d.   What 93/4.    15. li. 11. shil. 4. d. At 3. d.   What 57/1.    7. li. 2. shil. 9 d. At 2. d   What 36/4.    3. li. 0. shil. 8. d. At 1. d.   What 66/5.    5●5. 00. shil. 4. d.  2. li. 16. shillings. 4. d.  
But if the number of pence, be not a certain part of 24. pence. Then must you bring them into the certain parts of 24. and make thereof divers productes, which must be added together, as shall hereafter appear.  
For 5. pence you shall first take for 3. pence, then for 2. pence, and add them together, according to the instruction of the last rule. Or else first take for 4. pence, and then for 1. d.  
For 7.d. first take for 4. d. then for 3. d. and add them together:  
For 9 d. first take for 6.d. then for 3. d. adding them together.  
For 10. d. first take for 6. d. then for 4. d. and add them together.  
For 11. d. take first for 8. d. then for 3. d. & add them together: as by these examples following doth appear.  
At 5. d   What 92/7.    11. .11. .9  7. .14. .6  19 li. 6. shil. 3. d. At 7. d.   What 51/2.    8. .10. .8  6. .8. .0  14. li. 18. shil. 8. d. At 9 d.   What 54/6.    13. .13. .0  6. .16. .6  20. li. 9 shil. 6. d. At 10. d.   What 27/3.    6. .16. .6  4. .11. .0  11. li. 7. shil. 6. d. At 11. d.   What 26/4.    8. .16. .0  3. .6. .0  12. li. 2. shil. 0. d.  
5. If you will multiply any number by shil. and pence, being both together you must take first for the shil. according to the instruction of the rule of this first chapter, then take for the pence after the order of the fourth rule before mentioned: but if there be any certain parts of 1. li. containing both shil. and pence, them for such parts you shall take the like part of the number that is to be multiplied, as the number is part of 1. li. the which certain parts are these, 6. s. 8. d: 3. s. 4. d: 2. s. 6. d: & 1. s. 8. d. For 6. s. 8. d. is the ⅓ of a li. 3. s. 4. d. is the ⅙ of a li. 2. s. 6. d. is the ⅛: & 1. s. 8. d. is the 1/12. then for 6. s. 8. d. you must take the ⅓ of the number that is to be multiplied: & if any thing do remain, they are thirds of a li. every one being worth 6. s. 8. d  
For 3. s. 4. d. you must take the ⅙ if any thing do remain, they are sixth parts of a li. every one being in value 3. s. 4. d.  
For 2. s. 6. d. you must take the ⅛: if any thing be remaining they are eight parts of a li. each one being worth 2. s. 6.  
For 1. shil. 8. d. you shall take the 1/12 if there do any thing remain, they are twelfth parts of a pound every one being valued at 1. shil. 8.d.  
At 6. shil. 8. d What 647. 215. li. 13. shil. 4. d. At 3. shil. 4. d What 220. 36. li. 13. shil. 4. d At 2. shil. 6. d What 47. 5. li. 17. shil. 6. d At 1. shil. 8. d What 400. 33. li. 6. shil. 8. d  
6. Hear shall you accustom yourself, to multiply by all sorts of sums, being compound of shillings, and pence, which may come to practise. As thus, for 1. s. 1. d. for 1. s. 2. d. 1. s. 3. d. for 1. s. 4d. Likewise for 2. s. 1.d. 2. s. 2.d. 2. s. 3. d. 2. s. 4. d. And so of all other: considering moreover, many subtle abbreviations, which happen oftentimes that are easy to be conceived. As thus at 11. s. 3. d. after that I have taken first the ½ for 10. s. Then for 1. s. 3. d. I take the ⅛ of the product, because 1. s. 3. d. is the ⅛ of 10. s. in taking the said ⅛ of the product. And by this means, when ye have taken one product, ye may oftentimes upon the same, take another more briefly than upon the sum that is to be multiplied, which thing you must foresee.  
At 11. shil. 3. d. What 53. 26. .10. .0 3. .6. .3 29. li. 16. shil. 3. d. At 6. shil. 3. d. What 58. 14. 10.. 3. 12. 6. 18. li. 2. 6. d. At 12. 8. d. What 64. 32. 0.. 6. 8.. 2. 2. 8. 40. li. 10. shil. 8. d.  
7. But if you will multiply, by pounds, shillings and pence being altogether. first you must wholly multiply by pounds. Then take for the shillings and pence, as in the fift rule of this chapter is plainly declared. And as by these examples following may apere.  
At 3. li. 6. shil. 8. d. What 49.   147. .0 .  16. .6 .8  163. li. 6. shil. 8. d. At 5. li. 18. shil. 4. d. What 543.    2715. .0. .  271. .10. .  135. .15. .  90. .10. .0  3212. li. 15. shil. 0. d. At 2. li. 7. shil. 4. d. What 927.    1854. .0. .  185. .8. .  154. .10. .  2193. li 18. shil. 0. d.  
8. So these rules do serve both to buy and sell, at such a price the elle, the yard, the piece, the pound weight, or any other thing: how much such a thing Likewise they are very necessary to convert all pieces of gold and silver into pounds: for I may as well say, at 4. shil. 8. d. the French crown, what are 135. crowns worth?  
9 When any one of the sums (which is to be multiplied) is compound of many denominations: & the other is of one figure alone: than shall ye multiply all the Denominations of the other sum, by the same one figure beginning first with that sum which is lest in value towards your right hand, and bring the product of those pence into shillings, and the product of the shillings into pounds, as by this example doth appear At 3. li. 9 shil. 8. d. What 7. 24. li. 7. shil. 8. d.  
10. But (if in any of the numbers which are to be multiplied) there be with it a broken number, you must (according to his denominator) take one or many parts of the other number, as need doth require: and set the number which cometh thereof, under the productes, adding the same together. As thus: At 5. li. 7. s. 8. d. the gross, what shall. 34. gross ½ cost? First At 5. li. 7. sh. 8. d What 34 1/7. 170. .0 11. .6. .8. 1. .14. .0 2. .13. .10 185. li. 14. shil. 6. d you shall multiply 5. li. 7. sh. 8. d. by 34. gross, saying 5. times 34. do make 170. li. then for 6. sh 8. d. take the ⅓ of 34 which is 11. li. 6. shil 8. d. thirdly, for 1 sh. take 34. shillings, which is 1. li. 14. shillings 0.  
Lastly, for the ½ gross, you must take ½ of the 5. li. 7. s. 8. d. which is 2. li. 13. s. 10 And then add them all together, so you shall find that the 34. gross ½ at 5. pound 7. shillings 8. pence is worth 185. pound, 14. shillings 6. pence, as appeareth in the margin.  
And as in this last example, you did take the half of the money, (which one gross was worth) for the ½ gross Because that 1. gross being worth 5. pound 7. shillings 8. pennies, the ½ gross must be worth half so much. So likewise, if you have ⅓ of a gross, or of any other thing, you must take the ⅓ of the price, that one gross is worth. Semblably, for the ¼ of any thing you shall take the ¼ of the price, also if you have ⅔, take the ⅔ of the price that one is worth, and of all other fractions, as by these examples following doth appear.  
At 4. li. 6. shil. 8. d. What 46. ½. 184. .0. .0 15. .6. .8 2. .3. .4 201. li. 10. 0. d.  
At 8. li. 0. shil. 9 d. What 54. ⅓ 432. .0. .0 1. .7. .0 0. .13. .6 2. .13. .7 43. li. 14. shil. 1. d. At 3. li. 16. shil. 8. d. What 17. ¾ 51. .0. .0 8. .10. .0 5. .13. .4 . .19. .2 68 li.. 00. shil. 10. d.  
11. If you will make the proof of these rules aforesaid, you must first abate the sum of money (which the fraction of the multiplication doth import) from the total sum. And divide the rest of the pounds of the said total sum, by the whole multiplicand, the fraction only accepted. And if any thing do remain after the division is made, that remain shall be multiplied by 20. and unto the product of that multiplication, you shall add the shillings which remained of the rest of the total sum Again, if any thing do remain after the same division, you must multiply the same by 12, & unto the product add the pence of the total sum that remained, if any be left. And thus if ye have truly wrought, you shall found again the higher sum of your question that is to say, the price that one gross or any other thing is worth, whereof you demand.  
Or otherwise reduce the remain of the total sum (the value of the money that the fraction is worth, being first deducted) all into pence, in multiplying the pounds by 20, and the shillings by 12. adding thereunto, the shillings and pennies, which are joined with the remain of the said total sum, if any such be, then divide those pence by the foresaid number that is to be multiplied, the fraction of the same number being also abated. So shall you find the price that one piece, one Gross, or any other thing is valued at. As in the first example going before, where the total sum is 201. pound 10. shillings, from the which I do first abate the price of the half gross, which is 2. li. 3. s. 4. d, the rest is 199. li. 6. s. 8. d. which being reduced into pens bringeth 47840. d. I divide the same by 46. and thereof cometh 1040. pence. Then I divide that 1040. pence by 12. and they bring 86. shillings 8. pence, that is to say, 4. li. 6. shillings eight pence, which is the price that one gross, or any other thing did cost, as in that first example doth appear.  
12. The like is to be done of any manner of thing that is sold by the hundred or by the Kyntall.  
As thus: at 12. pound 7. shillings. 6. d the 100 pound weight: what shall 374. pound weight cost? You shall first multiply twelve pound, seven shillings, six pence, by three: that is to say, by three hundredth. Then for 50. li. weight, you At. 12. li. 7. sh. 6. d What 3/74. 37. .2. .6 6. .3. .9 2. .9. .6 0. .9. .10. ⅘ 46. li. 5. sh. 7. d. ⅘ shall take the ½ of 12. li. 7. s. 6. d. because 50. li. is the ½ of 100 li. Likewise for 20. pound weight, which is the ⅕ of 100 li. take the ⅕ of 12. li. 7. shil. 6. d. lastly for 4. li. weight take the ⅕ of the last product. This done, you must add all these productes into one sum, which will make the sum of 64. li. 5. s. 7. d. ⅘, as by this example above written doth appear.  
The proof is made by reducing the total sum into pence. And to divide the product by the number that is to be multiplied, that to to say by 374. likewise divide the quotient produced of that first division by 12. so shall you find again the higher sum 12. li. 7. shil. 6. d. which is the price of 100 li. weight, as before.  
13. Also the like may be done of our usual weight here in England (which is 112. li. for every hundred pound weight) in case you know the certain parts of a hundred, that is to say, of 112. li. weight, which are these 56. li. 28. li. 14. li. 7. li, For 56. li. is the ½ of 112. 28. li: is the ¼ of 112. li: 14. li. is the ⅛, and 7. li. is the 1/16.  
Therefore, for 56. li. take the ½ of the sum of money, that the 112. pound weight is worth.  
For 28. li. take the ¼ of the sum of money that the 112. li. is worth.  
For 14. li. take the ⅛ of the sum that the C. is worth.  
For 7. li. take the 1/16 of the sum of money that the C. is worth.  
As thus: at 3. li. 6. s. 8. d. the hundredth pounds weight, that is to say, the 112. li. What shall 24. C. 3. quar. 21. li. cost after the rate?  
first, you shall multiply 24. hundredth by 3. which is the 3. li. & thereof cometh 72. li. then for 6. s. 8. d. which is the ⅓ of 20. s. you shall take the ⅓ of 24 which is 8. li. for At 3. li. 6. sh. 8 d. What 26. .3. 21. li. 72. .0. .0 8. .0. .0 1. .13. .4 . .16. .8 . .8. .4 . .4. .2 83. li. 2. sh. 6. d 24. nobles maketh 8. li. afterward, for the 3. quarters of the C. you shall first for the 56. li. take the ½ of 3. li. 6. s. 8. d. because 56. li is the ½ of the C. & thereof cometh 1. li. 13. shil. 4. d. then for 28. li. (which is the quar. of a C.) you shall take the ¼ of 3. li. 6. s. 8. d. or else the ½ of the product, which came of 56. li. which is 16. s. 8. d. likewise for 14. li. take the ⅛ of 3. li. 6. s. 8. d. which is 8. s. 4. d. or else the ½ of the product of 28. li. which is all one: lastly for 7. li take the 1/16 of 3. li. 6. s. 8. d. or else the ½ of the product, that came of 14 li. and thereof cometh 4. s. 2. d. Then add all these products together: & the total sum will be 83. li. 2. s. 6. d. so much are the 24. c. 3. quar. 21. li. weight worth after 3. li. 6. s. 8. d. the C. as appeareth in the margin.  
The proof hereof is made, like to the other proofs aforesaid, saving that where in those proofs, you abate the price of the money, that the fraction was worth, from the total sum: here in this example (and in such other like) you must abate the price of money, that the odd weight amounteth unto (over and above the just hundreds) from the said total sum, the rest thereof shall you convert into pence, dividing the product of the multiplication by the just number of the hundreds, so shall you find the pence the one hundredth is worth, which you shall bring into pounds by the order of division, & so all other.   

¶ The second Chapter treateth of the rule of three compound, which are four in number.  
THere belongeth to the first & second parts of the rule of three compound always five numbers: whereof (in the first part of the rule of three compound the second number and the fift, are always of one semblance, and like denomination: whose rule is thus, multiply the first number by the second, & that shallbe your divisor: then multiply the other three numbers the one by the other to be your dividend. Example, of this first part: if 100 crowns in 12. months, do gain 16. li. what will 60. crowns gain in 8. months? Answer, first multiply 100 crowns by 12. months, & thereof cometh 1200. for your divisor: then multiply 15. li. by 60. crowns, & by 8. months, & you shall have 7200. divide 7200. by 1200. & thereof cometh 6. li. so many li. will 6. crowns gain in 8. months: this question may be done by the double rule of 3. that is to say by the rule of 3. at 2 times: yet this rule of 3 compound is more brief Crowns. months. pounds. crowns. months. 100 .12. .15. .60. .8.  1     72 72 00     12 12 00 (6. li.    
2. In the second part of the rule of three compound, the 3. number is like unto the fift, whereof the rule is thus: multiply the 3. number by the 4, the product shallbe your divisor: them multiply the first number by the second, & the product thereof by the fift, the which number shall be your dividend, or number that is to be divided: as by example.  
When 60. crowns in 8. months do gain 6. li. in how many months will 100 crowns gain. 15. li. Answer: Multiply the third number 6. by the fourth number 100: & thereof cometh 600 then multiply the first number 60. by the second number 8. & by the fift number 15. thereof will come 7200. then divide 7200. by 600. & the quotient willbe 12: in so many months will 100 crowns gain 15. li. This question may likewise be done by the double rule of 3.  
crowns months. pounds. crowns. pounds. 60. .8.  .6. .100. .15.  1      72 00 months.    66 00 (12    
3. In the third part of the rule of 3. compound, there may be 5. numbers or more: & in this rule the first number & the last are always dissemblaunt the one to tother: & the question is from the last number unto the first, whereof the rule is thus: multiply that number which you would know by those numbers which do give the value, & divide the product of the same, by the multiplication of the numbers which are already valued, as by example. If 4. deniers Parisis, be worth 5. deniers Tournois, & 10. deniers tournois, be worth 12. deniers of Savoy, I demand how many deniers Parisis are 8. deniers of Savoy worth? Answer: Multiply 8. deniers of Savoy (which is the number that you would know) by 4. deniers parisis, & by 10 deniers tournois which are the number that give the value, & they make 320: then multiply 5. deniers tournois, by 12 deniers of savoy (which are the numbers already valued) & they make 60: lastly divide 320. by 60 and you shall find 5. deniers ⅓ parisis, so much are the deniers of Savoy worth.  
Parisis. tournois. tournois. savoy. savoy. 4. d. .5 d. .10 d. .12 d. .8 d.  32 32 0 par.    6 0 (5. d. ⅓.    
4. In the fourth part of the rule of three compound: the first number and the last are always semblant and of one denomination, and the question of this rule, is always from the last number to the last saving one. Whereof there is a rule which is thus. You must multiply that number which you would know, by the numbers that are already valued, and divide the product of the same, by the multiplication which cometh of the numbers that give the value, as by example.  
If 4. deniers Parisis, be worth 5. Deniers Tournois, and 10. Deniers Tournois, be worth 12. Deniers of Savoy, I demand how many Deniers of Savoy, are 15. Deniers Parisis' worth. Answer: Multiply 15. Deniers Parisis that you would know, by 5. Deniers Tournois, & by 12. Deniers of Savoy, which are the numbers already valued, and they make 900. Divide the same by 4. times 10. which are the numbers that do give the value, and you shall find 22. Deniers ½ of Savoy, so much are the 15. Deniers Parisis' worth.  
Parisis. tournois. tournois. Savoy. Parisis. 4. d. .5 d. .10 d. .12 d. .15 d.  12     90 0 Savoy.   44 0 (22. d. ½    

The third Chapter treateth of questions of the trade of Merchandise.  
IF 31. Deuonsh. dosens' do cost me 100 li. 15. shil. What shall 4. dosens cost? Answer: first bring the 100 li. 15. shill. all into shillings, in multiplying the 100 li. by 20. adding to the product the 15. shill. and thereof cometh 2015. shill. then multiply 2015. by the third number 4. and divide the product by 31. and the quotient willbe 260. s. The which divide again by 20. and thereof cometh 13. li.  
Dosens.   Dosens. 31. 100 li. 15. sh. 4. d.  20    2015    4    8060.    1    28    8060 (260.   3111    33    
If four Dosens be worth 13. pound. What are 31. dozens worth by the price? Answer: Multiply 31. by 13. and thereof cometh 403. The which you shall divide by 4. and thereof cometh and thereof cometh 100 li. ¾, which ¾ are 15. s. and so much are 31. dozens worth as before.  
Dosens. li. Dosens. 4. .13. .31.   13   393   1   403.  4 03   444 (100 li. ¼   
If 49. else be worth 2. li. 4. s. 11. d. what are 18. else worth by the price? First you must bring 2. li. 4. s. 11. d. all into pence, in multiplying 2. li. by 20. maketh 40. add thereto 4. shil. they make 44. s. the which multiply by 12. d & they make 528. d. whereunto add 11. d all is 539. d. the which 539. d. must be your second number in the rule of 3. then multiply 539. by 18. & thereof cometh 9702. divide the same by 49. & you shall have in your quotient 198. d. the which divide by 12. & you shall find 16. s. 6. d. so much are the 18. else worth.  
:  . 49. .2. li. 4. sh. 11. d. .18  20 539  44 162  12 54  99 90  44   539 9702 13 1 427 76 386 198 (16. sh. 6. d 9702 (198. 122 4399 1 44   
If 18. else be worth 16. s. 6.d. what are 49. else worth by that price? Auns. bring 16. s. 6.d. into pence, in multiplying 16. by 12. and thereof cometh 198.d. with the 6. d. added to it, then multiply 198 by 49. the product will be 9702. The which divide by 18. else and thereof cometh 539.d. Then divide 539.d. by 12. and the product thereof by 20. So shall you have 2. li. 4. sh. 11.d. so much are the 49. else worth.  
.  . 18. .16. sh. 6. d. .49  12 198  32 392  166 441  198 49   9702 17 1 446 151 9702 (539. ●39 (44. shill. 1888 122 ●1 1  
If a yard of Velvet cost 19. s. what shall ¾ of a yard cost? Answer: set down your numbers thus. If 1/1 19/1 ¾. Then multiply 1. times 19 by 3. and thereof cometh 57 for your dividend, or number to be divided. The which 57 you shall divide by 1. times 1, four times, which are 4, and your quotient will be 14. s. ¼, which ¼ is worth 3.d. so much are the ¾ of a yard worth after 19 shil. the yard, as by practice followeth.  
1/1 19/1 ¾ 1 1    57 (14. sh. ¼    44  
Or otherwise by the rules of practice: first for 2/4 of a yard which is ½ of a yard, you must take the ½ of 19. s. which is 9. s. 6.d. then for ¼, take the ½ of the product, that is to say, of 9. s. 6.d. and thereof cometh 4. s. 9.d. add these numbers together, & 19 shil. 9 sh. 6.d. 4. .9. 14.. 3.d. you shall have 14. s. 3.d. as above is said, and as appeareth here in the margin.  
If ¾ of a yard of Velvet do cost 14. shil. 3.d. What shall 1. yard cost set your numbers down thus: if ¾ 14 ¼ 1/1. Reduce 14. ¼ into a fraction, and they will be 57/4 them multiply 57 by 1.4. times, & thereof cometh 228. for your dividend. Likewise multiply 1. times 4.3. times, & thereof cometh 12. for your divisor: then divide 228. by 12. & your quotient will be 19 shil. so much is the yard of velvet worth.  
 57  1 ¾ 14 ¼ 1/1 10    228 (19 shil.    122    1  
Or otherwise by the rule of practice: you shall take the ½ part of 14. sh. 3.d. and add it with the same 14. sh. 3.d. and you shall have 19 shill. as before.  
14. shil. 3. d. 4. .9. d. 19 shil. .0. d.  
If one ell of Holland cloth be worth 5. s. what are ⅔ worth after the rate? Answer, say thus if 1/1 5/1 ⅔. Then multiply 2. times 5. one time, and thereof cometh 10. for your dividend: likewise multiply three times 1. one time, they make 3. for your divisor, then divide 10. by 3. & thereof cometh 30. s ½ which ⅓ is worth 4. pennies, & so much are the ⅔ of an ell worth.  
1/1 5/1 ⅔ 1     10 (3. shil. ⅓    3   
Or otherwise, by the rule of practice: take first the ⅓ of 5. s. for the ⅓ of an ell, which is 1. s. 8.d. Likewise, for the other ⅓ of an ell take again another ⅓ of 5. s. which is also 1. sh. 8.d. and add them together, and so shall you have 3. s. 4.d. as before.  
5. shill.  1. .8 1. .8 3. shill. 4 d.  
If ⅔ of an ell of Holland cloth do cost me 3. s. 4.d. what shall the el cost? Answer: set down your sum thus, if ⅔ 3 ⅓ 1/1. First reduce 3 ⅓ all into thirds, and it will be. 10/3. Then multiply 1. times 10.3. times, and thereof cometh 30. for your dividend. Likewise multiply 1. times 3.2 times, your quotient will be 6. then divide 30. by 6. & you shall have 5. s. so much is the ell of Holland cloth worth.  
 10  30 2 3 ⅓ 1/1 6 (5. sh.  
Or otherwise by practice, take the ½ of 3. s. 4.d. which is 1. s. 8.d. & add it to the same 3. s. 4.d. and thereof will come 5. s. as before. For the ⅓ of 5. s. is as much as the ½ of 3. sh. 4. 1. 8. 5. sh. 0.d. 3. s. 4. d. which was the price that the ⅔ of an elle did cost, as appeareth.  
If one ell cost me 17. s. what shall. 15. else ⅛ part cost? which ⅛ is half a quarter of an elle. Answer: say, of 1/1 17/1 15. ⅛.. First reduce 15 ⅛ into eight parts, and they make 121/8 then multiply 121. by 17.1 time, and thereof cometh 2057. for your dividend. Likewise multiply 8. times 1. 1. time, and your quotient will be 8. for your divisor, then divide 2057. by 8. and you shall find 257. sh. ⅛, which is 12. li. 17. shil. 1.d. ½ and so much are the 15 else ⅛ worth, as by practice doth appear.  
  121 1/1 17/1 15 1/8  
Or otherwise, for 10 sh. take the ½ of 15 which is 7 li. 10 sh. then for 5 sh. take the ½ of 7. li. 10. s. which is 3. li. 15. s. thirdly for 2. s. take the ⅕ of 7. li. 10. s. because the ⅕ of 10. sh. is 2. sh. fourthly, for the ⅛ of the ell, you shall take the ⅛ of 17. s. which is 2. s. 1.d. ½. lastly, add all these sums together, and then shall you found 12. li. 17. s. 1.d. ½ as before, and as appeareth more plainly in the margin. 15. . ⅛.  17. s.   7. .10. . 3. .15. . 1. .10. .  2. 1 ½. 12. li. 17. s. 1.d. ½  
If 25. else be worth 2. li. 3. s. 4.d. what are. 18 else ¾ worth by that price? Answer: first put 3. s. 4.d. into that part of a li. and you shall have ⅙ then say, if 25/1 give me 2. li. ⅙ what shall 18 ¾ give: put the whole number into his broken, and then multiply 1. times 13. by 75. the product will be 975. the which you shall divide by 25. times 6.4. times, which maketh 600. Then divide 975. by 600. and your quotient will be 1. li. and 375. remaineth, the which 375. you shall multiply by 20. thereof cometh 7500. divide the same by 600. your quotient will be 12. s. and 300. remaineth the which abbreviated bringeth ½ which is 6.d: thus the 18 else ¾ are worth 1. li. 12. s. 6.d. as by practice appeareth.  
 13 75 25/1 2 ⅙ 18 ¾.  
Or otherwise by the rules of practice: for because that 12. else ½ is the ½ of 25. else, therefore take the ½ of 2. li. 3. s. 4.d. which is 1. li. 1. s. 8.d. then for 6. else ¼ take ¼ of 2. li. 3. s. 4.d. or else the ½ of the last product (that is to say of 1. li. 1. s. 8.d.) which is all one, & add them together, so shall you have 1. li. 12. s. 6.d. as before.  
.2. .3. .4. .1. .1. .8.  .10. .10. 1. li. 12. s. 6.d.  
If 15. yards be worth 32. s. what are half a yard or half a quarter or else ⅝ of a yard worth. Answer: say, if 15/1 give 32/1 what will ⅝ give? Multiply 1 times 32. by 5. and divide the product by 15. times 1. shil. and 4. remaineth, which is ⅓ of a shil. that is to say 4.d. and so much are the 5/2 of a yard worth.  
15/8 32/1 ⅝  
Or otherwise, see what the yard is worth after the manner aforesaid in the other examples, & you shall found that the yard is worth 2. s. 1.d ⅗ of the which number take first the ½ for 4/8 which is 1. s. 0 d. ⅘, of the which number, take the ¼ for the other ⅛ which is 3 d. ⅕, add these two numbers together, and you shall find the ⅝ to be worth 1. s. 4.d. as before is said 2. sh. 1.d. ⅗. 1. 0. ⅘. 1. sh. 4.d. 0.  
If 13. ells ⅚, be worth 27. s. what are 10. else ⅔ worth by that price? Answer: say if 13. ⅚ give 27/1, what shall 10. ⅔ give: put the whole numbers into their broken, & you shall find 83/6, 27/1, & 32/3. Then multiply 6. times 27. by 32. & thereof cometh 5184. the which number you shall divide by 83. times 1. three times, and you shall find 20. sh. 68/83 which is worth 9.d. 69/83 part of a penny.  
83  32 13 ⅚ 27/1 10 ⅔  
If two yards ½ be worth 4. s. 8.d. what are 8. yard's ¼ worth? Answer: put the 8.d. into the part of a shilling, which willbe ⅔ then reduce the whole numbers into their broken, and they will stand thus. 5/2, 14/3, 33/4, then multiply two times 14. by 33. and divide the product by 5. times 3.4. times, & you shall find 15. s. 4.d. ⅘, so much are the eight yards ¼ worth.  
5 14 33 2 ½ 4 ⅖ 8 ¼  
If one kersey be worth 2. li. 6. s. 8.d. how many kerseys shall I buy for 36. li. 3. s. 4.d. after that rate? Answer: put 6s. 8ds. into the part of a li. & you shall have 2. li. ⅓ for the first number in the rule of 3. and 1. ell for the second number: then put 3. s. 4.d. into the part of a li. and you shall find 36. li. ⅙. for the third number, then will your 3. numbers in the rule of 3. stand thus. 2. ⅓ 1/1 36 ⅙.. Therefore reduce the whole numbers into their broken, & you shall have 7/3 1/1 217/6. Then multiply 3 times 1. by .217. & thereof will come 651. for your dividend. Likewise, multiply 7 times 1. by 6. & the product thereof will be 42. Then divide 651. by 42. and you shall find 15. ½. So many kerseys of 2. li. 6. s. 8.d. the piece, shall you have for 36. li. 3. s. 4. d.  
7 217 2 1/7 36. ⅚   

¶ The 4. Chapter treateth of losses and gains, in the trade of Merchandise.  
If 13. yard's ⅓ be worth 22. li. 10. s. how shall I sell the yard to gain ⅓, or to make of 3.4? which is all one? Auns. say by the rule of 3. if 3. be come of 4. or if 3. yield 4. what will 22. ½ yield: multiply & divide and you shall find 30. li. Then say gain by the rule of 3. if 13. yards ⅓ do give 30. li. aswell of principal as of gain: what will 1. yard be worth by the price? Multiply and divide, and you shall found 2. li. 5. s. and for that price must the yard be sold to gain the ⅓, or to make of 3.4.  
  45 40.  5/1 4/1 22 ½ 13 ⅓ 30/1 1/1  
Or otherwise, take the ⅓ part of 22. li. 10. s. which is 7. li. 10. s. that shall you add with 22. li. 10. s. and you shall have 30. li. as before. 22. .10. s. 7. .10. 30. .00. Then divide 30. by 13. ⅓, & you shall find 2. li. 5. s. as above is said.  
If one yard be worth .27. sh. 6.d. for how much shall 16. yard's ⅔ be sold to gain 2. s. upon the pound of money, that is to say: upon 20. s. Answer, add 2. unto 20. and you shall have 22, then say: if 20. s. of principal, do give 22. s. as well of principal as gain: how much will 27. s. 6.d. principal yield. Multiply and divide & you shall find 30. s. ¼: then say again by the rule of 3. if one yard do give me 30. s.¼ (which is aswell the principal as the gain) what shall .16. yards ⅔ give me? Multiply and divide, and you shall find 25. li. 4. s. 2.d. For the same price shall the 16. yards ⅔ be sold to gain after the rate of 2. s. upon the pound of money, or in 20. s. which is all one.  
  55.  121 50 20/1 22/1 27 ½. 1/1 30 2/4 16 ⅔  
If 10. yards ⅔ be worth 25. li. 10. s. For how much shall 2 yards ¼ be sold to gain after 10. li. upon the 100 li. of money? Answer: say if 100 of principal yield 110. as well principal as gain, how much will 25. 10. s. yield me? Multiply & divide, and you shall find 28. li. 1. s. Then say if 10. yards ⅔ do yield me 28. li. 1. sh. aswell of principal as of gain, how much shall two yards ¼ yield me? multiply & divide & you shall find 5. li. 18. s. 4.d. 1/12, for so much shall the 2. yards ¼ be sold to gain after 10. li. upon the 100 li. of money.  
  51    100/1 110/1 25 ½ 10 ⅔ 28 1/20 2 ¼  
And although that in these questions of gain and loss, sometimes the first number is not like unto the third number, that is to say, of the same denomination: as one would say: if 20. s. gain 2. shil. what shall 50. li. gain? or 25. li. etc. Or if 20. li. do gain 2. li. What shall 25. s. gain me, or what shall 27. sh. ½ gain? Yet nevertheless, the rule is not therefore false. For if 20. s. do gain 2. s: 20. li. shall gain 2. li. & 20.d. shall gain 2.d. likewise 20. crowns shall gain 2. crowns, and so of all other: therefore it is to be understand, that the first number in these reasons is presupposed to be semblable to the third.  
When one Merchant selleth wares to another, and he giveth to the bier 2. upon 15: how much shall the bier gain upon the 100 after the rate? Answer: say if 15. give 17. what shall 100 give? Multiply and divide, and you shall find 113 ⅓, so the bier getteth after the rate of 13 ⅓ upon the 100  
15 17 100  
If one northern dozen cost me 3. li. 5. s. & I sell the same again for 3. li. 12. s. 6.d. how much do I gain upon the pound of money after that rate? Auns. say if 3. li. ¼ do give 3. li. ⅝ what shall 20/8 give, put the whole number into their broken & you shall have 13/4, 29/8, 20/1, then multiply 4. times 29. by 20. & thereof cometh 2320. for your number that is to be divided, likewise multiply 13. times 8. 1 time, & thereof cometh 104. Then divide 2320. by 104. & you shall find 22. s. 4/13. So I shall get 2. s. 4/13 upon 20. s. or upon the li. of money.  
13 29  3 ¼ 3 ⅝ 20/1  
If a yard of cloth cost me 7. s. 8.d. & afterward I deliver out 13. yards ¼, for 4. li. 13. s. 4.d. I would know whether I do win or loose, & how much upon the 100 li. of money?  
Answer: see first at 7. s. 8.d. the yard, what the 13. yards ¼ shall cost, and you shall find 5. li. 1. s. 7.d. And I sold them but for 4. li. 13. s. 4.d. so that I do loose upon the 13. yards ¼ the sum of 8. s. 3.d. Then for to know how much is lost upon the 100: say by the rule of three, if 5. li. 1. s. 7.d. do loose 8. s. 3.d. What will 100 loose? first, put 1. shil. 7.d. into the part of a li. and it will be 19/240. Likewise put 8. s. 3.d. into the part of a li. and it is 33/80. Then will your numbers stand thus 5 19/240, 33/80, 100/1, put the whole into his broken, and then multiply and divide, so you shall find 8. li. 1184/9752 which is worth .2. sh. 5. d. 169/1219 and so much is lost upon the 100 li. of money.  
 1219.   5 19/240. 38/80 100/1.  
Moore, if 12. yards ½ of scarlet be sold for 30. li. 15. s. upon the which is gained after the rate of 11 1/9 upon the 100 I demand what the yard did cost at the first. Answer: from 30. li. 15. s. subtract his 1/10 part which is 3. li. 1. s.6.d. and there resteth 27. li. 13. s.6.d the which number multiplied by 2. bringeth 55. li. 7. s. of the which is 11. li one shilling and four pence.  
Then take again the ⅕ of the said 11 pound 1. shil. 4. pence, which is 2. pound 4. shillings three pence. 9/25. And so much did the Elle cost at the first penny.  
30. li. 15. sh.   3. 1. sh. 6.d.  27. 13. 6.  2.    55. 7. 0.  11. 1. 4. ⅘. 2. 4. 3. 9/25.  
Moore, if 15. yards ¾ of scarlet do cost me 32. li. 13. s. 4.d. And I cell the yard again for 2. li. whether do I win or loose, and how much upon the pound of money.  
Answer: Look what the 15. yards ¾ are worth at 2. li. the yard, and you shall find that they are worth 31. li. 10.s. But they did cost 32. li. 13. s. 4.d. so that there is lost upon the whole 1. li. 3.s.4.d. Then, to know how much is lost upon the li. say by the rule of three, if 32. li. ⅔ do loose 1. li. ⅙: what will ⅛ loose? that is to say, what will 1. li. loose? reduce the whole numbers into their broken, & then multiply & divide, so shall you find 21/588. part of a li. Then multiply 21. by 240. because so many pence are in a li. & divide the product by 588. so shall you find 8.d. 336/588 which being abbreviated do make 7/4, & thus you see that 8.d. 4/7 is lost upon the li. of money.  
98 7  32 ⅔. 1 ⅙ 1/1  
If 1. yard of cloth of tissue be sold for 3. li. 15. s. whereupon is lost after the rate of 10. s. upon the 100 I demand what 12. yards ½ of the same tissue did cost? Answer: add unto 3. li. 15. his own 1/10 part, which is 7. s.6.d. and all amounteth to 4. li. 2. s. 6.d. then look what the 12. yards ½ will amount unto, after 4. li. 2. s. 6.d. & you shall find that they will come to 51. li. 11. s. 3.d. so much did the 12. yards ½ cost.  
3. li. 15. s. 12.  ½ 7. s. 6.d. 4. li. 2. s. 6.d. 4. li. 2s. 6ds. 48. .00. .0.  1. .10. .0.  2. .01. .3.  51. li. 11. s. 3.d.  
Moore, if I cell one wiltshire white for 6. li. 12. s. whereupon I do gain after the rate of 2. s. upon the li. of money, that is to say, upon 20. s. I demand what 11. pieces of the same whites did cost me? Answer: abate from 6. li. 12. s. (which is 132. s.) his 1/11 part, & thereof cometh 12. s. and there remaineth 120. s. or 6. li. Then see at 6. li the cloth, what the 11. clotheses are worth 66. li. so much did the 11. clotheses cost.  
132. sh. 11 12. sh. 6 120. sh. 66. li.  
If I cell 10. else ½ of Holland for 22 s.6.d. whereupon I do loose after the rate of 2. s. upon the li. of money. I demand what the ell did cost me? Answer: say by the rule of 3. if 18. give 20. s. what will 22. s.6.d. give? Multiply & divide, & you shall find 25. s. Then divide 25. s. by 10. ½, & thereof cometh 2s. 4 d. 4/7: So much did the el cost.  
11/1 20/1 22 ½  
If I cell one cloth for 5 li. where upon I do loose after 10 upon the 100, I demand how much I should loose or gain upon the 100, in case I had sold the same for 5 li. 10 shil. Answer: say, if 90 yield 100, how much will 5. li. give? Multiply & divide, & you shall find 5. li. 5/9: then say again by the rule of three, if 5.5/9 come to 5. ½, what will 100 come unto? Multiply & divide, & you shall find 99 li. which being abated from 100 there will remain 1. li. and so much is lost upon the 100  
90. 100 5. 5 5/9 5 ½ 100/1   

¶ The 5. Chap. treateth of lengths & breadthes of tapestry, and other clotheses.  
IF a piece of tapestry be 5 elles ¾ long, and 4 else ⅔ in breadth, how many else square doth the same piece contain? Answer: Multiply the length by the breadth, that is to say 5. ¾ by 4. ⅔, and thereof cometh 26. else ⅚ so many else square doth the same piece contain.  
Moore, if a piece of tapistry do contain 32. else square, and the same being in length 6. else ¼. I demand how many else in breadth the same piece doth contain. Ans. divide 32. else by 6 ¼ and thereof cometh 5. 3/ 25: So many else doth the same piece contain in breadth.  
Moore, a piece of cloth being 13. yards ⅓ in length, and 5 quarters 1/2 in breadth, how many yards of ⅔ & ½ broad will the same piece make? Answer: see what part of a yard, the 5/4 and ½ be, and you shall find that they make 1 yard ⅜. Then multiply 13. yards ⅓ by 1 yard ⅜ and you shall have 18. yards ⅓ in square the which you must divide by ⅔ & ½ that is to say by ⅚, (because that ⅔, ½ being brought into 1 fraction maketh ⅚) & you shall find 22. yards: So many yards of ⅔ & ½ large doth the same piece contain.  
Moore, a merchant hath bought 4. yards ⅔ of cloth being six quarter's ½ broad to make him a gown the which he will line throughout, with black Say of three quarters of a yard broad, I demand how much Say he must buy? Answer: Multiply the length of the cloth, by the breadth, that is to say 4 ⅔ by 1. ⅝, (which is the six quarter's ½) and thereof cometh 7. yards 7/12, the which divide by ¾ & you shall find ten yards 1/9. So many yards of Say must he have to line the same 4. yard's ⅔ of cloth of 6. quart. ½ broad.  
Moore, at 6. s. 8.d. the ell square, what shall a piece of tapistre cost me, which is five else ½ long and 4. else ¼ broad? Answer, multiply 5. ½ by 4. ¼ and thereof cometh 23. else ⅜ square: then say by the rule of three, if one ell square cost me 6. s. 8.d. what shall 23. ⅜ cost? Multiply and divide, and you shall find 7. li. 15. s. 10.d. so much the said piece of tapistry did cost.  
Or otherwise, by the rules of practice, take the ⅓ of 23. 3/8: and you shall find 7. li. 15. s. 10.d. as above is said.  
Moore, a piece of Holland cloth containing 42. else ⅔ flemish, how many elles english do they make? Here must you first note that 100 else flemish, do make but 60. else english, and so consequently five else flemish do make but 3. else english. Therefore say by the rule of 3. if 5. else flemish do make three else english, how many elles english will 42. ells ⅔ flemish make. Multiply & divide, so shall you find 25. else ⅗ english, and so many elles english doth 42. 2/3 flemishe contain, the like is to be done of all others.  
Moore, I have bought a piece of tapistry, being 5. else ¾ long, and 4. else ⅔ broad measure of Flaunders, I demand how many else square it maketh English measure?  
Answer. First, forasmuch as three else english are worth 5 else flemish, therefore put 3 elles english into his square, in multiplying 3. by himself which maketh 9: likewise multiply 5. in himself squarely, and it willbe 25. Then multiply 5 ¾ which is the length of the piece, by 4 2/3 which is the breadth, & thereof cometh 26 else ⅚ square: them say by the rule of three, if 25 else square of flemish measure, be worth 9 else square of english measure, what are 26 else flemish ⅚ worth? multiply & divide, and you shall find that they are worth nine else 33/80 square of english measure.  
Moore at 3 s. 6 d. the el flemish what is the english ell worth after that rate. Answer,: say if 5. else flemish be worth three else english, what is 1 ell flemishe worth? multiply and divide, & you shall find ⅗ of an english ell. Then say by the rule of 3, if ⅗ of an english ell, be worth 3 s. 6 d. what is 1. english ell worth? multiply and divide, and you shall find 5 s. 10d. so much shall the english ell be worth.  
Moore at 6 s. 8ds. the flemish ell square, what is the english ell worth. Answer, say by the aforesaid reason, if 25 else flemishe square, be worth 9 else square english, what is one ell square flemish worth? multiply and divide, & you shall find 9/25 of a square english ell: Then say, if 9/25 of an english ell be worth 6 s. 8 d. what is one square ell english worth? multiply and divide, and you shall find 18 s. 6ds. 2/9, so much shall one english ell square be worth.   

¶ The sixth Chapter treateth of the reducing of the paumes of Genes into english yards, whereof four Paumes maketh one english yard.  
I Have bought 97. paumes ½ of Genes velvet, & I would know how many yards they will make? Answer, Divide 97. ½ by 4. and you shall have 24. yards ⅜. So many yards do the 97. paumes ½ contain.  
Or otherwise, take some other number at your pleasure, as 20. paumes, which do make five yards, and then say by the rule of three, if 20/1 paumes, give 5/1 yards, what will 97. ½ give? Multiply and divide, and you shall find 24. yards 3/2 as before.  
Moore, at two shillings 7.d. the paume of Genes, what will the english yard be worth after the rate? Answer, say by the rule of three, if ¼ of an english yard be worth two shillings 7/12. What is 1/1 yard worth? Multiply & divide, and you shall find ten shillings 4.d. So much is the english yard worth.  
Or otherwise, multiply 4. paumes (which is one yard) by two shillings 7. pence, and you shall find 10. s. 4.d. as before.  
If 257. Paumes ½ be worth 20. li. 16. s. 8.d. What is one yard worth after the rate? Answer, say: by the rule of 3. if 257. ½ paumes be worth 20. ⅚, what are 4/1 paumes worth. Multiply and divide, and you shall find 100/309 part of a pound, which is worth 6. s. 5. pence, 5●/103: so much is one yard worth.   

¶ The. seven. Chapter treateth of merchandise sold by weight.  
AT 9.d. ½ the ounce, what is the li. weight worth? Answer, say if 3/2 give 9 ½ what will 16/1 give multiply and divide, & you shall find 12. s. 8.d. so much is the yard worth?  
Or otherwise, by the rules of practice for six pence, take the ½ of 16. which is 8. s. then for 3.d. take the ¼ of 16. s. which is 4. s. Finally, for the halpenye, take 16. ob. which are 8.d. add all these numbers together and you shall find 12. s. 8.d. as before.  
Moore, at 10 d. ½ the ounce, what are 112. li. weight worth after the rate? Answer: reduce. 112. li. into ounces, in multiplying. 112. li. by 16. ounces & you shall have 1792. ounces, them say by the rule of 3. if 1/1 10 ½ 1792/1: Multiply and divide, and you shall find 18816 d. which do make 78. li. 8 s. and so much are the 112. li. worth after 10.d ¼ the ounce.  
At 12. s. 8ds. the li. weight, what is the ounce worth? Answer: put 12. s. 8ds. into pence, and you shall have 152. pence: then say by the rule of 3. if 16. ounces cost 152d. what shall 1. ounce cost, multiply and divide, and you shall find 9.d. ½, so much is the ounce worth.  
Or otherwise, take the ¼ of 12 s. 8.d for 4 ounces, and thereof cometh 3. s. 2.d. then for one ounce, take the ¼ of 3. s. 2d. and you shall have 9.d. ½ as before.  
At 32. li. 10. s. the quintal, that is to say, the 100 li. weight: what is 1. li. weight worth after the same rate? Answer, Put 32. li. 10. s. all into shillings and you shall have 650. s.  
Then say, by the rule of three, if 100 650 1. multiply and divide, and you shall find, 6. s. 6.d. so much is the li. worth.  
If one pound weight of saffron do cost me 18. s. 8.d. what shall 355. li. 10. ounces cost me by yᵉ same price? Answer say by the rule of 3. if 1/1 18 ⅔ 355 ⅝. Multiply and divide, & you shall find 331. li. 18. s. 4.d. so much are the 355. li. ten ounces worth.  

Brief rules of weight.  
WHo that multiplieth the pence that 1. li. weight is worth by 5. and divideth the product thereof by. 12. he shall find how many pounds in money the quintal is worth, that is to say, how much the 100 li. weight is worth.  
And contrariwise he that multiplieth the pounds of money that the 100 weight is worth by 12. and divideth the product by 5. shall find how many pence the pound weight is worth.  

¶ Example.  
AT seventeen pence the pound weight, what is the 100 pound weight worth? Answer, Multiply 17. by 5. and thereof cometh 85. divide the same by 12. and you shall find 7. pound 1/12, which 1/12 is worth one shilling and eight pence. So much is the 100 pound weight worth.  
Moore, at 13. li. the 100 li. weight, what is one pound weight worth? Answer, Multiply 13. by 12. amounteth to 156. the which divide by 5. and you shall find 31.d. ⅕ which is 2. s. 7.d. ⅕ and so much is one pound weight worth.  
The like is to be done of yards, else, or of any other measure, when we reckon but five score to the hundred.    

Brief Rules for measure.  
Who that multiplieth the pence that one ell is worth, by 6. And divideth the product by 12. he shall find how many pounds in money the 120. else are worth, which 120. else we count but for a C.  
And contrariwise, he that multiplieth the pounds in money that the 120. else are worth by 12. and divideth the multiplication by 6. shall find how many pence the ell is worth.  

¶ Example.  
At ten pence the ell, what are 120. else worth? Answer, Multiply 10.d. by 6. and thereof cometh 60: The which divide by 12. and you shall found five pound, so many pounds in money are 120. else worth at 10.d. the ell.  
Moore, at 9 pound, the 120. else, what is one ell worth? Answer, Multiply nine pound by twelve, and thereof cometh 108. the which divide by 6. and you shall find 18.d. so much is one ell worth.  
The like is to be done of all manner of wares, which are sold after 120. for the hundred.    

¶ Brief Rules for our hundredth weight here at London, which is after 112. li. for the C.  
WHo that multiplieth the pence that one pound weight is worth by 28. and divideth the product by 60. shall find how many pounds in money the 112. li. weight is worth.  
ANd contrariwise, he that multiplieth the pounds in money that 112. li. is worth by 60. and divideth the product by 28. shall find how many pence one li. weight is worth.  

¶ Example.  
AT nine pence the pound weight, what is the 112. li. weight worth? Answer: multiply 9.d. by 28, and thereof cometh 252, the which divide by 60. & you shall find 4. li. 12/60 which being abbreviated is 1/5 of a pound, which is worth 4. s. And thus the 112. li. is worth 4. pound 4. shil.  
At 8. li. the 112. li. weight, what is 1. li. weight worth? Answer, Multiply 8. li. by 60. and thereof cometh 480, the which divide by 28. & you shall find 17.d. 1/7: so much is 1. li. weight worth.     

¶ The. viii. Chapter treateth of tars and allowances of merchandise sold by weight.  
AT 12. li. the 100 subtle, what shall 987. li. subtle be worth? in giving 4. li. weight upon every 100 for tret? Answer, add 4. li. unto 100 & you shall have 104. Then say by the rule of three, if 104 be worth 12. li. what are 987. li. weight worth? multiply & divide, & you shall find 113. li. 23/26 which is worth 17. s. 8.d. 4/13. So much shall the 987. li. weight be worth. 104. 12. 987.  
At 6. s. 8.d. the pound weight what shall 345. li. ½ be worth in giving 4. li. weight upon every 100 for the tret. Answer, see first by the rule of three, what the 100 pound is worth saying, if 1/1 6 ⅔ 100/1 Multiply and divide, & you shall find 33. li. ⅓ then add 4. li. unto 100 & they are 104. then say again by the rule of 3. if a 104. li. be sold for 33. li. ⅓ for how much shall 345. li. ½ be sold? multiply & divide, and you shall find 110. li. 14. s. 8.d. 12/13 So much shall the 345. li. ½ be worth, at 6. s. 8.d. the pound, in giving 4. upon the 100  
Moore, if 100 be worth 36. s. 8.d. what shall 780. li. be worth in rebating 4. li. upon every 100 for Tare & Cloffe? Answer, Multiply 780. by 4. and thereof cometh 3120. The which divide by 100 and you shall have 31. li. ⅕ abate 31. ⅓ from 780. and there will remain 748 ⅘. Then say by the rule of three, if 100/1 do cost 36. ⅔, what will 748. ⅘ cost after the rate? Multiply & divide so shall you find 274. s. 6.d. 18/25, and so much shall the 780. li. cost, in rebating 4. li. upon every 100 for Tare and Cloffe.  
Moore, whether doth he loose more that giveth 5. li. upon the 100 or he that rebateth 5. li. upon the 100 for tore and cloffe? Answer. first, note that he which giveth 5. li. upon the 100 giveth 105. for 100: and he which rebateth 5. li. upon the 100 giveth the 100 for. 95. Therefore say by the rule of 3. if 105. be given for 100 for how much shall the 100 be given? Multiply and divide & you shall find 95. 5/21: and he which rebateth 5. upon the 100 maketh but 95. of 100: so that he loseth 5. upon the 100 & the other which giveth 5. upon the 100 loseth but 4. 16/21 upon the 100 Thus he that rebateth 5. upon the 100 loseth more by 5/21 upon the 100 than the other which gave 5. upon the 100 for tore and cloffe.  
If 100 of Allam do cost me. 26. s. 8.d. how shall I cell the li. weight to gain after the rate of 10. upon the 100 Answer, put 26. s. 8.d. all into pence, & you shall have 320.d. Then say by the rule of 3. if 100 give 110. what shall 320. give, multiply and divide, and you shall find 352.d. Then say, if 100 li. be worth 352.d. what is 1. li. multiply & divide, and you shall have 3.d. 26/50 which 26/50 is worth ½, and 1/25 of ½. That is to say, the pound weight shallbe worth 3.d. ½. 1/25 of a half penny, in gaining 10. upon the 100  
If one pound weight do cost me, 6. s. 10.d. and I cell the same for 7. s. 2.d. I demand how much I should gain upon the 100 li. of money after the rate? Answer, say by the rule of 3. if 6. ⅚ yield 7. ⅙ what will 100/1 yield? Put the whole numbers into their broken, then multiply and divide, & you shall find 104. 36/41 from the which subtract 100 and there resteth 4. li. 36/41 so much is gained upon the hundred pound of money after the rate.  
Moore, if one pound do cost me 5. s. 4.d. and I cell the same again for 4. s. 9.d. I demand how much I shall loose upon the 100 pound of money? say, if 5. ⅓ do give but 4. ¾, what shall 100/1 give? Put the whole number into their broken. Then multiply and divide & you shall find 89. 1/16 the which you must subtract from 100 and there will remain 10. li. 15/16, so much is lost upon the 100 li. of money.  
Moore if the li. weight do cost me 3. s. 2.d. & I cell it again for 4. s. 4.d. how much shall I gain upon 20. s. Answer: say if 3. ⅙ give 4. ⅓ what shall 20/1 give, Multiply and divide & you shall find 27. s. 7/19: out of the which abate 20. s. and there will remain 7. shillings. 7/19, which is 4.d. 4/19: and so much is gained upon the pound of money that is to say upon 20. s.  
If the pound weight do cost me 4. s. 4.d. and I cell it again for 3. s. 2.d. I demand how much I shall loose upon the pound of money? that is to say upon twenty shillings.  
Answer: say, if 4. ⅓ give but 3. ⅙ what will 20/1 give, multiply & divide & you shall find 14. s. 8/13 the which you must abate from 20. s. & there will remain 5. s. 5/13 which 5/13, is worth 4.d. 8/13 of a penny and so much is lost upon the pound of money.   

¶ The. ix. Chapter treateth of certain questions, done by the double rule, and also by the rule of three compound.  
WHen the quarter of wheat, doth cost 6. s. 8.d. the loaf of bread weighing 20. ounces is sold for a ob. I demand that if the quarter of wheat did cost ten shillings, for how much shall the loaf of bread be sold, that weigheth 16. ounces?  
Answer: by the first part of the rule, of 3. compound which is mentioned in the third part of this book, and in the second Chapter of the same. Therefore say by the same first part of the rule of 3. compound, if 6. ⅔ 20/1 ½ 10/1 16/1.  
Then multiply the first number by the second, and the product thereof shallbe your divisor. Likewise multiply the other three numbers the one by the other, and the product thereof shallbe your dividend: as thus, first multiply 6. ⅔ by 20/1, and thereof cometh 400/3 for your divisor, then multiply ½ by 10/1 and the product thereof by 16/1, so you shall have 160/2 for your number that is to be divided, then divide 160/2 by 400/3, and thereof cometh 480/800 the which being abbreviated bringeth ⅗ of a penny: and for that price must the loaf of bread be sold, which weigheth 16. ounces, and the quarter of wheat being worth ten shillings.  
Or otherwise by the rule of 3. at two times. First say if 20/1 ounces give, ½ what will 16/1 ounces give? Multiply and divide, and you shall find ⅖ of a penny. Then say again, if 6. 2/7 do give me ⅖, what will 10/1 give? Multiply and divide, and you shall find ⅗ of a penny, as afore is said.  
When the carriage of one hundredth weight of merchandise 50. miles doth cost 5ss. what shall the carriage of 500 weight cost me for 16 miles?  
Answer, By the first part of the rule of 3 compound, saying if 100 50 5 500 16. Multiply 100 by 50 the product will be 5000, which shall be your divisor. Then multiply 5 times 500 by 16 and thereof cometh 40000 for your dividend. Therefore divide 40000 by 5000 and you shall find 8 s. so much shall cost the carriage of 500 weight 16 miles.  
Or otherwise by the double rule of three, that is to say by the rule of three at two times: first say if 50 miles do pay 5 s. what shall 16 miles pay? Multiply and divide, & you shall found 1 s. ⅗. Then say, again, if 100 weight do cost me 1 s. ⅗ what shall 500 weight cost? Multiply and divide, and you shall find 8 s. as before.  
When the carriage of 100 pound weight of Merchandise 84. miles, doth cost me six shillings, how many miles may I have 64. pound weight carried for three. s. 4.d. Answer, by the second part of the rule of three compound: say if 100/1 14/1 6/1 64/1 3 ⅓.  
Then multiply the fourth number 64/1 by the third number 6/1, and thereof cometh 304/1 for your divisor. Likewise multiply 3 ⅓ by 100/1, and by 14/1 and you shall have in the product 14000/3: then divide 14000/3 by 384/1 and you shall find 72. miles 11/12 of a mile. So many miles shall the 64. li. weight be carried, for three shillings 4.d.  
Otherwise by the rule of three, at two times: first say, if 100 weight do cost me 6. s. what shall 64. pound weight cost? Multiply and divide, and you shall find three shillings .21/25. Then say, if 3. 21/25 be paid for 84. mile's carriage: for how many miles shall 3. s. ⅓ be paid? Multiply & divide and you shall find 72. miles. 11/12.  
If 100 horses in 100 days do spend 180. quarters of oats: how many quarters of oats will 350. horses spend in 150. days? Answer: by the first part of the rule of three compound: multiply 180. times 350. by 150. and divide the product by 100 times 100: and you shall find 945. quarters. So many quarters of Oats will 350. horses spend in 150. days.  
Or otherwise by the rule of 3. at two times: first say, if 100 days do yield me 180. quarters of oats: what shall 150. days yield: multiply and divide, and you shall find 270. quarters: then say again, if 100 horses do spend 270. quarters of Oats, how many quarters of oats will 350. horses spend? Multiply and divide, and you shall find 945. quarters as before.   

¶ The tenth Chapter treateth of the rule of Fellowship, without any time limited.  
THe rule of fellowship is thus: you must set down each man's sum of money that he layeth into company, 
A Rule.  every one directly under the other, the which you shall add altogether, & the total sum of all their whole stock being thus assembled, shallbe your common divisor, to the finding out of every man's part of the gain. Then shall you multiply the gain, or the loss, by each man's portion of money that he laid in, & divide the products by the said divisor: so shall you have in your quotient every man's part of the gain, or else of the loss, if any thing be lost.  

¶ Example  
1. Two Merchants have made company together, the first laid in 500 li. The second put in 300. li. and with occupying they have gained 64. li. I demand how much each man shall have of the same gains according to the money that he laid in. Answer: Add 500 & 300. both together, which are the parcels that they laid in, and thereof cometh 800. for your divisor: then say by the rule of three, if 800. li. (which is their stock) do gain 64. li. what shall 500 li. gain? (which is the first man's money that he laid in) multiply & divide and you shall find 40. li. for the first man's part of the gain: then say if 800. give 64. what will 300. give? Multiply and divide, and you shall find 24. li. for the second man's part of the gain.  
500    300 800 64 500 800     800 64 300.  
Or otherwise, put 500 li. which is the first man's money that he laid in, over the 800. li. which is the whole, stock, and you shall have 500/800 which being abbreviated, do make ⅝, & such part of the gain shall the first man take, that is to say ⅝ of 64. li. which is 40. li. And consequently, by the same manner, the second shall take the ⅜ of 64. which is 24. pound for his part of the gain as before.  
5 00 3 00 8 00 8 00  
2. Two Merchants have companied together, the first put in 640. li. and he taketh ⅝ parts of the gain. I demand what the second Merchant laid in? Answer,  that the first Merchant taketh ⅝ of the gain, it followeth that the second must have ⅜ which is the rest, & therefore say by the rule of three, if ⅝ of the gain, which the first man taketh, did lay into the stock 640/1. How much shall the ⅜ of the gain lay in, which is the second man's gain? Multiply and divide, & you shall found 384. li. so much aught the second man to lay into company.  
3. Two Merchants have companied together, the first man laid in 640. li. and the second hath laid in so much, that he must have 60. li. for his part of 100 li. which they have gained. I demand how much the second man did say into company? Answer: seeing that the second man taketh 60. li. of the gain, it followeth that the first must have but 40. pound. Therefore say by the rule of three, if 40. li. do lay in 640. li. what shall 60. li lay in? Multiply and divide, and you shall find 960. pound, so much did the second merchant lay in.  
4. Two merchants have companied together, the first laid in 83. li. 6. s. 8.d. the second put in 170. ducats: & they have gained 100 li. of the which the first man must have 60. li. I demand what the ducat was worth? Answer, seeing that the first man must have 60. li. it followeth that the second must have 40. li. therefore say by the rule of three if 60. li. of gain that the first man taketh did lay in 83. li. 6. s. 8.d. of principal, how much shall 40. li. of gain put in, multiply & divide, and shall found 55. li. 5/9: so much are the 170. ducats worth. Then put 55. li. 5/9 into shillings, and you shall have 1111. s. 1/9 them to know what the ducat is worth, say by the rule of three, if a 170/1 give 1111. 1/9, what will 1/1 give? Multiply and divide, & you shall find 6. s. 6.d. 22/51, so much is the ducat worth.  
5. Two Merchants have companied together, the second man laid in more by 30. li. than did the first man: and they gained 120. li. of the which the first man aught to have 50. li. I demand what each of them did lay in. Answer, from 120. li. abate 50. li. and there resteth 70. li. for the second man's part: so that by this means the second man (because he laid in 30. li. more than the first man did) taketh 20. li. more of the gain: & therefore say by the rule of 3. if 20. li. of gain did lay in 30. li. of principal, how much shall 50. li. lay in?  
Multiply and divide, and you shall find 75. li. so much did the first man lay in, and consequently the second laid in 105 li.  
6. Two merchants have companied together, the second hath laid in twice so much as the first man did, and 10 li. more: and they gained 100 li. of the which, the first aught to have 32 li. for his part: I demand how much each of them did lay into company?  
Answer, If it were not for the 10 li. that the second man laid in more: he should have had but 64 li. of the gain which is the double of the first man's part. But because he laid in 10 li. more, he hath four pound more of the gain, and therefore say by the rule of three, if 4 li. of gain did say in 10 li. of principal, (which was over and above the double of the first man's laying in) what shall 32 li. of gains lay in? which is the first man's part of the gains that he taketh. Multiply and divide, and you shall find 80 li. for the first man's laying in: and consequently 170 li. for the second man's portion that he laid in.  
7. Two merchants have companied together, and they have gained 100 li. of the which the first must have after the rate of 10 upon the 100 li. and the second must have after the rate of 15 li. upon the 100 li. I demand how much each of them ought to have? Answer, Put 10 li. for the first man's laying in, and 15 li. for the second man's laying in. Add 10 li. and 15 li. together, and they make 25 li. Then put 10 over 25. and it is 10/25 which being abbreviated are ⅖. Therefore he that taketh 10 li. upon the 100 li. must have the ⅖ of the gain, which is 40 li. Then put 15 over 25. and it is 15/25 which being abbreviated are ⅕. Therefore the second must have ⅗ of the 100 li. which is 60 li.  
8. Two Merchants have companied together, the first laid in 46. li. 18. s. and the second laid in 33. li.2.s. so they have gained 30. li. I demand how much every man shall have for his part of the gain? Answer: Add 46. li. 18. s. and 33. li.2.s. both together and you shall find 80. li. for your common divisor: then say if 80. li. which is all their stock do gain 30. li. what will 46.9/10 gain, which is the first man's laying in: Multiply & divide, and you shall find 17. li. 11. s. 9.d. for the first man's part of the gain. Then say again, if 80. li. do gain 30. li. what will 33. li. 1/10 gain, which was the second man's, laying in: multiply and divide, and you shall find 12. li. 8. s. 3.d. for the second man's part of the gain.  
And after the same manner shall you do, in case there were three or four Merchants that would company together: Adding all their sums of money (which they lay into the stock) into one total sum: which shallbe your common Divisor: and then work with the rest, as is taught in the former Questions of the rule of company.  
9 Three Merchants have companied together, the first laid in I know not how much: the second did put in 20. pieces of cloth, and the third hath laid 500 pound. So at the end of their company, their gains amounted unto a thousand pound, whereof the first man aught to have 350. pound, and the second must have four hundred pound.  
Now I demand how much the first man did lay in, and for how much the 20. pieces of cloth were put into company?   

Answer.  
Seeing that the first and the second merchants must have 750. li. for their parts of the gain. Then the third man must have the rest of the thousand pound which is 250. li.  
And therefore say by the rule of three, if 250. of gain, become of 500 li. of principal: of how much shall come 350. li. of gain? which the first man taketh, multiply and divide and you shall find 700. li. So much did the first man lay in: then say if 250. li. of gain be come of 500 li. principal, of how much will come 400. li. which is the gain that the second man taketh. Multiply and divide, & you shall find 800. li. For so much were the 20. pieces of cloth laid into company.  
10. Three Merchants have gained 100 li. the first must have the ½, the second must have ⅓: And the third must have ¼. I demand how much every man must have of the gain? Answer, Reduce ½, ⅓, ¼, into a common denomination, after the order of the second reduction in fractions, & you shall find 12/24, for the ½: 8/24, for the ⅓: and 6/24, for the ¼: Then take 12 for the first man's laying in, 8. for the second man's laying in: and 6 for the third man's laying in. The which three numbers being added together shall be your common divisor, which do make 26. Then multiply 100 by 12, for the first man, by 8 for the second man, and by 6 for the third man. And divide every multiplication by 26. So shall you find 46 li. 2/13 for the first man's part of the gain. 30. li. 10/13 for the second man's part: and 23 li. 1/13, for the third man's part.  
11. Two merchants have gained 100 li. the first must have ½ and 5 li. more: the second must have ⅓ and 4 li more: I demand how much each of them shall have? Answer, From 100 abate 5 and 4. so there will remain 91. then take the ½ of 100 li. which is 50 li. for the first man's laying in: Likewise, take ⅓ of 100 li. for the second man's laying in, which is 33 li. ⅓. Then add 50 li. and 33 li. ⅓ together, and you shall have 83 li. ⅓ for your common divisor, then multiply 91. pound by 50. and divide by 83. ⅓: and thereof cometh 54. pound, ⅗ unto the which number add 5, and all is 59 li. ⅗ for the first man's part. Likewise multiply. 91. by 33. ⅓: and divide by 83. ⅓, & you shall find 36. li. ⅖ unto the which add 4: and you shall have forty pound, ⅖ for the second man's part.  
12. Two Merchants have gained a hundred pound, the first must have the ½ less by 4. pound, the second must have ⅓: less by 2. pound. I demand how much each of them shall have? Answer, Add 4. & 2. with 100 & they make 106. Then take as before is said 50. pound, for the first man, & 33. ⅓ for the second, add them both together, and they be 83. which shallbe your divisor. Then multiply 106. by 50. and divide the product by 83. ⅓, so thereof cometh 63. li. ⅗. From the which abate the four pound less that the first man taketh, and then is there remaining 59 pound, ⅗ for his part. Likewise multiply 106. by 33. ⅓ and divide by 83 ⅓ & you shall find 42. li. ⅖: from the which abate 2. li. less and there remaineth 40. pound, ⅖ for the second man's part.   

¶ The Rule of Fellowship with time.  
THe money that every man layeth in, must be multiplied by the time that it remaineth in company: and of that which cometh thereof you shall make their new layings in for each of them: and then multiply the gains by every one of them severally, the which you shall divide by all their new layings in added together, and you shall have proporcionally each man's part of the gain according to his laying in.  

¶ Example.  
1. Two Merchants have companied together, the first hath put in the first of january 450. pound, the second did lay in the 2. of May. 750. pound And at the years end, they had gained 100 li. I demand how much each of them shall have of the gain? Answer: forasmuch as the first did put 450 li. the first of january: his money remained in company 12. months, and therefore multiply 450. by 12 months, and thereof cometh 5400. for his new laying in. And the second laid in his 750 li. but at the first day of may: so that his money remained in company but 8 months. Therefore multiply his 750 li. by 8. and thereof cometh 6000 for his new laying in: Then add 5400. with 6000. and they make 11400 for your common divisor: Then multiply 100 li. which is the gains by 5400, and divide the product by 11400. and thereof cometh 48 li. 7/19 for the first man's part of the gain. Likewise multiply 100 by 6000, and divide the product by 11400. and you shall find 52 12/19 & so much must the second man have for his part of the gain.  
2. Two merchants have companied together, the first hath put in the first of january 640. li. The second can lay in nothing until the first of April. I demand how much he shall then say in, to the end that he may take half the gains? Answer, Multiply 640 li. by 12. months that his money abideth in the company, and thereof cometh 7680 li. for his laying in. And so much ought the second man's laying in to be, for because he taketh ½ of the gain: But for that, that he putteth in nothing until the first of April, his money can be in company no longer than 9 months. And therefore divide 8680 by 9, and thereof cometh 753 li. ⅓ So much aught the second merchant to say in the first of April, to the end that he may take the one moiety of the gains.  
3. Three Merchants have companied together, the first laid in the first of March 100 li. The second laid in the first of june so much money, that of the gain, he must have the ⅓ part: and the third laid in the first of November so much money, that of the gains he must have likewise ⅓ and they continued in company, until the next March following. I demand how much the second and the third Merchants did say in? Answer, Multiply 100 which the first man did lay in, by 12. months that his money continued in company, and thereof cometh 1200. for his laying in: and so much aught the second and the third merchant each of them to lay in: Because they part the gains by thirds. But for that, that the second Merchant putteth in nothing till the first of june, his money can be in company but nine months. Therefore divide 1200. by nine months, and thereof cometh 133. ⅓. And so much aught the second Merchant to say in: Then, forasmuch as the third Merchant, did say in nothing until the first of November: His money abideth in company but the space of four months. Therefore divide 1200. by 4. and thereof cometh three hundred pound. And so much aught the third merchant to lay into company.  
4. Three merchants have companied together, the first laid in the first of january a hundred Ducats. The second hath laid in fifty pound, the first of March: And the third put in a jewel the first of july: And at the years end, they had gained four hundred crowns: of the which, the first merchant must have fifty crowns, and the second must have 80. I demand what the Ducket was worth, and at what price the jewel was valued, which the third Merchant laid in?  
Answer: the first man's money is 1200 as afore is said, and he taketh 50 crowns of the gain: therefore say, if fifty crowns of gain be come of 1200, which was his stock, of how much shall come 80. crowns of gain that the second man taketh? multiply and divide, and you shall find 1920. for the second merchants laying in. Then say again, if 50 crowns be come of 1200. stock: of how much shall come 270. crowns, which the third man taketh of the gain? Multiply and divide, & you shall find 6480. for the third merchants laying in. Then divide 1920, which is the second man's laying in, by 10. months that his money did continued in company, and you shall find 192 Ducats, which are worth 50. li. because he laid in 50 li. Then divide 192 Ducats by the said 50. li. (being reduced into shillings) and thereof cometh 5. shillings 2. pence, ½. So much was the Ducat worth: finally, divide 6480. (which is the third man's laying in) by 6. months that his jewel remained in company, and you shall find 1080 Ducats: and for that price was the jewel put into company.  
5. Three Merchants have companied together: the first laid in the first of january 100 li. and the first of April he hath taken back again 20. li. The second hath laid in the first of March 60 li. and afterward he did put in more 100 li. the first of August. The third laid in the first of july 150 li. And the first of October he did take back again 50 li. And at the years end, they found that they had gained 160 li. I demand how much every man shall have? Answer, Multiply 100 li. which the first man laid, by 12 months, and thereof cometh 1200. li. from that number abate 9 times 20 which are 180. and there will remain 1020. for the first man's laying in. Then multiply 60. which the second man laid in, by ten & you shall have 600. unto the which add 5. times one hundred, which are 500 so all amounteth to 1100. for the second man's laying in: Afterwards, multiply 150. pound, which the third man hath laid in, by 6. months, and thereof cometh 900. from the which number abate three times 50. and they are 150: so there resteth 750. for the third man's laying in. Then proceed with the rest, as in the first Question of the rule of fellowship with time, in adding 1020, 1100. and 750. altogether, which shall be your Divisor: Then multiply 160 by 1020. by 1100. and by 750, & divide at every time by your Divisor, which is by all their layings in added together, and they make 2870, so you shall find 56. 248/287: for the first man, 61. 93/287 for the second, and 41. 233/287 for the third man.  
6. Two Merchants have companied together, the first hath put in 960. pound, for the space of 12. months, and he aught to have 8. pound upon the hundred pound of the gain. The second hath laid in 1120. li. for the space of eight months, & he aught to have after 12. pound upon the 100 pound of the gain.  
And at the years end, they have gained eight hundred pound. I demand how much each of them shall have of the gain. Answer, multiply 960. that the first man did lay in, by 12. months, and the product thereof, multiply again by 8. and you shall have 92160. for the first man's laying in: then multiply the 1120. that the second hath laid in by eight months, and that which cometh thereof you shall multiply again by 12. and you shall find 107520. for the second man's laying in: Then proceed with the rest, as in the first Question of the Rule of Fellowship, and as in the last example, and you shall find 399 3/13 for the first man: and 430. li. 10/13 for the second man.   

¶ The rule of company, between Merchants and their factors.  
7. The estimation of the body or person of a Factor, is in such proportion to the stock, which the Merchant layeth in: as the gain of the said Factor is unto the gain of the said Merchant. As thus: if a Merchant do put into the hands of his Factor 200 li. to employ, and he to have half the profit, the person of the said Factor shall be esteemed 200 li, And if the Factor do take but the ⅓ of the gain, he should have but ½ so much of the gain as the Merchant taketh, which should take ⅔ wherefore the person of the Factor is esteemed but the ½ of that which the Merchant layeth in, that is to say 100 li.  
And if the Factor did take the ⅖ of the gain, than the Merchant shall take the residue, which are ⅗ of the gain wherefore the gain of the Master unto that of the factor is in such proportion as 3 unto 2. Then if you will know the estimation of the person of the Factor, say if 3 give me 2 what will 200 give? Multiply 200 by 2 and divide by 3 so you shall find 133 ⅓ Otherwise, consider that the factor taketh the ⅔ of that which the Merchant taketh. And therefore take the ⅔ of 200, and you shall find 133 ⅓ as before: and so much is the person of the factor esteemed to be worth.  
8. And if the Merchant should deliver unto his factor 200. li. and the Factor would say in 40 li. and his person, to the end he might have the half of the gain: I demand for how much shall his person be esteemed Ans. abate 40 li. from 200 li. and there will remain 160. li. And at so much shall his person be esteemed.  
And if the factor would take the ⅔ of the gain, his person with his 40 pound, shall be esteemed twice as much as the stock that the merchant layeth in, which should have but ⅓ of the gain: for ⅔ unto ⅓, is in double proportion. Therefore double two hundred pound, thereof cometh 400. li. from the which abate 40. li. & there will remain 360. li. And if the Factor would take but the ⅓ of the gain, that shall be but the ½ of ⅔ which the merchant taketh: then the estimation of his person, with his laying in should be esteemed but the half of the which the merchant layeth in: take therefore the ½ of 200. li. which is 100 li. from the which abate forty pound, and the rest which is 60. li. is the estimation of his person.  
9 If it so chance that for to make traffic of 240. li. the person of the factor should be so esteemed, that he should have but the ¼ of the gain, & yet he would have the ⅔, I demand how much he should put in of ready money, besides his person? Answer, seeing that his person gaineth the ¼, all the whole laying in, shall gain the rest that is to say the ¾: now for because ¼ is the ⅓ of ¾ therefore his person shallbe esteemed the ⅓ of all the laying in. Take then the ⅓ of 240. and you shall have 80. for the estimation of his person, and for that, that he will have the half of the gain, you shall add 80. with 240. li. and thereof cometh 320. of the which take the half, which is 160. and from the same you shall abate the 80. and there will remain other 80. which he aught to lay in of ready money, and the merchant must lay in the overplus, which amounteth to 160. li.  
10. A merchant hath delivered to his Factor 1200. li. to govern them in the trade of Merchandise upon such condition that he for his service shall have the ⅓ of the gain if any thing be gained, or of the loss if any thing be lost: I demand for how much his person was esteemed? Answer, seeing that the factor taketh the ⅓ of the gain, his person aught to be esteemed as much as ½ of the stock which the Merchant layeth in, that is to say the ½ of 1200 li. which is 600 li. The reason is, because the ⅓ of the gain that the factor taketh, is the ½ of the ⅔ of the gain that the Merchant taketh.  
11. A Merchant hath delivered unto his Factor 1200 li. and the Factor layeth in 500 li. and his person: Now, because he layeth in 500 li. and his person, it is agreed between them that he shall take the ⅖ of the gain: I demand for how much his person was esteemed? Answer, Forasmuch as the Factor taketh the ⅖ of the gain, he taketh the ⅔ of that which the Merchant taketh, for ⅖ are the ⅔ of ⅗: and therefore the factors laying in aught to be 800. pound, which is the ⅔ of 1200. pound, that the merchant laid in: Then abate 500 pound, which the Factor did lay in, from 800. pound, which should be his whole stock and there remaineth three hundred pound for the estimation of his person.  
12. Moore, a merchant hath delivered unto his factor a thousand pound upon such condition, that the Factor for his pains and service, shall have the gains of 200. pound, as though he laid so much in of ready money: I demand what portion of the gain, the said Factor shall take? Answer: See what part the 200. li. (which the Factor laid in) is of 1200. which is the whole stock of their company, & you shall find that it is the ⅙, and such part of the gain shall the Factor take.  
But in case, that in making the covenants, it were agreed that the Factor should have the gain of two hundred pound of the stock, which the merchant layeth in, that is to say, of the thousand pound. Then should the Factor take the ⅕ part of the gain. For 200. li. is the ⅕ of a 1000 pound.     

¶ The xj. Chapter treateth of the Rules of barter.  
Two Merchants will change their merchandise, the one with the other. The one of them hath cloth of 7. s. 1.d. the yard to cell for ready money, but in barter he will cell it for 8. s. 4.d. The other hath Cinnamon of 4. s. 7.d. the li. to cell for ready money. I demand how he shall cell it in barter to the end he be no loser? Answer, say, if 7. 1/12 (which is the price that the yard of cloth is worth in ready money) be sold in barter for 8. ⅓ for what shall 4.7/12 be sold in barter which 4. 7/12, is the price that the li. of Cinnamon is worth in ready money, reduce the whole numbers into their broken, and then multiply and divide, and you shall find 5. s. 4.d. 12/17 parts, of a penny, and for so much shall he cell the pound of Cinnamon in barter.  
2. Two Merchants will change their merchandise the one with the other, the one of them hath Chaumlets of two pound 18. s. 4.d. the piece to cell for ready money, and in barter he will cell the piece for 4. li. 3. s. 4.d. the other hath fine caps of 35. s. 10.d. the dozen to cell in barter. I demand what the dozen of caps did cost in ready money? Answer, say if 4. li. 3. s. 4.d. which is the overprice of the piece of Chamlet, become of 2. li. 18. s. 4.d. which was the just price of the same, of what shall come 35. s. 10.d. which is the overprice of the dozen of caps? Multiply and divide, & you shall find 25. s. 1.d. and so much are the dozen of caps worth in ready money.  
3. Two Merchants will change their merchandise the one with the other: the one of them hath Fustians of 18. s. 4.d. the piece to cell for ready money, and in barter he will cell the piece for 26. s. 8.d. The other hath tapestry of 15.d. the ell to cell for ready money, and in barter he will cell it for 20.d. the ell: I demand which of them gaineth, and how much upon the hundred pound of money?  
Answer: say if 18. s. ⅓ (which is the just price of the piece of Fustian) be sold in barter for 26. s. ⅔: for how much shall 1. s. ¼ (which is the just price of the ell of tapestry) be sold in barter? Multiply and divide, & you shall find 21.d. 9/11. And he doth oversel it but for 20.d. so that of 21.d. 9/11: he maketh but 20.d. And therefore say by the rule of three, if the second merchant, of 21 9/11, do make but 20/1 how much shall he loose upon the 100/1? Multiply and divide, and you shall find 91. ⅔, the which being abated from a hundred there will remain 8. ⅓. And after the rate of 8. ⅓. doth the second merchant loose upon the 100 And consequently, the first merchant, of 20.d. maketh 21.d. 9/11: and therefore say again by the rule of three, if the first merchant of 20/11, do make 21.9/11 how much shall he gain upon 100/1? Multiply and divide, and you shall find 109. li. 1/11.  
Thus the first gaineth after the rate of 9 li. 1/11: upon the hundred pound of money.  
For your better understanding of these Questions, you must note that when one merchant gaineth of an other after the rate of ten pound upon the hundred pound he gaineth the 1/10 of his own principal, and the other which loseth after the rate of 9 1/11 upon the hundred he loseth the 1/11 of his principal. And it may be proved thus: When one merchant will cell his wares unto another, which wares stand him but in 100 li. & he will cell them for 110. li. he, of his 100 li. maketh 110. li. wherefore he gaineth after 10. li. upon the 100 which is the 1/10 of his principal, and the other which buyeth wares for 110. li. that cost but 110. pound of the 110. pound he maketh but 100 li. And therefore say by the rule of three, if 110. become of 100 of how much shall come 100? Multiply and divide, and you shall find 90.10/11, the which abate from 100: and there resteth 9.1/11 is the 1/11 of his principal that the second loseth upon the 100 as afore is said. And therefore, who so that will know what one Merchant gaineth of another, either after the rate of ten upon the hundred, which is the 1/10 of of his principal, or else after the rate of twenty upon the hundred which is the ⅕, or of any other part, and that he would likewise know what part the other loseth of his principal: he must take for the numerator of the broken number of him that loseth, as much as for him that gaineth, them add the numerator and the denominator (of the broken number of him that gaineth) both together, & make thereof the denominator of the broken number of him that loseth, & then shall you have the part of him that loseth, as by example, of him that gaineth after ten. li. upon the 100 li. which is the 1/10 of his principal: take the numerator which is 1. and make that the numerator of the broken number of him that loseth, then add 1. which is the numerator of the fraction of him that gaineth with ten, which is his denominator, & you shall have 11. for the denominatour of the fraction of him that loseth. Then put one over the 11. and so you shall have 1/11. Thus it appeareth when one merchant gaineth of another after ten upon the hundred, he gaineth the 1/10 of his principal, and the other loseth 9 1/11 which is the 1/11 of his principal. And if he would gain after 20. upon the hundred which is the ⅕ of his principal, the other should loose 16. ⅖ which is the ⅙ of his principal, and so is to be understand of all other fractions.  
4. Two merchants will change their merchandise the one with the other, the one of them hath Seys of 20. s. & 10.d. the piece, to cell for ready money, and in barter he will cell the piece for 23. s. 4.d. & yet he will gain moreover after ten pound upon the hundred pound. The other hath will of 50. s. the hundred to cell for ready money. I demand how he shall cell the C. of will in barter? Answer, say if 20. s. 10.d. which is the just price of the piece of Say, be sold in barter for 23. s. 4.d. for how much shall 50. s. (which is the just price of the C. of will) be sold in barter? Multiply & divide, & you shall find 56. s. Then for because the first merchant gaineth after 10. li. upon the C. li. he maketh of his C. li. 10. li. & consequently the second merchant maketh of 110. li. but 100 li. And therefore say, if the second merchant of 110. do make but 100 how much shall he make of 56: Multiply & divide, & you shall found 50. s. 10.d. 10/11 of a penny, and for so much shall he cell the hundred of will in barter.  
5. Moore, two Merchants will change their merchandise, the one with the other, the one of them hath Taffeta, of 16. crowns the piece to cell for ready money, and in barter he will cell the piece for twenty crowns, and yet he will gain moreover after ten pound, upon the hundred pound. The other hath ginger of 3. s. 9.d. the pound weight, to sell in barter. I demand what the pound did cost in ready money? Answer: say if twenty crowns which is the surprice of the piece of Taffeta, become of 16. crowns the just price, of how much shall come. 3. s. 9.d. which is the price of the overselling the pound of Ginger? Multiply & divide, and you shall find 3. s. Then, for because that the Merchant of Taffeta will gain after the rate of ten upon the hundred: say if 100 do give 110. what shall 3. s. give? Multiply and divide, and you shall find 3. s. 3.d. ⅗ and so much did the pound of ginger cost in ready money.  
6. Moore, two merchants will change their merchandise the one with the other, the one of them hath Worsteds of 25. s. the piece to cell for ready money, and in barter, he will cell the piece for 33. s. 4.d. and yet he loseth after ten upon the hundred: the other hath wax of 3. li. 6. s. 8.d. the hundred to cell for ready money. I would know how he should cell his wax in barter? Answer: say if 25. s. which is the just price of the piece of Worsted be sold in barter for 33. s. 4.d. for how much shall three pound 6. s. 8.d. be sold, which is the just price of the hundred of wax. Multiply and divide, and you shall find 4. li. 4/9 which is 8. s. ten pence, ⅔ then for because that the Merchant of Worsteds, loseth after ten upon the hundred: Of a hundred he maketh but 90. And therefore, say: If 90. give 100 what giveth 4. pound. 4/9? Multiply and divide, and you shall find 4. 76/81 which is worth 18. s. 9.d. 5/27, and for so much shall he cell the 100 of Ware in barter.  
7. Moore, two Merchants will change their merchandise the one with the other, the one of them hath Worsteds of 5. pound 6. shillings, eight pence the piece to cell for ready money, and in barter he will cell the piece for 6. pound, 13. shillings. 4.d. and yet he loseth after ten upon the hundred, and the other hath Musk of two shillings, nine pence ⅓, the pound weight, to cell in barter? I demand what the pound did cost in ready money? Answer: say if 6. pound. ⅔ which is the overprice of the piece of Worsted, become of 5. pound, ⅓ which is the just price of the same, of how much shall come two shillings 9 pence. ⅓. Multiply and divide, & you shall find 2. 2/9 which is 2.d. ⅔ then for because that the merchant of Worsteds loseth after ten upon the hundred, of a hundred he maketh but 90. and therefore say if 100 give but 90. how much shall 2. s. 2/9 give? Multiply and divide and you shall find 2. s. and so much cost the pound of Musk in ready money.  

Other Rules of Barter, wherein is given some part in ready money.  
WHen a Merchant over selleth his merchandise & he will give also some part of his overprice in ready money as the ½ the ⅓ or the ¼ etc. He must subtract the same part of money from the just price, and also from the over price of his merchandise: and the two numbers that remain after the substraction is made, shallbe that two first numbers in the rule of three and the just price of the second merchant shallbe the third, to know how much he shall oversell the part of his merchandise.  
8. Two Merchants will change their merchandise the one with the other, the one of them hath fine will at five pound the hundred, to sell for ready money, and in barter he will cell it for six pound, and yet he will have the ⅓ in ready money. The other hath cloth of 13. s. 4.d. to sell for ready money. I would know how he shall cell the same in barter? Answer: take the ⅓ of 6. li. which is the overprice of the 100 of wool, & you shall have 2. li. the which abate from 5. li. which is the just price of the 100 of wool & from 6. li. which is the overprice, and there shall rest 3. li. and 4. li. for the two first numbers in the rule of three, then take 13. s. 4.d. which is the just price of a yard of cloth for the third number: Then multiply and divide & you shall find 17. s. 9.d. ⅓: for so much shall the second cell his cloth in barter.  
9 Moore, two Merchants will change their merchandise the one with the other, the one of them hath wax of three pound 6. s. 8.d. the C. to sell for ready money, and in barter he will cell the same for 4. li. 3. s. 4.d. & yet he will have the ¼ in ready money: and the other hath fine Crimson satin of 15. s. the yard to sell in barter. I demand what it is worth in ready money. Answer, Take the ¼ of 4. li. 3. s. 4.d. and abate it from 4. li. 3. s.4.d. and from three pound 6. s. 8. pence, and there resteth 3. li. 2. s.6.d. & 2. li. 5. s. 10.d. for the two first numbers in the rule of three, and 15. s. for the third number which is the overprice of the yard of satin. Then multiply and divide, and you shall find 11. s. And so much did the yard of Satin cost in ready money.  
10. Two Merchants will change their merchandise the one with the other, the one of them hath tin of 50. shillings the hundred to cell for ready money, and in barter he will cell it for three pound 6. s. 8.d. and he will gain after ten upon the hundred, and yet he will have the one half in ready money: and the other hath lead of 3. halfpence the li. to sell for ready money. I demand how he shall cell the pound in barter? Answer: See first at ten upon the hundred, what the three pound ⅓ will come unto, and you shall find that they will come to 3. li. ⅔, which is 13. s. 4.d. of the which, the half which he demandeth in ready money, is 36. shillings and 8. pence, the which being abated from fifty shillings, and also from three pounds 13. shillings 4. pence, there shall rest 13. shillings 4. pence, and one pound 16. s. 8.d. for the two first numbers in the rule of three, which you must put all into halfpence, and three halfpence for the third number, and then multiply and divide, & you shall find 4.d. ⅛, and for so much shall he cell the pound of lead in barter.  
11. Moore, two merchants will change their merchandise the one with the other, the one of them hath steel of 16. s.8.d. the hundred weight to cell for ready money, & in barter he will cell it for 25. s. and yet he loseth after ten upon the hundred, but he will have the ½ in ready money, the other hath iron of 6. s. 8.d. the hundred to cell in barter, I demand what it did cost in ready money? Answer: say if a hundred come but to 90. how much shall 25. s. come to? Multiply and divide, and you shall find 22. s.6.d. of the which number, take the ½ which is 11. s. 3.d. & subtract it from 22. s.6.d. and from 16. s.8. pennies and there shall rest 11. s.3.d. and 5. s.5. pence, for the two first numbers in the rule of three, and 6. s. 8.d. which is the overprice of a hundred of iron for the third number, then multiply and divide, and you shall find 3. s. 2. pennies, 14/27: & so much did the hundred of iron cost in ready money.  
12. Moore, two merchants will change their merchandise, the one with the other, the one of them hath says of 20. s. 10.d. the piece to sell for ready money, & in barter he will cell the piece for 21. s. & he will have the ¼ in ready money: The other hath caps of 35. shillings the dozen to cell for ready money: but he will gain after ten upon the hundred. I demand how he shall cell the same caps in barter? Answer: say if a hundred be worth 110. What shall 35. s. be worth, which is the just price of the dozen of caps? Multiply and divide, and you shall find 38. shillings 6. pence. Then take the ¼ of 25. which is 6. s. 3.d. and subtract it from 20. s. 10.d. & from 25. s. and there shall rest 14. s. 7.d. and 18. s. 9.d. for the two first numbers in the rule of three, and 38. s 6.d. which is the just price with his gain of the dozen of caps, for the third number: then multiply and divide, and you shall find 49. s. 6.d. and for so much he shall cell the dozen of caps in barter.    

¶ The 12. Chapter treateth of the exchanging of money from one place to another.  
FIrst, you must note, that at Andwerpe they use to make their accounts by Deniers de gros, that is to say by pence Flemishe, whereof 12. do make 1. s. flemish, and 20, shillings flemish do make 1. li. de gros.  
1. If I deliver in Flaunders, 500 li. flemish, at 19. s.6. de gros that is to say at 19. s. 6.d. flemish, to receive 20. s. at London, I demand how much I shall receive sterling at London for the said 5. hundred pound Flemishe? Answer,: Say, if 19 ½ give 20/1, what will 500/1 give? Multiply & divide, and you shall find 512. li. 16.s. 4. pence 12/13 of a penny. And so much sterling shall I receive in London for my 500 li. flemish.  
2. If I deliver in London 375. li. sterling, to receive in Andwerp 21. s.9.d. de gros, that is to say flemish, for every pound sterling. I demand how many pounds Flemish I shall receive in Andwerpe, for the said 375. li. sterling? Answer, say if 20/1 give 21. ¾: what will 375/1 give? Multiply & divide, and you shall find 407. li. So many pounds Flemish shall I receive for the said 375. li. star. in Andwerpe.  
3. If I take up money at Andwerp after 19. s. 6.d. flemishe to pay for the same at London 20. s. star. and when the day of payment is come, I am forced to rechaunge the same, and to take up money again here in London to repay the same, so that for twenty shillings, which I take up here, I must repay, 19 shillings 9 d. at Andwerpe. I demand whether I do win or loose, and how much upon the 100 li. of money? Answer, Say. if 19 3/4 give 19 ½, what will 100/1 give?  
Multiply and divide, and you shall find 98. 58/79, the which being abated from a hundred there will remain 1. 21/ 79. And so much do I loose upon the 100 pound of money.  
4. If I take up at London 20. shill. sterling to pay at. Andwerpe 21. s. 8. d. flemish, and when the day of payment is come. I am constrained to take up money again at Andwerpe wherewith to repay the foresaid sum: and there I do receive 22. shillings. Flemishe to pay 20. shill. at London. Now I demand whether I do win or loose and how much upon the 100 li. of money after the rate? Answer, say if 21. ⅔ give 22/1. What will 1000/1 give? Multiply & divide, and you shall find 101. 7/13, from the which abate 100 and there will remain 1. 7/13, and so much shall I gain upon the 100 li. of money.  
The exchange from London into France, is not like as it is into Flaunders but is delivered by the French crown, which is worth 50. sauce Tournois the piece.  
And here must you note that in France they make their account by Deniers Tournois, whereof 12. maketh one sauce Tournois, and 20. sou. Tournois maketh 1. li. Tournois, which they call a Liver, and the French Crown is currant among Merchants for 51. sauce Tournois, but by exchange it is otherwise, for they will deliver but 50. sou. Tourneys, which is. 2. li. 10. sou. Tournois for a crown, or at such price as the takers up of money can agreed with the deliverer. As by Example.  
5. If I deliver 340. li. star. here in London after 6. s. 4. d. sterling the crown, to receive at Rouen, or at Paris 50. sou. Tournois for every crown, I would know how many livres Tournois I shall receive there for my 340. li. star. Answer: say if 6. s. ⅓ do give me 2. li. ½. Tour. what will 6800/1 shil. give (which is the 340. li. reduced into shillings) multiply and divide, and you shall find 2684. livres 4/19 which is worth 4. s. 4/19 Tournois, and so much shall I receive in Rouen or Paris for my 340. li. sterling.  
6. If I deliver in Paris or Rouen, or elsewhere in France 1250. livres Tournois, at 50. sou. Tournois the crown to receive for every such crown, 6. s. 3. d. sterling at London. I demand how much sterling money I shall receive at London for my 1250. pound Tournois. Answer: say, if two pound, ½ do give me 6. shil. ¼, what will 1250/1 give? Multiply & divide, and you shall find 3125. shil. sterling, which maketh 156. pound, 5. shillings sterling. And so many pounds shall I receive at London, for the said 1250. pound Tournois, after 6. shillings three pence for every crown.   

¶ The 13. Chapter treateth of the Rule of Alligation.  
THe Rule of Alligation is so named, for that it teacheth to alligate or bind together divers parcels of sundry prices, and to know how much you must take of every parcel, according to the numbers of the Question.  

¶ Example.  
1. A Goldsmith hath three sorts of Gold. The first is worth thirty Crowns the pound weight: The second is worth 36. Crowns. And the third is worth 45. crowns, and of these three sorts he will make a Sceptre of six pound weight, which shall be worth forty Crowns the pound. I demand how much he must take of every sort?  
Answer: first you must set down the numbers whereof you shall make the Alligation (which are 30.36.38. & 45. orderly the one under the other, as if you should make of them an addition: and the common number whereunto you will reduce them, shall you set on the left hand, which common number in this example is 40. Then mark what sums be lesser than that common number, and which be greater, and with a draft of your pen, evermore lynke two numbers together, so that the one be lesser than that common number, and the other greater than he. For two greater nor two smaller numbers may not be linked together, for they will either be lesser, or else greater than the common number: but one greater number, and one smaller may be so mixed, that they will make the common number. And two greater or two smaller numbers, can never make the common number in due order, as hereafter shall appear.  
After that you have thus linked them, then mark how much each of the lesser numbers is smaller than the common number, and that difference shall you set against the greater numbers, which be linked with those smaller, each of them with his match still on the right hand. And likewise you must set the excess of the greater numbers against the lesser which Bee combined with them. Then shall you add all those differences into one sum, which shallbe the first number in the rule of three, and the second number shallbe the whole massy piece that you will have of all the particulars, the third sum shall be each difference by itself, and by them shall you find out the fourth number, declaring the just portion of every particular in that mixture, as now by the former example, I will make it plain.  
The prices several. The differences.  The common price or number. 40 30 5. A 36 2. B 42 4. C 45 10 D    21.   
21. .6. .5. 21. .6. .2. 21. .6. .4. 21. .6. .10.  
Here in this former example, you see that I have set down the several prices, which be 30.36.42.45. and have linked together 30. with 45. and 36. with 42. The common price 40. I have set on the left side, & the difference of it from every several price, I have set on the right hand, against that sum with the which it is linked. So the difference of 30. from forty is ten, which I set against 45. that he is linked with all, and the difference of 45. about 40 is 5. which I have set against. 30. So likewise, the difference of 42. above 40. is 2. that I have set against 36. And the difference between 36. and 40. (which is 4.) I have set against 42. Then I add all those differences together & they make 21. which I make the first number in the Rule of three, and two the second number, which is the weight of the Sceptre of Gold, and the third number shall be every particulars difference. Then I work by the Rule of three: saying if twenty and one (which is the differences added together) do give me six pound, which 'tis the weight of the Sceptre, what shall five give, which is the first difference?  
Multiply and divide, and you shall find one pound 3/7: so much must I have of the first price.  
Then do likewise with the rest and you shall find 4/7 of the second price, one pound, 1/7 of the third price, and 2. li. 6/7 of the fourth, the which four sums being added together, do make 6. li. which is the total that I would have. And now to prove if the prices do agreed, you shall do thus: first multiply this total sum 6. by the common price 40. and it will make 240. crowns, which you shall keep by itself. And afterward multiply every several sum of weight by the price belonging to the same weight, and if that sum do agree with the first that you kept by itself, then is your work well done, as here one pound, 3/7 is the weight of the sort of gold which is of 30. crowns price. Then multiply 30. by 1. li. 3/7, & it maketh 42. crowns 6/7, which you shall set down. Then multiply 4/7 (which is the weight of the second sort of gold) by 36. which is the price of the same & thereof cometh 20. crowns 4/7: so again 1. li. 1/7 multiplied by 42. doth make 48. crowns. And last of all 2. li. 6/7 multiplied by 45. maketh 128. crowns 4/7. All these added together doth make 240. crowns, agreeable to the former sum of 40. multiplied by 6 And thus I may affirm that this work is well done.  
2. A Taverner hath four sorts of wine, of four several prices, the first of eight pence the Gallonde, the second of ten pence the gallonde, the third of 15. pence and the fourth of 18. pennies. And he will mingle one punchen with all these sorts, so that the Gallonde shall be worth but twelve pence. I demand how many Gallondes he must take of every sort? Answer: First suppose the punchen to hold some certain measure, as to contain 84. gallonds and then the form will be after this sort, as you see hereafter following.  
12 8 3 10 6 15 4 18 2   15  
If 15. do give 18.  
What william. 3. They make 16 ⅘ of the first. What william. 6. 33 ⅗ of the second What william. 4. 22 ⅖ of the third. What william. 2. 11 ⅕ of the fourth   84    

¶ The 14. chapter treateth of the Rule of falsehood, or false positions.  
THe rule of falsehood is so named, not for that it teacheth any deceit or falsehood, but that by feigned numbers taken at all adventures, it teacheth to find out the true number that is demanded. And this (of all the vulgar Rules which are in practice) is the most excellent: this rule hath two parts the one is of one false position alone: the other is of two positions as hereafter shall appear  
Those questions which are done by false positions, have their operations, in a manner like unto that of the rule of three, but only that in the rule of three, we have three numbers known, and here in this rule we have but one (I mean that cometh in operation) unto the likeness whereof we must devise two other, the one multiplying, and the other dividing, as by example.  
1. I have delivered to a banker a certain sum of pounds in money, to have of him by the year 6. li. upon the 100 li. And at the end of 10. years, he paid me 500 li. for all both principal and gain. I demand how much was the principal sum that I delivered at the first. Here you see that there are divers terms: but the chief to work with all is 500 pound which cometh of the other numbers, that is to say, of 10. and 100 for of them is compound the tenor of the question, the practice whereof is thus.  
Let us feign a number at pleasure, and with the same let us make our discourse, even as though it were the principal sum that we seek for. As by Example. Suppose that I delivered him at the first 200. li. the which were worth to me in ten years. 120. li. after the rate of 6. li. upon the hundred pound. Then 120. li. added with 200. li. Do make but 320. li. and I must have 500 pound. Thus you see that I have three terms for the rule of three: the one which shall contain the Question the other two, which I have form artificially, which are 200. and 320: in such sort, that 320. aught to have such proportion to 200, as 500 hath unto the number that I seek: that is to say, unto the true principal sum, then must I have recourse unto the rule of three, after this sort, saying.  
If 320. li. become of 200. li. of how much shall come 500 li. Multiply 500 by 200. and they are 100000. the which you shall divide by 320. li. and thereof cometh 312. li. ½ which is the sum that I deliver at the first. And thus, this rule hath some congruence with the double rule of three.  
2. I have a Cistern with 3. unegal cocks containing 60. pipes of water: And if the greatest cock be opened, the water will avoid clean in one hour, at the second it will avoid in two hours, and at the third it will require three hours. Now I demand in what space will it avoid, all the cocks being set open. Suppose that it will avoid in half an hour, that is to say, in 30. minutes. Then must there avoid at the first cock the ½, which is thirty pipes, and by the second cock the ¼ which is 15, pipes, and by the third cock the ⅙, that is ten pipes, all the which sums being added together do make 55. pipes, but it should be 60. pipes. Therefore I say by the rule of three, if 55. pipes do void in 30. minutes: in how many minutes will 60. pipes void? Multiply and divide, & you shall find 32. minutes 40/55. And in that space will the water avoid if all the cocks be set open.  

¶ Of the Rule of two false positions.  

A Rule.  THe sum of this Rule of two false positions is thus, when any question is proponed appertaining to this rule. First imagine any number at your pleasure, which you shall name the first position, and with the same shall you work in stead of the true number, as the question doth import, and if you see that you have miss. Then is the last number of the work either to great or to little, that shall you note for to be the first error, in the which you have miss with the sign of more, or less, which signs shallbe noted with these figures, 4:—. This figure 4: betokeneth more, & this plain line— signifieth less, that is to say the one signifieth to much, & the other to little: than begin again, & take another number, which shall be the second position, and work by the question as before, if you have miss again, note the excess or want, for that is the second error. Then shall you multiply the first position by the second error crosswise, and again the second position by the first error (& this must always be observed) & keep the two products: then if the signs be both like, that is to say, either both to much, or both to little, abate the lesser product from the greater, and likewise, you shall subtract the lesser error from the greater, & by the remain of those errors, you shall divide the residue of the products: the quotient shall be the true number that you seek for. But if the two signs be unlike, that is to say the one to much and the other to little, than shall you add those productes together so shall you also add both the errors together, and by the sum of those errors, divide the total sum of both the products: the quotient shall be likewise the true number that the question seeketh, and this is the whole rule, as by example.  
3. A man lying at the point of death, said that he had in a certain Coffer a hundred Ducats, the which he bequeathed to three of his friends by him named, after this sort. The first must have a certain portion, the second must have twice so many as the first abating eight Ducats: and the third must have three times so many as the first, less by 15. Ducats. Now I demand how many every of them must have. Answer: first I do imagine that the first man had 30. Ducats, then by the order of the question the second should have 52: & the third 75. These three sums being added together do make 157: & I should have but 100: so that this first error is to much by 57 then I note a part the first position 30. with his error 57 to much after this sort 30.457. Therefore I prosecute my work and I suppose that the first had 24. then by the order of the question, the second should have 40. and the third 57: these three sums being added together, do make 121. & I must have but 100 so the second error is to much by 21. Therefore I note 24.421. under the 30.457. as you may see in the margin of the next side following.  
Then I multiply crossways, 30. (which is the first position) by twelve which is the second error, and thereof cometh 630. likewise I multiply twenty & 4. (which is the second position) by 57 which is the first error, and I find 1368: then because the signs of the errors are both like: that is to say to much, I must therefore subtract 630. from 1368. & there will remain 738 which is the dividend: again I must subtract the lesser error from the greater, that is to wit, 21. out of 57 and there will remain 36. which shall be my divisor. This done I divide 738. by 36. and the quotient will be 20. ½. The which 20. ½ is the just number of the Ducats that the first man had for his part, so consequently the second man had 33. Ducats, and the third 46. ½, as by proof in the margin, may appear.  1 1.  73 8. 20 ½ 36 6. (20. ½ 33 3 46. ½  100   
The like number will also appear, in case the errors were both to little, as in making the two positions by 18. and 20. where you shall find the two errors both to little, the first will be to little by 15. and the second to little by 3. as by perusing this work in the margin you shall well perceive.  
Again if one of the errors were to much, and the other to little, yet shall I have the true number, as before: As if the two positions were 24. and 20. I shall find that the first error will be 21. to much, and the second will be three to little: Therefore I multiply 24. by 3. crossways, thereof cometh 72.  
Likewise I multiply twenty by 21. the product will be 420: These two sums 72. and 420, I add both together because the signs of the errors be unlike, & they make 492. the which shall be my dividend, and again, I add the the lesser error 3. with the greater error 21. and they make 24. for my divisor, them dividing 492 by 24. the quotient willbe 20 ½: as in the margin doth plainly appear.  
And now because you shall not forget this part of the rule, learn this brief remembrance following.  
The signs both like substraction do require.  
And unlike signs addition will desire.  
The meaning whereof is thus if both the errors have like signs, them must the dividende and the divisor be made by substraction, as is taught before, and if those signs be unlike, then must I by addition gather the dividend, and the divisor, as I have done in this last example.  
4. A man hath two silver cups of unegal weight, having to them both, but one cover, the weight whereof is 5. ounces, if the cover be put to the lesser cup, it will be in double proportion unto the weight of the greater, and the cover being put to the greater cup, will be in triple proportion, unto the weight of the lesser. I demand what was the weight of every cup?  
Suppose that the lesser cup did wayghe seven ounces, then with the cover it must weigh twelve, and this weight should be in double proportion unto the greater, therefore the greater must weigh six ounces add unto it 5. ounces for the cover, all will be 11. ounces, but it should be 21. for to have it in triple proportion, unto 7. which representeth the weight of the lesser cup: So that this first error is to little by 10. which you shall note after 7. in this sort 7.— 10.  
After you shall suppose some other number, as 9 & make the like work as before, so shall you found 15. to little, for the second error, which you shall put behind 9 and then work with the rest as above is said, and you shall find that the lesser cup weighed three ounces, and consequently the greater four ounces.  
5. One man demanded of another in a morning what a clock it was, the other made him this answer, if you do add (saith he) the ¼ of the hours which be passed since midnight, with the ⅔ of the hours which are to come until noon, you shall have the just hour, that is to say, you shall know what a clock it was: Suppose that it was 4. a clock in the morning, so should there remain 8. until none: then I take the ¼ of 4. which is 1. and the ⅔ of eight which is 5. ⅓, and I add them together, so I find 6. ⅓ and supposed but 4. therefore this first error is to much by 2. ⅓, which I note after my position thus, 4.42. ⅓: then again I suppose another number, that is to say 9 so should remain but 3. hours until none, I take the ¼ of 9 and the ⅔ of 3. which is 2. ¼ & 2. these I add together and they make 4. ¼: but I supposed that it was 9 therefore the second error is 4. ¾ to little which I note behind my position thus 9.44. ¾.  
And then I multiply crosswise, as before is taught, and because the signs of the errors are unlike, that is to say, the one to much, and the other to little, therefore in this work I must add the productes, and they will be 40. Likewise I add the errors, and they be 7.1/12. Then I divide 40. by 7.1/12, and thereof cometh 5. hours 11/17, and that hour it was in the morning.    

¶ The 15. Chapter treateth of sports, and pastime, done by number.  
IF you would know the number that any man doth think or imagine in his mind, as though you could divine.  
Bid him triple the same number, then of the product let him take thee, ½ if the number be even, or else the greater half, if the same be odd, then bid him triple again the said ½: after say to him that put away, if he can 36.27.18. or 9 from the last number being tripled: that is to say, 'cause him subtellye to put away 9 as many times as is possible & keep the number secretly: & when he can no more take away 9 than to know if that yet there remain any number, bid him abate 3.2. or one, if he can: this done, see how many times 9 you have caused him to abate, for the which keep you in mind so many times 2. & if that you know that he had any thing remaining beside the nines, the same shall also note unto you 1.  
Suppose that he thought 6. which being tripled is 18. whereof the ½ is 9 the triple of that is 27: now 'cause him to abate 18, or 9 or 27. and again 9: but then he will say unto you that he cannot, bid him then abate 3. or 2. or 1. he will say also that he can not: wherefore considering that you have made him to abate three times 9 justly, you shall tell him that he thought 6. for 3. times 2. maketh 6. If he had thought 5. the triple thereof is 15. whereof the greater ½ is 8. the triple of that maketh 24. which containeth two times 9 they are worth 4. & the remain signifieth 1. the which added together make 5. which is the number that he thought.  
2. If in any company, one of them hath a ring upon his finger, and you would know by manner of divining, who hath the same & upon what finger & what joint: cause the persons to sit down in order, & keep likewise an order of their fingers: then separate yourself from them in some certain place, and say unto one of the lookers on, that he double the number (marking the order) of him that hath the ring: and unto the double bid him add 5. and then 'cause him to multiply this addition by 5. & unto the product bid him add the number of the finger of the person which hath the ring: be it that the same last sum did amount to 89. then afterward say to him that he put after the same last number toward his right hand a figure signifying upon which of the joints he hath the ring. As if it be upon the third joint, let him put 3. after 89. and it will be 893: this done, you shall ask him what number he keepeth, from the which you shall abate 250. & you shall have three figures remaining at the lest. The first toward your left hand shall signify the number of the person which hath the ring. The second or middle figure shall represent the number of the finger. And the last figure toward your right hand, shall betoken the number of the joint. As if the number which he did keep were 893. from that you shall abate 250. and there will remain 643: Which do note unto you, that the sixth person hath the ring upon the fourth finger, and upon his third joint.  
But note that when you have made your substraction, if there do remain a cipher in the place of tens, that is to say in the second place, you must then abate from that figure which is in the place of hundreds, that is to say from the figure which is next your left hand, & that shallbe worth ten tenths, signifying the tenth finger: as if there should remain 703. you must say that the sixth person (upon his tenth finger & upon his third joint) hath the ring.  
3. And after the same manner, if a man do cast three dice, you may know the points of every one of them, for if you do cause him to double the points of one die, & unto the double to add 5. & the same sum to multiply by 5. & unto the product add the points of one of the other dice, and behind the number toward the right hand, to put the figure which signifieth the points of the last die, & then shall you ask him what number he keepeth from the which abate 250. & there will remain 3. figures, which do note unto you the points of every die.  
4. Likewise if 3. of your companions, to say, Peter, james, and john that would (in your absence) give themselves every one a contrary name: as for example: Peter would be called a king, james a duke, and john a county: And I would divine which of them is called a king, which the duke, and which the county. Take 24. stones, or other pieces whatsoever, & give unto Peter 1. unto james 2. and unto john 3. or otherwise. But mark well unto which of them you have given 1, unto which 2, and unto whom 3. Then leaving the 18. stones (before them) that are remaining, you shall absent yourself from their sight, or else turn your face from them, saying thus unto them, whosoever nameth himself a king: for every stone that I gave him let him take one of the residue, and he that nameth himself a duke for every stone that I gave him let him take 2 of them that remain, and he that calleth himself a county, for every stone that I gave him let him take 4: this being done approach near them, and mark how many stones are remaining: and know this, that there can not remain any other number, but one of these six, 1, 2, 3, 5, 6, 7, for the which six numbers we have chosen to every of them a several name, which are these: Angeli, Beati, Qualiter, Messiah, Israel, Pietas: each of them containing three Vowelles, a, e, i, which do show the names by order: That is to say, A, 1 2 1 2 3 3 2 1 3 3 1 2 3 3 2 1 2 1 a e a e i i e a i i a e i i e a e a 1 2 3 5 6 7 A B Q M I P showeth which is the king, E, telleth which is the duke and, I, showeth which is the county: in following the order how, & to whom you have given one stone, to which 2, and to which 3. Then if there do remain but one stone, the first name Angeli, (by these three vowels a, e, i,) showeth that Peter is the king, james the duke, and john the county. And if there do remain 2 stones, the second name Beati, shall show you by these three vowels, e, a, i, that Peter is the Duke, james the King, and john the county. And so of the other as by this Table doth plainly appear.    
FINIS.   

Here beginneth the Table of this Book.  

¶ The contents of the Chapters of the first part.  

THe definition of number. fol. 1.  
Numeration in whole number fol. eodem. Cap. 1.  
Addition in whole number. Cap. 2. fol. 6.  
Substraction. ca 3. fol. 9  
Multiplication. ca 4. fol. 12.  
Division. ca 5. fol. 21.  
Progression Arithmetical and Geometrical. ca 6. fol. 32.  
The Rule of three called the Golden rule. And the backer rule, unto all these is added their proofs. ca 7. fo. 34    

The Contents of the second part.  

¶ The first Chapter showeth what a fraction or broken number is. fol. 43.  
Reduction of fractions. ca 2. fo. 44.  
Abbreviation. ca 3. fol. 50.  
Addition in fractions. ca 4. fol. 54.  
Substraction. ca 5. fol. 58.  
Multiplication in fract. ca 6. fol. 61.  
Division in broken numb. ca 7. fo. 64.  
Duplation, Triplation, and Quadruplation. ca 8. fo. 68  
All the proofs of fractions. ca 9 fol. 69.  
Questions done by Reduction, by Addition, Substraction, Multiplication, and by Division in broken numbers chapter. 10. fol. 71.    

The Contents of the third part.  

Rules of Practice called brief rules. chap. 1. fol. 79.  
Rules of three compound, being 4. in number. ca 2. fol. 99  
Questions of the trade of Merchandise. ca 3. fol. 101.  
Questions of loss and gain in the trade of merchandise. ca 4. fo. 108.  
Questions of divers breadthes and lengths of tapestries. ca 5. fol. 113.  
Questions of the reducing of paumes of Genes into yards. ca 6. fo. 115.  
Questions of merchandise sold by weight, with brief rules for the same. chapter. 7. fol. 116.  
Questions of tars and allowances. ca 8. fol. 119.  
●●●stions done by the double rule of 〈◊〉 ca 9 fo. 22.  
Questions 〈◊〉 〈◊〉 Rule of fellowship. chap. 10. fol. 124.  
Questions of bartering. ca 11. fol. 138.  
Questions of the exchanges▪ chapter 12. ●●l. 146.  
Questions of the Rule of Alligation. chapter. 13. fo. 146.  
Questions of the Rule of falsehood. chapter. 14. fol. 152.  
Questions of sport and pastime, done by number.    
Finis.