Of Numeration.

NVmeration containeth the maner how to expresse the valewe of anie somme or number. Whiche occa­sion may present, beeyng small or great, and is furthered by tenne Carecters or Figures followyng, to say.

The whiche nine Figures in proper si­gnification of value equall to the Woordes and Letters vnder theym sette, beyng sepa­rate with pricke or lyne. Howbeit, beeyng sette together and mixed without pricke or line of separation: than an encrease of va­lue they receiue, by vertue and propertie of the place: wherein they stande, which places being of number infinite, doe yelde vnto eue­rie vnity of any Figure: ten tymes so much in any place towarde the lefte hande, as that same vnitie is worthe in place next to it towardes the right hande, the ef­fect wherof more plainly may appeare by the Table folowyng, for the same pur­pose furthered.

THE nine vnities set aboue the vpper lyne of the Table, doe signifie the value of eue­ry vnitie in the Figures against any of the same vnder the lyne, and that by helpe of Ci­phers made like the Letter. 0. The whiche being of no value in proper signification, the same notwithstandyng, they are of necessarie vse in practise of Arithmetique, only to kepe the places, wherby is expressed infinite nom­bers, [Page] which without helpe of theim, the other Figures could not performe, as by the fore­named vnities, with Ciphers before them, the effect may appeare for the figure of one, in the first place is there but one: But in the second place is tenne, by helpe of the Ci­pher set thus. 10. So in the thirde place a hundreth thus 100. And so vnderstand of all the rest infinitely.

The titles written vnder the Table, serue also to shewe the value of euery vnitie found in the figures standyng in the places aboue the titles, as 9 in the first place is but nine, in the second place it is nine tymes ten, in the thirde place nine hundreth, and so forthe infi­nitely. And thus muche may seeme sufficient for an introduction for the vnderstandyng of Numeration, which is to shewe the value of any nomber, which occasion may procure to be knowen.

Howbeit, it may séeme necessarie heare to make distinction of certaine Termes, belon­gyng to Nomber. Not for vse of any thereof in this parte, but for helpe in other partes, by the saide termes furthered, as in place of their neede hereafter will appeare, whiche termes are to say, Nombers Diget, Article, [Page 2] and compound, or Mixed.

The Diget nombers are not onely euery one of the nine Caracters or Figures stan­dyng alone: but also sometyme are founde a­mongst mixed or compound nombers remai­ning in workes vnder ten. Betwéene 10 and 20. Betweene 30 and 40. & so forth 100. &c.

Article nobers are suche as are furthered by Ciphers, & no mixed figures with theim, as 10. 20. 30. 40. and so for the infinitely.

The mixed nombers are set together thus 123. 542. 3045. and euery suche like, either sondrie figures together, or figure together and Ciphers betweene. But if a Cipher bee founde in the firste place towarde the right hande of any nomber, then euery suche nom­ber is an Article nomber. &c.

Of Addition.

ADdition containeth the maner how to assemble, and ioyne sondrie perti­cular sommes or numbers, into one totall. As if three sondrie menne should owe vnto a Marchaunte, three seuerall sommes. The firste 548. li. the seconde 1346. li. and the third 15. li. The which to bryng into one totall, ye shall set the said three seuerall per­ticulers [Page] together, one right vnder an other, to saie: vnitie vnder vnitie, tenne vnder tēne, hundreth vnder hundreth, and in like maner infinitely in this order hereafter apperyng.

The whiche Perticulars set in order accordynglie, you shal drawe a line vnder them, and then resorte vnto the vnities, placed euer in the first place towardes the right hande, all those vnities added together as 5. and 6. make 11. and therto 8. make 19. the whiche founde, sette the digette number, whiche is all aboue 10. beyng 9. vnder the line, as you see, and for 10. the Article, you shall retaine one in memorie, to bee borne to the seconde place, for 10. in the firste place, is but one in the seconde place, and tenne in the seconde place, is but one in the thirde, and so from place to place infinitely. Thus hauyng ended the worke of the first place, findyng 5. 6. and 8. to make 19. whereof the 9. sette vn­der the line in the firste place, and for the ar­ticle 10. one kepte in memorie, then saie one in memorie, and one founde in the seconde place make 2. the whiche added to 4. standing ouer 1. make 6. wherewith 4. standyng ouer [Page 3] 1. make together 10. the whiche beyng an Article, set a Cipher vnder the line in the se­conde place, and for 10. there founde, beare one to the thirde place, the whiche put to 3. and fiue there standyng, makyng together 9. to bee sette vnder the line in the third place, then commyng to the fowerth and last place findyng one, set for the same 1. vnder the line in the fowerth place, and so the worke is en­ded, and the totall is found 1909. li. of equal value to the perticulars.

In the practice whereof is to bée seen the order of Addition, in euery occasion thereof furthered to bée performed. And for to am­plifie the effecte, take here a fewe lines in verse.

Hereafter is sette sunderie examples of whole numbers in practise, whereof maie bée seen the effecte before taught, with the order of proofe of the same.

These fower examples maie giue the lear­ner occasion, to examine his vnderstandyng in the preceptes before giuen of Addition. And also to note the order of proofe, of the same in workes of the whole numbers, wherof the effecte (I meane of proofe) consisteth in castyng awaie euery 9. founde in the sim­ple figures of any example, without respecte of place: first the perticulars aboue the line, and the remaine aboue euery 9. caste awaie, set at the vpper ende of a Burgunion crosse, in maner before apperyng. Then so many ti­mes as 9. is to bée founde in the totall of the same example, caste theim awaie also, and the remaine sette at the lower ende of the saied Crosse. Then if the 2. remaines, the one at the heade, and the other at the foote of the crosse bée equall, then the worke is true, and els not, as practise of the firste example maie more at large manifest.

To proue the first of the foure former ex­amples, repaire to the first place, where stan­deth the figures of 4. 2. and 9, in the perti­culers, whereof 9. cast awaie, there is 6. to be ioyned with 8. 4. 8. in the second place, which makyng 26. and 18 thereof cast away, the re­maine is .8. to be ioyned with 5. 8. and 4. in the thirde place, which makyng 25 and ther­of twise 9 caste away, the reste is 7 to be ioy­ned with 6. 5. & 6 in the fourth place, which makyng 24. and thereof twise 9 caste of, the remaine is 6. to be ioyned with 3. 6 and 5. in the fift and last place which makyng 20. and thereof twise 9. cast awaie, the remaine is 2. to be set at the toppe of the Crosse as remain for the perticulars. Then resort to the to­tall, where is founde 519851 and make to­gether 29. whereof thrise 9 cast away, which is 27. the remaine is 2. to set at the foote of the Crosse for remaine of the totall, and for that the figures in the toppe and bothom of the Crosse are like and equall, therefore the Addition is well and truely made, and so for all other examples wrought in whole Nom­bers.

Thus muche may seeme to suffice for the woorke of whole nombers, howbeit there is [Page 5] some varietie of worke in the diminute par­tes, of many and innumerable thyngs of son­dry Denominations. But for that it is not possible to write of all matters, I haue fur­thered some examples of Moneys, waights and Measures, as moste apte for the purpose in commune: referryng all men to th' appli­yng the same order to matters in priuate vo­cation, seruyng their occasions.

Firste is to be noted, that in sommyng of many perticuler sommes of Money, contay­nyng Poundes, Shillings, Pence, Farthin­ges and Mytes, firste giue heede howe many Mites make one Farthyng, and that beeyng 6. you shall for euery 6 Mytes eary one Far­thyng to the place of farthynges, and the re­main in Mytes vnder 6. you shall set vnder the lyne againste the Mytes, whiche stande next the right hand. Also when you come to the place of Farthinges, consider that as 4 Farthinges make one penie: so for euerie 4 Farthinges cary one penie to the place of pence, and set the remaine vnder 4. vnder the line against the Farthinges likewise. As 12. Pence make one shillyng: so in the place of Pence cary for euery 12. one shillyng to the place of shillynges, and the remaine vnder 12. [Page] set vnder the lyne in the place of pence. Also for euery 20 shillinges, carry one pounde vn­to the place of poundes, and the remaine of shillinges vnder 20 set vnder the line against the Shillynges, and so with pounds beeyng whole nombers, carry for euery 10 one from place to place, as before is taught, &c. and for the further vnderstandyng of the effect, here­after is set doune sondry exāples of Moneis.

Other examples wherein Farthings are o­mitted, and the mytes are borne for euery 24. one peny.

Here may bee seene in the formar of the 2 [Page 6] last examples, that the myts being 22. 9. 11. and 17. make together. 59. whereof 48. for 2 pence taken away, the remaine is 11. to set vnder the line, then the 2 pence ioyned with 8. 11. 7. and 3. d. make together 31. d. wher­of 24 for 2 s. taken away, the rest is 7 to set vnder the Lyne. Then the 2 still borne to the place of shillinges, with the other there stan­dyng, make 56. shilligns, from the which 40 for 2 poundes taken away, reste 16. s. to set vnder the lyne, and the 2 li. borne to the first place of the poundes, and ioyned with the o­ther Figures make 19. li. wherof the Diget 9. is set vnder the line, and for the Article. 10. one is caried to the seconde place, and with the Figures there standyng make 19. wher­of the Diget 9 is set vnder the lyne in the se­cond place, and for the Article 10. one is ca­ried to the thirde place, and so the woorke is ended, wherein appeareth the varietie of worke betweene whole nombers and broken in the practise of Addition.

For proofe of adding the Diminute parts there is no better, then double perusing the examples or Additions made. Howbeit whā you come to the whole nombers, you haue to consider what vnities are borne from the [Page] place next before the whole, and with theim ioyned, for those borne vnities make a parte of the totall, of the said whole Numbers, and therefore in making the proofe, must be par­cell of the perticulars, when the nienes are cast awaie, for otherwise the remaine of the perticulars, after the nienes cast awaie wil­be so muche lesse than the remaine of the to­tall, as by profe of the former and last practi­zed example the effect may appeare.

The totall of the foresayd example being 199. the Figures make together 19. wher­of twise 9 cast away, rest one to put vnder the crosse of proofe, as doth appeare. Then adde all the Figures of the perticulers together, and they make 35. wherof thrise 9 cast away, rest. 8. and agreeth not with the remayne of the totall Wherefore to that 8 put 2 whiche in the Addition was brought from the place of shillinges, and that maketh 10. whereof 9 cast away rest one, equall to the rest of the to­tall. And so the worke found true.

Hereafter is set sondrie examples of Ad­dition of weightes and measures, referryng the learner to the maner heretofore shewed, giuyng good heede to the nomber of vnities in a smaller denomination contained in an [Page 7] vnitie of a greater, and accordyngly to beare from place to place in former order.

¶ Substraction.

SVbstraction containeth the maner how to deduct or take away a smallar somme or Nomber from a greater, by practise wherof is found and brought forth a remain sought for & desired, as if one man owe vnto an other 356 li. wherof he hath paid 234 li. and would knowe what rested vnpaid. Then when the paiment is rebated from the debte, the remaine will appeare as practise by the same sommes the effect will manifest.

Here is to bee perceaued the debt beyng the grea­test somme is placed vp­permost, and the paimēts vnder the same. Vnity vn­der vnitie, cen vnder ten, and hundreth vnder hundreth, and a line dra­wen vnder all and so made apt for the worke.

Then to performe the Substraction, resort to the first place, which is of vnities and say. 4 payde taken out of 6. of debt, the remayne is 2. to set vnder the lyne in the firste place, then say 3 payd out of 5 of debt in the second place rest. 2. to be set vnder the line in the se­cond place also, 2 out of 3 in the place of hun­drethes rest. 1. set vnder the line in the third place, and so the worke ended, wherein ap­peareth that 234 li. taken out of 356. the remaine appeareth 122 li. &c.

When you haue made a substraction, and would proue, whether you haue made a true reste or not, then adde together the reste, and the paimente, and if the totall agree with the firste debte, then the Substraction is true, or els not. Whereof the practise hereafter she­weth the effecte by the same numbers, wher­of the former substraction was made.

Here doeth appere that 2. of rest, added to 4. of paiment, maketh 6 vn­der the line in the firste place: also 2. reste with 3. paied, maketh 5. vn­der the line in the se­conde place, and so 1. reste with 2. paied, is 3. to sette vnder the line in the thirde and laste place: and so the totall beyng 356. li. equall to the debte, proue the worke true, whiche o­therwise would not bée.

Thus muche maie seem sufficiente for the woorke of substraction, where the figures of the lesser number, are smaller then the figu­res standyng right ouer them in the greater noumber, but when the contrary is founde, then the woorke is of more difficultie, as by example.

Here resortyng to 9. in the firste place of paiment, to be taken out of 6. ouer it, whiche can not bee doen, therefore borowe [Page] an vnitie of 7. in the seconde place of the debt to ioyne with 6. in the first place, and so haue you 16. from the whiche 9. paied, rebated, reste 7. vnder the line in the first place. Then not forgettyng the vnitie, borrowed of 7. in the seconde place, to make the woorke of the first, saie one that was borowed with 8. in the seconde place of paiment, maketh 9. to be ta­ken out of 7. aboue, whiche can not be, wher­fore in former order, borrowe an vnitie of 5. in the third place of debte, to make 17. in the seconde, from the whiche 9. aforesaied reba­ted, the remaine is 8. vnder line in the se­conde place. Then saie as before, 1. borro­wed of 5. in the thirde place ioyned with 9. vnder 5. maketh the 10. to bée rebated from 5. ouer 9. whiche can not be, but by the helpe of an vnitie, borrowed of 3. in the fowerth place, and so 10. from 15. reste 5. vnder the line in the third place. Lastlie saie one boro­wed of 3. with 2. paied in the fowerth place, make thrée to bée taken out of thrée of debte in the same place, and so remaineth nothyng, wherefore a Cipher is sette vnder the line, in the fowerth and laste place, and so the woorke ended. Wherein doeth appeare that 2989. li. Substracted from 3576. li. the [Page 10] rest vnpaied is 587. li. The proofe whereof is by addyng the paimentes and remaine to­gether, and the totall thereof agréeyng with the debte, proueth the woorke true, as before is taught.

A like or more difficultie is founde in woorke of Substraction, when the places in the debte haue fewe or no figures, but Sup­plied with Ciphers, for that the woorke re­quireth a borrowyng of an vnitie in euery place of wante, from one place to an other, vnto the ende, as by example the effecte maie appeare.

Here 6. out of 13 made by helpe of one borrowed in the seconde place of the debt rest 7. vnder the line for the firste woorke. Then to paie that was borrowed saie, one and 7. make 8. to bée taken out of 10. in the seconde place, by the helpe of one borrowed in the third place, and so remaineth 2. vnder line in the seconde place. Againe, one with 9 in the thirde place, make 10. to bée taken out [Page] of 10 aboue made by one borrowed in the 4. place, and the reste is nothyng, and therefore a Cipher vnder line in the thirde 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. vnder line in the fourth place. So again one and 3. make 4. out of 10. and so reste 6. in the fifte place. Lastlie one borowed with one in the sixt and laste place, make 2. to bée taken out of three ouer, and so reste 1. in the same place, and the worke finished, and as aforesaid, the paimen­tes and restes together, makyng againe the debte, proue the worke true.

Thus muche maie séem sufficiente for the practise of Substraction in whole numbers: howbeit to further the vnderstandyng of the learner, take these fewe lines in verse.

When occasion presenteth workes of Sub­straction, of diminute partes, of what deno­mination so euer. Then like consideration is to bée had (as was noted in Addition) what quantitie of vnities in one denomination, is conteined in an vnitie of an other denomina­tion: and accordynglie make the Substracti­on, whereof the effecte in sondrie examples followyng maie appeare.

Multiplication.

MVltiplication conteineth the maner how to finde the number of vnities, of a smaller denomination, in an o­ther number of vnities, in a greater denomi­nation conteined. The effect whereof is bet­ter to vnderstande with fewe examples, then with many wordes. And for that it is neces­sarie for euery learner, to vnderstande the [Page] contente or somme, produced by multiplica­tion of one diget by an other, before he can muche profite without the same, therefore is prepared a Table for the effecte thereof: and notes giuen for vnderstanding, and vse of the same, as hereafter appeareth.

To vnderstand the vse of this Table, note that the Digettes Multiplicatours are set in 2 Collombes, to saye, in the highest side of the table, in places distinct from one vnto. 9. and likewise in the left ende of the Table, al­so [Page 14] from 1 vnto 9. And all the rest of the table, except those 17. places, wherein the Digets do stand, are places for the seuerall products of Multiplication of one of the sayd 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 collumne, and the other in the other, and in the place where the 2 Collomnes méete, wherein the saide figures doe stande: the one proceedyng from one ende to the other, and the other descendyng from the higher side to the lougher, there shall you finde 35. for pro­duct or somme sought for, and in like order may you finde any other desired.

The effect furthered in the former Table, Let euery desirous learner haue perfectly in memory: for that the same had the workes of multiplication are performed with more fa­cilitie, then by wante thereof is possible. And for aide of memorie note, whē 2 digets are to be multiplied together, consider if any of the same may bee parted in halfes, and so found multiply the contrary diget by the one halfe, and the double of that product is the somme ye would haue.

For example when 7. is to be multiplied [Page] by 6. you see 6. may be parted into twise 3. wherfore say 3 tymes 7. the contrary Diget maketh 21. then by former note the double thereof beyng 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 thrise 3. and whith one of the sayd partes multiply the contrarie diget as 5 beyng admitted for example, saiyng 3 tymeg 5 is 15. the whiche taken 3 tymes, maketh 45. which is the pro­duct of 9 and 5 multiplied together.

Likewise for 7 by 9 say 3 tymes 7. is 21. the whiche trebled, maketh 63. the product desired.

Also for 9 by 9. saye 3 tymes 9 is 27. the which trebled 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 halfe a­ny of the same, then to tripertate or put 9 in­to three partes.

Wherefore when 9 is to be multiplied by 8. say 4 beeyng the halfe of 8 and multiplied with 9 maketh 36. The double whereof be­yng 72. is the product of 8 and 9 beyng mul­tiplied together.

So for 9 by 6. say 3 the halfe of 6 multi­plied in 9. maketh 27. the double whereof beyng 54. is the product of 6. tymes 9. &c.

More to be sayd for the vnderstandyng of the former Table, or maner to finde the pro­duct of any 2 Digetts, multiplied the one by the other, might seeme superfluous, where­fore now I will shewe the order of Multipli­cation of one number of figures by an other.

First there is to bee noted in Multiplica­tion three Nombers by seuerall names. Di­stinct, that is to say, the Multiplicande, which is the nomber to bee multiplied, the Multi­plicatour, whiche is the multiplier, and the Product whiche is brought foorthe by the worke, the effect whereof to be shewed by ex­ample, may be performed in sondry and infi­nite matters, whereof take this that follo­weth, to mee seemyng very apte for the pur­pose, for an entraunce thereunto.

If you bye 5678. Yardes of Clothe, co­styng 86. pence euery yarde, & would know how many pence the whole amounteth, then the nomber of yardes shalbe Multiplicande, and stande vppermost in worke, and the nom­ber pence shalbe set vnder the same for Mul­tiplicatour. Vnitie of pence vnder vnitie of [Page] yardes, tenne vnder tenne, and so forth, when both partes haue the places supplied, with Figures. Hundreth vnder hundreth, thou­sandes vnder thousandes, and so infinitely in maner, as in examples heare may appeare.

The whiche 2 Nombers sette downe as you see, and a Lyne drawen vnder theim, then the Product of the worke wilbe so many Pence, as will paye for 5678 yardes of clothe, at 86 Pence the Yarde, whiche is 488308 pence, as appea­reth in the totall of the perti­culer Productes added together, as the or­der of woorke requireth, whereof the man­ner followeth. Firste you shall resorte to the place of Vnities, and saie, 6 tymes 8. is 48. whereof the Diget 8 put vnder the lyne in the place of vnities, and for the article 40. you shall retaine in memory 4 vnities to bee borne to worke of the second place. Then say 6 tymes 7 is 42. and 4 retained in memory, maketh 46 in the seconde place, whereof the Diget 6. is set vnder lyne, and for 40 retain 4 in memory to bee borne to the worke of the thirde place. Then saie, 6 tymes 6 is 36. and [Page 16] 4 in memory maketh 40 an Article nomber, and therefore put a Cipher vnder the line in the third place, and retain 4 in memory (that is) for euery 10. of the Article one, to bee borne to the worke of the fourth place. Then say, 6 times 5 is 30. and 4 in memory is 34. wherof the Diget 4 is put vnder line in the fourth place, and for the Article 30. beare three to the fift place, and because the Mul­tiplicand hath no Figure in that place, ther­fore put 3. retained in memorie vnder the line in the same place, and so the woorke is ended, for. 6. the firste Figure in the Multi­plitour, whereof the Product perticuler, is 34068. as doth appeare. Then resorte to 8. in the second place of the Multiplicatour, and therewith multiply euery Figure of the multiplicande in former order saiyng. 8 ty­mes 8. is 64. wherof put the Diget 4 vnder the line right vnder 8. the multiplicatour, and beare 6. for the Article 60. in minde for the seconde worke, and say, 8 tymes 7. is 56. and 6 in memory maketh 62. whereof put 2. vnder line in the second place of that seconde ranke, & beare 6 for the next worke, saiyng, 8 tymes. 6. is 48. and 6. in minde maketh 54. whereof put 4 vnder lyne and beare. 5. [Page] for the Article to the next woorke saiyng, 8 tymes 5 is 40. and 5 in memory maketh 45. whereof put 5 vnder line, and for the Article carry 4 to the next place where no Figures founde to make further woorke, set it vnder the line, and so the multiplication is ended. Then adde together the 2 perticuler Pro­ductes, and the totoll thereof will containe so many Pence as dooeth amounte of 5678. yardes of clothe at 68 d. euerie yarde, which is the effect desired in the example furthered, and so of other woorkes.

Here is to be noted, that for euery Figure in the multiplicatour of any example, there is a perticuler Product, and euery Diget made in the firste woorke of any of the same shalbe set vnder the Figure Multiplicatour, in what place soeuer it stande, and the Arti­cle nombers to bee transpozed for euery 10. in any place founde one to be carried by me­mory to the next place toward the left hand, to bee ioyned with the vnities made by the woorke in the sayde place. All and euery the whiche preceptes well vnderstanded, are suf­ficient for the practise of Multiplicatiō, how­beit hereafter is set doune sondry examples, wherein the effect aforesaide doth appeare, [Page 17] and for a further ayde to the learner hereaf­ter are furthered a few lynes in Verse.

Hereafter are set doune sondrie examples for the practise of multiplication, by 3. 4. and 5. Figures in the Multiplicatours.

The proofe of Multiplication is made by casting away all the Neines first in the mul­tiplicand & the remaine set at the one side of a crosse, thā the remaine of the multiplicator set at the other side therof. The whiche 2 re­maines multiplie together and from the re­sult thereof caste away all the nyenes and set the remaine at the vpper ende of the crosse. Lastly caste away all the nienes in the pro­duct, & set the remain at the foote of the same crosse, the whiche performed, if the remaines at the toppe and foote of the crosse be equall, the worke of that multiplication is true, and else not, as by an example may appeore.

In the multiplicande of this example, the Figures make 27. whiche is 3 tymes 9 [Page] and nothyng remaineth, wherefore I set a Cipher at the right side of the Crosse as you see. Likewise the Figures of the multiplica­tour make 14. whereof 9 caste away, the re­maine is 5. at the left side of the crosse. Then saiyng 5 tymes nothyng is nothyng, where­fore I set a Cipher at the vpper ende of the Crosse. Lastly the Figures of the product to­gether make 45. which is 5 tymes 9. and no­thyng remaineth, wherefore I set a Cipher at the foote of the Crosse. And for that the toppe and foote of the crosse are like, I know thereby the woorke of that multiplication to be good, and so of all other, whereof the effect appeareth in the former examples. Howbeit for that in the proofe of the former example, the Ciphers are to many to shewe the whole effect of the order of proofe, here is giuen an other example to amplifie the same.

In the multiplicande of this example the figures together make 25. whereof twise 9 cast awaie, reste 7 put at the right side of the Crosse. The figures of the multiplicatour make 17. whereof once 9 cast away rest 8 at the left side of the crosse. Then 7 and 8 being multiplied together make 56. whereof 54 cast awaye, for 6 times 9. reste 2 to set at the vpper ende of the Crosse. Lastly the Figures of the product make together 47. wherof 45 for 5 tymes 9 caste away reste 2 to set at the foote of the Crosse. And the Figures of the toppe and foote of the Crosse being like and equall, proueth the worke true, as afforesaid.

An other perfect and sure order of proofe, of Multiplication is made by Diuision, the which here I omit, till I haue shewed the practise of Diuision, which hereafter followeth.

¶ Diuision.

DIuision cōtaineth the maner how to shewe the Nomber of tymes, that a small somme or nomber is contained in a greater, and the ef­fect is procured in occasions infinite. And to [Page] the practise therof belongeth three nombers by seuerall names distinct, that is to say. The Diuidend, whiche is the nomber to be diui­ded. The Diuisor, whiche is euer in whole nombers lesser then the Diuidende. And the Quotient whiche sheweth the nomber of ty­mes, that the Diuisor is contained in the Diuidende. As for example if occasion pro­cured to bee knowen how many Poundes were contained in 396 Nobles. Then 396 is diuided and the nomber of Nobles con­tained in one pounde, whiche is 3. muste bee Diuisor. The whiche Diuidend diuided by the said Diuisor, the Quotient wilbe found 132. Whiche are so many tymes as 3 No­bles whiche is 1 Pounde, are contained in 396 Nobles, whereof the effect by example hereafter is practised.

[...] Here the Diuidend 396 beyng sette doune, then the Diuisor 3. is set vnder 3. in the Diuidende, and the worke begonne in the last place towarde the left hand, for that is the order in woorkes of Diuision, though therein it bee contrary to the other partes whiche euer beginne at the [Page 21] right hande: then is to be sought how many times 3. the diuisor is found in 3, y^e diuidend. and that being one tyme therfore 1 is put in a place separate from the rest as you see, & so the first worke ended and nothing remaining in 3. the diuidend, and therefore it and the di­uisour is canselled with a dashe of a Penne, thereby to signifie the woorke to be ended in that place. For the second woorke 3 the di­uisour is set vnder 9. in the diuidend, and is founde to be contaided therein 3 times, and therfore 3. is put in the Quotient, and so the seconde woorke ended, ane therefore 9 and 3 canselled as in the former woorke. Lastly, thre the diuisour is put vnder 6. in the diui­dend, and is found to be contained therein 2 tymes, and nothyng remaineth, wherefore 2 is put in the Quotient, and the whole worke is ended. And by the Quotient is found that 1 Pounde beyng 3 Nobles, is contained in 396 Nobles 132 tymes, whiche is the ef­fect required in the worke.

Here is to bee noted, that the Ciphers set ouer euery figure of the deuidende, are there sett to signifie nothyng to remaine, after the Woorke in the place, made vnder any of [Page] the saied Ciphers, and oft tymes putte so in workes, more for helpe of memorie, then for other neede.

Note also, that when the laste Figure of any diuidende, is lesser then the diuisor, then the diuisor shall bee sette vnder the Figure in the laste place saue one of the diuidende, and so worke in former order. How bee it, to make the matter more plaine, the effecte shal appeare in an example followyng.

If you would knowe how many poundes are contained in 2758. Crounes, then for that 4. Crounes make one pounde, therefore 4. muste bee diuisor, and sette vnder the diui­dende, in the last place sauyng one, for that in 2. in the laste place 4. is not conteined, and so the worke practised as followeth.

[...]

In this example you maie see 4. the diui­sor sette vnder 27. wherein it is conteined 6. tymes, and 3. remainyng: therefore 6. is put in the Quotient, and 3. the remaine is set ouer 7. and so the firste worke ended, and [Page 22] the diuisor in that place, and the 27. ouer it canselled, as afore taughte. Then for the se­conde woorke, the diuisor is set vnder fiue in the diuidende, the whiche with 3. remainyng of the former woorke, maketh 35. wherein the diuisor 4. is contained 8. tymes, and 3. remainyng, wherefore 8. is putt in the quo­tient, and 3. remainyng set ouer 5. and so the seconde Woorke ended, and therefore the diuisor, and 35. ouer it canselled in former order. Againe for the thirde and laste worke, the diuisor 4. is set vnder 8. the whiche with 3. remainyng in the former woorke, maketh 38. wherein the diuisor 4. is contained 9. tymes, and 2. remaineth putte ouer 8. and so the whole woorke ended, the diuisor and the 38. beyng canselled as the other, and the 2. remainyng of the whole woorke, is separate from the reste, to signifie the same to bée a re­maine of the worke, and not sufficient to cō­taine 4. the diuisor. By the whiche woorke doeth appeare, that in 2758. Crounes, 4. of the same makyng one pounde, is contained in the whole somme 689. tymes, and twoo Crounes remainyng, whiche is the effecte sought for by the worke, wherein the perfect order of diuision is shewed, where the diuisor [Page] is one figure onely.

Howbéeit, when the diuisor containeth a number of Figures, as more then one bee it fewe or many: Then the quotient shall euer bee made with that Figure of the diuisor, whiche standeth next towarde the left hande, and none other. And the quotiente so made, shall bée multiplied by the reste of the Figu­res of the diuisor, one after an other, and eue­ry producte shall bée rebated out of the diui­dende, standyng right ouer the Figure of the diuisor, whiche maketh any of the saied pro­ductes from place to place throughout, for e­uery figure of the quotiente made. And the quotient shall bée made no greater then that a remaine maie bée lefte in euery worke, out of whiche the saied productes maie bée taken accordyngly, as in example practized hereaf­ter, the effecte more plainly maie appeare.

If you would knowe how many pence are contained in 56847. mites. Then the nū­ber of mites makyng one penie, shall bée de­uisor whiche is 24. and sette vnder the diui­dende thus.

[...]

In this example are 4. Figures in the quotient, the whiche are made by 4. seuerall workes in the diuidend. And for the first you shall set 24. the diuisor, vnder 56. in the diui­dende, and saie, how many tymes 2. in 5. and that is 2. tymes, whereof 2. you shall put in the quotient, and sette 1. remainyng ouer 5. Then the 2. in the quotient, multiplied by 4. in the diuisor, produceth 8. to bée taken out of 16. remaining in the diuidend ouer 4 and so the first worke ended 8. remainyng of 16 set ouer 6. then shall you cancell your diui­sor 24 and 56 in the diuidend with 1 remai­ning ouer 5. and so you haue finished all thin­ges belonging to the first worke. Then shall you set your diuisor 24. vnder 88 and saye, howe many tymes two the diuisor in 8. ouer it, and that is 3 tymes, and 2 remaineth to set ouer 8. The which 3 put in the quotient, and multiply the same by 4 in the diuisor, and the product being 12. rebated out of 28. rest 16 [Page] ouer 28. then cancell all the Figures of the Diuisor the diuidende and of the remaines vnder & behinde 16. and so the second worke is ended. Thirdly you shall put your diuisor 24 vnder 164 and say, how many tymes 2 in 16. and that is 6 tymes, & 4 remaining to set ouer 6. 16 beyng cancelled, then 6 multi­plied by 4 in the Diuisor, the product is 24 to bee taken out of 44 so resteth 20 ouer 4 the diuisor, and the thirde woorke ended, and then is to be cancelled all figures of the diui­sor diuidende, and remaines vnder and be­hinde 20. Lastly, you shall put the Diuisor 24 vnder 207 and saye how many tymes 2 in 20 ouer it, and that is 8 tymes, and 4 re­maineth ouer the cipher in the second place: then 8 in the quotient multiplied by 4 in the Diuisor produceth 32. to bee taken out of 47. and there remaineth 15. and the whole woorke ended, and therefore the sayde 15 is to be separated from all the other figu­res of the worke plainly to appeare, all the other beyng cancelled. And so is found that in 56847 mites is cōtained 2368 pence, and 15 mites remainyng, whiche is the ef­fect in the worke required.

Because that it is harde for a learner to [Page 24] vnderstande the woorke of Diuision where the example is practised in one place. Ther­fore in an other example the woorkes shall haue so many seuerall distinctions as there shalbe figures put in the quotient: for euerie figure of the saide Quotient doth require a perticuler worke, whiche is not easie to bee perceiued in the former example, as in an o­ther folowyng may appeare.

If you would diuide 856942. by 354. you shall set doune the diuidend first, and the diuisor vnder, as before is taught, and as he­reafter appeareth.

[...]

Then say how many tymes 3 the diuisor in 8. ouer it, and that is 2 tymes and 2 re­maineth, wherefore 2 for the tymes is put in the quotient, and the 2 of the remaine is put ouer 8. and so the woorke for the Quotient makyng in that place is ended. Then 2 in the Quotient muste bee multiplied by 5 in the diuisor, and that maketh 10 to bee taken out of 25 ouer it, and so will remain 15 ouer [Page] 5. and all the figures vnder 15 to bee cansel­led. Also you must multiply 2 in the quotient by 4 in the diuisor, and that maketh 8. to bee taken out of 6 ouer 4 whiche can not be but by borowyng an vnitie out of 5 to make 16. by order in Substraction, from whiche 8 a­foresaid rebated, the rest is 8 ouer 6, and 4 o­uer 5. and so the whole woorke for makyng the first figure put in the quotient is ended.

To procede in the woorke, you shall sette doune the deuidende, whiche is 148942. and set the diuisor vnder it thus.

[...]

Then saie, how many tymes 3. in 14. that is 4. to bee set in the quotient, and 2. re­mainyng ouer 14. the saied 14. and 3. in the diuisor canselled. Then saie 4. in the quotiēt, multiplied by 5. in the diuisor, maketh 20. to be taken out of 28. ouer 5. & there remaineth 8. Likewise saie, 4. in the quotiente, multi­plied by 4. in the diuisor, produceth 16. to be taken out of 9. whiche can not bee, but by helpe of an vnitie, borowed of 8. to make 19. them 16. out of 19. reste 3. ouer 9. and 7. ouer [Page 25] 8. and so the woorke for makyng the seconde Figure in the quotient ended, all the figures cancelled, vnder and behinde 73.

For the thirde worke, you shall set doune the diuidende remainyng, whiche is 7342. and the diuisor vnder it, thus.

[...]

Likewise saie, how many tymes 3. in 7. that is 2. to bee set in the quotient, and there remaineth 1. ouer 7. beyng canselled, and al­so 3. the diuisor vnder it. Then multiplie 2. in the quotiente, by 5. in the diuisor & that is 10 to be taken out of 13. and so remaineth 3 the 1 ouer 7. beyng cancelled. Also multi­plie 2. in the quotiente, by 4. in the diuisor, and the resulte is 8. to bée taken out of 4. whiche can not bée but by helpe of an vnitie, borowed of 3. and so 8. out of 14. reste 6. o­uer 4. and 2. ouer 3. all the figures vnder and behinde 26. beeyng cancelled, and so the woorke for the third figure in the quotiente ended.

For the fowerth and laste worke, you shal set doune the diuidende remainyng, whiche [Page] is 262. and the diuisor vnder it, thus.

[...]

Finallie saie, how many tymes 3. in 2. o­uer it, and that is no tyme, wherefore sette a Cipher in the quotiente, to supplie a place, and cancell 3. Then saie 5. tymes nothyng is nothyng to bée taken out of 6. and there­fore 6. remaineth ouer 5. beyng cancelled. Likewise saie 4. tymes nothyng is 0. to bee taken out of 2. ouer 4. and therefore 2. re­maineth, and 4. to bée cancelled, and so the whole worke ended, the quotient made at fo­wer seuerall woorkynges, and founde to bee 2420. and so many tymes is the diuisor 354 conteined in the diuidende, 856942. whiche is the effecte in the diuision sought for, and required. And the remaine of the woorke is but a parte of a tyme, and therefore to bee set ouer the diuisor thus 262/354, whiche signifieth that the diuisor is not contained in the same, and therefore appeareth a remaine, and ac­cordyngly made to appeare.

And where for findyng of the former quotiente, there hath been made 4. particuler workes, to saie, for euery Figure or Cipher, [Page 26] one seuerall practise, you shall vnderstande, that suche maner of distinctions is not fur­thered, but onely for helpe in teachyng, but the diuision to bee made in one place, and the quotiente to bee brought forthe in one prac­tise, in all diuisions generally, whereof the maner is hereafter practized in example, by the former diuidende, diuisor, and quotiente, in whiche woorke is to be seen the former 4. particular workes all in one.

[...]

The sunderie preceptes and practises of diuision before shewed, well noted and vn­derstanded, thereby any diuision is to bee made with facilitie: howbeit, further to note that when the diuisor containeth a greater nomber of figures, then hath been in any e­xample before practised. Then euery of the same shall multiplie the quotiente, and the product taken out of the remaine ouer it, and [Page] other difficultie is not founde, wherefore I will practise sunderie examples, wherein the effecte maie appeare, with order of proofe of the worke, and so proceede to the next parte, and for aide to the learner, here is furthered a fewe lines in verse.

¶ Of Reduction.

REduction is no proper part of Arithmeticke, for howbeit that the change of one denomination vnto another, or the alteration of thinges from one title to an other may well bee termed Reduction. The same notwithstandyng, the effect is perfor­med by Multiplication or Diuision, or else both. Neuerthelesse, for that the learner may haue experience howe thinges are reduced and altered in name and propertie: the sub­staunce or value remainyng perfectly, great­ly to his contentation and commoditie. I therefore thinke conuenient to shewe some examples therof in such place as other haue furthered it as a parte of Arithmetique, though as you may perceiue the effect fur­thered by Multiplication and Diuision, as aforesaide.

¶ Progression.

PRogression Arithmeticall is a short and bréefe maner, addyng sonderie figures or noumbers sett doune, euery one (after the firste) encreasyng by equalle quantitie, as 1. 2. 3. 4. 5. 6. 7. 8. 9. there is encrease by an vnitie: also 2. 4. 6. 8. 10. 12. the encrease is by 2. Againe 3. 6. 9. 12. 15. 18. &c. euery noumber of the Progression is augmented by 3. more then an other. The whiche progressions, and all other like, are to bee sommed by rule of Progression with muche more facilitie then by Addition, as by example the effecte maie appeare.

There is to bee noted, that if the tymes of the Progression bee odde, then the firste and laste Noumbers added together, and the halfe of that totall, multiplied by the Noum­ber of tymes of the progression, the pro­duct thereof will be the iust somme of the saied progression, as by example plainely may appeare.

AS whole numbers compounde of vnities maie bee augmented and encreased to infinite effect: so an vnitie maie bee diuided into sondrie and infinite dimi­nute partes, whiche partes in name and na­ture are agreable. For a fraction is a parte of one vnitie, and not of many: for howbeeit that whole Numbers maie bee diuided into partes, to seuerall effectes, the same notwithstandyng, suche diuided partes are no proper fractions, but improperly shewe the parts of whole nombers, and not of an vnitie, as 40. pounde to bee parted amongst three menne: the firste to haue ¼, whiche is one fowerth parte: the seconde ⅖, whiche is twoo fift par­tes: and the third 7/20, whiche is seuen twentie partes. The saied Partes maie bee shewed in whole noumbers, not needyng the vse of fractions for the same. For the ¼ is 10. lib. the ⅖ is 16. pounde, and the 7/20 is 14. pounde, whiche together maketh 40. pounde, and all [Page 45] suche numbers expressed in maner like frac­tions, are not proper fractions, but impro­perly borowyng the propertie of fractions, whiche as aforesaied, are partes of one vni­tie onely, and not aboue.

Here is to bee noted, that a fraction is ex­pressed by twoo figures, set the one ouer the other, with a line betwene, thus ⅔, whiche si­gnifieth twoo thirde partes of an vnitie, and that vnder the line, is called the denomina­tour, because it doeth euer represent the par­tes, wherein the vnitie is diuided: and that aboue the line is called the numeratour, be­cause it sheweth the number of partes, by oc­casion required, not needyng the whole vni­tie. As when a manne hath right to ⅔ partes of a pounde in money, whiche is twoo No­bles, then 2. ouer the line, sheweth the par­tes of his right, and 3. vnder the line, she­weth wherein the vnitie is diuided, and re­presenteth one pounde of money, diuided in­to three partes.

Here note, that euery fraction abstract or free from denomination, maie bee applied and made contracte to any denomination, by occasion required, and more easie for the learner to vnderstande, what the nature and va­lewe [Page] of a fraction is, when the Carracter of denomination is ioyned with it, then when it is without the same, as by example the ef­fecte maie appeare.

Euery of the denominatours of the 10. former fractions, doeth represente an vnitie diuided into so many partes, as the figure or figures of the same doeth demonstrate: how­beeit, not to bee knowen of what thyng, for want of a Carracter, to signifie the denomi­nation. Likewise euery Numeratour, is so muche lesse thē an vnitie, as the difference appeareth, betwene it and the denominatour thereof, and maie bée applied to sondrie thyn­ges, ioynyng a Carracter for the same, bee it of weightes, Measures, Moneyes, or other thynges whatsoeuer, as hereafter appeareth by Carracters for Monies, seemyng moste apte for the purpose.

The firste and seconde of these 10. frac­tions, hath the letter d. for carracter, signifi­yng [Page 46] of a peny, and halfe a penie, and 2. thirde partes of a peny. The third and fowerth haue the letter s. for carracter, signifiyng to bee of a shillyng: as 3. fowerth partes, and fower fifte partes of a shillyng. All the reste haue the letter l. for carracter, signifiyng euery of the same to bee a fraction, or a parte of a pound of Money, as 5. sixt partes of a pound 7. eight partes of a pounde, and so the reste 9. tenth, 17. twenteth, 25. thirtie twoo, 162 twoo hundreth and fourtie partes of a poūde and accordyng to the former saiyng, so much as any of the numeratour (whiche is aboue the line) is lesser then the Denominatour of the same: so muche it wanteth of the valewe of the vnitie by the Denominatour, and car­racter represented, be it of a peny, a shillyng, or a pounde: and so to vnderstand of all other fractions, of what denomination so euer, and for the learners better vnderstandyng here­after the same fractions are applied to other denominations, signified by wordes, for wāt of vsuall carracters, as for moneys is foūde.

½ Inche, ⅔ Foote, ¾ Yarde, ⅘ Elle, ⅚ Ounce or vnz, ⅞ lib or pounde weight, 9/10 C. or hundreth weight, 17/20 Hower, 25/32 Moneth, 162/240 Yere. &c. Thus maie you see a fraction to [Page] bee a parte of an vnitie, whereof knowen by Carracter, or woorde of denomination, and not hauyng denomination, maie bee applied to any thyng, by occasion required, and thus muche maie seme sufficiente to giue vnder­standyng how to expresse a fraction, whiche is a parte of Numeration: howbeit, now re­maineth to shewe how to finde the valewe of a fraction, whereof the effecte hereafter fol­loweth by examples in the former fractions, applied to seuerall denominations.

½ Bushell signifieth a Bushell to bée diui­ded in twoo partes, and the halfe thereof the fraction representeth: whereof to finde the valewe to bee expressed in common and kno­wen partes, you shal consider what diminute partes the Bushell containeth, and that is 4. Peckes. Then multiplie the numerator 1. by 4. Peckes in the Bushell, and the producte diuided by the Denominator 2. the quo­tient will shewe 2. Peckes to bee contained in the fraction, and is the valewe of halfe a Bushell desired to bee knowne, and this take for a generall rule, to bryng a fraction into common and knowne partes, the effect more at large appearyng in sondrie examples fol­lowyng.

⅔ Foote signifieth twoo thirde parts of a foote, whereof to finde the valew in common knowen partes, consider what Diminutiue partes a foote containth, and that is 12 In­ches, by the whiche multiplie the Numera­tor. 2. maketh 22 Inches, the which diuided by the Denominator. 3. yeldeth inquotient 8 Inches for ⅔ partes of a foote. &c.

Likewise 2/4 lib. is 3 quarteas of a Pounde weight, and to knowe the value thereof, you must consider what knowen partes the same containeth, the whiche beeing haberdipoys waight is 16 ounxes. Wherby multiply the Numerator 3, and the product is 48 ounzes, the whiche diuided by the Denominator 4. the quotient sheweth 12. ounzes to bee ¾ of the lib. habardipoys.

Howbeit if the Pounde waight bee Troy weight wherby Gold Siluer and Precious stones are waied, then 12 ounzes maketh the lib. the whiche multiplied by the Numera­tor 3. produceth 36. ounze, the whiche diui­ded by the Denominator 4 yeldeth in Quo­tient 9. ounzes for ¾ of the lib. Troye.

Also 4/6 ell representeth foure fift partes of an ell, the whiche to bring into common and [Page] knowen partes, consider what knowen par­tes an ell containeth, and that is founde in 3 sondrie sortes. Firste it containeth 4. q. pro­per to the same. Also 5 q. of the yarde, and thirdly 45 Inches. To haue it in parts pro­per to it, multiplie the Numerator 4 by the quarters in the ell, whiche is also 4, maketh 16. and that diuided by the Denomitor 5. yeldeth 3 q. of the ell, and ⅕ of one of the same q. To haue it in quarters of the yarde, multi­plie the Numerator by 5. quarters of a yard in an ell contained, the product wilbe 20, and the same diuided by 5 the Denominator, she­weth 4. q. of a yard in the fraction contained.

Lastly to bring it into Inches, multiplie the Numerator 4 by 45 Inch in the ell con­tained, the Product wilbe 180. The which diuided by the Denominator 5. yeldeth in quotient 36 Inches for ⅘ of an ell.

To bryng ⅚ s. into apte knowen partes, multiply the Numerator 5 by 12. d. in a shil­ling contained, maketh 60. and the same di­uided by 6. the denominator, yeldeth in quo­tient 10. d. for ⅚ of a shillyng. &c.

To bryng ⅞ d. into knowen partes, mul­tiplie the Numeratour 7. by the Mytes of a Penny, whiche are 24, and the Product wil [Page 48] be 168, and the same diuided by the denomi­nator. 8. the quotient will shew 21. Mytes to be the ⅞ partes of a penny.

To bring 9/10 Crowne into knowen partes, multiplie the numerator 9. by 15 grotes, or 60. Pence in a Crowne contained, and you shall haue produced 135. for groates, and 540 for Pence, the whiche products diuided by the denominator 10. yeldeth in quotient 13. groates and a halfe, and 54 d. euery of whiche is 4. s. 6. d. for 9/10 of a Crowne.

To bryng 17/20 Noble into knowen partes, multiply the numerator 17. hy 20. greates or 80. pence in a Noble contained, and the Producte will be 340. for the groates, and 1360. for pence, the whiche diuided by the Denominator 20. yeldeth in Quotient 17 groates and 68. d. euery of whiche is 5. s. 8. d. for 17/20 of a noble.

To bring 25/32 l. into knowē partes, multiply the numerator 25. by the shillyngs in a poūd whiche is 20. and the Product wilbe 500. the which diuided by the Denominator 32. yeldeth in quotient 15. s. and 20/32 partes of a shillyng, and to knowe the value of that later fraction multiplie the numerator 20. by the [Page] pence in a shilling, which is 12. and the pro­duct wilbe 240. the whiche diuided by 32. sheweth in quotient 7. d. ob. and so haue you 15. s. 7. d. ob. for the 25/32 partes of a pounde.

Lastly to bryng 162/240 li. into knowen parts, multiplie the Numerator 162. by 20. s. in a Pound, and the Product will be 3240. the whiche diuided by the Denominator 240. yeldeth in quotient 13. s. 6. d. for 162/240 li. and so of all other.

Progression of Fractions.

PRogression of Fractions is in 2 sortes, the one of property contrary to y^e other. for the first which is ½. ⅓. ¼. ⅕. ⅙. and so in­finitely, the greater that the denominator is, so much the smaler is the value of the fractiō, for ⅙ li. whiche is 3. s. 4. d. is of smaller va­lue then ⅕ li. whiche is 4. s. also ¼ li. is 5. s. and of smaller value then ⅓ li. whiche is 6. s. 8. d. and so ⅓ 6. s. 8. d. is smaller thā ½ li. which is 10. s.

But to the contrary in the seconde sort of progression, whiche is ½. ⅔. ¾. ⅘. ⅚. &c. the grea­ter that the Denominator is, the more is the value of the fraction. For ⅚ li. beeyng 16. s. 8. d. is more then ⅘ li. whiche 16. s. Also ¾ li. [Page 49] whiche is 15. s. is greater then ⅔ li. whiche is 13. s. 4. d. and so of all other like vnder­stande.

And here note in the first Progression, the greater that the Denominator is, so muche the more is the vnity decreased, whiche may bee to infinite effect and euer to be somwhat, and to the contrary in the seconde Progres­sion, the greater that the denominator is, the nearer to the whole vnitye the value of the fraction doth approche, how be it can neuer attaine to make the vnity.

Reduction of Fractions.

REduction of Fractions contai­neth the maner how to bryng 2 or more Fractions into one, ei­ther such as be of one Deuomi­nation, as other whiche are of contrary denominations, the effect whereof more easie to vnderstand by a fewe examples then in many wordes.

To reduce ⅓ li. ¾ li. into one, that is to make one Fraction to containe the value of them both. You shall by a generall Rule mul­tiplie the 2 denominators, the one by the o­ther [Page] saiyng, 3 tymes 4 is is 12. to bee sette doune twise for 2. newe denominators thus. 12. 12. then multiplie the numerator of the firste fraction by the Denominator of the se­conde, that is one by 4. maketh 4 for a newe Numerator to stande ouer the common and newe denominator thus 4/12 li. Also multiplie the numerator of the seconde fraction by the denominator of the first which is 3. by 3 is 9 to set ouer 12 thus 9/12 li. & so haue you 2 new fractions of one Denomination, containing the value of 2 first. For ⅓ 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 firste beyng of contrary Denominations reduced into the 2 later beyng of one denomination. And to make one fraction of them both, adde together the 2 Numerators 4 and 9 is 13. to set ouer 12 thus 13/12. and so you haue 1 fra­ction 13/12 li. containyng the iuste value of ⅓ 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 suche forme by néede in worke or otherwise, and then by Rule generall diuide the nume­rator by the Denominator, and the quotient [Page 50] will shewe the vnitie or vnities in the sayde fraction, and the remaine if there be any will be a proper fraction. Wherfore to ende this reduction, diuide the Numerator 13. by the Denominator 12. and the quotient will bée 1 and 1/12 li. whiche is 21 s. 8 d. the iust value of the 2 firste fractions ⅓ li. whiche is 6 s. 8 d. and ¾ li. whiche is 15 s. and together maketh 21 s. 8 d. as the Reduction hath brought forthe.

Sometimes occasion may require reduc­tion of 3. 4. or more seuerall Fractions of sondry Denominations to bee brought into one denomination, and to make one fraction of many, and then you shall multiplie the first denominator by the seconde, and that product by the thirde and the seconde product, by the fourth &c. And so many Fractions as there are to be reduced, so many newe Denomina­tors shal you set downe in former order. And to finde numerators to euery of the same you shall multiply euery numerator into all the denominators of the other fractions not be­longing to the numerator multipliar, and so finde to euery newe denominator a newe nu­merator, as example will declare.

To reduce ½ li. ⅔ li. ¾ li. and ⅘ li. into one [Page] denomination, and so to one Fraction firste multiplie the denominators, one into ano­thers product, as 2 by 3. is 6. and that by 4. is 24. the whiche by 5. is 20. for common denominator to be set downe 4 tymes. Then by the numerator of the firste, whiche is 1. multiplie the Denominators of the other, whiche is 3. 4. and 5. the product is 60. to set ouer the common Denominator 120. thus 60/120 li. and is in value equall with the first fraction ½ li.

Then by the Numerator of the seconde, whiche is 2. multiplie the denominators of the other, whiche is 2. 4. and 5. and the pro­duct is 80. to set ouer the common denomi­nator thus, 80/120 li. equall to the seconde fra­ction ⅔ li.

Likewise by the numerator of the thirde fra­ction, whiche is 3. multiply the denomina­tors of the other whiche is 2. 3. and 5. the product is. 90. to set ouer 120 thus 90/120 li. and is in value equall with the third fraction ¾ li.

Lastly by 4. the Numeratour of the fo­werth fraction, multiplie the denominators of the other, whiche is 2. 3. and 4. and the product is 96. to sette ouer the common de­nominator [Page 51] thus 96/120, and is in valewe equal with ⅘ li. and so you haue fower newe frac­tions of one Denomination, for the fower firste of contrary Denomination, whiche is the effecte causyng the reduction.

Then accordyng to former instruction, adde together the Numeratours of all the newe fractions, beyng of one Denominati­on, whiche is 60. 80. 90. 96. and make 326. to sette ouer the common Denomina­tour, thus 326/120, the whiche appearyng to bée a fraction improper, diuide the Numerator by the Denominatour, and the quotient will shewe the vnities in the same, and the proper fraction, all whiche is 2. and 86/120 li. whiche is 2. li. 14. s̄. 4. d. the iuste valew of the first fower fractions: For ½. li. is 10. shillynges, ⅔. li. is 13. s̄. 4. d. ¾. li. is 15. s̄. and ⅘. li. is 16. s̄. and make together, 2. li. 14. s̄. 4. d. as the woorke of Reduction hath brought forthe. &c.

If the number of Fractions bee so many, that the Reduction of them would bee tedi­ous to bee made at one tyme, then you maye reduce parte of theim at one tyme, and the reste at an other, and so make twoo newe Fractions of all the firste. Then reduce the [Page] saied twoo newly made both into one, and so you haue doen, as example maie declare.

To bryng ½. ⅔. ¼. ⅖. ⅚. ⅞. li. all into one fraction, would séeme tedious to a lear­ner to performe. Wherefore reduce three of the firste together, and thei will make 12/24. 26/24. 6/24. and makes in one 3/2 4/4. li. Then reduce the three laste figures together, and you shall haue 96/240. 200/240. 210/240. and makes in one 506/240. and so haue you twoo newe Fractions for all the other 6. Lastly reduce the twoo newly made into one and you shall finde 20304/5760 li. whiche is worthe 3. li. 10. s̄. 6. d. the iuste valewe of the sixe firste. &c.

Thus muche maie séeme sufficient for re­duction of proper fractions, whiche are par­tes intire of an vnitie, and neither greater, nor so muche as the saied vnitie, and I ac­coumpte suche improper, whiche are either greater then an vnitie, or lesse then an intire parte, as some other kinde bee, whiche are but partes one of an other, and bee called Fractions of Fractions, whereof the reduc­tion followeth.

To reduce fractions of fractions, whiche are partes one of an other, all not makyng so muche as an vnitie, you shall multiplie all [Page 52] the Denominatours together, and so haue one Denominatour, for an newe and proper Fraction: then ye shall multiplie all the Nu­meratours together, and haue one Numera­tor to sette ouer the newe Denominator, and finde one proper Fraction for many other, as by example maie appeare.

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 newe Deno­minator. Then multiply the numerators to­gether, as 2. by 3. is 6. the whiche by 4. ma­keth 24. to bee sette ouer the Denominator thus, 24/60 li. whiche is the value of 8. s̄. re­presented by the three firste fractions.

Sometymes occasion maie procure a re­duction of proper Fractions, and improper of bothe sortes all together (that is to saie) whole numbers, proper fractions, and frac­tions of fractions to bee brought into one, as by example.

To bryng 3. and ⅔. li. with ¾. li. and ½. of ⅔. of ¾. li. into one, you shall firste bryng the whole number, and the Fraction thereto be­longyng, into one Fraction improper, the whiche to performe, you shall multiplie the whole number 3. by the denominator of the [Page] fraction thereto belongyng, whiche is also 3. and the product is 9. wherevnto adde 2 the Numerator of the same fraction, so haue you 11/3. li. for the whole number and first fractiō. Then bryng the 3. fractions of fractiōs into one, as before is taught, whiche will make 6/24. li. so shall you haue 3. Fractions for all, whiche are 11/3. li. ¾. li. and 6/24 li. and reduced make 1344/288. li. whiche is 4. li. 13. s̄. 4. d. and so muche representes the Figures of the Example.

¶ Of diuision in broken Numbers, and firste of Abreuiation of greater termes into smaller.

TO abreuiate a fractiō of great termes (that is of many Figu­res) into an other of smaller termes, or fewer Figures, you shall consider what Digette is moste apte to diuide aswell the Numerator, as also the Denominator of any suche frac­tion, as is to bee abreuiated, and sette the 2. quotientes one ouer the other, and you shall haue a newe fraction of smaller termes then the firste, as by example.

To abreuiate 54/72 li. giue regard what Di­get or figure, will diuide both the numerator and denominator, and that may be doen by 4 sondrie digettes, as by 9. by 6. by 3. and by 2 and the moste apte of them is 9. And as you make your diuision, sette the quotiente of the numerator aboue the same, and the quotient of the denominator vnder the same denomi­nator thus. Wherein you maie perceiue that in 54, 9. is contained 6/54 li. 6. tymes and so 72. 8. tymes, and so 72/8 you haue a frac­tion of twoo Figures 6/8. li. for the other of 4. Figures 54/72. the greater termes abreuiated into smaller, and the value not channged.

Likewise by the same order, consider that 2. beyng made diuisor of 6/8. li. you shall haue that fraction abreuiated to 8/4. li. whiche in smallest tearmes that maie bee, is of equalle valewe with the twoo other: for euery of the same representeth 15. shillynges, 3/6. li. and thus is the practise. 8/4.

When the learner findeth a fraction to be abreuiated, whiche beyng of greater termes then with facilitie to knowe the Digette, moste apte for the abreuiation, then let hym examine the example by mediation thus, 2. beyng alwaies Diuisor, as in this Fraction [Page] 48/96. li. for example.

Hereby three mediations the Fraction 48/96. pounde, is brought to 6/12. pounde, where it is with facilitie perceiued, that 6. is halfe of 12. and therefore ½. li. is settedoune for it, and so the abreuiation ended.

Sometyme a fraction maie require 2. 3. or more digettes, to bryng the same to smal­lest termes, as by sundrie examples the effect maie appeare by this fraction 160/240. pounde.

Here note, that when a Fraction hath e­quall number of Ciphers, in the place or pla­ces towarde the right hande, then the abre­uiation may bee made the shorter, by cuttyng awaie the Ciphers of bothe sides, in equall number, thus.

Wherein doeth appeare, that the ciphers of euery of the three Fractions, separated from the Figures, then euery of the same is 16/24. and diuided by 8. sheweth ⅔. li. for smalleste termes, and so in al other like vnto thesame.

To abreuiate 75/120. pounde, there is requi­red the vse of twoo figures, whiche is 3. and 5. to beginne with the one at pleasure to bee taken, and to ende with the other by conse­quence, as in example, practise doeth shewe.

To abreuiate 112/192, there is required the vse of one figure onely, for the moste apte, whiche is 4. and maie bee doen by twoo Fi­gures, whiche is firste by 8. and then by 2. as practise will manifeste.

To abreuiate 128/160. lib. moste aptly there is required twoo figures, 8. and 4. and with more circumstaunce by 4. and 2. as by ex­ample.

To abreuiate [...]75/500 li. seeke for the moste apt diget to diuide by, and that is 5. by the which at 3 tymes is brought forthe ¾ li. whiche is 15. s. as by example.

Thus muche may seeme to suffice for A­breuiation of Fractions, whiche is perfor­med by Diuision practized in whole noum­bers. Howbeit, diuision of fractions is much contrary, as by examples the effect may ap­peare.

When one fraction is to be diuided by an other, that is to say, when you would knowe how many tymes I fraction is contained in an other, set the diuisor one the [...] lefte hande the other, & a crosse betwéene theim thus, which re­quireth by Diuision to make knowen how many tymes ⅖ li. is contained in ⅞ li. Then multiplie the numerator of the diuisor by the denominator of the diuidende, and that product shalbe Diuisor. Likewise [Page] multiplie the numerator of the diuidende by the denominator of the diuisor, and the pro­duct thereof shalbe diuidende, and the same diuided by the laste diuisor sheweth in Quo­tient that is required in the woorke, as pra­ctise may more amply deelare.

In this practise appeareth that 2. nume­rator of the diuisor, multiplied by 8. the De­nominator of the diuidende, produceth 16. set vnder the Crosse for diuisor, and 5 deno­minator of the Diuisor multiplied by 7. the Numerator of the diuidende produceth 35. for diuidende to set ouer the Crosse. The whiche 35 diuided by the diuisor thereof 16. sheweth in quotient that ⅖ li. whiche is 8. s. is contained in ⅞ li. whiche is 17. s. 6. d. 2 tymes, and 3/16 of a tyme, and note that ⅖ lib. twice is 16 s. and 3/16 of 8 s. or 1 tyme is 1 s. 6 d. and maketh together 17 s. 6 d. the iuste value of the fraction diuided.

To diuide ¾ lib. by ⅓ lib. by the same order [Page 57] the practise foloweth.

Wherein appeareth that in ¾ li. whiche is 15 s. ⅓ li. beyng 6 s. 8 d. is contained 2 ty­mes, and ¼ in value equall to the diuidende whiche is also 15 s. &c.

As in woorkes by whole numbers a sma­ler somme cannot bee diuided by a greater, but is set ouer the greater, to shewe in pro­portion a part of a tyme, geuen so in diuision of fractions, when the diuisor is greater thā the diuidend. Then the Diuidend produced will be lesse then the produced Diuisor, and therefore to stande ouer the diuisor, to shewe the proportionall part of a tyme, sought for by the woorke, the effect by example, made more plaine.

If I demaunde how many tymes ⅚ lib. is contained in ⅔ li. reason doth perswade that no tyme in the Quotient will appeare. Ne­uerthelesse procéedyng in the woorke, the di­uidend will shewe such part of a tyme as pro­portion [Page] will allowe, & in practise appearing.

Hereby doth appeare that 12/15 parte of the diuisor is the value of the diuidend, and ther­fore wanteth of a tyme, and 12/15 of 16 s. 8 d. the diuisor is 13. s. 4 d. the iuste value of the diui­dend ⅔ li. the effecte sought for by the worke.

To shew the effect in fractions improper, may satisfie the desires of suche as the same would knowe, and therefore is furthered the example followyng.

If it be demaunded howe many tymes ¾ li. is contained in 4 li. and ⅘. lib. You shall firste make 4. li. and ⅘ in fraction, and it will bee 24/5 which is the diuidend to be diuided by ¾ the diuisor, and so in the Quotient will appeare, that is sought for as by example practized.

Here the effect sought for, beyng how ma­ny [Page 58] tymes 15 s. whiche is ¾ li. is contained in 4 li. 16 s̄. whiche is 24/5 li. the same in Quo­tient appeareth to be 6. times & 6/15 of a tyme, and euery tyme containeth the value of the Diuisor, whiche is 15 s̄. maketh in all 4 li. 10 s̄. and therwith 6/15 of a tyme which is 6 s̄. maketh 4. li. 16 s. the iuste value of the Di­uidende 24/5 li. whiche is the effect the woorke requires.

Multiplication of fractions.

THe worke of Multiplication of Fra­ctions, is in nature contrary to the workyng by whole noumbers, for as the one increaseth a Noumber of Vnities: so the other increaseth a Dimunition of a Fraction. For in multipliyng 3 lib. by 2 li. you saye 2 tymes 3 maketh 6 lib. in whole Noumbers, but ¾ li. by ⅔ lib. you muste vn­derstand the saiyng ¾ li. taken ⅔ of a tyme ma­keth halfe a Pounde 10 s. the which to bring forthe by order of Multiplication of Fracti­ons, you shall multiplie the Denominators together, and the product thereof is an newe Denominator. Then must you multiply the 2 Numerators together, and the product is the numerator to the foresaid denominator, [Page] and you haue done, as by example the effect may appeare.

Here note, that as ¾ lib. taken ² of a tyme maketh 6/12 li. which is 10 s. yeuen so ⅔ li. ta­ken ¾ of a tyme maketh also 6/12 lib. whiche 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 noumber is to bee brought into forme of fraction, and so multiplied by for­mer order, the Producte will shewe that is sought for, as example may manifest.

To multiplie 5 li. ⅘ by ¾ li. reduce the first fraction, and the whole noumber into a fra­ction improper, and it will be 29/5 li. the which multiplie by ¾ li. produceth 87/20 lib. as in pra­tice.

29/5 by ¾ yeldeth 87/20 whiche is 4. li. 7 s.

The whiche product wanteth so muche of the firste value, as ¾ li. wanteth of an Vnitie, [Page 59] whiche is ¼ part, and for the vnderstandyng of the reason therof, you shal note, that if 5 li. 16 s. bee multiplied by 1 lib. it will not chaunge the value, if by 2 li. the value will double of by 3. it wilbe treble, and so forth in­finitely. But to the contrary, if you multiplie 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 ⅓ in value. If by ½ lib. halfe the va­lue diminisheth, and so infinitely, accordyng to the value of the fraction multiplicator.

To multiplie 3 li. ⅔ with 4 li. ⅚ reduce eue­ry 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 whiche multiplied to­gether, produceth 39/18 li. whiche is 17 li. 13/18 or 14 s. 5 d. ⅓.

The trueth whereof by reason to witnes, consider, that 3 li. by 4 li. produceth 12 lib. then 3. li. by ⅚ the contrary fraction yeldeth 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. whiche is 11 s. 1 d. ⅓. 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 seue­rall partes wil appeare hereafter set doune.

Hereby is to be perceiued aswell the order of the rule, as also the reason of the producte, herevnto hid, from many whiche can mul­tiplie broken numbers.

Substraction of fractions.

TO substraie one fraction from an other, there is required that both the boken nombers be of one denominatiō, and then the lesser numerator rebated from the greater, the rest will appeare to be set o­uer the common Denominator, and so the worke is ended, as by example the effect may appeare.

To substraie ⅜ li. from ⅞ li. rebate 3. from 7. rest 4. to sette ouer 8. the commom deno­minator thus 4/8 li. whereby to vnderstande if you take ⅜ whiche 7 s. 6 d. from ⅞ li. whiche [Page 60] is 17 s. 6 d. the reste will bee 4/8 li. whiche is 10 s. and so of all other when both be of one denomination, as the sondry herereafter set downe.

⅖ lib. from ⅗ lib. reste ⅕ lib. and ¼ lib. from ¾. lib. reste 2/4.

Likewise 4/10. lib. from 7/10. lib. reste 3/10. lib. so 4/12. lib. from 11/12. lib. reste 7/12. lib. &c.

Notwithstandyng, when occasion procu­reth Substraction, the Fractions beyng of contrary Denomination, then you muste re­duce theim into one Denomination, and so made apt for the worke as the former, wher­of 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 beyng brought to one denomination Substraie 8/12. from 9/12. and the reste is 1/12. li. and so you haue doen, wherein vnderstande, if you take ⅔. or 8/12. lib. from ¾. or 9/12. lib. the reste is 1/12. li. the whiche is 20. d. as 13. s̄. 4. d. from 15. s̄. the reste is 20. d. as aforesaied.

The like effecte taketh place in fractions improper of bothe kindes, as firste by exam­ple [Page] of Fractions of Fractions shall appere.

To Substraie ½. of ⅔. lib. from ¾. of ⅘. lib. firste reduce the twoo firste into one proper Fraction, whiche is 2/6. lib. and the twoo laste also into one, maketh 12/20. pounde, whiche beeyng of contrary Denomination, muste bee brought into one, and you shall for the firste haue 40/120. lib. and for the last 72/120. lib. of the whiche lesser Numerator 40. substraied from the greater 72. the remaine is 32/120. whiche is 5. s̄. 4. d. and the same to vnder­stande so to bee note, that 2/6. pounde, beyng 6. s̄. 8. d. taken out of 12/20. lib. whiche 12. s̄. the reste is 5. s̄. 4. d. as aforesaied, whiche is 30/120. li. &c.

Likewise of Fractions improper, greater then an vnitie, here followeth an example.

To Substraie 2. lib. ⅔. from 4. lib. ¾. firste reduce 2. lib. ⅔. in one, maketh 8/32. pounde, and 4. lib. ¾. also in one is 19/4. pounde, and brought to one Denomination, will bee 32/112. li. and 57/12. pounde, whereof the lesser Numera­tour 32. taken from the greater 57. the reste is 25/12. lib. whiche is 2. pounde 1. s̄. 8. d. and easely perceiued in former maner. For 2. lib. 13. s̄. 4. d. taken from 4. lib. 15. s̄. [Page 61] the reste is twoo pounde, one shillyng eight pence, as by the worke doeth appeare.

¶ Addition of Fractions,

FOr addition of Fractions there is to bee considered, as was in Sub­straction: that the broken nombers bee of one Denomination before thei bée ad­ded, & then put the Numerators into one, to sette ouer the common Denominator, and so the woorke is ended. But if thei bee of con­trary Denomination, thei must bee brought into one, and so made apte for woorke: as by example more at large, you maie perceiue.

To adde ¾. 2/4. and ¼. li. into one Fraction, adde together all the Numeratours, as 3. 2. and 1. make 6. to sette ouer the common De­nominatour 4, thus. 6/4. whiche is 1. pounde 10. shillynges, and so the worke is ended.

Howbeit, if the Fractions bee of sonderie Denominations, as ¾. ⅘. and ⅚. pounde, then thei muste bee reduced, and will bee 90/120 96/120. and 100/1200. and the Numeratours added to­gether, as aforesaied make 280/120, whiche is 2. lib. 7. s̄. 8. d. And the same to vnderstande [Page] 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. pounde 7. shillynges 8. pence, as the worke hath brought forthe.

If you adde together Fractions of Frac­tions, as ½. of ⅔. lib. to ¾. of ⅘. lib. then re­duce the twoo firste into one maketh 2/6. lib. and the twoo laste is 12/20. pounde, and in one Denomination is 40/120. li. and 72/120. lib. whiche make 112/120. and is 18. s. 8. d. and by memorie to witnesse the truthe, consider that ½. of ⅔. li. is 6. s. 8. d. and ¾. of ⅘. pounde is 12. shil­lynges, the whiche together maketh 18. s. 8. d as the worke findeth.

Likewise, if you adde sonderie whole Noumbers, ioyned with Fractions into one, you muste either reduce all into Fracti­ons improper, and so to one Denomination, addyng the Numeratours together, to sette ouer the common Denominatour by former order, or els you maie adde the whole nom­bers firste together, and reduce the Fracti­ons onely, and so ende the worke, as in exam­ple followyng appeareth.

To adde 2. lib. ¾. to 5. lib. ⅗. first reduce 2. lib. ¾. yeldeth 11/4. and againe 5. lib. ⅗. ma­keth [Page 62] 28/5. li. and in one denomination is 55/20. and 212/20. and added maketh 167/20. li. whiche is eight pounde 7. s.

Otherwise, adde the whole noumbers 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. whiche is 27. s. and put to the foresaied 7. lib. make together 8. lib. 7. s. as before.

Such as in readyng of Substraction and Addition do not well vnderstande the effect. Let them labour well to vnderstand Nume­ration and Reduction of fractions, for ther­in is taught all thinges néedefull to make the rest easie. &c.

THe Rule of thrée is framed of the former partes of Arithmetique, especiallie of Multiltiplication, and Diuision. And is called the Rule of three, for that by three Noumbers knowen, and set doune in order as the worke requireth, is founde a fourth number, fought for and desired, and the commoditie growing by vse of the saied Rule procured Learned writers doe name it the Golden Rule, excel­lyng all other, as Golde doth other mettals. It is also called the Rule of Proportion, for that euer the fowerth and vnknowen Num­ber found by the worke, shall beare suche pro­portion vnto the thirde of the knowen num­bers, as the second beareth to the firste. The effect better appearyng in fewe Examples, then in many wordes.

If 2. clothes cost 16 li. what 15. clothes.

Here you see 3 Noumbers knowen, as 2 [Page 63] Clothes bought or prised at 16 lib. and 15 Cloathes to bee bought or prised after the same rate the price of whiche 15 Clothes is the fowerth Number sought for and desired, founde by the woorke in order as followeth. First you shall multiply the second and third Numbers the one by the other, and the pro­duct thereof diuide by the first Number, and so shall you haue in Quotient the fowerth Number sought for & desired, as by example.

By this example appeareth that 15. the third number, multiplied by 16. the seconde nomber, produceth 240. the whiche diuided by the firste noumber 2. yeldeth in quotiente 120. lib. for the price of 15. clothes, and in suche proportion as 16. beareth to 2. that is to saie, 8. lib. for euery clothe.

The proofe of this rule is made chāgyng [Page] the places of 3. of the 4. numbers, and so one of the 2. firste will bee founde in quotient, if the worke bée true. As by example.

Here is to bee noted, that aswell in prac­tise of the rule, as also in the proofe, the firste and thirde numbers, muste bee of one Deno­mination and nature, & then of consequence, the fowerth number will bee of denomina­tion and nature, as is the seconde, the effecte [Page 64] whereof as in the former examples plainly appeareth, so in other followyng more at large may be seene.

If 1 pounde waight of Pepper coste 2 s. 8 d. what 9 ounzes of Peper.

Here the first and thirde Numbers are of one nature, but not of one Denomination. Wherefore before you woorke you must re­duce the first number into onzes, and so made apte for the worke. Likewise for that the se­conde Number is in 2. Denominations, as shillinges and pence, therefore you must re­duce the shillinges into pence, and then your 3 Numbers beyng apt for the woorke, will stand thus.

If 16. ounzes cost 32. d. what 9. ounzes, [...]

If 1. yarde coste 3 lib. 7. s. 6 d. what 75. yardes.

The midle number in this example is not apt for the woorke, till the whole be brought [Page] into pence, whiche is the smallest Denomi­nation of 3. in the same seconde Noumber, wherefore it must be reduced, and will make 810 pence, and will stand thus apt for the worke.

If 1 yard cost 810 d. what 75 yardes. [...]

Here the diuision is made more for plain­nes in obseruyng the Rule, then for any ne­cessitie. For one the first number cannot any thing diminishe in Diuision, nor any thyng augment in Multiplication, as by the Diui­sion before may appeare, and in the multipli­cation required in the proofe followyng is manifested.

If 75. yardes cost. 60750. d. what 1. yard [...]

Thus you maie perceiue, that the Quo­tient in the firste woorke, is equall with the diuidende in the same, and nothyng dimini­she by the diuision. And likewise the Pro­ducte of the seconde woorke, is equall with the multiplicande, and nothyng augmented by the multiplication: wherefore it is good to note, that suche diuisions and multiplica­tions maie bee cutte of, when 1. is one of the 3. knowen numbers in this rule, as by some examples the effecte maie appeare.

If the C. weight of Currance coste 33. s̄. 4. d. what 1. lib.

The sayd C. reduced into pound waights to agree in Denomination with the thirde number, and the seconde number reduced in­to pence, the smaller denomination of twoo in the same, then all the three are made apte for the whole worke, and will stande thus.

If 112 li. coste 400 d. what 1 lib. [...]

Here the multiplication is omitted, for [Page] that 1. the Multiplicatour can nothynge augmente in Multiplication, as aforesaide, and therefore the seconde Noumber is diui­dende where it standeth, and beeyng diuided by the firste, the quotiente is 3. d. 4/7. whiche fraction is halfe a penie, and somthyng more wherefore alwaies when 1. is the third num­ber, diuide the seconde by the firste, and the quotient will bee that you seeke for.

Here the diuision is omitted, because that 1. the firste Number, can nothyng diminishe in diuision. Wherefore in all workes where one is the firste number, the Producte made by multiplication of the seconde by the third is that you seeke for in the quotient, whiche in this worke is 960 d. as appeareth.

If 4. s. 8. d. buye one ounze of siluer, how many ounzes buyeth 100. li.

After you haue made the firste and thirde noumbers to agree, in denomination by re­duction, bryngyng bothe into pence, as the [Page 66] rule teacheth, then the question will stande thus.

[...]

Here the multiplication is omitted in for­mer respecte, and the thirde number the true diuidende, and diuided by 56. yeldeth in quo­tient 428 ounces 4/7. and so all other woor­kes, where 1. is the seconde number.

The seconde parte of the Rule of three, is of effecte contrary to the former, and is na­med the Backer Rule of three, vppon cause reasonable: for as in the former Rule, the fo­werth number is euer so muche greater thē the thirde, as the seconde is aboue the firste, so in the Backer Rule, the fowerth number is euer so muche lesser then the seconde, as the third is greater then the firste: As to the contrary, so muche greater then the seconde, as the third is lesser then the firste.

And the order of this Backer Rule is suche, that when the three knowen numbers [Page] are sette doune, then you shall multiplie the firste and seconde, the one by the other, and the Producte thereof diuided by the thirde number, and so finde in the quotiente, that is desired, & sought for by the worke: as by exā­ples the effecte more amplie maie appeare.

When the bushell of wheat is worth 3 s. 4 d. the Wheaten Lofe waiyng 20. ounzes for 1 d. what shall the penny Wheaten Lofe way when the bushell of wheat is worth 5. s.

Herein touchyng the woorke, you shall giue no respect to the Bushell of wheate, but to the price thereof, to be made the first num­ber neither to the wheaten Lofe, but to the weight thereof, for the seconde number, and accordyngly of the bushell of wheate and the price thereof, for the thirde number, and then the thrée numbers agréeyng in denomi­nations apt for worke, as was taught in the former part, then the example is thus to bee set doune and wrought.

d.

If 40. admit 20. onz what 60 d. [...]

Here you may see, that as the Bushell of wheate is augmented in price, a thirde parte in 5 s. so the Lofe of a peny is diminished in waight a thirde parte of 20 ounzes. Where­in appeareath the nature of the rule, and the effect of that was taught before touchyng the same

Likewise, if 32 d. admit 24. onz what 20 d.

[...]

So that as the peny Lofe waieth 24. oun­zes, when wheat is at 2 s. 8 d. the Bushell, it shall way 38 ounzes ⅕, when wheate is at 20 d the Bushell.

This backer rule may be applied to son­dry effectes of greater consequence then eue­ry man vnderstandeth. Wherefore I will set doune a fewe examples whiche to some men may seeme not superfluous.

The lode of Hay at 13 s. 4. the bottell of ob. waiyng 6 lib. what shall the bottle waye when the like Lode of Hay is worth 20 s.

If 13 s. 4 d. admit 6 li. what 20 s.

[...]

If the lode 15 s. admit 5. li. bottell what 10 s. the lode.

[...]

The ounze of fine Golde worth 55 s. The Crowne of 5 s. waiyng 2 d. waight Troye, what shall the saide Crowne waye when fine Golde is at 3 li. by ounze.

If 55 s. admit 2. d. waight, what 60 s.

[...]

If 60 s. admit 1 d. waight 20. Graines, what 45 s. Reduce and it will stande thus.

If 60 s. admit 44. graines, what 45 s.

[...]

The ounce of Starlyng at 2. s. 8. d. the Englishe grote waiyng 2. d. ½. waight, what ought the Grote to waie, when the ounce of starlyng is at 5. s.

Reduce and it will stande thus.

If 32. d. admitted 60. grain, what 60 d.

[...]

The ounce of Starlyng at 5. s. the En­glishe Grote waiyng 32. graines Troye. What shall the said grote waie, when Star­lyng is at 3. s. 4. d. the ounce.

Reduce and it will stande thus.

If 60. d. admitt 32. graines, what 40. d.

[...]

THE double Rule is so called, for that the aunsweres of suche questions, as the same requireth, are founde at the double workyng of the Rule of three directe whereof the order followeth.

If the 100. lib. waight cost carriage 20. miles 18. d. what will 1500. lib. waight coste 60. miles.

In this question and all other like, you maie note, that the firste and thirde number, must bee of one denomination and kinde: as herein bothe miles, or bothe waight to bee taken at pleasure for the firste woorke. And then of consequence the other shall serue in the saied firste and thirde noumber in the se­conde worke: as by examples the effect maie appeare.

The Rule of 3 Compound.

TO the Rule of 3 Compound, belon­geth 5 knowen numbers, for the firste parte of the same, whereof the seconde and first must euer be of one Denomination, and for practise thereof you shall multiplie the first and second Numbers, the one by the other, and the product thereof shalbe your di­uisor. Then multiply the other thrée (that is) the third by the fourth, and the product ther­of by the fifte, and that laste product shall be the diuidende, and diuided by the forenamed Diuisor yeldeth in quotient that whiche is[Page] sought for and desired.

¶ The Rule of Companie with­out tyme limitted.

TWoo menne in Companie, the firste put into stocke 45. li. and the other put in 68. li. who gained 32. li. Que­stion, what portion of the gaine groweth to either partie.

To aunswere this Question, and all o­ther suche like, how many soeuer are ioyned in Companie, their whole stocke shall euer [Page] bee the firste number in the Rule of three di­recte, and that whiche hath been gained by their saied stocke, shall euer bee the seconde number in the same Rule, and euery mannes proper and particular stocke shall bee the third number, and so woorkyng euery Quo­tiente will shewe the portion of hym, vnto whom the particular stocke doeth belong, as by example the effecte more plainly ap­pearyng.

[...]

[...]

Thus appeareth the gaine for the firste man is 12 li. 84/113, & for the seconde 19 li. 29/11 [...]. The whiche two sommes together makyng the iuste gaines, which is 32. li. proueth the worke true or els not.

Note, that menne in Companie, hauyng losse by Traffique vpon the Seas, or other­wise, their seuerall portions to bee borne, is founde by this Rule also. and so many parti­culare menne as are in Companie, so many seuerall Quotientes shall bee made, and all together make the gaine or losse, euery man­nes [Page] portion, accordyng to his stocke, where­of to giue many examples were superfluous, but onely to shewe how to applie the Rule, whereof a fewe examples followe hereafter.

Three menne laded a ship, the aduenture of the first was 546. li. of the second 628. li. and of the thirde 732. li. By tēpest vpon the sea, the master was forsed to caste ouerborde, to the value of 640. li. Question, what por­tion of the losse euery manne ought to beare?

Aunswere. The whole aduentures added together, make 1906. li. for firste number in proportion, and the losse 640. li. must bee the seconde number, and euery particular portion of the stocke the thirde number. The whiche multiplied and diuided, accordyng to the Rule, yeldeth three seuerall quotientes, shewyng the losse of euery manne, whiche for the firste is — 183. li 642/1906. for the seconde 210. li. 1660/1906, and for the thirde 245 li. 510/19 [...]. And all added together make the iuste losse, whiche is 640. li.

Three men in Companie gained 100. li. whereof for 32. lib. whiche the firste manne putte in, he had of the gaine 25. li. and of the reste the thirde manne had ¼, more then the seconde. Question, what the seconde and thirde putte into stocke? Aunswere. Firste, consider the 25. li. taken out of the gaine, there reste 75. li. for the seconde and third whereof the thirde muste haue 5. li. for 4. to the seconde. Wherefore adde 5. li. and 4. li. together, and that maketh 9. li. for firste number in the Rule of three, and saie, if 9. li. require 75. li. what 4 li. and for the seconde man you shall finde in gaine 33. li. 6. s. 8. d. and for the thirde 41. li. 13. s. 4. d. as the practice sheweth.

Then hauyng founde euery mannes par­ticular parte of the gaine, you shall saie. If 25. li. of gaine come of 32. li. in stocke for the firste manne, whereof commeth 33. li. 6. s. 8. d. for the seconde, and 41. li. 13. s. 4. for the thirde woorke, and you shall finde the [Page] seconde man put in 42. li. 13. s. 4. d. and the third 53. li. 6. s. 8. d.

¶ The rule of Companie with tyme.

THree Marchauntes in Companie, the firste putte in 50. li. for fo­wer Monthes, the second 65. li. for seuen Monethes, and the third 72. li. for nyne Monethes, who gained 85 Question. What euery mannes portion of the gaine?

Here is to bee noted, that euery mannes money muste bee multiplied, by his tyme of continuaunce in the Companie, and the thre productes added together, shall bee first num­ber in the rule of three, the gaine the seconde and euery particulare producte the thirde noumber, and so proceadyng the woorke, you shall finde three seuerall quotientes she­wyng euery mannes parte of the gaine to [Page 68] hym due, accordyng to his stocke, and tyme of continuaunce, as by example will appere.

Thus hauing found the thrée seuerall pro­ductes to be 1303. then you shall go to the Rule and say.

Here is to bee noted, that so many men as are in company, so many seuerall productes must be made, and so many seueral quotients muste manifest the gaine to euery one belon­gyng. &c.

❧ The Rule of Aligation.

THE Rule of Aligation requi­reth a certaine circumstaunce, for gatheryng of differences, of thynges of sondry Prices, wherof part may be better and parte worse then a common price, whereas a quantitie of euery sorte occasion may re­quire to bee taken, and the saide differences added together, shalbe firste noumber in the Rule of three. The whole quantitie of the matter desired the second, and euery perticu­lar difference the thirde noumber, and so ma­ny perticulars as are in the woorke, so many seuerall quotients will make the Quantitie of the matter sought for. The effect more plainly appearyng in fewe examples, then in many wordes or great discourse thereof, as hereafter you may see.

An Appoticary for recouery of health in a noble man is charged to compose an ingre­dience of 4 sondrie sortes of riche 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 euery sort a quantity. Question. [Page 77] How muche of euery sorte is to be taken?

Aunswere. Firste set doune the seuerall prises one vnder an other, the highest vpper­most, with the common price at the lefte side thus.

Then you muste linke together one aboue the common price with one of the other vnder the common price, and the difference of euery one aboue the common price shalbe set against the other, linked with it vnder the common price, and to the contrary, the difference of euery one vnder the common price shalbe set against the other, linked with it aboue the common price, the whiche differences found and set accordingly, as aboue appearing, the totall makyng 19. Then by the Rule of thrée is to be sought the 4 Quotients, to make 8 ounzes of 40 s. the ounze of euery price a quantitie, whiche is as by practice hereafter appeareth.

Here you may perceiue, 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 truely brought forthe, as the question requireth, the same notwithstan­dyng, the same quantities maye shewe like trueth, if the differences chaunge their pla­ces, as by linkyng the vppermost price with the lowest sauyng one, and the lowest with [Page 70] the vppermost sauyng one thus.

A Marchaunt hath bought Canuas of 22 d. 19 d. 15 d. 10 d. 9 d. and 8 d. the Ell. A frende requireth to haue a thousande elles (of euery sort a parcell) to stand hym in 12 d. the ell one with an other, the Marchaunt to gaine nothing by him, but to haue giuen him a Satten Doublet for his frendship? Que­stion. How muche of euery sort to bee taken. Neither partie to haue wrong. Aunswers. Firste finde the differences by former order thus.

The whiche being found procéede in the Rule of thrée, & so you shall haue 6 Quotientes, whiche shewe the quantitie of e­uery sort of Canuas, to be taken as practise will shewe.

A Marchaunt hath 4 sortes of Golde, of seuerall finenes (to saye) of 23 Carratz ¾ of 22 Carratz ⅔ of 21 Carratz ⅚, and of 20. Carratz. ½. fine Question, what quantitie of euery sort is to be taken, to haue 100. oun­zes of 22 Carratz fine iust. Aunswere.

First note, that forasmuche, as the Fracti­ons, fine aboue, and vnder the common fine­nes, are of sondry Denominations: therfore they must be reduced, and made of one Deno­mination, and will stand thus.

Here you may per­ceiue, that the finer sorts aboue the com­mon finenes 21/12 and 8/12 are sett againste the parcells linked with theym, whiche are vnder the common [Page 79] goodnes, and for the common finenes therof 2. is set against the percell, linked therewith Also 20 Carratz 6/18. is set against the parcell linked therewith, and so is found 49/12 for firste nomber in the Rule of thrée, and euery par­ticular difference, the thirde noumber with the 100. onze desired the second, with which noumbers procéedyng in the Rule of three you shall finde 4 quotient which will declare the quantity of euery sort of Gold, to be ta­ken to haue 100. ounces of 22 carrratz fine iust, the effect in example appearyng.

An Assaie master hath fiue sortes of siluer of sondrie finesse: that is to saie, of 11. onzes 14. d. 11. vnzes 10. d. 10. vnzes. 5. d. 9. vnzes 16. d. and 9. vnzes. 12. d. waight fine, and would haue 100. lib. weight of 11. vn­zes 2. d. fine. Question. What quantitie to be taken of euery sorte? Aunswere.

Firste reduce your seuerall denominatiōs [Page] into one, and then it will stande thus.

The which dif­ferences found procede with y^e totalle for the firste noumber. the 100. lib. the second: and eue­ry particular the third, and so shall you haue the Quotiente of euery sorte to bee taken, to make 100. lib. weight of 11. vnzes 2. d. fine, the effecte in example appearyng.

Howbeeit that these woorkes are to bee proued by the common order, of prouyng the Rule of three: The same notwithstandyng, there are other sonderie orders of prooues, for the commixions of Goldes and Siluers [Page 72] whiche here I omitte, in respecte of seuerall causes, referryng suche as by vocation, maie desire knowledge therein, to priuate confe­rence who maie bée satisfied to effecte extra­ordinarie.

The Quintall containing 100 lib. subtill.

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.

The 100 lib. at 29. li. 10. s. what 90. lib. Take the whole lacke 1/10.

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.

The 100 lib. at 8 li. 2 s. 6 d. what 14. lib. Take 1/10 and ⅖ thereof

The 100 lib. at 5. 13. 4. what 9. lib. Take 1/10 lacke 1/10 thereof.

The 100 lib. at 3 li. 6 s. 8 d. what 6. lib.

Take ½ of 1/10 and ⅕ thereof

As euery of former examples are profita­ble, and of many the vnderstandyng maie bee desired to finde the sonderie partes of the 100 lib the same to value after the rate of y^e C. so of all the other it maie seme moste ne­cessary to vnderstāde, how by the price of the C. to finde the value of the pounde waight, & the onze with moste facilitie: wherefore for a generall rule, to finde the value of 1. pounde waight by the price of the C. take euer 1/10 of 1/10 of the price that the C. lib. is valued at, and that is the true value of the pounde waight, as before appeareth, and by an other exam­ple folowyng, the effecte is manifeste.

Likewise to finde the value of the vnze be­yng 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. waight Troye, then take 1/12 of 1/10 of 1/10 and ac­cordyngly haue your desire.

Of the C. waight containyng 112 li.

The C. waight at 36. li. what 96 lib. Take the whole price of the C. lackyng ⅛ and 1/7 therof.

For proofe, the C. at 36. l. what ¾ & 12 lib. Take ½ and ½ thereof, and ½ thereof, lacke 7/2

For profe the C. at 32. li. 10. s. what 84. lib.

Take the whole lacke ¼ thereof.

The C. at 28. li. 5. s. what—72. li.

Take ½ and ¼ thereof, and 1/7 thereof.

The C. at 25. li. 13. s. 4. d. what 6 6. li.

Take ½ and ⅛ thereof, and 3/7 thereof.

The C. at 22 li. 12 s. what—60.

Take ½ and 1/7 of ½ thereof.

The C. at 6 li. 13 s. 4 d. what 20. li,

The C. at 3. li. 6 s. 8 d. what 10. lib. Take ¼ of ¼ and 3/7 thereof.

The C. at 30 s. 6 d. what 4 lib. Take 1/7 of ¼

The C. at 26 s. 8 d. what 2. lib. take ½ of 1/7 of ¼

The C. at 20 s. what 1. lib. Take ¼ of 1/7 of ¼ or 1/7 of 1/16

To find the value of the ounze by the price of the C. waight, you must first finde the va­lue of the pound waight, 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 profe, the C. at 24 lib. what 1. ounze.

Take 1/16 of ¼ of 1/7 of ¼

In like maner may bée taken any parte of the pounde waight, accordyng to the pro­portion it beareth to the whole.

Of the C. containyng 120. for the C.

The C. of Canuas at 7. li. what 90. elles.

Take ½ and ½ thereof.

The C. 16 s̄. 8 d. what—68. Take ½ and ⅕ of ⅓.

The C. at 13 s̄ 8 d. what 51. l. Take ⅓ and ¼ thereof ond 1/10 thereof

The C. at 12 s̄. 6 d. what—45.—Take—½ and ⅛ thereof.

The C. at 11 s̄. what—36.— Take ¼ and ⅕ thereof

The C. at 8 s. what—25. Take ⅙ and ¼ therof, or ¼ lacke ⅙ thereof

The C. at 6 s. 8 d. what—16.— Take 3/6 lacke ⅕ therof or 1/10 & ⅓ thereof

The C. at 5 s̄. what—10. Take 1/12, or ½ of ⅙, or ¼ of ⅓

The C. at 4.s. what—6.— Take ½ of 1/10, or ⅕ of ¼.

The C. at 3 s. 4 d. what—2.— Take ⅙ of 1/10, or ⅓ of ⅕ of ¼ or ⅕ of ½ of, ⅙ or ⅕ of ⅓ of ¼.

The C. at 3.li. 6.s. 8. what—1.— Take 1/22 of 1/10, or ⅙ of ⅕ of ¼.

For proofe the C. at 3. li. 6. 8. what—1— Take ⅙ of ⅕ of ¼.

Here note for a generall rule, that suche proportion as the giuen nomber beareth to the C. the same proportion beareth the price of the giuen nomber, to the price of the C. and therein consisteth the difficultie, that to any learner maie appeare.

Thus is brought to ende gentle reader, the effecte by my trauaile pretended herein. The whiche beyng so well accepted of thee, [Page] as it hath been willyngly furthered, to pro­cure contentation to all suche, as maie take profite or delectation by the same: so I maie bee incouraged to augmente my good will, in furtheryng of other workes of grea­ter consequence, therin assisted by the fauour of the almigh­tie, into whose han­des I committe thee fare­well.