❧ The store-house of brevity in woorkes of Arithemetike, containing as well the sundry partes of the Science in whole and broken numbers, with the Rules of proportion, furthered to profitable use: As also sunderie rules of brevity of work, of rare, pleasant, and commodious effect, set forth by Dionis Gray of London Goldsmith. 1577.  
¶ Imprinted at London for William Norton, and john Harison, dwelling in Paules Church-yard.   

To the right honourable Sir Jhon Langley knight, lord Maior of London, & the other worshipful Maisters, wardens and whole assistaunts of the mystery of Goldsmithrie, Dianis Gray, a member of the same, wisheth virtuous prosperity.  
AS I had conference with myself, right honourable and worshipful, of the great utility, delectation and estimation, procured to every commonweal, by the sciences mathematical, and how much more unto those countries, wherein knowledge and understanding of the said sciences doth flourish & abound over and above other the same wantyng. And also noting the diligence of sundry authors of most Nations in their vulgar languages in writing of the premises, to profit their countries. I of good will not inferior to any other, to profit the commonweal, whereof I am a member, so far forth as with most diligence and understanding, by the goodness of God I might haue habilitie: I was moved thereby to employ some endeavour, wherein my good will in part might appear, treatyng of the premises, not without great hope to prefer many things, in every of the said Sciences, as may bee found of rare and commodious effect, for most vocations and degrees of people. And for that arithmetic is the ground, direction, and producer of the most parte of such harvest, as in the fertile fields of the said sciences is to bee reaped. I haue therefore framed this rude discourse of the Arte of numberyng, the first fruits of my good will, dedicated to your wisedoms, containing as well the sundry partes of the said science, in whole and broken Numbers, the same appliyng to several uses, for furtherance of common utility, as also many and sundry Rules of breuetie of work, no less profitable, then rare to bee seen in any author, Englishe or other. beseeching your wisedoms, to haue more regard to my good meaning herein, then either to my boldness, or rude order in penning of the same: & as the effect of my diligence, may procure contentation, or benefit to any vocation in the commonweal, so I may haue cause, not onely to rejoice of my travail, but also encouraged to further other works of greater consequence, therein assisted by the goodness of the almighty, who increase your honour and usurps with grace, wisdom, and godly felicity.   

To the Reader.  

DElightyng who it bee,  
in Sciences Mathematical,  
every princely practise,  
in order to define:  
Note that arithmetic,  
of all the rest is principal,  
joined with the other,  
in sisters loving line,  
So pleasyng divine sapience,  
the effect to assign.  
Gained is thereby most,  
in the rest desired,  
refusal else is made,  
of that might bee required.   

¶ All earth, Plattes, and Edifices,  
by measure to advance,  
In the circuit of the world,  
how so ever it bee framed:  
Of sides, ends, Angles, and points,  
number sheweth distance,  
Formed in Globe, Square, and Cube,  
or other title name,  
geometry and astrology,  
confess and not ashamed.  
arithmetic your derection,  
in most ye do pretend,  
Without whose secret ingeny,  
your praise were half at end.   

¶ The describyng of the Sphere,  
with marks celestial,  
The placyng of the signs,  
the zodiac round about,  
The passing of the Planetts,  
the great and eke the small:  
By number hath distinction,  
no cause therein to doubt:  
The course of son and moon,  
in several race and rout  
Of tropics and other zones,  
the Artique and Australl:  
By number is showed the distance,  
and of each Meridianall.   

From horizon to Pole,  
from Pole to equal line:  
From each of them the zenith,  
true distance for to see:  
By instrument mathematical,  
and in proportion fine:  
By number is brought forth,  
in high and low degree:  
The Astrolabe, Quadrant,  
staff and rule or compass what it be:  
Are not of right perfection,  
to serve without excuse:  
Except the partes divided be,  
by number to show the use.   

¶ To show the aspects of Planetts,  
within the ecliptic line:  
Whereby the health of man,  
the learned doth procure:  
Coniunction, Opposition,  
Quadrant, sextile and trine:  
By number is the mean,  
most certainly and sure:  
Hath been, is and shal be,  
for ever to endure:  
An Ephemirides for to frame,  
no man can, or make well:  
Except in science of numberyng,  
such as do excel.   

¶ Howe of the world the time doth pass,  
to make true computation:  
By the course of Phaebus,  
both violent and natural:  
The one by day the other by year,  
in sundry sorts and fation.  
By number is dilated,  
for knowledge universal:  
So by the race of Luna,  
for a sure memorial:  
Of the floods, fulles, and faules  
of Seas at time and tide,  
By number is made known,  
how for ever to abide.   

¶ All Armony in Musicque,  
to memory recreatife:  
By voice of men or Instruments,  
to further and to frame:  
With mood, tense, Ray and Note,  
Minnom, Long and brief:  
Or other divided part,  
what ever it haue to name:  
By proportion is appointed,  
the service of the same:  
discords to disappoint,  
and in Musicque to difface:  
By number is performed,  
concords to put in place.   

¶ The coniunction of Billion,  
by quantety propotionall:  
Of Gold, silver and their Alloyes,  
to every apt degree.  
By Arithemetique is furthered,  
in orders many and several,  
And some of them more admirable,  
then credible seem to bee,  
As such whose vocation shall,  
theffecte procure to see,  
may find the pen a loadstone,  
an assaier to direct,  
And not of less perfection,  
then the fire to correct.   

¶ Likewise of the premises,  
to make true valuation,  
Aswelt in things mysterious,  
as other more in general:  
There is no mean so near to proche,  
by any maner of fashion,  
As by rule most intricate,  
of Arithemetique especial,  
far hide from many which do it want,  
for whom it right effectual:  
which if were known with perfectness,  
as truth doth say it is:  
Would it esteem accordynglie,  
and not such knowledge mis.   

¶ The Treasure of traffics trades,  
who wisheth to procure,  
With accoumptes must bee acquainted,  
his doings to address:  
Or else the things he hopeth of,  
most times shall find unsure.  
And not haue mean as else he might,  
to mend it more or less,  
For Companies and exchanges,  
to make a sure access,  
And Moneis, receipt, and Measures,  
in order to reduce,  
Of several rules Arithemeticall,  
required is the use.   

¶ Men, Money, goods, and debts,  
or charge what else it bee,  
To bring in Debitour and creditor,  
as most men do require:  
The state of all things, how it stands,  
most needful for to see,  
By Arithemetique is accomplished,  
even as ye can desire,  
Wherefore they heed it busily,  
affairs therewith t'attire.  
In such order of account kept  
as other all excel:  
As to all such refer report,  
which knoweth the'ffecte right well.   

¶ The time to me which appointed is,  
though it were many daies,  
Could not suffice me to direct,  
each thing in order right:  
which might with truth be furthered,  
te'xtoll the noble praise,  
Of this most singular Science,  
the load star of great light:  
The truth whereof is witnessed,  
to many mens learned sight.  
Wherefore I end with that is paste,  
wherein who would haue skill:  
Procure to win Arithemetique,  
and finds the rest at will.   

¶ And here to God give laud and praise,  
blessing his holy name,  
For all his gifts of nature and grace,  
received from above:  
Who is the author of all goodness,  
and giver of the same,  
Employed vpon all yearthly wights,  
by his most tender love:  
He gives us grace to render thankes,  
as duty doth behove.  
Who save our queen her state, and realm,  
for whom also give praise,  
And pray her grace may reign in rest,  
long time in joyful daies.   
Vale. {quod} D. Graye.   

The book to youth.  
ALL little imps in commonweal,  
which wisdom would attain,  
apply your mindes with Apollo his train,  
and so to honour win:  
And reap the wealth more worth then gold  
renown shall bee your gain:  
With other lores as you frequent,  
Arithemetique begin,  
And every part from first to last,  
to memory call you in:  
which hear appears in order set,  
much profit to enlarge:  
To such as gain them as they go,  
and heedyng this my charge.   

The contents.  
THe first parte containeth sundry partes of arithmetic, that is to say.  

1. Numeration.  
2. Addition.  
3. Substraction.  
4. Multiplication.  
5. division.  
6. Reduction.  
7. Progression.  
For practise by whole Numbers.   
THe second parte, containeth the said partes serving for practise of broken Numbers or fraccions, that is to say.  

1. Numeration.  
2. Progression.  
3. Reduction.  
4. division.  
5. Multiplication.  
6. Substraction.  
7. Addition.   
THe third part containeth the sundry Rules of proportion, furthered by use of the foresaid partes, that is to say.  

The Rule of three. 

1. Direct.  
2. Backer.  
3. Double.  
4. Compound.  
5. Of Company, with time and without  
6. Of Aligation  
7. And Position.     
THe fourth parte containeth sundry Rules of breuetie, whereof the number is more, then needful perticulerlie to bee set down, wherefore I refer the Reader to the whole matter, which to many may be found of rare and profitable effect, pleasant and change of practise.   


Of Numeration.  
NVmeration containeth the maner how to express the value of any somme or number. which occasion may present, being small or great, and is furthered by ten Carecters or Figures following, to say.  
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. one two three four five six seven eight nine ten j. ij. iij. iiij. v. vj. vij. viij. ix. x  
The which nine Figures in proper signification of value equal to the words and Letters under them set, being separate with prick or line. Howbeit, being set together and mixed without prick or line of separation: than an increase of value they receive, by virtue and property of the place: wherein they stand, which places being of number infinite, do yield unto every unity of any Figure: ten times so much in any place toward the left hand, as that same unity is worth in place next to it towards the right hand, the effect whereof more plainly may appear by the Table following, for the same purpose furthered.  


The Table of Numeration.  100000000 10000000 1000000 100000 10000 1000 100 10 1 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 Hundreth Millions. ten Millions. Millions. Hundreth thousands. ten thousands. Thousands. hundreds. gins. unities.   
THE nine unities set above the vpper line of the Table, do signify the value of every unity in the Figures against any of the same under the line, and that by help of cyphers made like the Letter. 0. The which being of no value in proper signification, the same notwithstanding, they are of necessary use in practise of arithmetic, only to keep the places, whereby is expressed infinite numbers, which without help of them, the other Figures could not perform, as by the forenamed unities, with cyphers before them, the effect may appear for the figure of one, in the first place is there but one: But in the second place is ten, by help of the Cipher set thus. 10. So in the third place a hundreth thus 100. And so understand of all the rest infinitely.  
The titles written under the Table, serve also to show the value of every unity found in the figures standing in the places above the titles, as 9 in the first place is but nine, in the second place it is nine times ten, in the third place nine hundreth, and so forth infinitely. And thus much may seem sufficient for an introduction for the understanding of Numeration, which is to show the value of any number, which occasion may procure to be known.  
Howbeit, it may seem necessary hear to make distinction of certain terms, belonging to number. Not for use of any thereof in this parte, but for help in other partes, by the said terms furthered, as in place of their need hereafter will appear, which terms are to say, numbers Diget, Article, and compound, or Mixed.  
The Diget numbers are not onely every one of the nine characters or Figures standing alone: but also sometime are found amongst mixed or compound numbers remaining in works under ten. between 10 and 20. between 30 and 40. & so forth 100. &c.  
Article nobers are such as are furthered by cyphers, & no mixed figures with them, as 10. 20. 30. 40. and so for the infinitely.  
The mixed numbers are set together thus 123. 542. 3045. and every such like, either sundry figures together, or figure together and cyphers between. But if a Cipher bee found in the first place toward the right hand of any number, then every such number is an Article number. &c.   

Of Addition.  
ADdition containeth the maner how to assemble, and join sundry particular sums or numbers, into one total. As if three sundry men should owe unto a merchant, three several sums. The first 548. li. the second 1346. li. and the third 15. li. The which to bring into one total, ye shall set the said three several particulars together, one right under an other, to say: unity under unity, ten under ten, hundreth under hundreth, and in like maner infinitely in this order hereafter appearing.   li. Particulars. 548 15 1346 total. 1909 The which particulars set in order accordynglie, you shal draw a line under them, and then resort unto the unities, placed ever in the first place towards the right hand, all those unities added together as 5. and 6. make 11. and thereto 8. make 19. the which found, set the digette number, which is all above 10. being 9. under the line, as you see, and for 10. the Article, you shall retain one in memory, to bee born to the second place, for 10. in the first place, is but one in the second place, and ten in the second place, is but one in the third, and so from place to place infinitely. Thus having ended the work of the first place, finding 5. 6. and 8. to make 19. whereof the 9. set under the line in the first place, and for the article 10. one kept in memory, then say one in memory, and one found in the second place make 2. the which added to 4. standing over 1. make 6. wherewith 4. standing over 1. make together 10. the which being an Article, set a Cipher under the line in the second place, and for 10. there found, bear one to the third place, the which put to 3. and five there standing, making together 9. to bee set under the line in the third place, then coming to the fourth and last place finding one, set for the same 1. under the line in the fourth place, and so the work is ended, and the total is found 1909. li. of equal value to the particulars.  
In the practise whereof is to be seen the order of Addition, in every occasion thereof furthered to be performed. And for to amplify the effect, take here a few lines in verse.  

¶ Of sundry sums particulars,  
one total for to frame,  
Set them down right orderly,  
as work doth best require:  
What place ye give to any one,  
the rest let haue the same,  
So may you well perform the'ffecte,  
of that you do desire.   

¶ To the figures in first place set,  
first see ye do resort,  
And of the somme which they do make,  
set diget under line:  
And for each ten in article found,  
one shall ye thence transport,  
unto the next and second place,  
by memory right fine.   

¶ And so all unities joined in one,  
by right of every place,  
And for more order duly kept,  
from first unto the end:  
So is the work at full performde,  
required in this case,  
What so ever circumstance,  
some other may pretend.   
Hereafter is set sunderie examples of whole numbers in practise, whereof may be seen the effect before taught, with the order of proof of the same.  
These four examples may give the learner occasion, to examine his understanding in the precepts before given of Addition. And also to note the order of proof, of the same in works of the whole numbers, whereof the effect( I mean of proof) consisteth in casting away every 9. found in the simplo figures of any example, without respect of place: first the particulars above the line, and the remain above every 9. cast away, set at the vpper end of a Burgunion cross, in maner before appearing. Then so many times as 9. is to be found in the total of the same example, cast them away also, and the remain set at the lower end of the said cross. Then if the 2. remaines, the one at the head, and the other at the foot of the cross be equal, then the work is true, and else not, as practise of the first example may more at large manifest.  
To prove the first of the four former examples, repair to the first place, where standeth the figures of 4. 2. and 9, in the particulars, whereof 9. cast away, there is 6. to be joined with 8. 4. 8. in the second place, which making 26. and 18 thereof cast away, the remain is .8. to be joined with 5. 8. and 4. in the third place, which making 25 and therof twice 9 cast away, the rest is 7 to be joined with 6. 5. & 6 in the fourth place, which making 24. and thereof twice 9 cast of, the remain is 6. to be joined with 3. 6 and 5. in the fift and last place which making 20. and thereof twice 9. cast away, the remain is 2. to be set at the top of the cross as remain for the particulars. Then resort to the total, where is found 519851 and make together 29. whereof thrice 9 cast away, which is 27. the remain is 2. to set at the foot of the cross for remain of the total, and for that the figures in the top and bothom of the cross are like and equal, therefore the Addition is well and truly made, and so for all other examples wrought in whole numbers.  
Thus much may seem to suffice for the work of whole numbers, howbeit there is some variety of work in the diminute partes, of many and innumerable things of sundry Denominations. But for that it is not possible to writ of all matters, I haue furthered some examples of Moneys, weights and Measures, as most apt for the purpose in commune: referryng all men to th' appliyng the same order to matters in private vocation, serving their occasions.  
first is to be noted, that in sommyng of many particular sums of Money, containing pounds, Shillings, Pence, farthings and mites, first give heed howe many Mites make one farthing, and that being 6. you shall for every 6 mites eary one farthing to the place of farthynges, and the remain in mites under 6. you shall set under the line against the mites, which stand next the right hand. Also when you come to the place of farthings, consider that as 4 farthings make one penny: so for every 4 farthings carry one penny to the place of pence, and set the remain under 4. under the line against the farthings likewise. As 12. Pence make one shilling: so in the place of Pence carry for every 12. one shilling to the place of shillings, and the remain under 12. set under the line in the place of pence. Also for every 20 shillings, carry one pound unto the place of pounds, and the remain of shillings under 20 set under the line against the shillings, and so with pounds being whole numbers, carry for every 10 one from place to place, as before is taught, &c. and for the further understanding of the effect, hereafter is set down sundry examples of Moneis.  
li s d q mi.   li s d q mi. 12. 15. 7. 1. 4. 25. 11. 10. 3. 4 23. 16. 7. 2. 5 48. 9. 11. 2. 3. 34. 17. 9. 3. 3 59. 13. 9. 3. 2 71. 10. 1. 0. 0 133. 15. 8. 1. 3  
Other examples wherein Farthings are omitted, and the mites are born for every 24. one penny.  
li s d mits           54. 12. 3. 17 24. 16. 3. 23 36. 15. 7. 11 32. 13. 4. 18 42. 10. 11. 9 53. 15. 9. 13 65. 17. 8. 22 41. 18. 2. 16 199. 16. 07. 11 153. 03. 8. 22  
Here may bee seen in the former of the 2 last examples, that the myts being 22. 9. 11. and 17. make together. 59. whereof 48. for 2 pence taken away, the remain is 11. to set under the line, then the 2 pence joined with 8. 11. 7. and 3. d. make together 31. d. whereof 24 for 2 s. taken away, the rest is 7 to set under the line. Then the 2 still born to the place of shillings, with the other there standing, make 56. shilligns, from the which 40 for 2 pounds taken away, rest 16. s. to set under the line, and the 2 li. born to the first place of the pounds, and joined with the other Figures make 19. li. whereof the Diget 9. is set under the line, and for the Article. 10. one is carried to the second place, and with the Figures there standing make 19. whereof the Diget 9 is set under the line in the second place, and for the Article 10. one is carried to the third place, and so the work is ended, wherein appeareth the variety of work between whole numbers and broken in the practise of Addition.  
For proof of adding the Diminute parts there is no better, then double perusing the examples or Additions made. Howbeit whan you come to the whole numbers, you haue to consider what unities are born from the place next before the whole, and with them joined, for those born unities make a parte of the total, of the said whole Numbers, and therefore in making the proof, must be parcel of the particulars, when the nienes are cast away, for otherwise the remain of the particulars, after the nienes cast away will be so much less than the remain of the total, as by proof of the former and last practised example the effect may appear.  
The total of the foresaid example being 199. the Figures make together 19. whereof twice 9 cast away, rest one to put under the cross of proof, as doth appear. Then add all the Figures of the particulars together, and they make 35. whereof thrice 9 cast away, rest. 8. and agreeth not with the remain of the total Wherefore to that 8 put 2 which in the Addition was brought from the place of shillings, and that maketh 10. whereof 9 cast away rest one, equal to the rest of the total. And so the work found true.  
Hereafter is set sundry examples of Addition of receipt and measures, referryng the learner to the maner heretofore shewed, giving good heed to the number of unities in a smaller denomination contained in an unity of a greater, and accordingly to bear from place to place in former order.  

¶ Examples of Additions of receipt.  
C. q. li. onz.   C. q. li. onz. 34. 3. 16. 13 25. 3. 22. 15 52. 2. 18. 11 28. 1. 17. 9 24. 3. 12. 9 84. 3. 25. 14 112. 1. 20. 1 139. 1. 10. 6  
C. q. li. onz.   C. q. li. onz. 53. 1. 21. 6 35. 2. 18. 11 62. 3. 25. 11 46. 3. 20. 13 58. 3. 23. 14 57. 2. 12. 10 44. 1. 17. 8 68. 3. 22. 14 219. 3. 4. 7 209. 0. 19. 00  
To make these former Additions of receipt and such like. first it behoveth the learner to understand, that the hundreth weight at the Common beam in London containeth 112. lib. haberdepoiz, the half hundreth 56. lib. the q. 28. lib. & the pound 16. onz. The which known, carry in Addition for every 16. onz one pound to the place of lib. for every 28. lib j. q. to the place of quarters, for every 4. quarters one hundreth to the place of hundreds, and so the work is well performed.   

Examples of Addition of Measures.  
yards q. nails.   yards q. nails 31247. 3. 2 7568. 1. 2 57689. 2. 3 6756. 2. 3 68754. 3. 3 8573. 3. 2 157692. 2. 0 22898. 3. 3  
Yards foot Inch.   yards foot Inch. 656 2. 7 7869. 1. 5 645 1. 9 6543. 2. 8 784 2. 8 9586. 2. 10 978 1. 3 8594. 1. 11 3065 2. 3 32594 2. 10  
To make the former Additions of measures and such like, it behoveth the learner to understand that the yard is divided into sundry Diminute partes, that is to say. For the measuryng of velvet. silks. cloth, Lace, and sundry other things. The yard is divided into 4 quarters, and every quarter into 4 nails, and accordingly the additions of such partes furthered, as before appeareth. And for the measuryng, of Timber, Wainscottes, Séelings, pavements, Land, and such like things, the yard is divided into 3. foot, the foot into 12 Inches, and the inch into 3 barley cornes ordained by Statute for Standard measure of England, and according to such Diminute partes, the Additions of those denominations are furthered, as before likewise may appear.    

¶ Substraction.  
SVbstraction containeth the maner how to deduct or take away a smallar somme or number from a greater, by practise whereof is found and brought forth a remain sought for & desired, as if one man owe unto an other 356 li. whereof he hath paid 234 li. and would know what restend unpaid. Then when the payment is rebated from the debt, the remain will appear as practise by the same sums the effect will manifest.  
  l. debt 356 paid 234 rest 122 Here is to bee perceived the debt being the greatest somme is placed uppermost, and the payments under the same. unity under unity, cen under ten, and hundreth under hundreth, and a line drawn under all and so made apt for the work.  
Then to perform the Substraction, resort to the first place, which is of unities and say. 4 paid taken out of 6. of debt, the remain is 2. to set under the line in the first place, then say 3 paid out of 5 of debt in the second place rest. 2. to be set under the line in the second place also, 2 out of 3 in the place of hundreds rest. 1. set under the line in the third place, and so the work ended, wherein appeareth that 234 li. taken out of 356. the remain appeareth 122 li. &c.  
When you haue made a substraction, and would prove, whether you haue made a true rest or not, then add together the rest, and the paimente, and if the total agree with the first debt, then the Substraction is true, or else not. Whereof the practise hereafter sheweth the effect by the same numbers, whereof the former substraction was made.  
debt. 356 paid. 234 rest. 122 proof. 356 Here doth appear that 2. of rest, added to 4. of payment, maketh 6 under the line in the first place: also 2. rest with 3. paid, maketh 5. under the line in the second place, and so 1. rest with 2. paid, is 3. to set under the line in the third and last place: and so the total being 356. li. equal to the debt, prove the work true, which otherwise would not be.  
Thus much may seem sufficient for the work of substraction, where the figures of the lesser number, are smaller then the figures standing right over them in the greater number, but when the contrary is found, then the work is of more difficulty, as by example.  
debt. 3576. li. paid. 2989. rest. 0587. proof. 3576.  
Here resorting to 9. in the first place of payment, to be taken out of 6. over it, which can not bee doen, therefore borrow an unity of 7. in the second place of the debt to join with 6. in the first place, and so haue you 16. from the which 9. paid, rebated, rest 7. under the line in the first place. Then not forgetting the unity, borrowed of 7. in the second place, to make the work of the first, say one that was borrowed with 8. in the second place of payment, maketh 9. to be taken out of 7. above, which can not be, wherefore in former order, borrow an unity of 5. in the third place of debt, to make 17. in the second, from the which 9. aforesaid rebated, the remain is 8. under line in the second place. Then say as before, 1. borrowed of 5. in the third place joined with 9. under 5. maketh the 10. to be rebated from 5. over 9. which can not be, but by the help of an unity, borrowed of 3. in the fourth place, and so 10. from 15. rest 5. under the line in the third place. lastly say one borrowed of 3. with 2. paid in the fourth place, make three to be taken out of three of debt in the same place, and so remaineth nothing, wherefore a Cipher is set under the line, in the fourth and last place, and so the work ended. Wherein doth appear that 2989. li. Substracted from 3576. li. the rest vnpaied is 587. li. The proof whereof is by adding the payments and remain together, and the total thereof agreeing with the debt, proveth the work true, as before is taught.  
A like or more difficulty is found in work of Substraction, when the places in the debt haue few or no figures, but Supplied with cyphers, for that the work requireth a borrowyng of an unity in every place of want, from one place to an other, unto the end, as by example the effect may appear.  
Debt. 302003. l. paid. 135976 rest. 166027 proof. 302003 Here 6. out of 13 made by help of one borrowed in the second place of the debt rest 7. under the line for the first work. Then to pay that was borrowed say, one and 7. make 8. to be taken out of 10. in the second place, by the help of one borrowed in the third place, and so remaineth 2. under line in the second place. again, one with 9 in the third place, make 10. to be taken out of 10 above made by one borrowed in the 4. place, and the rest is nothing, and therefore a Cipher under line in the third place. Also in the same maner, one and 5. maketh 6. to be taken out of 12. made by one borrowed in the fifte place, and so resteth 6. under line in the fourth place. So again one and 3. make 4. out of 10. and so rest 6. in the fifte place. lastly one borrowed with one in the sixth and last place, make 2. to be taken out of three over, and so rest 1. in the same place, and the work finished, and as aforesaid, the payments and rests together, making again the debt, prove the work true.  
Thus much may seem sufficient for the practise of Substraction in whole numbers: howbeit to further the understanding of the learner, take these few lines in verse.  
¶ When diget of debt, is not so great,  
as that in payment made,  
Then next place lend, to wantyng friend,  
to help this pleasant trade.  
And in repair, one with thee bear,  
to paiyng second place:  
So with his fear, which standeth there,  
pay thou that borrowed was.  
Thus to procede, in work with speed,  
from place to place I say:  
The rests in fine, set under line,  
agreeing to thy pay.  
When resting due, with payments true,  
the debt again do make.  
Then is well doen, which was begon,  
that dare I undertake.  
When occasion presenteth works of Substraction, of diminute partes, of what denomination so ever. Then like consideration is to be had( as was noted in Addition) what quantity of unities in one denomination, is contained in an unity of an other denomination: and accordynglie make the Substraction, whereof the effect in sundry examples following may appear.  

Of Money.  
  li. s. d. mites. debt. 65. 17. 16. 19. paid. 52. 12. 9. 15. rest. 13. 05. 7. 4. proof. 65. 17. 16. 19.  
  li. s. d. mites. debt. 8764. 12. 7. 11. paid. 5897. 17. 9. 18. rest. 2866. 14. 9. 17. proof. 8764. 12. 7. 11.  
In the former of these two examples, the work is performed with great facility, howbeit in the second there is found more difficulty, for that the figures in the payments are for the most parte, greater then in the debt: wherefore in the place of mites wantyng, borrow one penny, which is 24. mites, and then perform the work, so borrow one shilling, which is 12. d. to supply the want of pence. Likewise borrow 1. li. which is 20. s. to supply the want of shillings, and then your rests set down, and the unities borrowed, born in memory truly, to answer every one in his place, then you can not fail to make good work.  

¶ Other examples where in the debt is no figures, but one in the last place.  
  li. s. d. mites. debt. 500. 0. 0. 0. paid. 368. 11. 9. 16. rest. 131. 8. 2. 8. proof. 500. 0. 0. 0.   li. s. d. mites. debt. 4032. 0. 0. 0. paid. 2978. 15. 10. 17. rest. 1053. 4. 1. 7. proof. 4032. 0. 0. 0.    

Examples of waightes.  
  C. q. li. onz. Bought. 52. 3. 16. 11. received 37. 2. 12. 8. rest. 15. 1. 04. 3. proof. 52. 3. 16. 11.  
C. quar. li. onz. 40. 0. 0. 0. 25. 3. 16. 14. 14. 0. 11. 2. 40. 0. 0. 0.   

Examples of Measures.  
  shepherds. quar. nailes. sold. 5684. 3. 2. delivered. 3879. 2. 1. rest, 1805. 1. 1 proof. 5684. 3. 2. shepherds. quart. nailes. 3000. 0. 0. 1978. 3. 2. 1021. 0. 2. 3000. 0. 0.  
  shepherds. foot. inches. Bought. 6523. 2. 7. hermits 4879. 2. 5. rest. 1644. 0. 2. proof. 6523. 2. 7. shepherds. foot. inches. 8000. 0. 0. 5684. 1. 10. 2315. 1. 2. 8000. 0. 0.    

Multiplication.  
MVltiplication containeth the maner how to find the number of unities, of a smaller denomination, in an other number of unities, in a greater denomination contained. The effect whereof is better to understand with few examples, then with many words. And for that it is necessary for every learner, to understand the content or somme, produced by multiplication of one diget by an other, before he can much profit without the same, therefore is prepared a Table for the effect thereof: and notes given for understanding, and use of the same, as hereafter appeareth.  

¶ The Table of Multiplication of digettes,  1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81  
To understand the use of this Table, note that the Digettes Multiplicatours are set in 2 Collombes, to say, in the highest side of the table, in places distinct from one unto. 9. and likewise in the left end of the Table, also from 1 unto 9. And all the rest of the table, except those 17. places, wherein the Digets do stand, are places for the several products of Multiplication of one of the said Digets by an other. And when you would know the somme or product of any such multiplication, as for example, of 5 and 7. take the one in the one column, and the other in the other, and in the place where the 2 Collomnes meet, wherein the said figures do stand: the one proceedyng from one end to the other, and the other descending from the higher side to the lougher, there shall you find 35. for product or somme sought for, and in like order may you find any other desired.  
The effect furthered in the former Table, Let every desirous learner haue perfectly in memory: for that the same had the works of multiplication are performed with more facility, then by want thereof is possible. And for aid of memory note, when 2 digets are to be multiplied together, consider if any of the same may bee partend in halves, and so found multiply the contrary diget by the one half, and the double of that product is the somme ye would haue.  
For example when 7. is to be multiplied by 6. you see 6. may be partend into twice 3. wherefore say 3 times 7. the contrary Diget maketh 21. then by former note the double thereof being 42. is the product of 6. and 7. multiplied together, and so of all other.  
When 9. is to be multiplied by 9. 7. or 5. then shal you put 9 in 3 partes, as into thrice 3. and whith one of the said partes multiply the contrary diget as 5 being admitted for example, saying 3 tymeg 5 is 15. the which taken 3 times, maketh 45. which is the product of 9 and 5 multiplied together.  
Likewise for 7 by 9 say 3 times 7. is 21. the which treble, maketh 63. the product desired.  
Also for 9 by 9. say 3 times 9 is 27. the which treble maketh 81. the product sought for.  
Note that when 9 is to bee multiplied by 8. 6. or 4. it is better to mediate or half any of the same, then to tripertate or put 9 into three partes.  
Wherefore when 9 is to be multiplied by 8. say 4 being the half of 8 and multiplied with 9 maketh 36. The double whereof being 72. is the product of 8 and 9 being multiplied together.  
So for 9 by 6. say 3 the half of 6 multiplied in 9. maketh 27. the double whereof being 54. is the product of 6. times 9. &c.  
More to be said for the understanding of the former Table, or maner to find the product of any 2 Digetts, multiplied the one by the other, might seem superfluous, wherefore now I will show the order of Multiplication of one number of figures by an other.  
First there is to bee noted in Multiplication three numbers by several names. Distinct, that is to say, the multiplicand, which is the number to bee multiplied, the multiplicator, which is the multiplier, and the Product which is brought forth by the work, the effect whereof to be shewed by example, may be performed in sundry and infinite matters, whereof take this that followeth, to me seemyng very apt for the purpose, for an entrance thereunto.  
If you bye 5678. shepherds of cloth, costyng 86. pence every yard, & would know how many pence the whole amounteth, then the number of shepherds shal be multiplicand, and stand uppermost in work, and the number pence shal be set under the same for multiplicator. unity of pence under unity of shepherds, ten under ten, and so forth, when both partes haue the places supplied, with Figures. Hundreth under hundreth, thousands under thousands, and so infinitely in maner, as in examples hear may appear.  
5678 86 34068 45424 488308 The which 2 numbers set down as you see, and a line drawn under them, then the Product of the work willbe so many Pence, as will pay for 5678 shepherds of cloth, at 86 Pence the yard, which is 488308 pence, as appeareth in the total of the particular Productes added together, as the order of work requireth, whereof the manner followeth. first you shall resort to the place of unities, and say, 6 times 8. is 48. whereof the Diget 8 put under the line in the place of unities, and for the article 40. you shall retain in memory 4 unities to bee born to work of the second place. Then say 6 times 7 is 42. and 4 retained in memory, maketh 46 in the second place, whereof the Diget 6. is set under line, and for 40 retain 4 in memory to bee born to the work of the third place. Then say, 6 times 6 is 36. and 4 in memory maketh 40 an Article number, and therefore put a Cipher under the line in the third place, and retain 4 in memory( that is) for every 10. of the Article one, to bee born to the work of the fourth place. Then say, 6 times 5 is 30. and 4 in memory is 34. whereof the Diget 4 is put under line in the fourth place, and for the Article 30. bear three to the fift place, and because the Multiplicand hath no Figure in that place, therfore put 3. retained in memory under the line in the same place, and so the work is ended, for. 6. the first Figure in the Multiplitour, whereof the Product particular, is 34068. as doth appear. Then resort to 8. in the second place of the multiplicator, and therewith multiply every Figure of the multiplicand in former order saying. 8 times 8. is 64. whereof put the Diget 4 under the line right under 8. the multiplicator, and bear 6. for the Article 60. in mind for the second work, and say, 8 times 7. is 56. and 6 in memory maketh 62. whereof put 2. under line in the second place of that second rank, & bear 6 for the next work, saying, 8 times. 6. is 48. and 6. in mind maketh 54. whereof put 4 under line and bear. 5. for the Article to the next work saying, 8 times 5 is 40. and 5 in memory maketh 45. whereof put 5 under line, and for the Article carry 4 to the next place where no Figures found to make further work, set it under the line, and so the multiplication is ended. Then add together the 2 particular Productes, and the totoll thereof will contain so many Pence as doth amount of 5678. shepherds of cloth at 68 d. every yard, which is the effect desired in the example furthered, and so of other works.  
Here is to be noted, that for every Figure in the multiplicator of any example, there is a particular Product, and every Diget made in the first work of any of the same shalbe set under the Figure multiplicator, in what place soever it stand, and the Article numbers to bee transpozed for every 10. in any place found one to be carried by memory to the next place toward the left hand, to bee joined with the unities made by the work in the said place. All and every the which precepts well vnderstanded, are sufficient for the practise of Multiplication, howbeit hereafter is set down sundry examples, wherein the effect aforesaid doth appear, and for a further aid to the learner hereafter are furthered a few lines in Verse.  
By Multiplicator, and Multiplicand,  
a Product out to find:  
give each his place, as taught thee was,  
that bear thou well in mind:  
What doth amount, in multiplied account,  
set Diget under line:  
Of Figure together. 2. one by the other,  
in first place do assign:  
The Articles convey, by memory I say,  
to next and second place:  
To work there made thou shalt them add,  
and so hold on thy race.  
Thus mayst thou haue a Product brave,  
pretended by thy pain:  
If thou proceed as work doth need,  
the end till thou attain:  
Of Figures just, thou mayst me trust,  
in multiplier to be seen:  
So many will there productes appear,  
in order as I ween:  
The which found out, then go about,  
in one them all to frame:  
So hast thou ended that was pretended,  
without suspect of blame.  
Hereafter are set down sundry examples for the practise of multiplication, by 3. 4. and 5. Figures in the Multiplicatours.  
6547382 345 32736910 26189528   19642146   2258846790 47869524   132 95739048 143608572 47869524   6318777168    
7654321 5462 15308642 45925926   30617284   38271605   41807901302 12435264   8643   37305792 49741056 74611584 99482112   107477986752    
  987654 51423 2962962 1975308   3950616   987654   4938270   50788131642 92837451   15263   278512353 557024706 185674902 464187255   92837451   1416978014613    
The proof of Multiplication is made by casting away all the Neines first in the multiplicand & the remain set at the one side of a cross, than the remain of the multiplicator set at the other side therof. The which 2 remaines multiply together and from the result thereof cast away all the nyenes and set the remain at the vpper end of the cross. Lastly cast away all the nienes in the product, & set the remain at the foot of the same cross, the which performed, if the remaines at the top and foot of the cross be equal, the work of that multiplication is true, and else not, as by an example may appeore.  
345627   4532   691254 1036881 1728135 1382508 1566381564    
In the multiplicand of this example, the Figures make 27. which is 3 times 9 and nothing remaineth, wherefore I set a Cipher at the right side of the cross as you see. Likewise the Figures of the multiplicator make 14. whereof 9 cast away, the remain is 5. at the left side of the cross. Then saying 5 times nothing is nothing, wherefore I set a Cipher at the vpper end of the cross. Lastly the Figures of the product together make 45. which is 5 times 9. and nothing remaineth, wherefore I set a Cipher at the foot of the cross. And for that the top and foot of the cross are like, I know thereby the work of that multiplication to be good, and so of all other, whereof the effect appeareth in the former examples. Howbeit for that in the proof of the former example, the cyphers are to many to show the whole effect of the order of proof, here is given an other example to amplify the same.  
47356   2573   142068 331492 236780 94712 121846988    
In the multiplicand of this example the figures together make 25. whereof twice 9 cast away, rest 7 put at the right side of the cross. The figures of the multiplicator make 17. whereof once 9 cast away rest 8 at the left side of the cross. Then 7 and 8 being multiplied together make 56. whereof 54 cast away, for 6 times 9. rest 2 to set at the vpper end of the cross. Lastly the Figures of the product make together 47. whereof 45 for 5 times 9 cast away rest 2 to set at the foot of the cross. And the Figures of the top and foot of the cross being like and equal, proveth the work true, as afforesaid.  
An other perfect and sure order of proof, of Multiplication is made by division, the which here I omit, till I haue shewed the practise of division, which hereafter followeth.   

¶ division.  
division containeth the maner how to show the number of times, that a small somme or number is contained in a greater, and the effect is procured in occasions infinite. And to the practise therof belongeth three numbers by several names distinct, that is to say. The dividend, which is the number to be divided. The divisor, which is ever in whole numbers lesser then the Diuidende. And the Quotient which sheweth the number of times, that the divisor is contained in the Diuidende. As for example if occasion procured to bee known how many pounds were contained in 396 Nobles. Then 396 is divided and the number of Nobles contained in one pound, which is 3. must bee divisor. The which dividend divided by the said divisor, the Quotient willbe found 132. which are so many times as 3 Nobles which is 1 pound, are contained in 396 Nobles, whereof the effect by example hereafter is practised.  
Here the dividend 396 being set down, then the divisor 3. is set under 3. in the Diuidende, and the work begun in the last place toward the left hand, for that is the order in works of division, though therein it bee contrary to the other partes which ever begin at the right hand: then is to be sought how many times 3. the divisor is found in 3, the dividend. and that being one time therfore 1 is put in a place separate from the rest as you see, & so the first work ended and nothing remaining in 3. the dividend, and therefore it and the diuisour is canselled with a dash of a pen, thereby to signify the work to be ended in that place. For the second work 3 the diuisour is set under 9. in the dividend, and is found to be contaided therein 3 times, and therfore 3. is put in the Quotient, and so the second work ended, one therefore 9 and 3 canselled as in the former work. Lastly, three the diuisour is put under 6. in the dividend, and is found to be contained therein 2 times, and nothing remaineth, wherefore 2 is put in the Quotient, and the whole work is ended. And by the Quotient is found that 1 pound being 3 Nobles, is contained in 396 Nobles 132 times, which is the effect required in the work.  
Here is to bee noted, that the cyphers set over every figure of the deuidende, are there set to signify nothing to remain, after the work in the place, made under any of the said cyphers, and oft times put so in works, more for help of memory, then for other need.  
Note also, that when the last Figure of any diuidende, is lesser then the divisor, then the divisor shall bee set under the Figure in the last place save one of the diuidende, and so work in former order. How bee it, to make the matter more plain, the effect shal appear in an example following.  
If you would know how many pounds are contained in 2758. crowns, then for that 4. crowns make one pound, therefore 4. must bee divisor, and set under the diuidende, in the last place saving one, for that in 2. in the last place 4. is not contained, and so the work practised as followeth.  
 
In this example you may see 4. the divisor set under 27. wherein it is contained 6. times, and 3. remaining: therefore 6. is put in the Quotient, and 3. the remain is set over 7. and so the first work ended, and the divisor in that place, and the 27. over it canselled, as afore taught. Then for the second work, the divisor is set under five in the diuidende, the which with 3. remaining of the former work, maketh 35. wherein the divisor 4. is contained 8. times, and 3. remaining, wherefore 8. is put in the quotient, and 3. remaining set over 5. and so the second work ended, and therefore the divisor, and 35. over it canselled in former order. again for the third and last work, the divisor 4. is set under 8. the which with 3. remaining in the former work, maketh 38. wherein the divisor 4. is contained 9. times, and 2. remaineth put over 8. and so the whole work ended, the divisor and the 38. being canselled as the other, and the 2. remaining of the whole work, is separate from the rest, to signify the same to be a remain of the work, and not sufficient to contain 4. the divisor. By the which work doth appear, that in 2758. crowns, 4. of the same making one pound, is contained in the whole somme 689. times, and two crowns remaining, which is the effect sought for by the work, wherein the perfect order of division is shewed, where the divisor is one figure onely.  
howbeit, when the divisor containeth a number of Figures, as more then one bee it few or many: Then the quotient shall ever bee made with that Figure of the divisor, which standeth next toward the left hand, and none other. And the quotient so made, shall be multiplied by the rest of the Figures of the divisor, one after an other, and every product shall be rebated out of the diuidende, standing right over the Figure of the divisor, which maketh any of the said productes from place to place throughout, for every figure of the quotient made. And the quotient shall be made no greater then that a remain may be left in every work, out of which the said productes may be taken accordingly, as in example practised hereafter, the effect more plainly may appear.  
If you would know how many pence are contained in 56847. mites. Then the number of mites making one penny, shall be deuisor which is 24. and set under the diuidende thus.  
 
In this example are 4. Figures in the quotient, the which are made by 4. several works in the dividend. And for the first you shall set 24. the divisor, under 56. in the diuidende, and say, how many times 2. in 5. and that is 2. times, whereof 2. you shall put in the quotient, and set 1. remaining over 5. Then the 2. in the quotient, multiplied by 4. in the divisor, produceth 8. to be taken out of 16. remaining in the dividend over 4 and so the first work ended 8. remaining of 16 set over 6. then shall you cancel your divisor 24 and 56 in the dividend with 1 remaining over 5. and so you haue finished all things belonging to the first work. Then shall you set your divisor 24. under 88 and say, howe many times two the divisor in 8. over it, and that is 3 times, and 2 remaineth to set over 8. The which 3 put in the quotient, and multiply the same by 4 in the divisor, and the product being 12. rebated out of 28. rest 16 over 28. then cancel all the Figures of the divisor the diuidende and of the remaines under & behind 16. and so the second work is ended. Thirdly you shall put your divisor 24 under 164 and say, how many times 2 in 16. and that is 6 times, & 4 remaining to set over 6. 16 being canceled, then 6 multiplied by 4 in the divisor, the product is 24 to bee taken out of 44 so resteth 20 over 4 the divisor, and the third work ended, and then is to be canceled all figures of the divisor diuidende, and remaines under and behind 20. Lastly, you shall put the divisor 24 under 207 and say how many times 2 in 20 over it, and that is 8 times, and 4 remaineth over the cipher in the second place: then 8 in the quotient multiplied by 4 in the divisor produceth 32. to bee taken out of 47. and there remaineth 15. and the whole work ended, and therefore the said 15 is to be separated from all the other figures of the work plainly to appear, all the other being canceled. And so is found that in 56847 mites is contained 2368 pence, and 15 mites remaining, which is the effect in the work required.  
Because that it is hard for a learner to understand the work of division where the example is practised in one place. Therfore in an other example the works shall haue so many several distinctions as there shalbe figures put in the quotient: for every figure of the said Quotient doth require a particular work, which is not easy to bee perceived in the former example, as in an other following may appear.  
If you would divide 856942. by 354. you shall set down the dividend first, and the divisor under, as before is taught, and as hereafter appeareth.  
 
Then say how many times 3 the divisor in 8. over it, and that is 2 times and 2 remaineth, wherefore 2 for the times is put in the quotient, and the 2 of the remain is put over 8. and so the work for the Quotient making in that place is ended. Then 2 in the Quotient must bee multiplied by 5 in the divisor, and that maketh 10 to bee taken out of 25 over it, and so will remain 15 over 5. and all the figures under 15 to bee canselled. Also you must multiply 2 in the quotient by 4 in the divisor, and that maketh 8. to bee taken out of 6 over 4 which can not be but by borowyng an unity out of 5 to make 16. by order in Substraction, from which 8 aforesaid rebated, the rest is 8 over 6, and 4 over 5. and so the whole work for making the first figure put in the quotient is ended.  
To procede in the work, you shall set down the deuidende, which is 148942. and set the divisor under it thus.  
 
Then say, how many times 3. in 14. that is 4. to bee set in the quotient, and 2. remaining over 14. the said 14. and 3. in the divisor canselled. Then say 4. in the quotient, multiplied by 5. in the divisor, maketh 20. to be taken out of 28. over 5. & there remaineth 8. Likewise say, 4. in the quotient, multiplied by 4. in the divisor, produceth 16. to be taken out of 9. which can not bee, but by help of an unity, borrowed of 8. to make 19. them 16. out of 19. rest 3. over 9. and 7. over 8. and so the work for making the second Figure in the quotient ended, all the figures canceled, under and behind 73.  
For the third work, you shall set down the diuidende remaining, which is 7342. and the divisor under it, thus.  
 
Likewise say, how many times 3. in 7. that is 2. to bee set in the quotient, and there remaineth 1. over 7. being canselled, and also 3. the divisor under it. Then multiply 2. in the quotient, by 5. in the divisor & that is 10 to be taken out of 13. and so remaineth 3 the 1 over 7. being canceled. Also multiply 2. in the quotient, by 4. in the divisor, and the result is 8. to be taken out of 4. which can not be but by help of an unity, borrowed of 3. and so 8. out of 14. rest 6. over 4. and 2. over 3. all the figures under and behind 26. being canceled, and so the work for the third figure in the quotient ended.  
For the fourth and last work, you shal set down the diuidende remaining, which is 262. and the divisor under it, thus.  
 
finally say, how many times 3. in 2. over it, and that is no time, wherefore set a Cipher in the quotient, to supply a place, and cancel 3. Then say 5. times nothing is nothing to be taken out of 6. and therefore 6. remaineth over 5. being canceled. Likewise say 4. times nothing is 0. to bee taken out of 2. over 4. and therefore 2. remaineth, and 4. to be canceled, and so the whole work ended, the quotient made at four several woorkynges, and found to bee 2420. and so many times is the divisor 354 contained in the diuidende, 856942. which is the effect in the division sought for, and required. And the remain of the work is but a parte of a time, and therefore to bee set over the divisor thus 262 / 354, which signifieth that the divisor is not contained in the same, and therefore appeareth a remain, and accordingly made to appear.  
And where for finding of the former quotient, there hath been made 4. particular works, to say, for every Figure or Cipher, one several practise, you shall understand, that such maner of distinctions is not furthered, but onely for help in teaching, but the division to bee made in one place, and the quotient to bee brought forth in one practise, in all divisions generally, whereof the maner is hereafter practised in example, by the former diuidende, divisor, and quotient, in which work is to be seen the former 4. particular works all in one.  
 
The sunderie precepts and practices of division before shewed, well noted and vnderstanded, thereby any division is to bee made with facility: howbeit, further to note that when the divisor containeth a greater number of figures, then hath been in any example before practised. Then every of the same shall multiply the quotient, and the product taken out of the remain over it, and other difficulty is not found, wherefore I will practise sunderie examples, wherein the effect may appear, with order of proof of the work, and so proceed to the next parte, and for aid to the learner, here is furthered a few lines in verse.  
¶ When the diuidende and divisor,  
are known and how to stand:  
Then to the place, make thou repair,  
next toward thy left hand.  
So oft as the divisor there,  
in diuidende is found:  
By one figure the same declare,  
in quotient art thou bound.  
The which thy quotient newly made,  
with figure most behind:  
Thou shalt it multiply with the rest,  
throughout unto the end.  
And every result see thou rebate,  
from rests in diuidende:  
over the figure made multipliar,  
so rule may thee defend.  
Then see all figures canceled be,  
except remainers made:  
Of every one thou wroughtest vpon,  
whereby true quotient had.  
The first work finished, again begin,  
divisor removed a place:  
By like order as erst was used,  
and alter not the case.  
until last figure of divisor,  
under last of diuidende seen:  
For there is made an end of all,  
so truth doth say I ween.  
What doth remain when work is doen,  
set over divisor fine:  
In severed place from all the rest,  
between them both a line.  

¶ Examples.  
  

¶ Examples.  
 
 
These 4. Examples are set down, as well to give the learner occasion to examine his skill in the practise of division, as also to see th'order of proof of the work by sume men allowed, which is an uncertain proof, by casting away the nynes by order used in the other partes. For howbeit that in three of the examples, the effect of the said order of proof agreeth with truth, the same notwithstanding in one of the four, the uncertainty of that order of proof doth plainly appear: which order followeth.  
first they cast away all nynes in the divisor, and the rest they set at the one side of a cross, then they cast away all the nines in the quotient, and set the remain at the other side of the cross, and multiply those two remaines together, and add the result to the remain of the same division( if there be any) of which total all nynes cast away, the remain is set at the vpper end of the cross: and lastly all nynes cast away in the diuidende, and the remain set at the foot of the cross, and found to agree with the figure in the top of the cross, then the work is allowed to be good, or else not. The which appeareth true, in three of the former examples, but one of the four is found contrary, and therefore the Rule not worthy to be allowed.  
But when you desire to prove any division, then multiply your quotient with the figure or figures that was divisor, and to the result add the remain of the division if any be found, and that total making the dividend or Somme that was divided, then the work is true and else not.  
Likewise if you would prove the truth of any multiplication, divide the result by the Multiplicator, and the quotient making again the Multiplicand, the work is true or else not. So that the most certain proof of multiplication is by division, and of division by multiplication, of the which the effect hereafter may appear in Reduction, by examples several for both those partes, wherefore that is aforesaid may seem sufficient for the practcse of division.    

¶ Of Reduction.  
REduction is no proper part of arithmetic, for howbeit that the change of one denomination unto another, or the alteration of things from one title to an other may well bee termed Reduction. The same notwithstanding, the effect is performed by Multiplication or division, or else both. nevertheless, for that the learner may haue experience howe things are reduced and altered in name and property: the substance or value remaining perfectly, greatly to his contentation and commodity. I therefore think convenient to show some examples therof in such place as other haue furthered it as a parte of arithmetic, though as you may perceive the effect furthered by Multiplication and division, as aforesaid.  

Reduction of Money by Multiplication.  
If you would reduce 586 li. into Pence, that is to say, if you would know how many Pence are contained in 586 li. the same you may perform by 2 manners. The one is by Multipliyng 586. by so many pence as are contained in one pound which are 240. The other is by bringing 586 li. into shillings, by multipliyng the same by 20 which are the number of shillings in one pound, and so brought into shillings the same to be multiplied by the pence of one shilling, which are 12. and so at 2 works the said 586 li. brought into pence of every the which orders the effect is hereafter practised by example.  

Example practised by the first Order.  
  li. 586   240 d. 23440   1172   140640 d.   

Example practised by the second Order.  
  li. 586   20 s. 11720 s. 12 d. 23440   11720   140640 d.  
Thus appeareth plainly by 2 maner of practises of Multiplication, that in 586 li. are contained 140640 d. and so the denomination changed from pounds to pence, and therefore said to be reduced.  
If you would reduce 140640 pence into pounds, that is to know how many pounds are contained in the said number of pence. Then you shall divide the said pence by so many as maketh one pound, or else first bring the same into shillings, dividing by 12 which are the number of pence in one shilling, and to bring the said shillings into pounds by 20 which are the shillings of one Pound, and so the said pence by 2 manners are brought into pounds, whereof the effect hereafter is practised by example.   

Example practised by the first maner.  
  

Example practised by the second Order.  
 
Thus appeareth also that in 140640 d. are contained 586 li. by 2 maner of practises of division, by the which may appear not onely the efecte of reducing things of one denomination to an other. But also the perfect order of proof of Multiplication, and division the one by the other as aforesaid.  
A further difficulty is found, when sundry denominations are to bee reduced into one, as if occasion required to bring 749 li. 15 s. 5 d. all into mites, for then the most convenient order is to multiply the pounds with the shillings of 1 pound which is 20. and to the result is to be added the 15 s. appearing alone, and then the total of shillings to bee multiplied with 12 d. in one shilling contained and to the product is to be added the 5 pence standing alone, and that total multiplied by 24 mites in 1 penny thereby is brought forth the whole number of mites in the foresaid somme contained. whereof the effect hereafter by practise appearing.   

¶ Example.  
li. s. d. 749. 15. 5 20 s.   14980 s.   15 s.   14995 s.   12 d.   29990     149955     179945 d.   24 mites. 719780     359890     4318680 mites.  
Here appeareth that in 749. li. 15 s. 5 d. is contained 4318680 mites, the sundry denominations reduced into one by Multiplication.  
Likewise to bring 4318680 mites into pounds that is to bee performed by division as in example practised the effect may appear.  
 
Here you may see that. 4318680. mites divided by the mites of one penny, which is 24. yieldeth in quotient 179945 pence, the which also divided by pence of one shilling, which is 12. yieldeth in the Quotient 14995 shillings and 5 pence remaining. The which shillings also divided by 20. contained in one Pound yieldeth in quotient 749 li. and 15 s. remaining. The which 749 li. 15 s. 5 d. is the original of the former examples of Reduction, first reducing the same from great denomination and small terms, into a small denomination and great terms by practise of multiplication. And contrariwise reducyng the same again, from small denomination and great terms into the first kind, of great denomination and small terms, by division wherein appeareth how to understand of Reduction, and the same to be performed by Multiplication or division as aforesaid, whereof to give further precepts needeth not, howbeit to show the learner wherein partly the effect to apply, here followeth sundry examples of Reductions of weights, measures moneys, by Exchunge for sundry countries.    

¶ Reduction of weights.  
C. quart. In 52. 3. 24 lib. What the whole in pound weights.  
C. qu. li. 52. 3. 24. 4     208     3     211. quart. 28     1688. li.   422     24     5932. li.    
To reduce receipt from one denomination to an other, requireth an understanding of the several denominations, belonging to the kind of receipt, procuryng a reduction: wherefore note, that the hundred weight at the common beam of London, containeth 112. li. the half hundreth 56. li. the quartern 28. li. The pound weight containeth 16. onzes, and are called receipt Haburdepoise. By which kind of receipt, the former example furthered, note that 52. li. is multiplied by 4, quarters, and yieldeth 208. quart. to the which is added 3. quart parcel of the example, the total whereof multiplied by 28. li. contained in one quart. yieldeth 1688. l. to the which is added 24. li. parcel of the example: the total whereof being 5932. pound, is the effect sought for in the example, which is the number of pound receipt, contained in 52. hundred 3. quart. 24. pound wrought by multiplication. The which to transpose again into the first kind by division: hereafter the effect appeareth by practise.  

¶ Example.  
In 5932. pound, what hundreth receipt Habardipoise.  
   

Reduction of Measures.  
In 568. shepherds, what Inches are contained?  
  shepherds. 568   36   3408   1704   20448 Inches.  
In 20448. Inches, what shepherds are contained?  
 
Thus you may see that 568. first multiplied by 36. Inches, contained in one yard produceth 20448. Inches, the which again divided by 36. yieldeth in the quotient 568. shepherds, agreeing with the former declaration.  
Here it is necessary to note, that when occasion requireth reduction of one denomination into an other, when neither of the same are of greatest, nor smallest denomination, belonging to the quality of that thing, which requireth the reduction, then the denomination to be reduced, requireth multiplication thereby, to bee brought into the smallest denomination needful, that by division of the same, it may bee brought into the other denomination, which the occasion searcheth: for any thing in small denomination, may bee turned into sundry sorts of greater, as sundry occasions may require, as by several examples hereafter the effect may appear.  
In 364. Nobles, of 6. s. 8. d. the piece, what crowns of 5. s. piece.  
These forenamed Nobles, being multiplied by 80. d. contained in one Noble, produceth the whole number of pence in those Nobles contained, the which pence divided by 60. contained in one crown, yieldeth in quotient so many crowns, as are contained in the said Nobles, the which being the effect of the former note, hereafter appeareth in example practised.  

Nobles.  368.     80     29440. d.       crowns. 490    
In, 490. crowns, and 40. d. remaining what Nobles?  

crowns,  490. 3. s. 4. d. 60     29400.     40.     29440. d.   Nobles. 368.    
By the same maner, when Englishe money is to be reduced into french crowns, spanish ducats, flemish Gildrens, or-Dolars, the somme of money being brought into pence, then it is denomination apt to bee divided by the number of pence, being price by exchange of the said crown, ducat, Guildren, or dolor, the effect likewise hereafter appearing by example.  
To make over by exchange 100. lib. starling for france, at 4. s. 9. d. every crown, for spain at 5. s. 10. d. every ducat: For flanders at 3. s. 11. d. the Guildren: or other place at 4. s. 3. d. the dolor.  
first, reduce the said 100. l. into pence which maketh 24000. the which divided by 57. d. the price of the crown for france, yieldeth in the quotient so many crowns as 100. li. maketh at that price, or for spain divided by 70. d. the price of a Duckett, yieldeth in the quotient so many ducats as 100. li. maketh at that price, or for flanders divided by 47. yieldeth in the quotient so many Guildrens as 100. l. maketh, and lastly divided by 51. d. yieldeth in quotient so many Dolars as 100. lib. maketh, whereof the effect hereafter by examples appeareth.  
lib. crowns The pence of 100 421 5 / 57 The price of the   french crown.     d. Pence of 100 li. duke. price of the Spanish duket 342 ● / 7 li.   Pence of 100 Guildrens price of the Guildren 510 30 / 47 li.   Pence of 100 Dolars price of the Dolar 470 3● / 5●  
By these examples appeareth that 100 li. made into france by exchange at 4. s. 9 d. the crown, maketh 421 3 / 57 Crownes.  
Also 100 li. made into spain by exchange at 5 s. 10 d. the ducat, maketh 342 ducats 6 / ●  
Likewise made into flanders at 3 s. 11 the Guildren, maketh 510 Guildrens 3● / 47  
And at 4 s. 3 d. the Dollar maketh 470. collars 30 / 5●.  
Here note that the exchange for flanders is for the most parte furthered by the pound, as well flemish as starling, whereof some examples follow.  
To make over to Andwarpe 100 li. starling at 24 s. 8 d. Flemish the pound starling, reduce the said 100 li. into flemish money by Multipliyng the same by 296 d. which is the price flemish of the pound starling, and the product willbe so many flemish Pence as the said 100 li. starling is worth at the price, the which Pence divided in order as afore taught, for English money yieldeth in quotient so many flemish pounds as the said 100 li. starling amounteth to by exchange, whereof the effect by example practised hereafter appeareth.  
li.   100 starling. 296 d. flemish. 600   900   200   29600 pence.  
 
Thus appeareth that 100 li. starling reduced into flemish Pence by multipliyng the same by the price of the li. maketh 123 li. 6 s. 8 d. flemish  
But when Flemish Money is to be made from thence into England, then you shall reducd the same into pence, and divide the total by the price of the Englishe pound, and so find in the Quotient the Englishe Money desired as in practise may appear by example by 100 li. Flemmishe at 4 s. 10 d. starling.  
li. 100 Flemmishe. 240 4000 200 24000  
li. s. d.     80. 10. 8. 253 / 298 starling  
Here appeareth that 100 li. Flemmishe multiplied by 240 d. contained in one pound produceth 24000 d. The which divided by 298 d. the price of the li. starling yieldeth in Quotient 80 li. 10. 8 d. and a parte of a penny, and is the value of a 100 li. Flemmishe at 24 s. 10 d. the li. starling.  
Likewise weights of what denomination soever being reduced into the smallest denomination needful, may be changed into any other Denomination required, as by examples may appear.  
If you would reduce Quintalles, containing 100 li. weight simplo or subtle into hundreth weights containing 112. lib. or to the contrary the C. at the beam in London into Quintalles. Then bring the denomination to be reduced into pound weights by multiplication, and being in pound weights, they are apt to bee brought into the other denomination by division.  

¶ Example.  
In 54. Quintalles. What C. weighs.  
  

¶ proof.  
In 48. C. 24. lib. what Quintals.  
112     96     48     4824     5400 54 Quintals.  
Likewise for measures, to turn shepherds into elles, or elles into shepherds: either of the same brought into quarters of a yard by multiplication, the other may bee brought to the denomination required with facility.   

¶ Example.  
In 568 shepherds, what elles.  
4   2272. quarters.  
  Elles. 454 ⅖  
In, 568 Elles, what shepherds?  
5     2840     2840 710. shepherds. 444      
Also Inches may be brought into feet, by dividing 12. into shepherds, by 36. into elles, by 45, and so of all other things.  
Thus much may seem sufficient, to give understanding for the effect of reduction, the which as aforesaid, not to bee accounted a parte proper of arithmetic, but rather an application of multiplication, and division to sundry things, whereof the practise profitable to bee known unto learners, wherein occasions growing to infinite effect.     

¶ Progression.  
PRogression arithmetical is a short and brief maner, adding sundry figures or numbers set down, every one( after the first) increasing by equal quantity, as 1. 2. 3. 4. 5. 6. 7. 8. 9. there is increase by an unity: also 2. 4. 6. 8. 10. 12. the increase is by 2. again 3. 6. 9. 12. 15. 18. &c. every number of the Progression is augmented by 3. more then an other. The which progressions, and all other like, are to bee sommed by rule of Progression with much more facility then by Addition, as by example the effect may appear.  
There is to bee noted, that if the times of the Progression bee odd, then the first and last numbers added together, and the half of that total, multiplied by the number of times of the progression, the product thereof will be the just somme of the said progression, as by example plainly may appear.  

¶ Example.  
 
Here appeareth the times of the progression to bee 9. and the first number 1. Added with the last which is 9. maketh 10. the half whereof being 5. multiplied with the times of the progression, which is 9. produceth 45 the just somme of the whole progression as by Addition is proved, and this is the perfect rule of Progression when the times be odd.  
Howbeit, when the times of Progression be even, then add the first and last together and multiply that total with half the number of times of the Progression, and the product will be the just somme of the Progression, as may appear likewise by example.   

¶ Example.  
 
Here the times of the progression being 8. whereof the first and last making 9. and multiplied by 4. the half of the times, produceth the somme desired, which is 36. as by Addition is proved.  
It may seem necessary to note one general Rule for both, the former which is to multiply the one hole, with the half of the other. As the first and last being odd, multiply the same by half the times of the Progression which then is ever even, and if the first and last be even, then with half thereof multiply the times of the Progression being even or odd, and so find the just somme desired.  
Some may by reading understand these former Rules, and yet want experience how to apply them, wherefore not amiss to give some example such to content. Wherefore somewhat thereof followeth.  
A Lordship is offered to sale, to be paid the first day of. 45. next following 20. shillings: the second 40. shillings: the third 60. shillings, and so every day 20. shillings more then an ather, till 45. might be ended, the question is, what the somme will amount unto.  
according to the first of the former rules, add the first and last numbers of the Progression together, as 1. pound with 45 pound, and that maketh 46. the half whereof 23. multiplied by the times of the Progression, 45. produceth 1035. pound, the somme required in the question, as by addition may appear.   

¶ Example.  
1. 45 1 11 1 2 12 46 3 13   4 14 23 5 15 45 6 16   7 17 115 8 18 92 9 19 1035 10 20 55 155 21 31 41 22 32 42 23 33 43 24 34 44 25 35 45 26 36 215 27 37 355 28 38 255 29 39 155 30 40 55 255 355 1035  
These examples, as well teach the practise of Progression, as also sheweth the difference of facility of the same, from the tedious use of Figures in Addition, the effect whereof well noted may suffice, for progression Arithemeticall.  
A Lapidarie sold, a Iuell to bee paid the first week of 52. in one year 1 crown the second, 2 Crownes, and so every payment one crown more then an other. 52. times. It is demanded what number of Crownes the whole Progression amounteth.  
according to the second of the former rules add 1 crown the first number with 52. the last, and that maketh 53. the which being multiplied with. 26. half the times of the Progression produceth. 1378. Crownes the just somme of the Progression, as by Addition will appear.   

Example.  
1 52   1 11 21 31 41   51 1 2 12 22 32 42 52 53 3 13 23 33 43 103 4 14 24 34 44 455 26 5 15 25 35 45 318 6 16 26 36 46 355 106 7 17 27 37 47 255   8 18 28 38 48 155 1378 9 19 29 39 49 55   10 20 30 40 50 1378   55. 155. 255. 355. 455    
A merchant sold 100. shepherds of cloth to bee paid in 40 weeks, to pay the first week 2 s. the second 4 s. the third 6 s. so every payment 2 s. more then an other, till 40 weeks expired. It is demanded what money the said 100. shepherds of cloth doth amounnt unto.  
according to the former general Rule add 2 s. the first number of the progression to 40 the last of the same, and that maketh 42. the which multiplied by 20. the half of the times of the Progression yieldeth 840. shillings.  
Or otherwise, multiply the whole number of times of the Progression with 21. the half of the Addition of first and last numbers of the Progression, and the result willbe also 840 shillings as by example.   

Examples.  
2. 40. 2. 40 2. 21 42 840 20   840    
There is an other kind of Progression, and that is geometrical, wherein every time containeth the next before it, so often as the second containeth the first, as 1. 2. 4. 8. 16. 32. 64. 3. 9. 27. 81. 243. 719.   4. 16. 64. 256. 1024.      
Here you may perceive 64. in the first Progression, containeth 32. so often as 2. containeth 1. Also in the second 719. containeth 243. so often as 9. containeth 3. Likewise in the third 1024. containeth 256. so often as 16. containeth 4.  
The which Progressions or such like, to show the whole, you shall multiply the last number of the Progression, by the first common multiplicator, and from the result you shall divide by one less, then was the multiplier, and so haue the quotient the just total of that Progression, as by examples the effect may appear.  
A testator giveth in legacy to eight of his friends, a certain somme of money: To the first 4. pound, to the second 4. times as much as the first, which is 16. pound: To the third 4. times as much as the second, and so every of the other 4. times as much as he before him. The question is, what amounted the whole legacy.  
As before is shewed, set down all the 8. terms: Thus.  
4. 16. 64. 256. 1024. 4096. 16384. 65536.  
Then multiply the last somme by the first, and the product is. 262144. from which rebate the first 4. so resteth 264140 to be divided by 3. which is, 1. less thy e the multiplier and the quotient, is the somme of the whole legacy, which is 87380. li. as by example, proved by addition.  
65536     4 li. 262144 87380 4   262140      
Thus much to understand 16 64 256 1096 4096 16384 65536 87380 is sufficient for the fommyng of any progression geometrical, where the first number is the roote in any work whatsoever the roote be.     

¶ The second parte containing the work of Fractions, or broken numbers, and first of Numeration.  

AS whole numbers compound of unities may bee augmented and increased to infinite effect: so an unity may bee divided into sundry and infinite diminute partes, which partes in name and nature are agreeable. For a fraction is a parte of one unity, and not of many: for howbeit that whole Numbers may bee divided into partes, to several effects, the same notwithstanding, such divided partes are no proper fractions, but improperly show the parts of whole numbers, and not of an unity, as 40. pound to bee partend amongst three men: the first to haue ¼, which is one fourth parte: the second ⅖, which is two fift partes: and the third 7 / 20, which is seven twenty partes. The said Partes may bee shewed in whole numbers, not needyng the use of fractions for the same. For the ¼ is 10. lib. the ⅖ is 16. pound, and the 7 / 20 is 14. pound, which together maketh 40. pound, and all such numbers expressed in maner like fractions, are not proper fractions, but improperly borowyng the property of fractions, which as aforesaid, are partes of one unity onely, and not above.  
Here is to bee noted, that a fraction is expressed by two figures, set the one over the other, with a line between, thus ⅔, which signifieth two third partes of an unity, and that under the line, is called the denominatour, because it doth ever represent the partes, wherein the unity is divided: and that above the line is called the numerator, because it sheweth the number of partes, by occasion required, not needyng the whole unity. As when a man hath right to ⅔ partes of a pound in money, which is two Nobles, then 2. over the line, sheweth the partes of his right, and 3. under the line, sheweth wherein the unity is divided, and representeth one pound of money, divided into three partes.  
Here note, that every fraction abstract or free from denomination, may bee applied and made contract to any denomination, by occasion required, and more easy for the learner to understand, what the nature and value of a fraction is, when the character of denomination is joined with it, then when it is without the same, as by example the effect may appear.  
½. ⅔. ¾. ⅘. ⅚. ⅞. 9 / 10. 17 / 20. 25 / 32. 161 / 240.  
every of the denominatours of the 10. former fractions, doth represent an unity divided into so many partes, as the figure or figures of the same doth demonstrate: howbeit, not to bee known of what thing, for want of a character, to signify the denomination. Likewise every numerator, is so much less thē an unity, as the difference appeareth, between it and the denominatour thereof, and may be applied to sundry things, joining a character for the same, bee it of receipt, Measures, moneys, or other things whatsoever, as hereafter appeareth by Carracters for moneys, seemyng most apt for the purpose.  
d. d. s. s. li. li. li. li. li. li. ½ ⅔ ¾ ⅘ ⅚ ⅞ 6 / 10 17 / 20 25 / 32 127 / 240  
The first and second of these 10. fractions, hath the letter d. for character, lying of a penny, and half a penny, and 2. third partes of a penny. The third and fourth haue the letter s. for character, lying to bee of a shilling: as 3. fourth partes, and four fifte partes of a shilling. All the rest haue the letter l. for character, lying every of the same to bee a fraction, or a parte of a pound of Money, as 5. sixth partes of a pound 7. eight partes of a pound, and so the rest 9. tenth, 17. twentieth, 25. thirty two, 162 two hundreth and forty partes of a pound and according to the former saying, so much as any of the numerator( which is above the line) is lesser then the Denominatour of the same: so much it wanteth of the value of the unity by the Denominatour, and character represented, be it of a penny, a shilling, or a pound: and so to understand of all other fractions, of what denomination so ever, and for the learners better understanding hereafter the same fractions are applied to other denominations, signified by words, for want of usual carracters, as for moneys is found.  
½ inch, ⅔ foot, ¾ yard, ⅘ ell, ⅚ Ounce or vnz, ⅞ lib or pound weight, 9 / 10 C. or hundreth weight, 17 / 20 hour, 25 / 32 month, 162 / 240 year. &c. Thus may you see a fraction to bee a parte of an unity, whereof known by character, or word of denomination, and not having denomination, may bee applied to any thing, by occasion required, and thus much may seem sufficient to give understanding how to express a fraction, which is a parte of Numeration: howbeit, now remaineth to show how to find the value of a fraction, whereof the effect hereafter followeth by examples in the former fractions, applied to several denominations.  
½ bushel signifieth a bushel to be divided in two partes, and the half thereof the fraction representeth: whereof to find the value to bee expressed in common and known partes, you shal consider what diminute partes the bushel containeth, and that is 4. pecks. Then multiply the numerator 1. by 4. pecks in the bushel, and the product divided by the Denominator 2. the quotient will show 2. pecks to bee contained in the fraction, and is the value of half a bushel desired to bee known, and this take for a general rule, to bring a fraction into common and known partes, the effect more at large appearing in sundry examples following.  
⅔ foot signifieth two third parts of a foot, whereof to find the value in common known partes, consider what diminutive partes a foot containth, and that is 12 Inches, by the which multiply the Numerator. 2. maketh 22 Inches, the which divided by the Denominator. 3. yieldeth inquotient 8 Inches for ⅔ partes of a foot. &c.  
Likewise 2 / 4 lib. is 3 quarteas of a pound weight, and to know the value thereof, you must consider what known partes the same containeth, the which being haberdipoys weight is 16 ounxes. whereby multiply the Numerator 3, and the product is 48 ounzes, the which divided by the Denominator 4. the quotient sheweth 12. ounzes to bee ¾ of the lib. habardipoys.  
Howbeit if the pound weight bee Troy weight whereby Gold silver and Precious stones are weighed, then 12 ounzes maketh the lib. the which multiplied by the Numerator 3. produceth 36. ounze, the which divided by the Denominator 4 yieldeth in Quotient 9. ounzes for ¾ of the lib. troy.  
Also 4 / 6 ell representeth four fift partes of an ell, the which to bring into common and known partes, consider what known partes an ell containeth, and that is found in 3 sundry sorts. first it containeth 4. q. proper to the same. Also 5 q. of the yard, and thirdly 45 Inches. To haue it in parts proper to it, multiply the Numerator 4 by the quarters in the ell, which is also 4, maketh 16. and that divided by the Denomitor 5. yieldeth 3 q. of the ell, and ⅕ of one of the same q. To haue it in quarters of the yard, multiply the Numerator by 5. quarters of a yard in an ell contained, the product will be 20, and the same divided by 5 the Denominator, sheweth 4. q. of a yard in the fraction contained.  
Lastly to bring it into Inches, multiply the Numerator 4 by 45 Inch in the ell contained, the Product will be 180. The which divided by the Denominator 5. yieldeth in quotient 36 Inches for ⅘ of an ell.  
To bring ⅚ s. into apt known partes, multiply the Numerator 5 by 12. d. in a shilling contained, maketh 60. and the same divided by 6. the denominator, yieldeth in quotient 10. d. for ⅚ of a shilling. &c.  
To bring ⅞ d. into known partes, multiply the numerator 7. by the mites of a Penny, which are 24, and the Product will be 168, and the same divided by the denominator. 8. the quotient will show 21. mites to be the ⅞ partes of a penny.  
To bring 9 / 10 crown into known partes, multiply the numerator 9. by 15 groats, or 60. Pence in a crown contained, and you shall haue produced 135. for groats, and 540 for Pence, the which products divided by the denominator 10. yieldeth in quotient 13. groats and a half, and 54 d. every of which is 4. s. 6. d. for 9 / 10 of a crown.  
To bring 17 / 20 Noble into known partes, multiply the numerator 17. hy 20. greates or 80. pence in a Noble contained, and the product will be 340. for the groats, and 1360. for pence, the which divided by the Denominator 20. yieldeth in Quotient 17 groats and 68. d. every of which is 5. s. 8. d. for 17 / 20 of a noble.  
To bring 25 / 32 l. into known partes, multiply the numerator 25. by the shillyngs in a pound which is 20. and the Product will be 500. the which divided by the Denominator 32. yieldeth in quotient 15. s. and 20 / 32 partes of a shilling, and to know the value of that later fraction multiply the numerator 20. by the pence in a shilling, which is 12. and the product will be 240. the which divided by 32. sheweth in quotient 7. d. ob. and so haue you 15. s. 7. d. ob. for the 25 / 32 partes of a pound.  
Lastly to bring 162 / 240 li. into known parts, multiply the Numerator 162. by 20. s. in a Pound, and the Product will be 3240. the which divided by the Denominator 240. yieldeth in quotient 13. s. 6. d. for 162 / 240 li. and so of all other.   

Progression of Fractions.  
PRogression of Fractions is in 2 sorts, the one of property contrary to the other. for the first which is ½. l⅓. ¼. ⅕. ⅙. and so infinitely, the greater that the denominator is, so much the smaler is the value of the fraction, for ⅙ li. which is 3. s. 4. d. is of smaller value then ⅕ li. which is 4. s. also ¼ li. is 5. s. and of smaller value then l⅓ li. which is 6. s. 8. d. and so l⅓ 6. s. 8. d. is smaller than ½ li. which is 10. s.  
But to the contrary in the second sort of progression, which is ½. ⅔. ¾. ⅘. ⅚. &c. the greater that the Denominator is, the more is the value of the fraction. For ⅚ li. being 16. s. 8. d. is more then ⅘ li. which 16. s. Also ¾ li. which is 15. s. is greater then ⅔ li. which is 13. s. 4. d. and so of all other like understand.  
And here note in the first Progression, the greater that the Denominator is, so much the more is the unity decreased, which may bee to infinite effect and ever to be somewhat, and to the contrary in the second Progression, the greater that the denominator is, the nearer to the whole unity the value of the fraction doth approach, how be it can never attain to make the unity.   

Reduction of Fractions.  
REduction of Fractions containeth the maner how to bring 2 or more Fractions into one, either such as be of one Deuomination, as other which are of contrary denominations, the effect whereof more easy to understand by a few examples then in many words.  
To reduce l⅓ li. ¾ li. into one, that is to make one Fraction to contain the value of them both. You shall by a general Rule multiply the 2 denominators, the one by the other saying, 3 times 4 is is 12. to bee set down twice for 2. new denominators thus. 12. 12. then multiply the numerator of the first fraction by the Denominator of the second, that is one by 4. maketh 4 for a new Numerator to stand over the common and new denominator thus 4 / 12 li. Also multiply the numerator of the second fraction by the denominator of the first which is 3. by 3 is 9 to set over 12 thus 9 / 12 li. & so haue you 2 new fractions of one Denomination, containing the value of 2 first. For l⅓ li. and 4 / 12 li. is of one value which is 6. s. 8. d. and ¾ li. and 9 / 12 li. is and of one value, which is 15 s. & the 2 first being of contrary Denominations reduced into the 2 later being of one denomination. And to make one fraction of them both, add together the 2 Numerators 4 and 9 is 13. to set over 12 thus 13 / 12. and so you haue 1 fraction 13 / 12 li. containing the just value of l⅓ li. and ¾ li.  
Note that when the Numerator of any fraction is greater then the Denominator, the same is a fraction improper, and made in such form by need in work or otherwise, and then by Rule general divide the numerator by the Denominator, and the quotient will show the unity or unities in the said fraction, and the remain if there be any will be a proper fraction. wherefore to end this reduction, divide the Numerator 13. by the Denominator 12. and the quotient will be 1 and 1 / 12 li. which is 21 s. 8 d. the just value of the 2 first fractions l⅓ li. which is 6 s. 8 d. and ¾ li. which is 15 s. and together maketh 21 s. 8 d. as the Reduction hath brought forth.  
Sometimes occasion may require reduction of 3. 4. or more several Fractions of sundry Denominations to bee brought into one denomination, and to make one fraction of many, and then you shall multiply the first denominator by the second, and that product by the third and the second product, by the fourth &c. And so many Fractions as there are to be reduced, so many new Denominators shal you set down in former order. And to find numerators to every of the same you shall multiply every numerator into all the denominators of the other fractions not belonging to the numerator multipliar, and so find to every new denominator a new numerator, as example will declare.  
To reduce ½ li. ⅔ li. ¾ li. and ⅘ li. into one denomination, and so to one Fraction first multiply the denominators, one into anothers product, as 2 by 3. is 6. and that by 4. is 24. the which by 5. is 20. for common denominator to be set down 4 times. Then by the numerator of the first, which is 1. multiply the Denominators of the other, which is 3. 4. and 5. the product is 60. to set over the common Denominator 120. thus 60 / 120 li. and is in value equal with the first fraction ½ li.  
Then by the Numerator of the second, which is 2. multiply the denominators of the other, which is 2. 4. and 5. and the product is 80. to set over the common denominator thus, 80 / 120 li. equal to the second fraction ⅔ li.  
Likewise by the numerator of the third fraction, which is 3. multiply the denominators of the other which is 2. 3. and 5. the product is. 90. to set over 120 thus 90 / 120 li. and is in value equal with the third fraction ¾ li.  
Lastly by 4. the numerator of the fourth fraction, multiply the denominators of the other, which is 2. 3. and 4. and the product is 96. to set over the common denominator thus 96 / 120, and is in value equal with ⅘ li. and so you haue four new fractions of one Denomination, for the four first of contrary Denomination, which is the effect causing the reduction.  
Then according to former instruction, add together the Numeratours of all the new fractions, being of one Denomination, which is 60. 80. 90. 96. and make 326. to set over the common Denominatour, thus 326 / 120, the which appearing to be a fraction improper, divide the Numerator by the Denominatour, and the quotient will show the unities in the same, and the proper fraction, all which is 2. and 86 / 120 li. which is 2. li. 14. s. 4. d. the just value of the first four fractions: For ½. li. is 10. shillings, ⅔. li. is 13. s. 4. d. ¾. li. is 15. s. and ⅘. li. is 16. s. and make together, 2. li. 14. s. 4. d. as the work of Reduction hath brought forth. &c.  
If the number of Fractions bee so many, that the Reduction of them would bee tedious to bee made at one time, then you may reduce parte of them at one time, and the rest at an other, and so make two new Fractions of all the first. Then reduce the said two newly made both into one, and so you haue doen, as example may declare.  
To bring ½. ⅔. ¼. ⅖. ⅚. ⅞. li. all into one fraction, would seem tedious to a learner to perform. Wherefore reduce three of the first together, and they will make 12 / 24. 26 / 24. 6 / 24. and makes in one 3 / 2 4 / 4. li. Then reduce the three last figures together, and you shall haue 96 / 240. 200 / 240. 210 / 240. and makes in one 506 / 240. and so haue you two new Fractions for all the other 6. Lastly reduce the two newly made into one and you shall find 20304 / 5760 li. which is worth 3. li. 10. s. 6. d. the just value of the six first. &c.  
Thus much may seem sufficient for reduction of proper fractions, which are partes entire of an unity, and neither greater, nor so much as the said unity, and I account such improper, which are either greater then an unity, or less then an entire parte, as some other kind bee, which are but partes one of an other, and bee called Fractions of Fractions, whereof the reduction followeth.  
To reduce fractions of fractions, which are partes one of an other, all not making so much as an unity, you shall multiply all the Denominatours together, and so haue one Denominatour, for an new and proper Fraction: then ye shall multiply all the Numeratours together, and haue one Numerator to set over the new Denominator, and find one proper Fraction for many other, as by example may appear.  
To reduce ⅔. of ¾. of ⅘ li. into one, multiply the denominators together, as 3. by 4. is 12 and that by 5. maketh 60. for a new Denominator. Then multiply the numerators together, as 2. by 3. is 6. the which by 4. maketh 24. to bee set over the Denominator thus, 24 / 60 li. which is the value of 8. s. represented by the three first fractions.  
sometimes occasion may procure a reduction of proper Fractions, and improper of both sorts all together( that is to say) whole numbers, proper fractions, and fractions of fractions to bee brought into one, as by example.  
To bring 3. and ⅔. li. with ¾. li. and ½. of ⅔. of ¾. li. into one, you shall first bring the whole number, and the Fraction thereto belonging, into one Fraction improper, the which to perform, you shall multiply the whole number 3. by the denominator of the fraction thereto belonging, which is also 3. and the product is 9. whereunto add 2 the Numerator of the same fraction, so haue you 11 / 3. li. for the whole number and first fraction. Then bring the 3. fractions of fractions into one, as before is taught, which will make 6 / 24. li. so shall you haue 3. Fractions for all, which are 11 / 3. li. ¾. li. and 6 / 24 li. and reduced make 1344 / 288. li. which is 4. li. 13. s. 4. d. and so much representes the Figures of the Example.   

¶ Of division in broken Numbers, and first of Abreuiation of greater terms into smaller.  
TO abreuiate a fraction of great terms( that is of many Figures) into an other of smaller terms, or fewer Figures, you shall consider what Digette is most apt to divide as well the Numerator, as also the Denominator of any such fraction, as is to bee abreuiated, and set the 2. quotientes one over the other, and you shall haue a new fraction of smaller terms then the first, as by example.  
To abreuiate 54 / 72 li. give regard what Diget or figure, will divide both the numerator and denominator, and that may be doen by 4 sundry digettes, as by 9. by 6. by 3. and by 2 and the most apt of them is 9. And as you make your division, set the quotient of the numerator above the same, and the quotient of the denominator under the same denominator thus. Wherein you may perceive that in 54, 9. is contained 6 / 54 li. 6. times and so 72. 8. times, and so 72 / 8 you haue a fraction of two Figures 6 / 8. li. for the other of 4. Figures 54 / 72. the greater terms abreuiated into smaller, and the value not channged.  
Likewise by the same order, consider that 2. being made divisor of 6 / 8. li. you shall haue that fraction abreuiated to 8 / 4. li. which in smallest terms that may bee, is of equal value with the two other: for every of the same representeth 15. shillings, 3 / 6. li. and thus is the practise. 8 / 4.  
When the learner findeth a fraction to be abreuiated, which being of greater terms then with facility to know the Digette, most apt for the abreuiation, then let him examine the example by mediation thus, 2. being always divisor, as in this Fraction 48 / 96. li. for example.  
3 6 12 24 By 2. 48 / 96 maketh 3 / 6. li. the which by 3. is 1 / ● 48 24 12 6  
Hereby three mediations the Fraction 48 / 96. pound, is brought to 6 / 12. pound, where it is with facility perceived, that 6. is half of 12. and therefore ½. li. is settedoune for it, and so the abreuiation ended.  
sometime a fraction may require 2. 3. or more digettes, to bring the same to smallest terms, as by sundry examples the effect may appear by this fraction 160 / 240. pound.  
2 4 80 160 By 2. 240 maketh ⅔. 120 6 3 By 3 160 it can not. 240 2 40 160 By 4 240 is ⅔. lib. 60 3  
Here note, that when a Fraction hath equal number of cyphers, in the place or places toward the right hand, then the abreuiation may bee made the shorter, by cutting away the cyphers of both sides, in equal number, thus.  
2 li. 16 0 By 8. 24 0 maketh ⅔. 3 2 li. 2 li. 16 00 16 000 24 00 24 000 3 3  
Wherein doth appear, that the cyphers of every of the three Fractions, separated from the Figures, then every of the same is 16 / 24. and divided by 8. sheweth ⅔. li. for smalleste terms, and so in al other like unto the same.  
To abreuiate 75 / 120. pound, there is required the use of two figures, which is 3. and 5. to begin with the one at pleasure to bee taken, and to end with the other by consequence, as in example, practise doth show.  
5. lib. 25 75 By 3. 120 or by 5. 40 8  
5 lib. 15 75 Or by 5. 120 by 3. 24 8  
To abreuiate 112 / 192, there is required the use of one figure onely, for the most apt, which is 4. and may bee doen by two Figures, which is first by 8. and then by 2. as practise will manifeste.  
7. li. 7 lib. 28 14 112 112 By 4. 192 or by 8. 192 and by 2 48 24 12 12  
To abreuiate 128 / 160. lib. most aptly there is required two figures, 8. and 4. and with more circumstance by 4. and 2. as by example.  
4 lib. 16 128 By 8. 160 and by 4. 20 5 4 lib. 8 32 128 By 4. twice 160 and by 2. 40 10 5  
To abreuiate ●75 / 500 li. seek for the most apt diget to divide by, and that is 5. by the which at 3 times is brought forth ¾ li. which is 15. s. as by example.  
lib. 3 15 75 375 By 5. 500 100 20 4  
Thus much may seem to suffice for Abreuiation of Fractions, which is performed by division practised in whole numbers. Howbeit, division of fractions is much contrary, as by examples the effect may appear.  
When one fraction is to be divided by an other, that is to say, when you would know how many times I fraction is contained in an other, set the divisor one the left hand the other, & a cross between them thus, which requireth by division to make known how many times ⅖ li. is contained in ⅞ li. Then multiply the numerator of the divisor by the denominator of the diuidende, and that product shalbe divisor. Likewise multiply the numerator of the diuidende by the denominator of the divisor, and the product thereof shal be diuidende, and the same divided by the last divisor sheweth in Quotient that is required in the work, as practise may more amply deelare.  
li. 35 li. times & 3 / 16 of a time.  
In this practise appeareth that 2. numerator of the divisor, multiplied by 8. the Denominator of the diuidende, produceth 16. set under the cross for divisor, and 5 denominator of the divisor multiplied by 7. the Numerator of the diuidende produceth 35. for diuidende to set over the cross. The which 35 divided by the divisor thereof 16. sheweth in quotient that ⅖ li. which is 8. s. is contained in ⅞ li. which is 17. s. 6. d. 2 times, and 3 / 16 of a time, and note that ⅖ lib. twice is 16 s. and 3 / 16 of 8 s. or 1 time is 1 s. 6 d. and maketh together 17 s. 6 d. the just value of the fraction divided.  
To divide ¾ lib. by l⅓ lib. by the same order the practise followeth.  
2 times and ¼  
Wherein appeareth that in ¾ li. which is 15 s. l⅓ li. being 6 s. 8 d. is contained 2 times, and ¼ in value equal to the diuidende which is also 15 s. &c.  
As in works by whole numbers a smaler somme cannot bee divided by a greater, but is set over the greater, to show in proportion a part of a time, given so in division of fractions, when the divisor is greater than the dividend. Then the dividend produced will be less then the produced divisor, and therefore to stand over the divisor, to show the proportional part of a time, sought for by the work, the effect by example, made more plain.  
If I demand how many times ⅚ lib. is contained in ⅔ li. reason doth persuade that no time in the Quotient will appear. nevertheless procéedyng in the work, the dividend will show such part of a time as proportion will allow, & in practise appearing.  
12 / 15 of a time.  
Hereby doth appear that 12 / 15 parte of the divisor is the value of the dividend, and therfore wanteth of a time, and 12 / 15 of 16 s. 8 d. the divisor is 13. s. 4 d. the just value of the dividend ⅔ li. the effect sought for by the work.  
To show the effect in fractions improper, may satisfy the desires of such as the same would know, and therefore is furthered the example following.  
If it be demanded howe many times ¾ li. is contained in 4 li. and ⅘. lib. You shall first make 4. li. and ⅘ in fraction, and it will bee 24 / 5 which is the dividend to be divided by ¾ the divisor, and so in the Quotient will appear, that is sought for as by example practised.  
6 times and 6 / 15.  
Here the effect sought for, being how many times 15 s. which is ¾ li. is contained in 4 li. 16 s. which is 24 / 5 li. the same in Quotient appeareth to be 6. times & 6 / 15 of a time, and every time containeth the value of the divisor, which is 15 s. maketh in all 4 li. 10 s. and therwith 6 / 15 of a time which is 6 s. maketh 4. li. 16 s. the just value of the Diuidende 24 / 5 li. which is the effect the work requires.   

Multiplication of fractions.  
THe work of Multiplication of Fractions, is in nature contrary to the working by whole numbers, for as the one increaseth a number of unities: so the other increaseth a diminution of a Fraction. For in multipliyng 3 lib. by 2 li. you say 2 times 3 maketh 6 lib. in whole numbers, but ¾ li. by ⅔ lib. you must understand the saying ¾ li. taken ⅔ of a time maketh half a pound 10 s. the which to bring forth by order of Multiplication of Fractions, you shall multiply the Denominators together, and the product thereof is an new Denominator. Then must you multiply the 2 Numerators together, and the product is the numerator to the foresaid denominator, and you haue done, as by example the effect may appear.  
lib.   lib. lib. lib. 2 by 3 yieldeth the 6 / 12 and abreuiated is ½ 3 4  
Here note, that as ¾ lib. taken 2 of a time maketh 6 / 12 li. which is 10 s. given so ⅔ li. taken ¾ of a time maketh also 6 / 12 lib. which is likewise 10 s. so that it forceth not which is set before the other.  
If occasion procure whole numbers and Fractions to bee multiplied together, then your whole number is to bee brought into form of fraction, and so multiplied by former order, the product will show that is sought for, as example may manifest.  
To multiply 5 li. ⅘ by ¾ li. reduce the first fraction, and the whole number into a fraction improper, and it will be 29 / 5 li. the which multiply by ¾ li. produceth 87 / 20 lib. as in pratice.  
29 / 5 by ¾ yieldeth 87 / 20 which is 4. li. 7 s.  
The which product wanteth so much of the first value, as ¾ li. wanteth of an unity, which is ¼ part, and for the understanding of the reason therof, you shal note, that if 5 li. 16 s. bee multiplied by 1 lib. it will not change the value, if by 2 li. the value will double of by 3. it willbe triple, and so forth infinitely. But to the contrary, if you multiply the said 5 li. 16 s. by ¾ lib. it diminisheth ¼ in value, as practise hath shewed. If by ⅔ lib it will want l⅓ in value. If by ½ lib. half the value diminisheth, and so infinitely, according to the value of the fraction multiplicator.  
To multiply 3 li. ⅔ with 4 li. ⅚ reduce every of the whole numbers into the fractions to it belonging, as 3 li. ⅔ reduced is 11 / 3 li. and 4 li. ⅚ maketh 29 / 6 li. the which multiplied together, produceth 39 / 18 li. which is 17 li. 13 / 18 or 14 s. 5 d. l⅓.  
The truth whereof by reason to witness, consider, that 3 li. by 4 li. produceth 12 lib. then 3. li. by ⅚ the contrary fraction yieldeth 2 li. 10 s. and 4 li. by ⅔ the contrary fraction is 2 li. 13 s. 4 d. Lastly 2 fractions, the one by the other produceth 10 / 18 li. which is 11 s. 1 d. l⅓. and together make the foresaid somme of 17 lib. 14 s. 5 d. ½ agreeyng with the product by the Rule, as the addition of the several partes will appear hereafter set down.  
      lib. s. d. 3 By 4 12.     3 ⅚ 2. 10.   4 ⅔ 2. 13. 4 ⅔ ⅚   11. 1. ⅔ Makes   17. 14. 5. l⅓  
Hereby is to be perceived as well the order of the rule, as also the reason of the product, hereunto hide, from many which can multiply broken numbers.   

Substraction of fractions.  
TO substraie one fraction from an other, there is required that both the boken numbers be of one denomination, and then the lesser numerator rebated from the greater, the rest will appear to be set over the common Denominator, and so the work is ended, as by example the effect may appear.  
To substraie ⅜ li. from ⅞ li. rebate 3. from 7. rest 4. to set over 8. the commom denominator thus 4 / 8 li. whereby to understand if you take ⅜ which 7 s. 6 d. from ⅞ li. which is 17 s. 6 d. the rest will bee 4 / 8 li. which is 10 s. and so of all other when both be of one denomination, as the sundry herereafter set down.  
⅖ lib. from ⅗ lib. rest ⅕ lib. and ¼ lib. from ¾. lib. rest 2 / 4.  
Likewise 4 / 10. lib. from 7 / 10. lib. rest 3 / 10. lib. so 4 / 12. lib. from 11 / 12. lib. rest 7 / 12. lib. &c.  
notwithstanding, when occasion procureth Substraction, the Fractions being of contrary Denomination, then you must reduce them into one Denomination, and so made apt for the work as the former, whereof some example followeth.  
To Substraie ⅔. li. from ¾. li. you must rndure them by order taught for reduction of proper Fractions: and so you shall haue for ⅔. li. 8 / 12. li. and for ¾. li. 9 / 12. li. and being brought to one denomination Substraie 8 / 12. from 9 / 12. and the rest is 1 / 12. li. and so you haue doen, wherein understand, if you take ⅔. or 8 / 12. lib. from ¾. or 9 / 12. lib. the rest is 1 / 12. li. the which is 20. d. as 13. s. 4. d. from 15. s. the rest is 20. d. as aforesaid.  
The like effect taketh place in fractions improper of both kindes, as first by example of Fractions of Fractions shall appear.  
To Substraie ½. of ⅔. lib. from ¾. of ⅘. lib. first reduce the two first into one proper Fraction, which is 2 / 6. lib. and the two last also into one, maketh 12 / 20. pound, which being of contrary Denomination, must bee brought into one, and you shall for the first haue 40 / 120. lib. and for the last 72 / 120. lib. of the which lesser Numerator 40. substraied from the greater 72. the remain is 32 / 120. which is 5. s. 4. d. and the same to understand so to bee note, that 2 / 6. pound, being 6. s. 8. d. taken out of 12 / 20. lib. which 12. s. the rest is 5. s. 4. d. as aforesaid, which is 30 / 120. li. &c.  
Likewise of Fractions improper, greater then an unity, here followeth an example.  
To Substraie 2. lib. ⅔. from 4. lib. ¾. first reduce 2. lib. ⅔. in one, maketh 8 / 32. pound, and 4. lib. ¾. also in one is 19 / 4. pound, and brought to one Denomination, will bee 32 / 112. li. and 57 / 12. pound, whereof the lesser numerator 32. taken from the greater 57. the rest is 25 / 12. lib. which is 2. pound 1. s. 8. d. and easily perceived in former maner. For 2. lib. 13. s. 4. d. taken from 4. lib. 15. s. the rest is two pound, one shilling eight pence, as by the work doth appear.   

¶ Addition of Fractions,  
FOr addition of Fractions there is to bee considered, as was in Substraction: that the broken numbers bee of one Denomination before they be added, & then put the Numerators into one, to set over the common Denominator, and so the work is ended. But if they bee of contrary Denomination, they must bee brought into one, and so made apt for work: as by example more at large, you may perceive.  
To add ¾. 2 / 4. and ¼. li. into one Fraction, add together all the Numeratours, as 3. 2. and 1. make 6. to set over the common Denominatour 4, thus. 6 / 4. which is 1. pound 10. shillings, and so the work is ended.  
Howbeit, if the Fractions bee of sundry Denominations, as ¾. ⅘. and ⅚. pound, then they must bee reduced, and will bee 90 / 120 96 / 120. and 100 / 1200. and the Numeratours added together, as aforesaid make 280 / 120, which is 2. lib. 7. s. 8. d. And the same to understand so to bee, note that ¾. lib. 15. s, with ⅘. lib. 16. s. added to ⅚. lib. 16. s. 8. d. make together 2. pound 7. shillings 8. pence, as the work hath brought forth.  
If you add together Fractions of Fractions, as ½. of ⅔. lib. to ¾. of ⅘. lib. then reduce the two first into one maketh 2 / 6. lib. and the two last is 12 / 20. pound, and in one Denomination is 40 / 120. li. and 72 / 120. lib. which make 112 / 120. and is 18. s. 8. d. and by memory to witness the truth, consider that ½. of ⅔. li. is 6. s. 8. d. and ¾. of ⅘. pound is 12. shillings, the which together maketh 18. s. 8. d as the work findeth.  
Likewise, if you add sundry whole numbers, joined with Fractions into one, you must either reduce all into Fractions improper, and so to one Denomination, adding the Numeratours together, to set over the common Denominatour by former order, or else you may add the whole numbers first together, and reduce the Fractions onely, and so end the work, as in example following appeareth.  
To add 2. lib. ¾. to 5. lib. ⅗. first reduce 2. lib. ¾. yieldeth 11 / 4. and again 5. lib. ⅗. maketh 28 / 5. li. and in one denomination is 55 / 20. and 212 / 20. and added maketh 167 / 20. li. which is eight pound 7. s.  
Otherwise, add the whole numbers together, that is 2. lib. and 5. lib. maketh. 7. li. Then ¾. lib. and ⅗, lib. reducted to one Denomination maketh 27 / 20. lib. which is 27. s. and put to the foresaid 7. lib. make together 8. lib. 7. s. as before.  
Such as in reading of Substraction and Addition do not well understand the effect. Let them labour well to understand Numeration and Reduction of fractions, for therein is taught all things needful to make the rest easy. &c.    

The third part containing the Rules of Proportion, and first of the Rule of 3.  

THe Rule of three is framed of the former partes of arithmetic, especially of Multiltiplication, and division. And is called the Rule of three, for that by three numbers known, and set down in order as the work requireth, is found a fourth number, fought for and desired, and the commodity growing by use of the said Rule procured Learned writers do name it the Golden Rule, excelling all other, as gold doth other metals. It is also called the Rule of Proportion, for that ever the fourth and unknown Number found by the work, shall bear such proportion unto the third of the known numbers, as the second beareth to the first. The effect better appearing in few Examples, then in many words.  
If 2. clothes cost 16 li. what 15. clothes.  
Here you see 3 numbers known, as 2 Clothes bought or prized at 16 lib. and 15 clothes to bee bought or prized after the same rate the price of which 15 Clothes is the fourth Number sought for and desired, found by the work in order as followeth. First you shall multiply the second and third Numbers the one by the other, and the product thereof divide by the first Number, and so shall you haue in Quotient the fourth Number sought for & desired, as by example.  
Clo. lib. C.   If 2. cost 16. what 15       16       90     15     240    
By this example appeareth that 15. the third number, multiplied by 16. the second number, produceth 240. the which divided by the first number 2. yieldeth in quotient 120. lib. for the price of 15. clothes, and in such proportion as 16. beareth to 2. that is to say, 8. lib. for every cloth.  
The proof of this rule is made changyng the places of 3. of the 4. numbers, and so one of the 2. first will bee found in quotient, if the work be true. As by example.  
li. clothes. li.   If 120. buy 15. what 16.       15       80     16     240 C. lib. C.   Or if 15. cost 120. what 2   2     240    
Here is to bee noted, that as well in practise of the rule, as also in the proof, the first and third numbers, must bee of one Denomination and nature, & then of consequence, the fourth number will bee of denomination and nature, as is the second, the effect whereof as in the former examples plainly appeareth, so in other following more at large may be seen.  
If 1 pound weight of Pepper coste 2 s. 8 d. what 9 ounzes of pepper.  
Here the first and third Numbers are of one nature, but not of one Denomination. Wherefore before you work you must reduce the first number into onzes, and so made apt for the work. Likewise for that the second Number is in 2. Denominations, as shillings and pence, therefore you must reduce the shillings into pence, and then your 3 Numbers being apt for the work, will stand thus.  
If 16. ounzes cost 32. d. what 9. ounzes,  
If 1. yard coste 3 lib. 7. s. 6 d. what 75. shepherds.  
The middle number in this example is not apt for the work, till the whole be brought into pence, which is the smallest Denomination of 3. in the same second number, wherefore it must be reduced, and will make 810 pence, and will stand thus apt for the work.  
If 1 yard cost 810 d. what 75 shepherds.  
Here the division is made more for plainness in observing the Rule, then for any necessity. For one the first number cannot any thing diminish in division, nor any thing augment in Multiplication, as by the division before may appear, and in the multiplication required in the proof following is manifested.  
If 75. shepherds cost. 60750. d. what 1. yard  
Thus you may perceive, that the Quotient in the first work, is equal with the diuidende in the same, and nothing diminish by the division. And likewise the product of the second work, is equal with the multiplicand, and nothing augmented by the multiplication: wherefore it is good to note, that such divisions and multiplications may bee cut of, when 1. is one of the 3. known numbers in this rule, as by some examples the effect may appear.  
If the C. weight of Currance coste 33. s. 4. d. what 1. lib.  
The said C. reduced into pound weights to agree in Denomination with the third number, and the second number reduced into pence, the smaller denomination of two in the same, then all the three are made apt for the whole work, and will stand thus.  
If 112 li. coste 400 d. what 1 lib.  
Here the multiplication is omitted, for that 1. the multiplicator can nothing augment in Multiplication, as aforesaid, and therefore the second number is diuidende where it standeth, and being divided by the first, the quotient is 3. d. 4 / 7. which fraction is half a penny, and somthyng more wherefore always when 1. is the third number, divide the second by the first, and the quotient will bee that you seek for.  
If 1. ell coste 20 d. what 48 960. d. 20   960    
Here the division is omitted, because that 1. the first Number, can nothing diminish in division. Wherefore in all works where one is the first number, the product made by multiplication of the second by the third is that you seek for in the quotient, which in this work is 960 d. as appeareth.  
If 4. s. 8. d. buy one ounze of silver, how many ounzes buyeth 100. li.  
After you haue made the first and third numbers to agree, in denomination by reduction, bringing both into pence, as the rule teacheth, then the question will stand thus.  
 
Here the multiplication is omitted in former respect, and the third number the true diuidende, and divided by 56. yieldeth in quotient 428 ounces 4 / 7. and so all other works, where 1. is the second number.   

The second parte of the Rule of three, is of effect contrary to the former, and is name the Backer Rule of three, upon cause reasonable: for as in the former Rule, the fourth number is ever so much greater thē the third, as the second is above the first, so in the Backer Rule, the fourth number is ever so much lesser then the second, as the third is greater then the first: As to the contrary, so much greater then the second, as the third is lesser then the first.  
And the order of this Backer Rule is such, that when the three known numbers are set down, then you shall multiply the first and second, the one by the other, and the product thereof divided by the third number, and so find in the quotient, that is desired, & sought for by the work: as by examples the effect more amply may appear.  
When the bushel of wheat is worth 3 s. 4 d. the Wheaten loaf weighing 20. ounzes for 1 d. what shall the penny Wheaten loaf way when the bushel of wheat is worth 5. s.  
Herein touching the work, you shall give no respect to the bushel of wheat, but to the price thereof, to be made the first number neither to the wheaten loaf, but to the weight thereof, for the second number, and accordingly of the bushel of wheat and the price thereof, for the third number, and then the three numbers agreeing in denominations apt for work, as was taught in the former part, then the example is thus to bee set down and wrought.  
d.  
If 40. admit 20. onz what 60 d.  
Here you may see, that as the bushel of wheat is augmented in price, a third parte in 5 s. so the loaf of a penny is diminished in weight a third parte of 20 ounzes. Wherein appeareath the nature of the rule, and the effect of that was taught before touching the same  
Likewise, if 32 d. admit 24. onz what 20 d.  
 
So that as the penny loaf waieth 24. ounzes, when wheat is at 2 s. 8 d. the bushel, it shall way 38 ounzes ⅕, when wheat is at 20 d the bushel.  
This backer rule may be applied to sundry effects of greater consequence then every man understandeth. Wherefore I will set down a few examples which to some men may seem not superfluous.  
The load of Hay at 13 s. 4. the bottle of ob. weighing 6 lib. what shall the bottle way when the like load of Hay is worth 20 s.  
If 13 s. 4 d. admit 6 li. what 20 s.  
 
If the load 15 s. admit 5. li. bottle what 10 s. the load.  
 
The ounze of fine gold worth 55 s. The crown of 5 s. weighing 2 d. weight troy, what shall the said crown way when fine gold is at 3 li. by ounze.  
If 55 s. admit 2. d. weight, what 60 s.  
 
If 60 s. admit 1 d. weight 20. grains, what 45 s. Reduce and it will stand thus.  
If 60 s. admit 44. grains, what 45 s.  
 
The ounce of starling at 2. s. 8. d. the Englishe groat weighing 2. d. ½. weight, what ought the groat to way, when the ounce of starling is at 5. s.  
Reduce and it will stand thus.  
If 32. d. admitted 60. grain, what 60 d.  
 
The ounce of starling at 5. s. the Englishe groat weighing 32. grains troy. What shall the said groat way, when starling is at 3. s. 4. d. the ounce.  
Reduce and it will stand thus.  
If 60. d. admit 32. grains, what 40. d.  
32 120 180 1920  
  

THE double Rule is so called, for that the answers of such questions, as the same requireth, are found at the double working of the Rule of three direct whereof the order followeth.  
If the 100. lib. weight cost carriage 20. miles 18. d. what will 1500. lib. weight coste 60. miles.  
In this question and all other like, you may note, that the first and third number, must bee of one denomination and kind: as herein both miles, or both weight to bee taken at pleasure for the first work. And then of consequence the other shall serve in the said first and third number in the second work: as by examples the effect may appear.  

¶ Example.  
C. d. C.   If 1. weight coste 18. what 15       18       120       15 d. s. d.     270 make 22. 6. s. d.   again if 20. mile cost 22. 6. what 60 60   1320   30 1350  
As you may perceive in the first work, the weight is used, and not the miles: and in the second the miles is used, and not the weight, which two denominations might bee changed in the said examples, and bring out the truth accordingly, as by other the effect may appear.  
If 1. C. weight coste 2. s. carriage 25. miles: what 8. C. weight carriage 100. miles.  
miles s. miles. say if 25. coste 2. what 100   2   200 C. s. C.   again if 1. coste 8. what 8       s. li. s.     64 maketh 3.4.  
By these examples it is manifeste, that as one hundreth costeth 2. s. for carriage 25. miles, so it costeth 8. s. for carriage 100 miles, by the first work brought forth. And as 1. C. costeth 8. s. so 8. C. costeth 64. s. for carriage 100. miles by the second work appearing, wherein is shewed the effect purposed, by furtheryng of the said examples, either of the same may bee taken to practise of the first work, and then the other of consequence must serve in the later.  
If 100 li. in 12 monthes gain 10 li. what 500 in 17 monthes.  
Say first, if 12 monthes gain 10 li. what 17 monthes.  
17 170  
again if 100. li. gain 14 li. 3. s. 4. d. what 500 li.  
Reduce, multiply and divide, and find. 70. l 16 s. 8 d..    

The Rule of 3 Compound.  
TO the Rule of 3 Compound, belongeth 5 known numbers, for the first parte of the same, whereof the second and first must ever be of one Denomination, and for practise thereof you shall multiply the first and second Numbers, the one by the other, and the product thereof shal be your divisor. Then multiply the other three( that is) the third by the fourth, and the product therof by the fifte, and that last product shall be the diuidende, and divided by the forenamed divisor yieldeth in quotient that which is sought for and desired.  

¶ Example.  
If one hundreth weight 20 Miles coste carriage 18 d. what 15 C. for 60 Miles.  
C. Miles d. C. Miles d. 810 1. 20. 18. 15. 60   1         20        
Herein appeareth that the first and second numbers multiplied together, the Product is 20 for divisor, also the fift, fourth, and third multiplied together, produceth 16200 the which divided by 20 the divisor, yieldeth in quottent 810 d. which is 3. li. 7. s. 6. d. for true answer, agreeing with the first example of the double Rule practised by the same question.  
Likewise as in the third Question of the double Rule.  
If 100 li. in 12 monthes grain 10 li. What 500 li. in 17 monthes.  
100 500 1200 5000   17   35000   5000   8500    
Here may you see the first & second numbers together, maketh the divisor 1200. And the other three maketh 85000 for dividend, and yieldeth in quotient 70 li. 16 s. 8 d. for answer, agreeing therein with the double Rule.  
The second part of the Rule of three compound is contrary to the first, for in this part the third and fourth numbers must be multiplied together, the product to bee divisor. Then the first, second, and fift to gether multiplied, the product shalbe the dividend, and so the Quotient will show that which is sought for and desired, and the third and first number is of one Denomination, the effect by example appearing.  
If 50 li. in 6 monthes gain 7 li. in how many monthes will 60 li. gain 10 li. multiply and divide, and you shall find 7 monthes 1 / 7 as by practise.  
li. monthes. li. li. li. 50 6 7 60 10 6     7   300     420   10         3000            
The third parte of the Rule of three compound, is contrary to the two former, for in the same, the first and fifte numbers, bee of contrary denomination: and you must multiply the numbers, whereupon the question dependeth, which is the fite number, by the first and third numbers, which give the value, and the product thereof must bee your diuidende, then multiply the second and fourth together, which are the numbers valued, and the product shall be divisor and so you shall find in quotient, that which is sought for and desired. as by example.  
If 4. d. starling bee worth 5. d. flemish, and 12. d. flemish bee worth 8. souse tourneys. Question, how many pence starling maketh 50. souse tourneys, which is the french crown by exchange.  
answer, multiply 50. souse tourneys( which is the number whereupon the Question dependeth) by 4. d. starling, and 12. d. flemish, which numbers give the value, and the product thereof shall bee your diuidende. Then multiply 5. d. flemish, and 8. souse Turnoys( which are the numbers valued) the one by the other, and that product shall bee the divisor, and so find in quotient 60. d. starling, the which is worth the crown of 50. souse tourneys, as by the practise may appear.  
d. d. d. souse, souse. 4. Star. 5. Fle. 12. Fle. 8. thou. 5. thou. 12 8       48 40       2400    
In the fourth parte of the Rule of three compound, the first and fiueth( or last of the known numbers) are of one denomination: and you must multiply the number whereupon the Question dependeth, by the numbers that haue valuation, and that product divided by the result of the numbers which give the valuation multiplied together, yieldeth in quotient that which is sought and desired. As by example.  
If 4. d. starling be 5. d. flemish, and 12. flemish bee 8. souse tourneys. Question, how many souse tourneys is 60. d. starling worth? answer. multiply 60. d. starling( which is the number whereupon the Question dependeth) by 5. d. flemish, & 8. souse Tourneys, the numbers valued, and the product being 2400 shalbe your dividend. Then multiply together the numbers which giveth the value, which are 4. d. and 12. and the product is 48. for divisor, & the division made yieldeth in Quotient 50. souse tourneys, as practise doth manifeste.  
d. d. d. souse. d. 4. Star. 5. Flem. 12. fle. 8. tower. 60. starl   8 4       40 48       60         2400              

¶ The Rule of company without time limited.  
two men in company, the first put into stock 45. li. and the other put in 68. li. who gained 32. li. Question, what portion of the gain groweth to either party.  
To answer this Question, and all other such like, how many soever are joined in company, their whole stock shall ever bee the first number in the Rule of three direct, and that which hath been gained by their said stock, shall ever bee the second number in the same Rule, and every mannes proper and particular stock shall bee the third number, and so woorkyng every quotient will show the portion of him, unto whom the particular stock doth belong, as by example the effect more plainly appearing.  
45 68 113 li. li. li. If 113. gain 32. what 45.   45     160     128     1440    
 
li. li. li. Likewise if 113. gain 32. what 68.   68     256     192     2176    
 
Thus appeareth the gain for the first man is 12 li. 84 / 113, & for the second 19 li. 29 / 11●. The which two sums together making the just gains, which is 32. li. proveth the work true or else not.  
Note, that men in company, having loss by traffic vpon the Seas, or otherwise, their several portions to bee born, is found by this Rule also. and so many particular men as are in company, so many several Quotientes shall bee made, and all together make the gain or loss, every mannes portion, according to his stock, whereof to give many examples were superfluous, but onely to show how to apply the Rule, whereof a few examples follow hereafter.  
Three men jaded a ship, the adventure of the first was 546. li. of the second 628. li. and of the third 732. li. By tempest vpon the sea, the master was forsed to cast ouerborde, to the value of 640. li. Question, what portion of the loss every man ought to bear?  
answer. The whole adventures added together, make 1906. li. for first number in proportion, and the loss 640. li. must bee the second number, and every particular portion of the stock the third number. The which multiplied and divided, according to the Rule, yieldeth three several quotientes, showing the loss of every man, which for the first is— 183. li 642 / 1906. for the second 210. li. 1660 / 1906, and for the third 245 li. 510 / 19●●. And all added together make the just loss, which is 640. li.  
    li.   li. If 1906 lib. lose 640 lib. what 546 facit. 183. 46● / 1900 628 210. 1000 / 190● 732 245. 1510 / 1906         640.  
Three men in company gained 100. li. whereof for 32. lib. which the first man put in, he had of the gain 25. li. and of the rest the third man had ¼, more then the second. Question, what the second and third put into stock? answer. first, consider the 25. li. taken out of the gain, there rest 75. li. for the second and third whereof the third must haue 5. li. for 4. to the second. Wherefore add 5. li. and 4. li. together, and that maketh 9. li. for first number in the Rule of three, and say, if 9. li. require 75. li. what 4 li. and for the second man you shall find in gain 33. li. 6. s. 8. d. and for the third 41. li. 13. s. 4. d. as the practise sheweth.  
li. li. li.   li. s. d. If 9. require 75. what 4 facit. 33. 6. 8. 5 41. 13. 4.  
Then having found every mannes particular parte of the gain, you shall say. If 25. li. of gain come of 32. li. in stock for the first man, whereof cometh 33. li. 6. s. 8. d. for the second, and 41. li. 13. s. 4. for the third work, and you shall find the second man put in 42. li. 13. s. 4. d. and the third 53. li. 6. s. 8. d.  
    .li   li. s. d. If 25 li. require 32. l. what 33. l⅓ facit. 42. 13. 4. 41. ⅔ 53. 6. 8.   

¶ The rule of company with time.  
THree merchants in company, the first put in 50. li. for four Monthes, the second 65. li. for seven months, and the third 72. li. for nine months, who gained 85 Question. What every mannes portion of the gain?  
Here is to bee noted, that every mannes money must bee multiplied, by his time of continuance in the company, and the three productes added together, shall bee first number in the rule of three, the gain the second and every particular product the third number, and so proceeding the work, you shall find three several quotientes showing every mannes parte of the gain to him due, according to his stock, and time of continuance, as by example will appear.  
1 2 3 50 65 72 4 7 9 200 455 648     455     200     1303  
Thus having found the three several productes to be 1303. then you shall go to the Rule and say.  
lib. 200 facit 13. 67 / 130● If 1303. gain 85 what 455 29. ●88 / 1383   648 42. 354 / 130●       85  
Here is to bee noted, that so many men as are in company, so many several productes must be made, and so many several quotients must manifest the gain to every one belonging. &c.   

❧ The Rule of Aligation.  
THE Rule of Aligation requireth a certain circumstance, for gathering of differences, of things of sundry Prices, whereof part may be better and parte worse then a common price, whereas a quantity of every sort occasion may require to bee taken, and the said differences added together, shal be first number in the Rule of three. The whole quantity of the matter desired the second, and every particular difference the third number, and so many particulars as are in the work, so many several quotients will make the quantity of the matter sought for. The effect more plainly appearing in few examples, then in many words or great discourse thereof, as hereafter you may see.  
An Appoticary for recovery of health in a noble man is charged to compose an ingredience of 4 sundry sorts of rich and costly drogges, to say, of 45 s. 42 s. 36 s. and 32 s. the ounze, and to haue 8. ounzes worth 40 s. the ounze of every sort a quantity. Question. How much of every sort is to be taken?  
answer. first set down the several prizes one under an other, the highest uppermost, with the common price at the left side thus.  
40 45 8 42 4 36 2 32 5     19 Then you must link together one above the common price with one of the other under the common price, and the difference of every one above the common price shalbe set against the other, linked with it under the common price, and to the contrary, the difference of every one under the common price shalbe set against the other, linked with it above the common price, the which differences found and set accordingly, as above appearing, the total making 19. Then by the Rule of three is to be sought the 4 Quotients, to make 8 ounzes of 40 s. the ounze of every price a quantity, which is as by practise hereafter appeareth.  
If 19. require 8. what 8 8 64  
If 19. require 8. what 4 8 32 If 19. require 8. what 2   8 16 0 16 / 19 16 19 If 19. require 8. what 5 8 40  
Here you may perceive, that the said Appoticary ought to take of 45 s. 3 ounzes 7 / 19, of 42 s. 1 ounze 13 / 19 of 36 s. 16 / 19 of an ounze, and of 32 s. 2 ounzes 2 / 10 the which together maketh 8 ounzes of 40 s. the ounze, the effect by the question required.  
Here is to be noted, that although the former quantities be truly brought forth, as the question requireth, the same notwithstanding, the same quantities may show like truth, if the differences change their places, as by linkyng the uppermost price with the lowest saving one, and the lowest with the uppermost saving one thus.  
40 45 4 If 19. require 8. what 4 2. 13 / 19 42 8 8 3. 7 / 19 36 5 5 2. 2 / 19 32 2 2 0. 16 / 19     16     8. oz  
A merchant hath bought canvas of 22 d. 19 d. 15 d. 10 d. 9 d. and 8 d. the Ell. A friend requireth to haue a thousand elles( of every sort a parcel) to stand him in 12 d. the ell one with an other, the merchant to gain nothing by him, but to haue given him a satin Doublet for his friendship? Question. How much of every sort to bee taken. Neither party to haue wrong. answers. first find the differences by former order thus.  
12 22 4 19 3 15 2 10 3 9 7 8 10     29 The which being found proceed in the Rule of three, & so you shall haue 6 Quotientes, which show the quantity of every sort of canvas, to be taken as practise will show.  
    Elles. If 29. require 1000 ells, what 4 137. 27 / 29 3 103. 13 / 29 2 68. 28 / 29 3 103. 1● / 29 7 241. 11 / 29 10 344. 24 / 29     1000.  
A merchant hath 4 sorts of gold, of several fineness( to say) of 23 Carratz ¾ of 22 Carratz ⅔ of 21 Carratz ⅚, and of 20. Carratz. ½. fine Question, what quantity of every sort is to be taken, to haue 100. ounzes of 22 Carratz fine just. answer.  
First note, that forasmuch, as the Fractions, fine above, and under the common fineness, are of sundry Denominations: therfore they must be reduced, and made of one Denomination, and will stand thus.  
22 22. ●● / 12 18 22. 8 / 12 2 21. 10 / 12 8 20. 6 / 12 21     49 Here you may perceive, that the finer sorts above the common fineness 21 / 12 and 8 / 12 are set against the parcels linked with them, which are under the common goodness, and for the common fineness therof 2. is set against the parcel, linked therewith Also 20 Carratz 6 / 18. is set against the parcel linked therewith, and so is found 49 / 12 for first number in the Rule of three, and every particular difference, the third number with the 100. onze desired the second, with which numbers procéedyng in the Rule of three you shall find 4 quotient which will declare the quantity of every sort of Gold, to be taken to haue 100. ounces of 22 carrratz fine just, the effect in example appearing.  
If 49. require 100. what 18 36. onz. 36 / 49. 2 4. 4 / 49. 8 16. 16 / 49. 21 42. 4● / 49.     100. ounces.  
An assay master hath five sorts of silver of sundry finesse: that is to say, of 11. onzes 14. d. 11. vnzes 10. d. 10. vnzes. 5. d. 9. vnzes 16. d. and 9. vnzes. 12. d. weight fine, and would haue 100. lib. weight of 11. vnzes 2. d. fine. Question. What quantity to be taken of every sort? answer.  
first reduce your several denominations into one, and then it will stand thus.  
222 234 30 230 26 205 26 196 11 192 12     105 The which differences found procede with the totalle for the first number. the 100. lib. the second: and every particular the third, and so shall you haue the quotient of every sort to bee taken, to make 100. lib. weight of 11. vnzes 2. d. fine, the effect in example appearing.  
    lib. If 105. require 100 li. what 30 28. 60 / 105. 26 24. 80 / 105. 26 24. 80 / 105. 11 10. 50 / 105. 12 11. 45 / 105.     100  
howbeit that these works are to bee proved by the common order, of proving the Rule of three: The same notwithstanding, there are other sundry orders of proves, for the commixions of golds and Siluers which here I omit, in respect of several causes, referryng such as by vocation, may desire knowledge therein, to private conference who may be satisfied to effect extraordinary.   


The rule of one false position.  
THE Rules of false Positions are so called( not that any untruths are furthered, or taught by the same, but that by a number supposed, though far from truth. The same put in use of the Rule, bringeth forth the truth, which of consequence is expected and desired, as by example the effect may appear.  
A merchant taketh a house, whereupon dependeth such yearly benefit, that he disburseth a somme of money, not name. A friend requestyng to haue the bargain, the merchant is content to take 10 by C. for his money, and at the end of seven yeres the time of his use therof, receiveth of his friend 606. lib. of money, for that he had disbursed and the intreste of the principal. The question is, what portion of money the Marchante disbursed for the said house. answer.  
The first number in the Rule of three, for answer of this must bee furthered by supposition, the which for example, take 300 li. supposed to bee the money first disburied, then of consequence, the intreste thereof seven yeres, being 210. lib. joined therewith is to bee made the second number in the work.  
Then to proceed, say if 510. lib. principal and gain come of 300. pound, whereof cometh 600. pound, work and find 352. pound. 1 / 1 6 / 7.  
If 500. come of 300. whereof 600? of 352. 19 / 17.  
Here note, that what number or somme of money soever bee taken for the supposition as first number, and the same with the interest thereof made the second. The 600. li. being third in the work, bringeth the truth to light: as by an other example above the truth supposed may appear.  
Suppose the merchant paid at the first 400. pound for the aforesaid house, the interest therof is 10. li. by C. for seven yeres, is 280. pound, which put to the principal maketh 680. pound, and is 80. pound more then should bee, if the supposition were true, wherefore say in former order.  
If 680. li. come of 400. li. whereof cometh 600. work saieth as before of 252. pound 16 / 17.   

The rule of ij. false positions.  
WHen any question is framed, found of such difficulty, as may require the practise of two false positions: you shall suppose any number at pleasure for the first position, and by consequence of work will appear an error either under or above the truth, the which being above, shall be noted with this character, lying more, & being under the truth, shal haue this note −, which signifieth less. And even so make a second position, to bring forth a second error with the like notes.  
Then you shall multiply the first position, by the second error, and the second position with the first error, and if the signs of the errors bee like, to say both more, or both less then the truth, then shall you substraie the lesser product from the greater. Also you shall substraie the lesser error from the greater, and with the remain thereof you shall divide the remain of the substraction of two productes: and the quotient of that division, will show the true number sought for.  
howbeit you shall note, that when the two errors haue signs unlike, as the one to much, and the other to little, then you shall add the two productes together, and divide the total by the somme made, by adding the two errors together, and the quotient will show the truth sought for also, as more plainly may appear by appliyng the use of the Rule, to the answer of some questions following.  
Three merchants gains 1000. lib. whereof the several portions are unknown saving that the second ought to haue double the portion of the first, and 5. pound more The third ought to haue double the portion of the second, and 10. li. more.  
The question is, what portion of the said gain belongeth to every man answer.  
You may suppose any Number at pleasure, as aforesaid, the which for example shall bee 150 li. supposed to bee the first mannes due. Then the double thereof with 5. pound more is 305. pound for the second. The double whereof with 10. pound more is 620 pound for the third, and the three portions together make 1075. li. wherein is found an error of 75. pound to much: wherefore for a second work, I suppose the first mannes portion to be 144. pound: then the second ought to haue 293. pound, and the third 596. pound, which together make 1033. pound, wherein is found an error of 33. pound to much also. Wherefore I set the first position 150 with the error 75. at the vpper end of a cross, with the sign to much thus, and the second position 144. with the error 33. at the nether end of the cross, with the sign to much, also as appeareth.  
Then the first position 150. multiplied by the second error 33. produceth 4950. also the second position 144. multiplied by the first error 75. produceth 10800. And because the signs of the errors bee like, as both to much I substraie the lesser product 4950. from the greater 10800 and there remaineth 5850. for diuidende. Likewise I substraie the lesser error from the greater 75. and the remain is 42. for divisor. Then dividing 5850. by 42. the quotient is 139 li. 2 / 7. which is the true portion for the first man, then the second of consequence hath 283. li. 4 / 7: and the third 577. li. 1 / 7. and together make 1000. li. the effect sought for by the work.  
Now to the end that the errors may bee both to little, as in the first work they were both to much, I will further an other supposition to show the agremente.  
Suppose the first mannes portion to bee 130. li. the double thereof and 5. li. more is 265. li. for the second, the double thereof and 10. li. more is 540. li. for the third, & maketh together 935. li. wherein is found the first error to bee 65. pound to little. −. and set at the vpper end of the cross, with the sign in former order.  
again suppose a biggar somme to bee his portion, as 135 pound. Then the second must haue 275. pound: and the third 560. pound, and maketh together 970. pound, wherein is found an error of 30. pound, −. to little also. Wherefore I set the position with the error at the foot of the cross as you see with the sign. −. to little.  
Then the first position 130. multiplied by the second error 30. yieldeth in product 3900. Likewise multipliyng the second position 135. by the first error 65. the product is 8775. And because the signs of the errors are both to little, I substraie the lesser product 3900. from the greater 8775. and the remain is 4875. for diuidende: also I substraie the lesser error 30. from the greater 65. and the remain is 35. for diuidende, Then dividing 4875. by 35. the quotient is 139. li. 2 / 7. as in the former work, and the portions of second and third follow of consequence, as before.  
The two former works with contrary positions show one truth, brought forth by one order, for that the errors in each work was like, though in the first both to much, and in the latter both to little: And now resteth the maner of work, when the errors haue signs unlike, as the one to much, & the other to little, where the productes and errors will require addition, as to the contrry before substraction: the effect appearing in a third work wherein the positions made so far under, and above the truth, that the rule may satisfy every mannes expectations, in bringing forth the truth, notwithstanding the distance of the suppositions from the same.  
Suppose the first mannes portion of gain in the former question to bee 3. pound: then the second having 5. pound more then the double thereof, hath 11. pound. And the third man 10. pound more then the double of the second, hath 32. pound, and maketh together 46. pound, which is 954. pound to little for the first error, set with the position at the head of a cross in former order, with the sign. −. to little.  
Then to haue the second position known. Suppose the said first mannes portion to be 500 pound, then consequence alloweth to the second 1005. li. and to the third 2020 pound, which together maketh 3525. pound, wherein is found an error of 2525 pound to much, the which with the sign and position, I set at the nether end of the cross as you see, and for that the signs bee unlike, as the one to little, and the other to great. You must add the two productes together for the diuidende, and the two errors for the divisor: and for your better understanding, note these few words in verse.  
The signs both like, substraction will haue: And contrary found, addition doth crave  
The which Addition made of the two productes, the total is 484575. to bee divided by the total of the two errors, which is 3479. The quotient thereof is 139. li. 994 / 3479 agreeing with the two former examples, the effect required in every work.  
Thus having passed through the common partes of Arithemetique, in whole and broken numbers, appliyng the same to the Rules of proportion, ordenarie to bee found in most authors. Now followeth other rules requiring further circumstances then in schools( I mean in universities been taught) to say of gain and loss upon the hundreth, of Barters, and of exchange for sundry nations.   

Of gain and loss by the. C.  
A merchant hath 100. Clothes, which coste 425. lib. he desireth to know how to sell every cloth to gain 8. li. vpon the hundreth. To answer this qustion and such like, you must use this circumstance, saying, by the Rule of three direct.  
If 100. lib. do gain 8. li. what gaineth 425. li. paid for the forenrmed 100. Clothes: work by the foresaid Rule, and find 34. li. gained, as practise will show.  
If 100. li. gain 8. li. what 425.   8   3400  
34 lib. gained.  
And having found the gain, add thereto the principal, which the Clothes coste, and you shall haue 459. lib. the value of the said Clothes, sold after the rate of 8. pound gained vpon the C. li.  
Then to find the price of every cloth, after the rate just, you shall say.  
If 100. Clothes sold, yield 459. lib. plincipall and gain, what one cloth, multiply and divide, and you shall find 4. pound 11. s. 9. d. ⅗: every cloth as practise sheweth.  
If 100. yield 459. li. what one 1 459  
4 lib. 59 / 100.  
Likewise if one piece of cloth containing 84. shepherds, coste 60. pound: how ought the yard to bee sold to gain 10. li. by C. li answer in former order saying, if 100 li. gain 10. li. what 60. li. work by the rule of three, you shall find 6. pound. Then say again if 84. shepherds yield 66. pound in principal and gain, what one, work accordingly, and you shall find 15. s. 8. d. 12 / 21. every yard.  
Clothes 60. costyng 53. pound, howe may 12. pieces thereof bee sold to gain 9. li. by C. answer former order, saying:  
If 100. li. gain 9. li. what getteth 53. pound: work and fide— 47. li. 14. s, the which gain with the principal maketh 577. li. 14. s. the which found, say again.  
If 60. Clothes sold, yield in principal and gain 577. li. 14. s. what 12? work by the rule, and find 115. li. 10. s. 9. d. 3 / 7. and so much ought 12 Clothes to bee sold to gain 9. li. by. C.  
If apiece of velvet coste every yard 18 shillings, howe may the yard bee sold again, to profit 9. li. by C:  
say if 100. li. gain 10. li. what 18. s? work and find 1. s. 9. d. ⅗. which makes for every yard to bee 19. s. 9. d. ⅗. to gain after 10 li. by C.  
If in sale of 100. shepherds of satin for 48. li. there bee gained 3. li. 10 s. I demand what coste every yard the first penny. answer. Rebate the gain 3. pound 10. s. from the total 48. li. and the principal will appear 44. pound 10. shillings wherefore say:  
If 100. shepherds coste first penny 44. li. 10. shillings, what one yard, work by rule, and find 8. s. 10. d. ⅘.  
200. ounces of gold taken in a shift for 645. li. and sold again to loss 10. li. in the hundred. I demand what is loss in every ounze? answer, say if 100. lose 10 pound, what 145. li? work by rule, and find 64. li. 10. s. then say gain. If 200. ounzes, lose 64. li. 10. s. what one ounze, work and find 6. s. 5. 4 / 10.  
If the pound of Saffron, which coste 18. s. bee sold again for 18. s. 6. d. I demand what is lost by the hundred pound in money? answer. If 18. s. lose 6. d. what 100. pound, work by rule, and find 2. li. 15 s. 6. d. in the C.  
If 100 shepherds of damask coste 65. li. and the buyer repentyng, would lose 5. li. in the hundred of money. I demand how the yard may bee sold, his loss to bee neither more nor leffe then after the rate aforesaid, of 5. by hundred. answer by rule and say.  
If 100. li. lose 5. li. what 65. li. work and find 3. li. 5. s. the which rebated from the principal 65. li. rest 61. li. 15. s.  
Lastly say, if 100 shepherds yield 61. li. 15. s. what 1. shepherds, work and find 12. s. 4. d. ⅕ every yard.   

Of the Rule of Barteryng.  
TWo merchants willing to change their merchandise together, the one having Carsies of 35. s. the piece redy Money will deliver them in Barter at 40. s. the piece. The other having Holland cloth worth 2. s. 6. d. the Ell ready money, would know how to put away an ell to make the Barter equal.  
To answer herein, and by like order all other, say by Rule, if 35. s. for a Carsey, make in barter 40. what will 2. s. 6. d. for an ell of holland yield in Barter, work and you shall find 2. s. 10. d. 10 / 35 the ell to equal the Barter.  
A merchant hath 100. Clothes to sell for ready money at 14 li. a piece, and in barter he will put them away at 15. li. 10. s. every cloth, an other will give for them in Barter, silks that are worth 9 li. a piece ready money. I demand at what price the silks are to be delivered in barter, & how many pieces payeth for the clothes, neither party to haue advantage of other. answer by former order and say.  
If 14 li. for a cloth redy Money yield 15. li. 10. s. in Barter, what giveth 9. li. for a piece of silk in Barter, to make the truck equal, work and find 9. li. 19. s. 3. d. ● / 7, the price of a piece of silk.  
Then say, if 9. li. 19. s. 3. d. 3 / 7, require 1. piece of silk? how many pieces of silk is bought with 1550 li. which is the value of the 100. pieces of clothes in truck, work by the Rule of 3. direct, and you shall find that, 155. pieces, and 10230 / 16734, at the former price payeth for the 100. Clothes, and neither party having advantage of the other.  
Two merchants desirous to change their merchandise together. The one having alum, worth 25. s. by C. ready Money, and will put it awoy for 30. s. by C. in truck, to take Pepper at 3. s. 4 d. the lib. which is worth but 3. s. the lib. The pepper merchant not of skill to equal the change, giveth Pepper for 100. Quintals of the alum at the prizes aforesaid. I demand what advantage the one hath of the other and who is the loosar, answer.  
First seek at what price a lib. of Pepper maketh the Barter equal saying. If 25 s. make 30. s. in alum, what 3. s. for Pepper, work and you shall find 3. s. 7. d. whereby appeareth 3. d. ⅕ lest in every lib. of Pepper delivered.  
Then to find the state of the change, say by the Rule of 3. If 3. s. 4. d. by 1. lib. of Pepper what bieth 150. li. the value of the 100. Quintalls of alum, work and you shall find 900 lib. of Pepper payeth for the said alum.  
again to find the loss, search what quantity of Pepper would haue paid for the alum, if the Bartar had been equal, saying. If 3. s. 7. d. ⅕ requireth 1. lib. of Pepper, what 150. lib. the value of the alum? work and you shall find 833, lib. l⅓ would haue paid for the alum, the Barter made equal. The which 833. lib. weight l⅓ rebated from 900. lib. delivered the rest, lost is 66. li. weight ⅔, and so much gained the alum of the Pepper. The effect sought for by the work.  
Here you shall understand, that if the one party require to haue a portion in redy Money, as ½. l⅓. ¼. or any other, you shall rebate the said such portion what it bee, as well from the price of his wears worth in ready Money, also ranted in Barter, and the 2. remaines shalbe the first and second numbers in the Rule of three, and the third shal be the price of the Ware of the contrary party, as hereafter by example the effect may appear.  
Two merchants willing to change merchandises, the one with the other, the first hath oils of 24. li. by tun, redy money, and in Barter he will put it away at 27. li. by tun, and will haue l⅓ in ready money. The other hath Bayies of 2. s. 6. d. the yard ready money. I demand how the yard of bays ought to bee ranted, to make the Barter equal? Aunsware. First rebate 9. li. which l⅓ of 27. li. from 24. li. the rest is 15. li. for first number in the Rule of three, also rebate the same 9. li. from 27. li. the value of the oil in Barter, and the rest is 18. li. for second number in the said rule, and the third number shal be 2. s. 6. d. to find howe the yard of bays shalbe delivered in Barter, the which to find, say, if 15. li. yield 18. li. a tun of oil, what 2. s. 6. d. for a yard of bays, work and you shall find 3. s. for every yard of bays in Barter, and so of all other.  
two merchants will change Marchaundizes, the one hath wines of 13. li. 6. s. 8. d. redy Money by tun, and in barter he will put them away at 16. li. 13. s. 4. d. by tun, and also will haue ¼ money content. The other hath tin at 3. li. the C. in barter. I demand what the C. of tin is worth ready money. answer. Take the one fourth parte of the price of a tun of wine in Barter, which is 4. li. 3. s. 4. d. from the price of a tun ready money, which is 13. li. 6. s. 8. d. so resteth 9. li. 3. s. 4. d. for second number in the Rule of 3.  
Likewise take the said 4. li. 3. s. 4. d. from the price of the Wine in Barter, and the rest will be 12 li. 10. s. for first number in the said Rule. Lastly, put the price of the C. of tin the third number, which is 3. li. in Barter, and work the Rule, and you shall find 44. s. the value of the C. of tin ready Money.  
two merchants will change their Marchaundizes the one with the other, one hath Cottons of 10. li. the pack ready money, and will put them away in Barter, at 13. li. 6. s. 8. d. the pack, and will gain 10. li. by C. and also haue the half in ready money. The other hath Burrace of 6. d. the li. ready money. I demand how the li. of of Burrace shall be put away in Barter. answer. First say, if 100. li. give 10. li. what giveth 13 li. 6. s. 8. d. the price of a pack of Cottons in Barter: work by the rule, and you shall find 14. li. 13. s. 4. d. whereof the one half demanded in ready money, rebated from 10. li. price of the Cottons ready money, the rest is 2. li. 13. s. 4. d. for first number in the Rule of three, and the same also rebated from 14. li. 13. 4. d. price of the Cottons in Barter, the rest is 7. li. 6. s. 8. d. for second number, then make the 6. d. price of Burrace the third number, and work by the Rule, and you shall find 16. d. ½ for the lib. of Burrace in Barter.   

Of the exchange of Moneys from one country to another.  
FOrasmuch as by the laws and statutes, of every or most nations it is defended to transport or carry out Gold and silver either in coin or Bullion, therefore was divised and ordained the exchange of moneis between country & country, that is to say. For a somme of money, the value great or small, in one nation delivered from one man to an other. The said deliuerar to receive the value therof in money of an other country wherewith to furnish his affairs for traffic or otherwise, in that place, the effect by examples more plainly apearyng.  
A merchant delivereth in London 100. li. starling to receive in Andwarp at sight of the bills made for exchange therof, for every li. starling 24. s. 9. d. Flemmishe. I demand what money Flemmishe payeth the bills in Andwarp. answer. Say, if 20. s. starling, bee worth 24. s. 9. d. Flemmishe, what 100. li. starling, work by the Rule of 3. direct and find 123. li. 15. s. flemish payeth the bills of the said 100. starling.  
A merchant delivereth in Andwarp 100 li. Flemmishe, to receive in London 20. s. starling for 24. s. 9. d. Flemmishe, I demand. what starling money payeth the Bills for the said 100 li. Flemmishe. answer and say  
If 24. s. 9. d. flemish, give 20. s. starling, what 100. li. Flemmishe, multiply and divide, and you shall find 80. li. 16. s. 1. d. 48 / 297, and so much starling money payeth the said bills of 100. li. Flemmishe.  
A merchant delivereth in london 100. li. starling, to receive in paris 50. s. Turnois for every french crown of 5. s. 3. d. starling. To say, valued at that price. I demand how much Turnois or french money payeth the bills for the said 100. li. starling. answer. Say by the Rule of 3. If 5. s. 3. d. starling, make 1. crown, what 100. li. multiply & divide, and you shall find 380. ▿ and 60 / 63 parte of a crown, and note that the character ▿ representeth the crown by exchange, and is ever 50. s. Turnois or french money. Then say again, if 1. ▿ be worth 50 s. what 380. 60 / 63 ▿ work by the Rule, and find 952 li. or franks 7. sowce, and 7. d. 47 / 6● payeth the bills for the said 100. li. starling.  
A merchant delivereth in paris 1000. li. or franks, the which frank or li. is 20. sounce or pound Turnois french money to receive in London 4. s. 10. d. starling for every ▿ of 50. souse Turnois. I demand how much starling money payeth the bills of exchange for the said 1000. li. Turnois. answer. say first, if 50. souse Turnois make 1 ▿ howe many Crownes maketh 1000 li. work by the Rule & find 400. ▿.  
Then say again, if 1. ▿ give 4. s. 10. d. starling, what 400. ▿. work accordingly, and find 96 li. 13. s. 4. d. starling, payeth the bills of exchange for the said 1000. li. Turnoys.  
A merchant delivereth in London 100. li. starling, to receive in Bayon for every 5. s. 10. d. 1. ducat of 374. Maruedies? I demand how many Maruedies payeth the bills for the said 100. li. starling. answer.  
Say first, if 5. s. 10. d. make 1. ducat, what 100. li. multiply and divide, and you shall find 342. ducats. 6 / 7.  
Then say again, if 1. ducat give 374. Maruedies, what giveth 342. ducats ● / 7. work accordingly, and find 128228. Maruedies 4 / 7.  
A merchant delivereth in Bayon 100. M. Maruedies, to receive in London 5. s. 10. d. for every ducat of 374. Maruedides: I demand how much starling money payeth the bills of exchange for the said 100. M. Maruedies. answer. Say if 374. Maruedies make one ducat, what 100. M. work by the Rule, and find 267. ducats. 331 / 374.  
Then say again. If one ducat give 5. s. 10. d. starling, what giveth 267. ducats 331 / 374. work and find 78. lib. 2. s. 7. d. 356 / 374. And so much payeth the bills of exchange for the said 100. M. Maruedies. &c.  
Thus having run over the several common partes of Arithemetique, as well in whole as broken numbers, now followeth the Rules of Breuetie, of rare and profitable effect, the original cause of furtherance of this my work.     

¶ The fourth and last part containing the Rules of Breuetie, of rare and singular effect.  
THe Rules of Breuetie in works of Arithemetique, are sundry and many, and to further a work, wherein to show all that might bee expected, would not onely be a tedious and superfluous toil, but also cunning might want in the beste learned the same to perform. Wherefore I mind not to enlarge my travail with such Rules, as men are ordinarily acquainted withall, neither so much to say of other,( which may seem more rare, and not in familiarity with many men) as might bee furthered to good and profitable purpose. notwithstanding as every man desireth the nearest way to end a weary journey: so I intend to show practise first, how to give thee some of any number, whereof the value of an unity, is an even parte of a pound, with as small circumstance and few figures as may be, therein, avoiding the tedious use of figures in Multiplication, and division, commonly practised in the Rule of three. And again, as many men in respect of benefit, or to withstand a detriment, may content themselves to take a compass out of the nearest way, to stop a breach in a hedge of a corn field, or to see his pasture void of other mennes cattle: so I think it both profitable and necessary, to further a more large walk by sundry orders, in searching the total of sundry sums, whereof the value of an unity, may be either some even parte of a pound, or sundry even, or odd partes of a pound. From one penny to twenty shillings, and so much above as may seem needful, the effect not so hard to understand in words, as with facility to bee perceived in example, which hereafter follow in plentiful maner, and so full of change as procured cause, wherefore this book is name the storehouse of Breuetie.  
It is good for every learner to print in memory the even partes of a pound of money, before he meddle with the brief rules. The which partes are put in the table following.  
d. l. 1. 1 / 24● 2. 1 / 120 3. 1 / 80 4. 1 / 60 6. 1 / 40 8. Take 1 / 30 At 10. 1 / 24 12. 1 / 2● 15 1 / 16 16 1 / 51 20. 1 / 12. or ½ of ⅙, or ¼ of l⅓ 2. s. 1 / 10 3. s. 4. d. ⅙ 4. s. ⅕ 5. s. ¼ s. d.   6. 8. take l⅓ 
¶ The price of an unity.  10. s. ½ 13. s. 4. d ⅔ 15. s. ¾ 16. s. 8. d. 5 / 6  
WHen occasion doth procure any number to be summed, whereof the unity beareth any even parte of a pound. Then the said number being divided by the Denominator of such parte as is the value of the said unity, the quotient of that division willbe the total, sought for of any such number, as by example the effect more plainly may appear.  
At 4. s. the yard, what 2156. shepherds? answer.  
forasmuch as 4. s. is the 1 / 5 of a pound, therefore if the number of shepherds being 2156. bee divided by 5. the Denominator of 1 / 5 li. which is 4. s. the price of every unity in the said number, the quotient will be 431. 4. s. the true total required as practise will manifest.  
 
Thus division hath brought forth the total of 2156. shepherds at 4. s. the yard, the effect whereof I further not as a work of brevity, though in some respects in deed it is a breuety, but rather for an example thereby the better to understand the maner of abreuiation of work in that example, and in all other like, by practising the same, and like division reteinyng in memory the divisor, and remain of every such division, and setting the total in one line under the number given, as by an other example of perfect breuetie the effect may appear.  
At 4. s. the yard what 2156. yards. Take ⅕. Makes 431. 4. s.  
Here I haue kept in memory the divisor. 5. the Denominator of ⅕ li. which is 4. s. the value of every unity in the number given for example, and haue found that the same is contained in 21. 4. times, and 1 remaining. Wherefore I set 4. under 21. and a line between. Then I find also that the said divisor 5. is contained in 15. 3 times, and nothing remaineth, wherefore I set 3. under 15. as the former. Then lastly I find that 5. the divisor is contained in 6. the last figure 1 time, and one remaining, wherefore I set one vnderline as the other, and the one remaining being 1 / 5 of a pound for the same I put 4. s. also in the Quotient, and so the work is ended with the use of as few figures set down, as can be, which is the effect ment by this last part, as in sundry examples following you may perceive.  
At 1. d. what 54368. elles. answer. Forasmuch as 1. d. is the 1 / 240 part of a pound, therefore, if this given number bee divided by the denominator 240. the quotient would be the pounds, containing the value of the said given number, as aforesaid, howbeit, for that I pretend the omitting use of the Figures, as well in Multiplication as division. I therefore imagine what even parte of a pound I may work by. Whereof a penny being a perfit parte, may be taken from it, and so my desire furnished. The which finding to be sundry, as 8. d. the 1 / 30 parte, whereof ⅛ parte serveth, and 6. d. 1 / 40 parte whereof ⅙ parte serveth. Also 4. d. 1 / 60 parte whereof ¼ parte serveth, the which last to me séemyng most apt, I further for example as followeth.  
At 1. d. what 54368. ells. Take ¼ of 1 / 60. for 4. d. 906. 2. 8. d. whereof ¼ facit 226. 10. 8. d.  
As I know that 4. d. is 1 / 60 part of a pound, so 1 / 60 part of the given number, is the pounds containing the value sought for at 4. d. the ell, howbeit, for that 1. d. is the value admitted for every unity, and is ¼ of 4. d. therfore ¼ of the quotient yielded for 4. d. is the total sought for, which as above appeareth is 226 li. 10. s. 8. d. found without use of more figures set down, then above appeareth  
And note that the divisor being 60. the Cipher is imagined to stand under 8 the last Figure of the dividend so that 6. dividing all the other Figures, yieldeth 906. li. and 8 / 60 remaining which is 2. s. 8. d. as also above you may see, whereof ¼ is the total sought for, as aforesaid.  
At 2. d. what 45682 foot Take ½ of 1 / 60. for 4. d. 761. 7. 4. d. whereof ½ facit. 380. 13. 8. d.  
This division is made in former order, and the truth shewed accordingly.  
At 3. d. pound weight what 4356. li.  
Take 1 / 80▪         facit 54. li. 9. s. 0. d.  
At 4. d. what   3986. Take 1 / 60.   facit 66. 8. s. 8. d.  
At 6. d. what 3245. Take 1 / 40. facit 81. 2. 6.  
At 8. d. what 2678. Take 1 / 30. facit 89. 5. s. 4. d.  
At 10. d. what 2576. Take ¼ of ⅙ for ⅙ 429. 6. 8. whereof ¼ facit 107. 6. s. 8. d.  
At 12. d. what 2432. Take 1 / 20. facit 121. 12. s. 0. d.  
At 15. d. what 2354. Take 1 / 16 or ½ of ⅛.   For ⅛ whereof ½ facit 147 li. 2. 6. d.  
At 16. d. what 2216. Bushelles. Take 1 / 15 or 1 / 20 and l⅓ thereof, For 1 / 20 110. 16. s. 0. For l⅓ thereof 36. 18 s. 8. facit 147. 14. 8. d.  
d.   At 20. what 1864. Take ½ of ⅙. For ⅙ Whereof ½ facit 155. 6. 8.  
At 2. shillings, what 1568. Take 1 / 10 facit 156 l.. 16. 0. d.  
At 2 s. 6. d. what 1453. Take ⅛. facit 181. 12. s. 6. d.  
At 3 s. 4 d. what 1263. Take ⅙ facit 210 li. 10. s. 0 d.  
At 4. shillings, what 1144. Take ⅕. facit 228. li. 16. 0. d.  
At 5 s. what 1123. Take ¼ facit 280. 15 s. 0. d.  
At 6 s. 8. what 1042, Take 1 / ● facit 347. 6. s. 8 d.  
If you mark the former practises, you may perceive that in every example, wherein the price of an unity in the given number is contained in a diget, or an article number, being either of shillings or pence. The total of that example is shewed in one line done, without use of mo figures, then in the same line doth appear: Howbeit, where the value of the unity is contained, in a mixed or compound number, then is required two lines, three lines, four or more, as the sundry partes in the example may proclure. For sometimes the total is given by one even parte, and that requireth but one line. sometimes by a parte of a parte, and that requireth two lines, the one substraied from the other: sometimes by even and sundry partes, and partes of partes also, which will require so many lines, as the sundry partes will procure. As in example following, the effect more at large appearing.  
At 1 d. the yard, what 24568 shepherds? Take ¼ of 1 / 60 or l⅓ of 1 / 80. or ⅙ of 1 / 40 or ⅛ of 1 / 30 or 1 / 1● of 1 / 20, or ¼ of ⅙ of 1 / 10 or ⅙ of ⅕ of ⅛.  
By every of which directions, the true total is brought forth, as by the several practises may appear.  
The given number. 24568. at 1. d. the yard. For 1 / 16 Whereof ¼ facit 102. 7. 4.  
Also the given number. 24568. at 1. d. the yard. For 1 / 80. Whereof l⅓ facit 102. 7, 4.  
Likewise the given number 24546. at 1. d the yard. For 1 / 40 614. 4. 0. d Wheref ⅙ facit 102. 7. 4.  
again the given number 24568. at 1. d. the yard. For 1 / 30 Whereof ⅛ facit 102. 7. 4 d.  
Accordingly the given number 24568 at 1. d the yard. For 1 / 20 1228. 8. s. 0. d Whereof 1 / 12 facit 102. 7. 4.  
Also the given number. 24568. at 1. d. the yard. For 1 / 10 2456. 16 s. 0. d. For ⅙ thereof. 409. 9. 4. Whereof ¼ facit 102. 7. 4.  
again the given number 24568. at 1. d. the yard. For ⅛ 3071. For ⅕ thereef 614. 4. s. 0. d. Whereof ⅙ facit 102. 7. 4.  
As necessity requireth, not such plenty of examples for one thing, so delectation in a desirous studente, may accept the good will of the quarreler herein. And nevertheless, for that every order is witness of truth one in an other, none of the same are without profit, for such as are exercised in accounts. And in respect as well thereof, as also to adorn the pearlesse Science( mathematical) of Arithemetique, with the jewels of her own closet: here after followeth sundry other examples of the same matter.  
At 1. d. the yard. what. 24586. shepherds. Take ¼ of 1 / 10 of ⅙ for ⅙ 4094. 13. 4.d. For 1 / 10 thereof 409. 9. 4. d. Whereof ¼ facit 102. 7. 4.d.  
Also at 1 d. the yard, what. 24568. shepherds Take 1 / 12 of ¼ of ⅕ for ⅕ 4913. 12. 0. For ¼ thereof 1228. 8. 0. Whereof 1 / 12 facit 102. 7. 4.  
Also at 1. d. the yard what. 24568 shepherds.  
Take ¼ of l⅓ of ⅕ of ¼.  
The given number. 24568. For ¼ 6142. li. For ⅕ thereof 1228. 8. s. 0 d. For l⅓ thereof 409. 9. 4 whereof ¼ facit 102. 7. 4. d.  
Thus appeareth that by 10. sundry orders of breuetie, without the use of the rules of 3. is brought forth the total of 24568. shepherds at 1. d. the yard, and because it may appear to the sight of every man, what difference of circumstance is between any of the said orders, and the said Rule of 3. here ●… olloweth the practise of the same by the said Rule.  
At 1. d. the yard what 14568. yards.  
Because 1. doth not increase or augment in Multiplication, I omit the said multiplication, and divide the given number by so many pence as is in a pound contained, which is 240. &c.  
 
At 2. d. the ell what. 23647 ells.  
Take for most brief ½ of 1 / 60 or ⅔ of 1 / 80 or l⅓ of 1 / 40 or ¼ of 1 / 30 or ⅙ of 1 / 20 or ½ of ⅙ of 1 / 10 or l⅓ of ⅕ of ⅛ or ½ of l⅓ of ¼ of ⅕.  
The given number. 23647. For 1 / 60 394. 2 s. 4 d. Whereof ½ facit. 197. 1 s. 2 d.  
And only for proof ½ of 1 / 10 of ⅙.  
The given number. 23647. For 1 / 06 3941. 3. s. 4. d. For ⅙ thereof 394. 2. 4. Whereof ½ facit. 197. 1. 2.  
At 3. d. lib. whaight what 4875. lib.  
Take for most brief 1 / 80  
The given number. 4875. facit. 60 li. 18 s. 9. d.  
And for proof Take ½ of 1 / 40 or ¼ of 1 / 20 or ⅛ of 1 / 10, or ½ of ⅕ of ⅛ or ¼ of ¼ of ⅕ or ½ of 1 / 60, and ½ thereof, or ½ of 1 / 40, &c.  
every of which facit 18. li. 17. s. 9. d.  
At 4. d. what. 457. Take for most bréef 1 / 60● facit 76. li. 5. s. 4. d.  
And for proof. Take ½ of l⅓ or l⅓ of 1 / 20 or ⅙ of 1 / 10 or ½ of ⅙ of ⅕ or ⅕ of ¼ of l⅓. &c.  
every of which facit 76 li. 5. s. 4. d.  
At 5 d. what. 4269. Take for brief 1 / 60 and ¼ thereof. For 1 / 60 71. 3. s. 0. d. For ¼ thereof 17. 15. 9. d. which together facit 88. 18. 9. d.  
And for proof Take 1 / 40 lacking ⅙ thereof as by example.  
At 5. d. what. 4269. For 1 / 40 106. 14. 6. From which ⅙ 17. 15. 9. per rest facit 88. 18. 9.  
Or for the same proof Take 1 / 20 and ½ of 1 / 60 or ¼ of l⅓ of ¼ or ¼ of ½ of ⅙ or ⅕ of ⅛ lacking ⅙ thereof. every of which facit 88 li. 18 s. 9 d.  
At 6. d. what 3896. Take for most bréef 1 / 40. facit 97. 8. s.  
And for proof and pleasure Take 1 / 60 and ½ thereof, or 1 / 80 double, or ½ of 1 / 20 or ¼ of 1 / 10, or ⅕ of 1 / ● or ½ of ⅕ of ¼.  
every of which facit 97. li. 8.  
At 7. d. what 3648. li.  
Take for brief 1 / 40 and ⅙ thereof  
The given number 3648. li. at 7 d. For 1 / 40 91. li. 4. s. 0. d. For ⅙ thereof 15. 4. 0. which together facit 106. 8. 0.  
And for proof. Take 1 / 30 lacking ⅛ thereof, or 1 / 60 and 1 / 80, or ½ of 1 / 20 and ⅙ thereof, or ¼ of 1 / 10 and ⅙ thereof, or 1 / 60 double lacking ¼ of 1 / 60.  
every of which ways facit— 106 li. 0. d.  
At 8 d. what 3579. Take for most ree1 / 3● facit 119. 6. 0.  
And for proof. Take 1 / 60 double or 1 / 40 and l⅓ thereof, or ⅔ of 1 / 02, or l⅓ of 1 / 10, or ⅕ or ⅙. &c.  
every of which ways facit 119. 6. 0  
At 9. d. what 3648. Take 1 / 40 and ½ thereof. For 1 / 40 91. 4. s. 0. d. For ¼ therof 45. 12 0. facit 136. 16. s. 0.  
And for the proof. Take 1 / 30 and ⅛ thereof, or ¾ of 1 / 20, or 1 / 60 do●ble, and ¼ of 1 / 60, or 1 / 80 triple, or ⅕ of ⅛ and ½ thereof. &c.  
every of which ways facit 136▪ li. 16 s.  
At 10 d. what 2973. Take ¼ of ⅙ for most brief. The given number 2973. For ⅙ 495. 10. s. 0. d. Whereof ¼ facit 123. 17▪ s. 6.  
And for proof. Take ½ of ¼ of l⅓, or ⅚ of 1 / 20, or 1 / 30 and ¼ thereof, or 1 / 40 and 1 / 60, or 1 / 40 and ½ thereof, and ⅕ thereof, or 1 / 80 triple, and l⅓ of 1 / 80.  
every of which ways facit 136. l. 16. s.  
At 11. d. the bushel what 2684. Take 1 / 20 lack 1 / 12 thereof   The given number 2684. at 11.d. For 1 / 20 134. li. 4 s. 0. d. From which 1 / 12 11. 3. 8. d. rest facit 123. 0. 4. d.  
And for proof. Take 1 / 30 and 1 / 80 as in example.  
At 11.d. what 2684.   For 1 / 30 89 9. 4▪ For 1 / 80 33. 11. Together facit 123 4  
Or for the same proof. Take ¼ of ⅙ and 1 / 10 thereof, or 1 / 40 and 1 / 60 and ¼ of 1 / 60 or 1 / 30, and the ½ thereof, and ⅔ thereof, or 1 / 30 and ¼ thereof and ½ thereof.  
every of which orders facit 120 l. s. 4. d  
At 12. d. what 2568. Take for most breee 1 / 20 facit 128. li. 8. s. 0.d.  
Foproofe at 12 d. what 2568. Take ⅕ of ¼ For ¼ 642. Whereof ⅕ faeit 128. 8. 0.  
which is sufficient, for such as will ●… ue onely the nearest way: howbeit such as vpon pleasure will range abroad. Take for the same proof ½ of 1 / 10, or 1 / 30 and ½ thereof, or 1 / 40 double.  
every of which facit— 128. li. 8. s. 0. d.  
At 13▪ d. what 2357. Take 1 / 20 and 1 / 1● therof.  
For 1 / 20 117. 17. 0.   For 1 / 12 therof 9. 16. 5. which facit 127. 13. 5.  
Here because 1 / 12 to bee taken from 1 / 10 requireth some difficulty. Therefore ⅙ is first taken, whereof ½ serveth, and the ⅙ canselled and the rest added, maketh the total.  
And for proof at 13. d. what 2357 Take 1 / 30 and ½ thereof, and ¼ thereof. The given number 2357. at 13 d. For 1 / 30 78. li. 11. s. 4.d. For ½ thereof 39. 5. 8. For ¼ thereof 9. 16. 5. which together facit 127. 13. 5.  
Or for the same proof Take 1 / 40 double, and ⅙ of 1 / 40 or 1 / 60 triple, and ¼ of 1 / 60 or 1 / 80 quadruple, and ⅕ of 1 / 80 &c.  
every of which facit 127. li. 13. s. 5. d.  
At 14 d. what 1926. Take ¼ of ⅕, and ⅛ thereof. For ⅕ 385. li. 4. s. 0 d. For ¼ thereof 96. 6. 0 d. For 1 / ● thereof 16. 1. 0 facit 112. 7. 0 d.  
And for proof at 14. d. what 1926. Take ½ of 1 / 10 and ⅙ therof   The given number at 14. d. 1926. For 1 / 10 192. 12. 0 For ½ thereof 96. 6. 0 For ⅙ thereof 16. 1. 0 facit 112. 7. 0  
Or for the same proof Take 1 / 20 and ⅙ thereof for most brief, take 1 / 30 and 1 / 40 or 1 / 60 triple, and ½ of 1 / 60 &c. li.  
every of which facit 112. 7. s. 0 d.  
At 15. d. what 1884. Take 1 / 20 and ¼ thereof   For 1 / 20 94. 4. s. 0 d. For ¼ thereof 23. 11. 0 facit 117. 15. 0  
And for proof at 15. d. what. 1884. Take ½ of ⅛   For ⅛ 235. 10. 0 d. Whereof ½ facit 117. 15. 0 d.  
Or for the same proof Take ¼ of ⅕ and ¼ thereof, or ½ of 1 / 10 and 1 / ● thereof, or 1 / 40 double and ½ of 1 / ●0, or 1 / 30 and 1 / 60 and 1 / ●0. &c.  
every of which facit 117. li. 15. s. 0 d.  
At 16. d. what 1468.  
Take for brief 1 / 20 and 1 / ● thereof.  
The given number at 16 d. 1468. take 1 / 20 〈◇〉 l⅓ of it.  
For 1 / 20 73. 8. 0 d. For l⅓ thereof 24. 9. 4 facit 97. 17 4  
And for proof at 16. d. what 1468. Take ⅕ of l⅓   For l⅓ 489. 6. 8 For ⅕ thereof 97. 17. 4  
Or for the same proof. Take ⅔ of 1 / 10 or 1 / 20 and l⅓ thereof or 1 / 30 double, or 1 / 40 double, and 1 / 60 or 1 / 60 quadruple.  
every of which facit. 97. 17. 4. d.  
At 17. d. what 1376. Take 1 / 20 and l⅓ thereof, and ¼ thereof The given number 1376. at 17 d. For 1 / 20 68. 16. 0 For— ½ thereof 22. 18. 8 For ¼ thereof 5. 14. 7 facit 97. 9. 0 3  
And for proof at 17. d. what 1376.  
Take ¼ of ⅕ and 1 / 60 and ¼ thereof.  
The given number 1376. at 17.d. For ⅕ 275. 4. 0 For ¼ thereof 68. 16. For 1 / 60 22. 18. 8 For ¼ thereof 5. 14. 7 which together facit 97. 9. 3  
Or for the same proof. Take l⅓ of 1 / 10 and l⅓ thereof and ¼ therof or 1 / 30 double and ⅛ of 1 / 30 or 1 / 30 add 1 / 40 and 1 / 80.  
every of which facit 97 li. 9 s. 3 d. At 18 d. what 1674. Take ¼ of ⅕ & ½ therof.   For ½ For ¼ thereof 83. 14. 0 For ½ thereof 41. 17. 0 facit 125. 11. 0  
And for proof at 18 d. what 1674.  
Take ½ of 1 / 10 and ½ thereof.  
The given number 1674. at 18d For 1 / 10 For ½ thereof 83. 14. 0 For ½ thereof 41. 17. 0 facit 125. 11. 0  
Or for the same proof. Take 1 / 20 and ½ thereof, or 1 / 30 double, and ¼ of 1 / 30, or 1 / 40 treb●e, or 1 / 30 and 40 and 1 / 60. &c.  
every of which facit 125. 11. 0 At 19 d. what 1735.  
Take 1 / 20 and ½ therof and ⅙ thereof  
The given number 1735. at 19 d. For 1 / 2● 86. 15. 0 d. For ½ thereof 43. 7. 6 For ⅙ thereof 7. 4. 7 facit 137. 7. 1  
And for proof at 19 d. 1735. Take 1 / 30 double and ⅛   For 1 / 30 Whereof the double 115. 13.4 For 1 / 80 21. 13. 9 facit 137. 7. 1  
Oor for the same proof. Take ¼ of ⅕ and ½ thereof, and ⅙ thereof, or ½ of 1 / 10 and the ½ thereof and the ⅙ thereof. &c.  
every of which facit 137 li. 7 s. 1 d. At 20 d. the piece, what 1876. Take ¼ of l⅓   For l⅓ Whereof ¼ facit 156. 6. 8 d.  
And for proof 20 d. what 1876.  
Take ½ of ⅙   For ⅙ Whereof ½ facit 156. 6. 8.  
Or for the same proof. Take l⅓ of ¼, or ⅔ of ½ or ⅚ of 1 / 30 o● 1 / 12, or 1 / 20 and ½ thereof and l⅓ thereof or 1 / 30 double and 1 / 60 &c.  
every of which facit 156 li. 6 s. 8 d. At 21 d. what 1289. Take ½ of ⅙ and ¼ of 1 / 60.   For ⅙ Whereof ½ 107. 8. 4 For 1 / 60 Whereof ¼ 5. 7. 5 And the vncanseld, facit 112. 15. 9  
And for proof at 21 d. what 1289.  
Take 1 / 20 and ½ thereof and ½ thereof.  
The given number 1289. at 21 d.  
For 1 / 20 64. 9. 0 d. For ½ therof 32. 4. 6 For ½ therof 16. 2. 3 which together facit 112. 15. 9  
Or for the same proof. Take 1 / 10 lack ⅛ therof, or 1 / 30 and 1 / 40 and 1 / 60 and 1 / 80, or 1 / 40 triple & 1 / 80.  
every of which facit 112. 15. 9  
At 22. d. The yard, what 1578. Take 1 / 10 lack ½ of 1 / 60.  
The given number 1578 at 22.d. For 1 / 10 157. 16. s. 0.d. For 1 / 60 For ½ thereof 13. 3. 0. which rest facit 144. 13. 0.  
For proof at 22.d. what 1578. Take ½ of ⅙ and 1 / 10 thereof.  
The given number. 1578. For ⅙ For ½ thereof. 131. 10. 0. For 1 / 10 thereof 13.03.0. facit 144. 13. 0.  
Or for the same proof. Take 1 / 20 and ½ thereof, and ½ thereof, and l⅓ thereof, or 1 / 20 and 1 / 40 and 1 / 60, or 1 / 30 double and 1 / 40, or 1 / 40 triple, and 1 / 60.  
every of which facit 144 l.. 13. s. 0. d.  
At 23.d. what 1627. Take 1 / 10 lack l⅓ of 1 / 80  
For 1 / 10 162. 14. 0. For 1 / 80 whereof l⅓ 6. 15. 7. Per rest facit 155. 18. 5.  
For proof at 23.d. what 1627. Take ½ of ⅙ and ⅛ of 1 / 10.    
The given number at 23.d. 1627.  
For— ⅙ For ½ thereof 136.11.8. For 1 / 10 For ⅛ thereof 20.9. 6. Whereof the uncanceled 155.18. 5.  
Or for the same proof take 1 / 20 and ½ therof and ½ thereof, and ⅔ thereof, o● 1 / 30 double, and 1 / 60 and 1 / 80. or 1 / 20 1 / 30 and 1 / 80 or 1 / 20, and 1 / 40 and 1 / 60 and ¼ thereof.  
every of which facit 155 li. 18. s. 5. At 2. s the ell what 1243. Take 1 / 10 most brief.   The given number at 2. s. 1243.. facit 124. 6.s. 0. d.  
For proof, at 2. s. what 1243. Take ½ of ⅕  
For ⅕ Whereof ½ facit 124. 6. 0.  
Or for the same proof. Take 1 / 20 double or 1 / 30 triple, or 1 / 40 quadruple, or 1 / 60 sextuple. &c.  
every of the which facit 124. li. 6.d. 0.d.  
At 2. s. and 1.d. what 1468. Take 1 / 10 and ¼ of ⅙ thereof.   The given number 146. at 2. s. 1.d. For 1 / 10 146. 16.s. 0.d. For ⅙ thereof For ¼ thereof 6. 2. 4. Whereof the uncanceled facit 152. 18. 4.d.  
For proof at 2. s. 1.d. what 1468. Take ½ of ⅕ and l⅓ of ⅛ thereof.  
The given number at 2. s. 1.d. 1468.  
    146.16.0.     6. 2.4.   152.18.4.  
Or for the same proof. Take 1 / 30 triple and ⅛ of 1 / 30, or 1 / 40 quadruple and ⅙ of 1 / 40 &c.  
At 3 s. 2 d. what 1248. elles. Take ⅛ and 1 / 30.    
At 3. s. 2. d. what 1248. elles. For 1 / ● 156. For ● / 30 41. 12.0 facit 197. 12.0  
And for proof. Take 1 / 10 and ½ thereof, and ⅙ thereof as by example.  
At 3 s. 2 d. what 1248. for 1 / 10 124. 16. for ½ thereof 62. 8. for ⅙ thereof 10. 8. facit 197. 12.  
Or for the same proof. Take ⅙ lacking 1 / 12 of 1 / 10 or ½ lack ⅙ of 1 / 10. &c.  
every of which facit 197 li. 12 s. 0 At 4 s. 3 d. what 1234. Take ⅕ and 1 / 10. for ⅕ 246. 16 s. 0 d. for 1 / 50 15. 8. 6 facit 262. 4. 6 d.  
For proof at 4 s. 3 d. what 1234. Take 2 / 10 double, and ¼ of 1 / 20 for 1 / 10 double 246. 16. 0 for— 1 / 20 for ¼ thereof 15. 8. 6 facit 262. 4. 6  
Or for the same proof. Take ⅕ & ⅛ of 1 / 10. &c. And all makes 262 li. 4 s. 6 d.  
At 5. s. 4. d. the ounze. what 1111 ounze.  
Take ¼ and ⅙ of 1 / 10.  
  277. 15. 0     18. 10. 4 Facit. 296. 5. 4  
And for proof, at 5 s 4. what 1111.  
Take ¼ and 1 / 60.  
For ¼ 277. 15. 0 For 1 / 60 18. 10. 4 facit 296. 5. 4  
Or for the same proof, ¼ and l⅓ of 1 / 20, or ¼ and ½ of 1 / 30 or ⅕ and ⅕ thereof, and l⅓ thereof.  
every of which facit— 296. li. 5. s. 4. d.  
At 6. s. 5. d.— 1231. Take ¼ and ⅕ therof, and l⅓ thereof and ¼ thereof.  
For ¼ 307. 15. 0 For— ⅕ thereof 61. 11. 0 For l⅓ thereof 20. 10. 4 For— ¼ thereof 5. 2. 7 which together facit 394. 18. 11  
And for proof at 6. s. 5. d. what 1231. Take ⅕ add ½ thereof and ⅙ thereof and ¼ thereof.  
The given number 1231. at 6. s. 5. d. For ⅕ 246. 4. 0 For ½ thereof 123. 2. 0 For ⅙ thereof 20. 10. 4 For ¼ thereof 5. 2. 7 facit 394. 18. 11  
Or for the same proof l⅓ lack 1 / 20, or ¼ & l⅓ therof lack 1 / 80 or ⅕ and ½ thereof, and 1 / 40 lack 1 / ● thereof.  
every of which facit 394 li. 18. s. 11 d.  
At 7. s 6.d what 9867. pieces. Take ¼ and ½ thereof.  
At 7. s 6.d. what 9867. pieces. For ¼ 2466. 15. 0.d For— ½ thereof 1233. 7. 6. facit 3700. 2. 6.  
And for proof at 7 s. 6. what 9867.  
Take ½ lack ¼ thereof.  
The given number at 7.6. d. 9867.  
For ½ 4933. 10. 0. From which ¼ 1233. 7. 6. Par rest facit 3700. 2. 6.  
Or for the same proof. Take l⅓ and 1 / 10 thereof and ¼ therof or ⅕ and ½ thereof, and ½ therof, and ½ thereof or ⅕ double lack ⅛ of ⅕. every of which facit— 3700. li. 2. s. 6. d.  
At 8. s. 7.d. what— 8976.  
Take l⅓ and ¼ thereof, and 1 / 80.  
The given number at 8 s. 7.d. 8976.  
For l⅓ 2992. 0. 0. For ¼ thereof 748. For— 1 / 80 112. 4. 0. which together facit 3852. 4. 0.  
And for proof, at 8. s. 7.d. what 8976. Take ⅕ double, and ⅛ of ⅕, and ⅙ thereof.  
The given number at 8 s. 7 d. 8976.  
For ⅕ 1795. 4. 0. Also for ⅕ 1795. 4. 0. For ⅛ thereof 234. 8. 0. For— ⅙ thereof 37. 8. 0. which together facit 3852. 4. 0.  
Or for the same proof. Take ¼ and ½ thereof, and 1 / 20 and 1 / 12 thereof, or ¼ and 1 / 10 and ½ thereof, and ½ thereof, and ⅙ thereof.  
every of which facit— 3852 li. 4. s. 0. d.  
At 9. s.8.d. what 8572. barrels. Take ½ lack 1 / 60.   For ½ 4286. 0. 0. From which 1 / 60 142. 17 s.4.d. rest facit 4143. 2. 8.  
And for proof. Take ¼ and ⅕ and ⅙ thereof.  
At 9. s. 8. d. what 8572. For ¼ 2143. 0. 0. For— ⅕ 1714. 8. 0. For ⅙ thereof 285. 14. 8. which together facit 4143. 2. 8.  
Or for the same proof. Take ⅕ double and ½ of ⅙, or ¼ and ⅕ and 1 / 30, or ⅕ double, and 1 / ●0 double.  
every of the which facit 4143. li 2 s. 8. d.  
At 10 s.9.d. what 7864. Take ½ & 1 / 40 & ½ thereof. For— ½ 3932.  
For 1 / 40 196. 12. 0. For ½ thereof 98. 6. 0. which facit 4226. 18. s. 0.  
And for proof. Take ¼ double and 1 / 30 and ⅛ thereof, or ⅕ double, and 1 / 1● and ¼ thereof, and ½ thereof, or ½ and 1 / 08 triple.  
every of which facit 4226. 18.  
At 11. s.10.d. what— 864. Hogsheades.  
Take ½ and ½ of ⅙, and 1 / 10 thereof.  
The given number 864 At 11. s.10. For ½ 432. For ⅙ For ½ thereof 72. For 1 / 10 thereof 7. 4. s. 0. d. The which uncanceled is 511. 4. 0  
For proof at 11s. 10 d. what 864. hogsheds. Take, and 1 / 30 double and 1 / 40.  
For ½ 432. 0. 0 For 1 / 30 28. 16. 0 For 1 / 30 again 28. 16. 0 For 1 / 40 21. 12. 0 which together facit 511. 4. 0 d.  
Or for the same proof. Take ½ and ⅕ therof, lacking 1 / 12 thereof, or ½ and 1 / 20, and 1 / 30 and ¼ thereof, or ½ and 1 / 20 and 1 / 40 and 1 / 60.  
every of which facit 511. li. 4. s. 0. d.  
At 12. s. 11. d. what 7853. Take ½ and 1 / 10 and 1 / 30 and 1 / 80.  
for ½ 3926. 10. 0 for 1 / 10 785. 6.0 for 1 / 30 261. 15. 4 for 1 / 80 98. 3. 3 which together facit 5071. 14. 7  
For proof at 12 s. 11 d. what 7854. Take ½ and ¼ thereof, and 1 / 60 and ¼ thereof.  
The given number 7853. at 12 s. 11 d for ½ 3926. 10. 0 for ¼ thereof 981. 12. 6 for 1 / 60 130. 17. 8 for ¼ thereof 32. 14. 5 which together facit 5071. 14. 7  
Or for more proof. Take ½ and ¼ thereof, and 1 / 40 lack ⅙ therof, or ¼ & ⅕ and ⅙ and 1 / 60 and ● / ●●. every of which facit 5071 li. 14.7  
At 13. s 1.d. the quarter, what 5938. Take ½ and 1 / 10 & ½ therof & 1 / 12 thereof.  
for ½ 1969. for 1 / 10 593. 16s. 0 d for ½ thereof. 296.18.0 for 1 / 12 thereof 24.14.10 which together facit 3884. 8. 10  
And for proof at 13 s. 1 d. what 5938. Take ½ and ⅛ and 1 / 60 and 1 / 80.  
for ½ 2969. for ⅛ 742.05.0 for 1 / 60 98.19.4 for 1 / 80 74. 4. 6 which together facit 3884. 8. 10  
Or for more proof. Take ½ and ⅙ lacking 1 / 80 or ¼ and ⅕ and ⅙ and 1 / 40 and ½ thereof, or l⅓ and ¼ and 1 / 20 and l⅓ thereof, a and ¼ therof.  
every of which facit 3884. li. 8. s 10 d.  
At 16 s. 4 d. what 2531. For— ½ 1265. 10. 0. For ½ thereof 632. 15. 0. For ⅙ thereof 126. 11. 0. For l⅓ thereof 42. 3. 0. which together facit 2066. 19. 8.  
For proof at 16 s. 4. d. what 2531. Take ½ and l⅓ lack 1 / 60.  
At 16. s. 4. what 2531. For— ½ 1265. 10. 0. For l⅓ 843. 13. 4. From which 1 / 60 42. 3. 8. Per rest facit 2066. 19. 8.  
Or for more proof. Take l⅓ double, and 1 / 10 and 1 / 20 thereof, or ¼ triple, and, 1 / 20 and l⅓ thereof, or 1 / 60.  
every of which— 2066.l ' 19. s. 8.d.  
At 17. s. 5.d. what— 9856. shepherds. Take ½ and ½ thereof, and 1 / 10 and ⅙ thereof, and ¼ thereof.  
At 17. s. 5.d. what— 9856. shepherds.  
for— ½ 4928. 0. 0.d. for ½ thereof 2464. 0. 0. for— 1 / 10 985. 12. 0. for ⅙ thereof 164. 5. 4. for— ¼ thereof 41. 1. 4. which together is 8582. 18, 8.  
For proof at 17 s. 5.d. what 9856. Take ½ and l⅓ and 1 / 40, and ½ thereof.  
At 17. s. 5.d what— 9856.  
for— ½ 4928. for 1 / ● 3285.6. 8. for— 1 / 40 246.8. 0. for ½ thereof. 123.4. 0. which together facit 8582. 18. 8.  
Or for the same proof. Take l⅓ double, and ⅕ and ¼ of 1 / 60, or ⅕ double and l⅓ and ⅛ and l⅓ of 1 / 40.  
every of which facit— 8582. 18. 8.  
At 18. s. 6. d. what— 896. Take the whole lack 1 / 20 and ½ thereof.  
At 18. s. 6.d. what— 896.  
  44. 16. 0.   22. 8. 0. Per rest facit 828. 16. 0.  
For proof at 18. s. 6.d. what— 896.  
Take ½ and ½ thereof, and ½ thereof and 1 / 20.  
At 18. s. 6. d. 896. for— ½ 448. for ½ thereof. 224. for— ½ thereof 112. for 1 / ●0 44. 16. 0. which together facit 828. 16. 0.  
Or for more proof. Take ½ and ⅖ and 1 / 40, or ½ and l⅓ and 1 / 10 lack 1 / 12 thereof, or ⅔ and ⅕ and 1 / 20, and 1 / 40 and 1 / 60, or ⅔ and ¼ lack ½ of 1 / 60.  
every of which facit— 828 li. 16. 0.  
At 19. s. 7. d. the piece. what— 746. Take the whole lacking 1 / 60 and ¼ thereof. At 19. s. 7.d. the piece. what 746. From which 1 / 60 12. 8. 8.d. and ¼ thereof 3. 2. 2 per rest facit 730. 9. 2.d.  
For proof at 19. s. 7. d. what 746.  
Take ½ and ¼ ⅕ & 1 / 60 and 1 / 80.  
for ½ 373. 0. 0. for ¼ 186. 10. 0. for ⅕ 149. 4. 0. for 1 / 60 12. 8. 8. for 1 / 80 9. 6. 6. which together facit 730. 9. 2.  
Or for more proof. Take ½ and l⅓ and ⅛ and 1 / 60 and ¼ thereof, or ½ and ⅖ and 1 / 20 and ½ thereof, and ⅙ thereof, or ⅔ and ¼ and 1 / 20 and ¼ thereof.  
every of which facit 730. li. 9. s. 2.d.  
At 20. s. and 8. d. what 694. Take the whole and 1 / 30   At 20 s. and 8. d. what 694. for the whole 694. for 1 / 30 thereof 23.2.7. facit 717. 2.8.  
For proof. At 20. s. 8.d. what 694. Take the whole and 1 / 40 and l⅓ therof 694. 17. 7 5. 15.8 717. 2.8  
Thus much may seem sufficient for the summyng of any number, whereof the price of the unity is under 20. s. and when in any number the price of the unity is 20. s. with some parte of a pound more, then the whole given number is to be taken, and the partes over and above the same to be taken, and added thereunto in former order. Whereof to give example, were superfluous, the effect easy to understand, and appearing in the last example. &c.  
As in the former examples, the price of an unity in every given number being under 20. s. division hath been practised memoratiuely, so in some other following, wherein the price of an unity being 40 s. or above, shall Multiplication bee furthered, as need shall require, accordingly sometime alone, & sometime with the like division also, where the partes of a pound the same may require, as in examples more at large you may perceive.  
At 40. the piece, what 568. Carseis Memoratiuely by 2 and take the product.  
At 40. s. the piece what 568. Carseis.  
facit 1136.  
At 41 s. 8. d. what— 546. By 2. and take ½ of ⅙ to be added with the product.  
At 41. s. 8. d. what 546. The product 1092. For ● / ● Where ½ 45. 10. Together facit 1137. 10.  
For proof at 41. s. 8.d. what 546. Take the product by 2. and 1 / 12 li.  
The product 1092. 0. 0. for 1 / 12 li. 45. 10. 0. facit 1137. 10. 0.  
At 3 li. what 642. Take the product by 3.  
facit 1926.  
At 3. li. 2. s. 4. d. what 465. Take the product by 3. and 1 / 10 li. and ⅙ thereof.  
At 3. li. 2. s. 4. d. what 465.  
The product 1395. for— 1 / 10 46. 100. for ⅙ thereof 7. 15.0. facit 1449. 5. 0.  
At 4. li. 3. s. 6. d.— what 572. Take the product by 4 & ½ li. and 1 / 20 li.  
At 4. li 3. s. 6. d. what 572. By 4 the product 2288. 0. 0. for ⅛ li. 71. 10. 0. for— 1 / 20 28. 12. 0. facit 2388. 2. 0.  
At 5. li. 4. s. 7. d. what— 346.  
Take the product by 5. and ⅕ li. and 1 / 40 and ⅙ thereof.  
At 5. li. 4. s. 7. d. 346. The product 1730. 0. 0. for ⅕ li. 69. 4. 0. for— 1 / 40 8. 13. 10. for ● / 6 thereof 1. 8. 10. facit 1809. 5. 10.  
At 6. li. 5. s. 8. what— 293.  
Take the product by 6. and ¼ li. and 1 / 30.  
At 6. li. 5. s. 8. d. what 293. The product 1758. for ¼ li. 73. 5. 0. for 1 / 30 9. 15. 3. facit 1841. 0. 4.  
At 7. li. 6. s. 9. d. what 278.  
Take the product by 7. and l⅓ li. and ¼ of 1 / 60.  
At 7. li. 6. s. 9. d. what 278. The product 1946. for l⅓ li. 92. 13. 4. for 1 / 60 for ¼ thereof. 1. 3. 2. which uncanceled facit 2039. 16. 6.  
At 8. li. 7. s. 10. d. what— 244.  
Take the product by 8. and l⅓ li. and 1 / 20, and 2 / 6 thereof.  
The given number 243. By 8. the product 1944. 0. 0. for l⅓ 81. 0. 0. for— 1 / 20 12. 3. 0. for ⅙ thereof. 2. 0. 6. d. facit 1944. 0. 0.  
At 9. li. what— 231.  
Take the product by 9.  
The given number 231. facit 2079.  
When the value of an unity is more then with a diget to bee expressed: Then the said value expressed by mixed figures, is urged by necessity, to bee set down for multiplicator, under the given number, and no parte thereof referred to memory, as in the former examples, and the partes taken by former order, as in some examples following, the effect also appearing.  
At 11. li. 12. s. 4. d. what 524.  
Take the product of 11. and ½ l. and ⅕ therof, and ⅙ thereof.  
The given number 524.   11.   524.   524.   262. 0. 0.   52. 8. 0.   8. 14. 8. facit 6087. 2. 8.  
At 23. li. 13. s. 6. d. what 234.  
Take the product of 23. and ½ li. and l⅓ thereof, and 1 / 20 thereof.  
The given number 234.   23.   702.   468.   117.   39.   1. 19. 0. facit. 5539. 19. 0.  
At 34. li. 14 s. 8 d. what 142.  
Take the product of 34, and ½ li. and ⅕ and ⅙ thereof.  
The given number 142.   34.   568.   426.   71. 0 0.   28. 8 0.   4. 14. 8. facit. 4932. 28.  
And according to the same order, every mannes occasion may be furnished infinitely, wherefore to give more examples of former effect, might seem superfluous. Howbeit to give the valuation accordingly, of the Quintall and several C. weights and partes of every of the same: sundry examples hereafter follow.  

The Quintall containing 100 lib. subtle.  
The quintall at 34. li. 13. s. 4. d. what 95 lib. Take the price of the 100. lacking 1 / 20 therof  
At 34. li. 13. s. 4. d. what. 95. lib.  
for the whole 100. 34. 13. 4. d. whereof 1 / 20 rebated 1. 14. 8. Per rest facit 32. 18. 8.  
The 100 lib. at 29. li. 10. s. what 90. lib. Take the whole lack 1 / 10.  
the whole. 29. 10. 0 d from which 1 / 10 2. 19. 0 per rest facit 26. 11. 0  
The 100 lib. at 26 li. 3. 8. d. what 86. lib. Take ¾ of the whole, and 1 / 10 and 1 / 10 thereof.  
At 12 li. 16 s. what 37. lib.  
  3. 4. 0   1. 05. 7. ⅕   05. 1. ⅖ & ⅕ of ⅕ facit 4. 14. 8. 16 / 21  
The 100 lib. at 8 li. 2 s. 6 d. what 14. lib. Take 1 / 10 and ⅖ thereof  
  0. 16. 3   06. 6 Facit. 1. 2. 9 d.  
The 100 lib. at 5. 13. 4. what 9. lib. Take 1 / 10 lack 1 / 10 thereof.  
for 1 / 10 0. 11. 4 from which 1 / 10 1. 1. ⅕ Per rest facit 10. 2. ⅖  
The 100 lib. at 3 li. 6 s. 8 d. what 6. lib.  
Take ½ of 1 / 10 and ⅕ thereof  
  0. 6. 8   3. 4 facit. 0. 4. 0 d.  
The 100 lib. at 2. li. 10. s. what 4 lib. Take ⅖ of 1 / 10. For 1 / 10. 0. 5. 0. d. Whereof ⅖ facit 0. 2. 0. d. The 100 lib. at 16 s. 8. d. what 3. lib. Take ⅕ of 1 / 10 & ½ thereof. For 1 / 10 For ⅕ thereof 0. 0. 4. For ½ thereof. 0. 0. 2. facit 0. 0. 6. d. The 100 lib. at 13 s. 4. d. what 2. lib. Take ⅕ of 1 / 10.   0. 0. 3.d. ⅕ The 100 lib. at 10 s. what 1. lib. Take 1 / 10 of 1 / 10. facit 0. 0. 1. ⅕ The 100 lib. at 8 s. 4. d. what 4. onzes. Take ¼ of 1 / 10 of 1 / 10. For 1 / 10. 0. 0 10 d. For 1 / 10 thereof Whereof ¼ facit 0. 0. 0 6. mits  
As every of former examples are profitable, and of many the understanding may bee desired to find the sundry partes of the 100 lib the same to value after the rate of the C. so of all the other it may seem most necessary to understand, how by the price of the C. to find the value of the pound weight, & the onze with most facility: wherefore for a general rule, to find the value of 1. pound weight by the price of the C. take ever 1 / 10 of 1 / 10 of the price that the C. lib. is valued at, and that is the true value of the pound weight, as before appeareth, and by an other example following, the effect is manifeste.  
The C. at 45. li. 17 s. 6. d. what 1 li. Take 1 / 10 of 1 / 10. For 1 / 10 4. 11. 9 d. Whereof 1 / 10 facit 09 2. d 1 / 10.  
Likewise to find the value of the vnze being 16. as in the haberdepoise. Take 1 / 10 of 1 / 10. of 1 / 10 and haue your desire.  
How be it, if the vnze bee of 12 in the lib. weight troy, then take 1 / 12 of 1 / 10 of 1 / 10 and accordingly haue your desire.  
Of the C. weight containing 112 li.  
The C. weight at 36. li. what 96 lib. Take the whole price of the C. lacking ⅛ and 1 / 7 therof.  
The whole 36. 0. 0. d. From which ⅛ 4. 10. 0. And 1 / 7 of it 0. 12. 10. 2 / 7. Per rest facit 30. 17. 1. 5 / ●.  
For proof, the C. at 36. l. what ¾ & 12 lib. Take ½ and ½ thereof, and ½ thereof, lack 7 / 2  
For— ½ 18. 0. 0. For ½ thereof 9. 0. 0. For— ½ thereof 4. 10. 0. From which 1 / 7 0. 12. 10. 2 / 7. Per rest facit. 30. 17. 1. 5 / 7.  
The C. at 32. li. what 8 4. lib. Take ¾ of the whole price. For ½ 16. li. 5. s. 0. d. For ½ thereof 8. 2. 6. Together facit 24. li 7. s. 6. d.  
For proof the C. at 32. li. 10. s. what 84. lib.  
Take the whole lack ¼ thereof.  
At 32 li. 10 s. what 84. li. For the whole 32. 10. From which ¼ 8. 2. 6. Per rest facit 24. 7. 6.  
The C. at 28. li. 5. s. what— 72. li.  
Take ½ and ¼ thereof, and 1 / 7 thereof.  
At 28 li. 5 s. what 72. li. For— ½ 14. 2. 6. d. For— ¼ thereof 3. 10. 7. ½. For— 1 / 7 thereof 0. 10. 1. 1 / 14 Together facit 18. 3. 2. 4 / 7.  
The C. at 25. li. 13. s. 4. d. what 6 6. li.  
Take ½ and ⅛ thereof, and 3 / 7 thereof.  
At 25 li. 13 s. 4 d. what 66. li. For— ½ 12. 16. 8. d. For ⅛ thereof 1. 12. 1. For— 1 / 7 thereof 4. 7. For the double therof 9. 2. Together facit 15. 2. 6.  
The C. at 22 li. 12 s. what— 60.  
Take ½ and 1 / 7 of ½ thereof.  
At 22 li. 12 s. what 60. For— ½ 11.6. 0. d. For ½ thereof For— 1 / 7 thereof 16. 1. 5 / 7 uncanceled facit 12. 2. 1. 5 / 7  
The C. at 18. li. 6. s. 8. d. what 50. li. Take ¼ and ½ thereof, & ½ therof, & 1 / 7 thereof. At 18 li. 6 s. 8 d. what 50. li. For— ¼ 4. 11. 8. For ½ thereof 2. 5. 10. For ½ thereof 1. 2. 11. For 1 / 7 thereof 3. 3. 2 / 7. Together facit 8. 3. 8. 2 / 7.  
The C. at 16. li. 13. s. 4. d. what 42. li. Take ¼ and ½ therof.   At 16 li. 13 s. 4 d. what 42. li. For— ¼ 4. 3. 4. d. For ½ thereof 2. 1. 8. Together facit 6. 5. 0. d.  
The C. at 12 li. 10. s. what 35. li. Take ¼ and ¼ thereof.   At 12 li. 10. s. what 35. li. For— ¼ 3. 2. 6. d. For ¼ thereof 15. 7. ½. Together facit 3. 18. 1. ½.  
The C. at 10. li. 15. s. what 30. lib. Take ¼ and 1 / 14 thereof.   At 10. li. 15. s. what 30. li. For— ¼ 2. 13. 9. For 1 / 14 thereof 3. 10. ¼ facit 2. 17. 7. 1 / 14.  
The C. at 8. li. 16. s. what 24. Take ¼ lack 1 / 7 thereof.   At 8. li. 16 s. what 24. li. For— ¼ 2. 4. 0. d. From which 1 / 7 6. 3, 3 / 7. Per rest facit 1. 17. 8. 4 / 7.  
The C. at 6. li. 13. s. 4. what 20. li. Take ⅛ and ½ thereof lack 1 / 7 thereof.    
The C. at 6 li. 13 s. 4 d. what 20. li, For— ⅛ 0. 16. 8. d. For— ½ thereof. 8. 4. From which— 1 / 7 1. 2. 2 / 7. Per rest facit 1. 3. 9. 5 / 7. The C. at 5. s. 4. what 16 li. Take ⅛ and and 1 / 7 thereof. For ⅛ 0. 13 0. d. For 1 / 7 thereof 1. 10. 2 / 7. facit 0. 14. 10. 2 / 7.  
The C. at 3. li. 6 s. 8 d. what 10. lib. Take ¼ of ¼ and 3 / 7 thereof.  
for ¼ for— ¼ thereof 0. 4. 2 for 1 / 7 thereof 0. 0. 7. 1 / 7 for the double therof 0. 1. 2. d. 2 / 7 facit 0. 5. 11 d. 2 / 7 The C. at 40 s. what 7. lib. Take ¼ of ¼ for ¼ whereof ¼ facit 0. 2. 6 d.  
The C. at 33 s. 4 d. what 6. lib. Take 1 / 7 of ¼ and ½ thereof   for ¼ whereof 1 / 7 0. 1. 2. 2 / 7 and ½ thereof 0. 0. 7. 1 / 7 facit 0. 1. 9. 3 / 7  
The C. at 30 s. 6 d. what 4 lib. Take 1 / 7 of ¼ for ¼ whstrof 1 / 7 facit 0. 1 s. 6 / 7 d.  
The C. at 26 s. 8 d. what 2. lib. take ½ of 1 / 7 of ¼ for ¼ 0. 06. s. 8 d. for— 1 / 7 thereof 0. 0. 11 d. 3 / 7 whereof ½ facit 0. 5 d. 5 / 7  
The C. at 20 s. what 1. lib. Take ¼ of 1 / 7 of ¼ or 1 / 7 of 1 / 16 for ¼ 0. 5. 0 for— 1 / 7 therenf 8. 4 / 7 whereof ¼ facit 0. 0. 2. 1 / 7  
To find the value of the ounze by the price of the C. weight, you must first find the value of the pound weight, as before, and then take 1 / 16 thereof, and that is the value of the ounze, as by example.  
The C. at 24. li. what 1. onz. Take 1 / 16 of 1 / 7 of 1 / 16 for— 1 / 16 for 1 / 7 thereof whereof 1 / 16 facit 0. 0. 3 3 / 14  
For proof, the C. at 24 lib. what 1. ounze.  
Take 1 / 16 of ¼ of 1 / 7 of ¼ for— ¼ 6. 0. 0 d. for 1 / 7 thereof 0. 17. 1 5 / 7 for— ¼ thereof 0. 4. 3. 3 / 7 whereof 1 / 16 facit 0. 0. 3. 3 / 14  
In like maner may be taken any parte of the pound weight, according to the proportion it beareth to the whole.  
Of the C. containing 120. for the C.  
The C. of canvas at 7. li. what 90. elles.  
Take ½ and ½ thereof.  
At 7 li. what 90. elles. For— ½ 3 li. 10 s. 0 d. For ½ thereof 1. 15. 0 Together facit 5. 5. 0 The C. at 8 li. what 85. Take ⅔ and 1 / 16 thereof.   For— ⅔ 5. 6. 8 d. For 1 / 16— thereof 0. 6. 8 Together is 5. 13. 4 The C. at 9 li. what 74. Take ½ and ⅙ thereof, and ⅖ thereof   far ½ 4. 10. 0 d. For— ⅙ thereof 0. 15. 0 For ⅖— thereof 0. 6. 0   5. 11. 0  
The C. 16 s. 8 d. what— 68. Take ½ and ⅕ of l⅓.  
For— ½ 0. 8. 4 d. For l⅓ For— ⅕ 0. 1. 1. l⅓ The uncanceled is 0. 9. 5 d. l⅓  
The C. at 13 s 8 d. what 51. l. Take l⅓ and ¼ thereof and 1 / 10 thereof  
For— l⅓ 0. 4 s. 6 ⅔ For ¼ thereof. 0. 1. 1 ⅔ For— 1 / 10 thereof 0. 0. 1 l⅓ & 1 / 10 of l⅓ Together facit 0.— 5. 9. 7 / 10  
The C. at 12 s. 6 d. what— 45.— Take— ½ and ⅛ thereof.  
For— l⅓ 0. 4. 2 d. for ⅛ thereof 0. 0. 6. ¼ facit 0. 4. 8. ¼  
The C. at 11 s. what— 36.— Take ¼ and ⅕ thereof  
For— ¼ 0. 2. 9. For 1 / 5 thereof 0. 0. 6. 3 / 5 facit 0. 3. 3. ⅗ The C. at 10 s. what 30.— Take ¼ for ¼ facit 9. 2. 6  
The C. at 8 s. what— 25. Take ⅙ and ¼ therof, or ¼ lack ⅙ thereof  
For— ⅙ 0. 1. 4. For ¼ therof 0. 0. 4. facit 0. 1. 8 d.  
The C. at 6 s. 8 d. what— 16.— Take 3 / 6 lack ⅕ therof or 1 / 10 & l⅓ thereof  
For— 1 / 10 0. 0. 8 d. For l⅓ thereof 0. 0. 2. ⅔ facit 0. 0. 10. ⅔  
The C. at 5 s. what— 10. Take 1 / 12, or ½ of ⅙, or ¼ of l⅓  
For 1 / 12 facit 0. 0. 5. d.  
The C. at 4. s. what— 6.— Take ½ of 1 / 10, or ⅕ of ¼.  
At 4 s. what 6.— For— 1 / 10 Whereof ½ is 0. 0. 2. ⅖.  
The C. at 3 s. 4 d. what— 2.— Take ⅙ of 1 / 10, or l⅓ of ⅕ of ¼ or ⅕ of ½ of, ⅙ or ⅕ of l⅓ of ¼.  
The given number what 2— For— 1 / 10 For ⅙ thereof is 0. 0. 0. ⅔ d.  
The C. at 3. li. 6. s. 8. what— 1.— Take 1 / 22 of 1 / 10, or ⅙ of ⅕ of ¼.  
The C. at 3 li. 6 s. 8 d. 1.— For— 1 / 10 Whereof— 1 / 12 is 0. 0. 6. ⅔.  
For proof the C. at 3. li. 6. 8. what— 1— Take ⅙ of ⅕ of ¼.  
At 3 li. 6 s. 8 d. what 1.— For— ¼ For ⅕ thereof Whereof ⅙ facit 0. 0. 6. ⅔.  
Here note for a general rule, that such proportion as the given number beareth to the C. the same proportion beareth the price of the given number, to the price of the C. and therein consisteth the difficulty, that to any learner may appear.  
Thus is brought to end gentle reader, the effect by my travail pretended herein. The which being so well accepted of thee, as it hath been willingly furthered, to procure contentation to all such, as may take profit or delectation by the same: so I may bee encouraged to augment my good will, in furtheryng of other works of greater consequence, therein assisted by the favour of the almighty, into whose hands I committe thee farewell.