❧ Robert Norton to the courteous Readers.
ALthough I haue too often been an vnwilling witnesse of the ouer-rash disposition of diuers vnaduised censurers, that would haue themselues esteemed skilfull, and yet either will not, or rather cannot doe any thing of worth themselues, not sparing to cauill, detract, and iniuriously to burthen other mens well pretended indeuours, with vnworthy and vndeserued scoffes and scandalls: but in stead of reading to vnderstand, and then to examine their true validitie, that so with iudgement they might censure them, haue crittically plaid the right Momes: And though I hope not as Iacke alone, to escape that [Page]which few or none haue done before me: yet the respect I haue to the publike good, that you my Countrymen, such as either want leisure or language, may become partakers of these excellent inuentions of that famous forraigne Authour, more preuailing with mee, then the carelesse regard I haue of such iniuries could hinder, I haue, as you see, aduentured to prouide for this worthy stranger, this English welcome, and haue preferred some few of mine owne friends (though vnworthy) to accompany him: And so commending him to your courteous entertainements, doe bid you hartily farewell.
DEFINITIONS appertaining to Arithmeticall whole Numbers.
The first Definition.
ARithmeticke is the Science of Numbers.
The second Definition.
NVmber is that which expresseth the quantitie of each thing.
The third Definition.
THe Characters by which Numbers are denoted, are ten; namely, 0 signifying the beginning of Number? and 1, and 2, and 3, and 4, and 5, and 6, and 7, and 8, and 9.
The fourth Definition.
EVery thrée Characters of a Number is called a Member, whereof the first are the thrée first towards the right hand, the second, the thrée Characters next following towards the left hand: And so by order, for the third Member and others following, as many as there shall be found thrées in the Number propounded.
The Explication.
AS in the number 357876297, the 297 is called the first Member: and 876 the second: and 357 the third.
The fift Definition.
THe first Character of the first Member, beginning from the right hand to the left, doth simply signifie his owne value: the second, so many times ten, as that containeth vnities: the third, so many times a hundred, as that containeth vnities: and the first Character of the second Member so many times a thousand, as that containeth vnities: and so by the tenth progression of all the rest of the Characters contained in the number proposed.
Explication.
LEt the Number propounded be 756871387130789276. Then according to this definition, the first Character 6, maketh sixe: and the 7 following [...] and the 2 following, two hundred: and the 9, nine thousand and so of the rest. To expresse thi [...] [...]umber, place ouer euerie first Character of each Member (except the first Member) a pricke or point, as you sée aboue: then say, seuen hundred fiftie sixe thousand thousand thousand thousand thousand, (namely, so many times thousand, as there are prickes or points from 7 to the end) eight hundred seuentie one thousand thousand thousand thousand, thrée hundred eightie seuen thousand thousand thousand, one hundred thirty thousand thousand, seuen hundred eightie nine thousand, two hundred seuenty sixe.
The sixt Definition.
A Whole number is either a vnitie, or a compounded multitude of vnities.
The seuenth Definition.
THe Golden Rule, or Rule of thrée, is that by which to three tearmes giuen, the fourth proportionall tearme is found.
❧ The operation of Arithmeticall whole Numbers.
Of the Addition of whole Numbers. The first Probleme.
ARithmeticall whole numbers being giuen to finde their Summe, Explication propounded, let the Numbers giuen to be added, be 379, and 7692, & 4545, Explication required, to find their summe. Construction: the Numbers giuen, shall be disposed as followeth: [...] so as their first Characters towards the right hand, stand directly one vnder another: and likewise their second Characters, and so also the rest following, drawing vnder them a line: then shall all the Characters of the first ranke towards the right hand be added, saying, 9 and 2 make 11, and 5 make 16, whereof the 6 shall be placed vnder the first ranke, and the 1 of the same 16, shall be added to the second ranke, saying, 1 and 7 make 8, and 9 make 17, and 4 make 21. of which the 1 shall be placed directly vnder the second ranke, and the 2 shall be added to the third ranke, saying, 2 and 3 make 5, and 6 make 11, and 5 make 16, whereof the 6 shall be placed vnder the third ranke, and the 1 shall be added [Page]to the fourth, saying, 1 and 7 make 8, and 4 make 12, which shall be wholly placed in their ranke thus.
I say, 12616 is the summe required. [...]
Demonstration: if from the thrée Numbers giuen, the two first be taken away, and there remaineth 4545. And if from the Summe 12616, the two first giuen be substracted also, there remaineth likewise 4545: But by the common Axiom, if from things equall, equall things bee substracted, their rests shall be equall: And things substracted equall to things substracted, all shall be equall. Therefore, 12616 is equall to thrée Numbers giuen, which is the thing required. Conclusion: Arithmeticall whole Numbers being giuen to be added, we haue found their summe as was required.
Substraction of whole Numbers. The second Probleme.
AN Arithmeticall whole Number being giuen, out of which to substract, and another Arithmeticall whole Number to bee substracted: to finde their Rest.
Explication propounded, bee the Number out of which to substract, 238754207: And the number to be substracted 71572604 giuen Explication required to finde their Rest. Construction: the Number to be substracted, shall be to placed vnder the Number out of which it is to bee substracted, [Page]as that the 4 stand directly vnder the 7, and the 0 vnder the 0, and so of the rest, drawing a line betweene the numbers giuen, and another vnder the number which is to be substracted, as hereunder appeareth. [...] Then beginning at the right hand, substract 4 from 7, and there resteth 3, which shall be set directly vnder the 4, and then say, 0 out of 0 resteth 0, placing 0 vnder the 0: then 6 from 2, which being impossible, say, 6 from 10, and 2 (which is 12) resteth 6, placing that vnder the 6: then 2 from 3, (true it is that you should haue said 2 from 4, had it not béen that you borrowed 1 from the 4 to make the other 2 to value 12) resteth 1, placing that vnder the 2: and so of all the other. The disposition of their Characters are as héere appeareth. I say that 167181603 is the Rest required. Demonstration: adding the Rest 167181603 to the number to be substracted 71572604, the summe shall bee equall to the number from which the substraction was made: wherfore séeing that 167181603 is the difference betwéene the number from which the substraction was made, and the number to be substracted; therfore that is their Rest which was to be demonstrated. Conclusion. An Arithmeticall whole Number from which to be substracted, and another to substract, being giuen, wee haue found their Rest which was required.
Multiplication of whole Numbers. The third Probleme.
AN Arithmeticall whole Number giuen to be multiplied, and another to multiply, to find their product. Explication propounded: Be the Multiplicand [Page]or Number to be multiplied 546, and the Multiplicator or number to multiply 37. Explication required: To finde their product, Note, that for the more easie solution of this proposition, it were necessary to haue in memory the multiplication of the 9 simple Characters among themselues, learning them by rote out of the Table here placed, séeking the Multiplicand in the superior line of squares, and the Multiplicator in the diagonall or slope line of squares: and in the common Angle answering them both, you shall find their product.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
| 3 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | |
| 4 | 16 | 20 | 24 | 29 | 32 | 36 | ||
| 5 | 25 | 30 | 35 | 40 | 45 | |||
| 6 | 36 | 42 | 48 | 54 | ||||
| 7 | 49 | 56 | 63 | |||||
| 8 | 64 | 72 | ||||||
| 9 | 81 |
As we would know y e product of 3 and 8, seeke 8 in y e vpper line, and 3 in the slope or diagonall: and in the common Angle you shall find 24 their product, and so of all the rest, as by the Table will plainely appeare.
Construction: place the first numbers on the right hand (of the giuen) one directly vnder another, and then draw a line, as heere-under is done. Then say, 7 times 6 make 42, place 2 vnder the 7, and retaine the 4 (because of the 4 tenths) in memorie: then say, 7 times 4 make 28, and the 4 which you had in minde, make 32, whereof place the 2 vnder the 3, and retaine 3, and say, 7 times 5 make 35, and 3 which was borne in minde, make 38, [Page]which shall be placed in order vnder the line, as you sée. [...] In the like sort shall the 546 bee multiplied by the 3 of the multiplicator, saying, 3 times 6 make 18, placing the 8 vnder the 3: and so of the rest. Then shall bee drawne a line, adding all that is betwéene y e two lines in this sort.
I say, that 20202 is the Product required.
Demonstration. The 20202 containeth the 37 so many times as there is vnities in the 546: therefore 20202 is the product which was to be found. Conclusion. An Arithmetical whole number being giuen to be multiplied, and another to multiply, we haue found their required product.
❧ Diuision of Arithmeticall whole Numbers. The fourth Probleme.
AN Arithmeticall whole Number being giuen to be diuided, and another to diuide, to finde their Quotient.
Explication propounded: Be the number to bee diuided, 995, and the number to diuide, 28 giuen. Explication required: to finde their Quotient. Construction: The number to be diuided (or diuident) and the number to diuide (or diuisor) shall bee placed in order, drawing a crooked line, as hereunder followeth, saying, how many times 2 in 9? thrée times, (true it is that there are 4 times 2 in 9, and 1 remaining) but wee will shew y e reason [Page]hereafter why we must say but three times) set downe 3 for the first Character of the Quotient, behynd the crooked line, and the 3 remayning of y e 9 cancelling the 2 & 9: then multiply 8 by the diuisor, by 3, the Quotient it maketh 24, which substract frō 39 (here appeareth the occasion why we sayd that 2 is but onely 3 times in 9: for if wee had sayd 4 times, resting of the 9, and had multiplied 8 by 4 it would haue bene 32 which should be substracted from 19 which then remayned of the diuident, which is impossible; therefore there must be such a number taken, & placed behind the crooked line, as that the product thereof may be substracted from the remaynder) resteth 15, which place ouer 39, cancelling the 39, and the 8, so shall the disposition of the Characters be in this manner. [...] Now to find the second Character of the Quotient, the diuisor must againe be set vnder the diuident, placing the 8 of the diuisor vnder the 5 of the diuident, and the 2 vnder the 8, saying how many times 2 in 15? fiue times, which 5 shalbe placed neere the 3 at the oblique line, for the second Character of the Quotient resteth 5 which shalbe placed ouer the 5 of the 15 cancelling the sayd 15 and 2: then multiplying the diuisor 8 by the Quotiēt 5 maketh 40, which substract from 55 remayneth 15, cancelling the 55 and the 8 and distinguishing the 15 with crooked lines from the other Characters: then draw a line neere the Quotient 35, placing ouer the same the sayd remaynder, and vnder the same the diuisor 28, and the disposition of the Characters wilbe as appeareth aboue, I say that 35 15/28 is the Quotient required.
Demonstration: the 35 15/28 contayneth the vnity so often as the 995 contayneth the diuisor 28: therefore 35 15/28 is the Quotient required which was to be demonstrated, Conclusion: an Arithmeticall whole nomber for diuident, [Page]and one for diuisor giuen, we haue found their Quotient required.
❧ The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. The fift Probleme.
THrée Termes of Arithmeticall Numbers, being giuen to finde their proportionall Terme.
Explication propounded: Be the thrée termes giuen 234. Explication required: To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2. Construction: Multiply the second terme 3, by the third terme 4, y t giueth the product 12: which diuiding by the first terme 2, giueth the Quotient 6: I say that 6 is the fourth proportionall terme required. Demonstration: there is from 6 to 4, Reason sesquialter, and the same Reason is there from 3 to 2: therefore 6 is the fourth proportionall terme to be demonstrated. Conclusion: thrée Arithmeticall numbers being giuen, wee haue found their fourth proportionall terme required.
The Preface of Simon Steuin.
MAny seeing the smalnes of this Book, and considering your worthynes to whō it is dedicated, may perchance esteeme this our conceyte absurd: But if the proportion be considered, the small quantity hereof compared to humane imbecility, and the great vtility vnto high and ingenious intendiments, it will be found to haue made comparison of the extreame tearmes, which permit not any conuersion of proportion. But what of that? is this an admirable inuention? No certainly: for it is so meane, as that it scant deserueth the name of an inuentiō: for as the coūtryman by chance sometime findeth a great treasure, without any vse of skill or cunning, so hath it hapned herein. Therefore if any will thinke, that I vaunt my selfe of my knowledge, because of the explicatiō of these vtilities, out of doubt, he sheweth himselfe to haue neyther iudgemēt, vnderstanding, nor knowledge to discerne [Page]simple things from ingenious inuentions, but he (rather) seemeth enuious of the common benefite: yet howsoeuer, it were not fit to omit the benefit hereof, for the inconuenience of such calumny. But as the Mariner hauing by hap found a certaine vnknowne Island, spareth not to declare to his Prince the riches and profits thereof; as the fayre fruits, precious mineralls, pleasant champions, &c. and that without imputation of Philautry: euen so shall we speake freely of the great vse of this inuention; I call it great, being greater then any of you expect to come from me. Seeing then that the matter of this Disme (the cause of the name whereof shalbe declared by the first difinition following) is number, the vse and effects of which, yourselues shall sufficiently witnes by your continuall experiences, therefore it were not necessary to vse many words thereof: for the Astrologer knoweth, that the world is become by computation Astronomicall (seing it teacheth the Pilot the eleuation of the Equator and of the Pole, by meanes of the declination of the Sunne, to describe the true Longitudes, Lātitudes, situatiōs & distances of places, &c.) a Paradise, aboūding in some places with such things as the Earth cannot bring forth in other. But as the sweet is neuer without the sowre: so the trauayle in such computations cannot be vnto him hidden, namely, in the busy multiplications and diuisions which proceed of the 60 progression of degrees, minutes, seconds, thirds, &c. And the Surueyor or Land-meater knoweth, what great benefite the world [Page]receyueth from his science, by which many dissensions and difficulties are auoyded, which otherwise would arise by reason of the vnknowne capacity of Land: besides, he is not ignorant (especially whose busines and imployment is great) of the troublesome multiplications of Roods, Feete, and oftentimes of ynches, the one by the other, which not onely molesteth, but also often (though he be very well experienced) causeth error, tending to the dāmage of both parties, as also to the discredit of Land-meater or surueyor, and so for the Money-masters, Marchants and each one in his busines: therefore how much they are more worthy, and the meanes to attayne them the more laborious, so much the greater and better is this Disme, taking away those difficulties: But howe? it teacheth (to speake in a word) the easy performance of all reckonings, computations, & accounts, without broken numbers, which can happen in mans busines, in such sort, as that the foure Principles of Arithmetick namely, Addition, Substractiō, Multiplication, & Devisiō, by whole numbers, may satisfie these effects, affording the like facility vnto those that vse Coūters. Now if by those meanes wee gaine the time which is precious, if hereby that be saued which otherwise should be lost, if so, the paines, controuersy, error, dammage, and other inconueniences commonly hapning therin, be eased, or taken away, then I leaue it willingly vnto your iudgements to be censured: and for that, that some may say that certaine inuentions at the first seeme good, which when they come to be practized, [Page]effect nothing of worth, as it often hapneth to the serchers of strong mouing, which seeme good in small proofes and modells, when in great, or comming to the effect, they are not worth a Button: whereto we answere, that herein is no such doubt: for experience dayly sheweth the same: namely, by the practize of diuers expert Land-meaters of Holland, vnto whom we haue shewed it, who (laying aside that which each of them had, according to his owne manner, inuented to lessen their paines in their computations) do vse the same to their great contentment, and by such fruit as the nature of it witnesseth, the due effect necessarily followeth: The like shall also happen to each of your selues vsing the same as they doe: meane while liue in all felicity.
The Argument.
THe Disme hath two parts, that is, Definitions & Operations: by y e first definition is declared what Disme is, by the second, third, and fourth, what Comencement, Prime, Second &c. and Disme numbers are: the Operation is declared by foure propositions, The Addition, Substraction, Multiplication and Deuision of Disme numbers. The order whereof may be successiuely represented by this Table.
| The Disme hath two parts. | Definitions, as what is | Disme, |
| Comencement, | ||
| Prime, Second &c. | ||
| Disme nomber. | ||
| Operations or Practize of the | Addition, | |
| Substraction, | ||
| Multiplication, | ||
| Deuision. |
[Page]And to y • end the premises may y e better be explaned, there shalbe hereunto an Appendix adioyned, declaring the vse of the Disme in many things by certaine examples, and also definitions and operations, to teach such as doe not already know the vse and practize of Numeration, and the foure principles of common Arithmetick, in whole numbers, namely, Addition, Substraction, Multiplication, & Diuision, together with the Golden Rule, sufficient to instruct the most ignorant in y e vsuall practize of this Art of Disme or Decimall Arithmeticke.
The first Part. Of the Definitions of the Dismes.
The first Definition.
DIsme is a kind of Arithmeticke, inuented by the tenth progression, consisting in Characters of Cyphers; whereby a certaine number is described, and by which also all accounts which happen in humane affayres, are dispatched by whole numbers, without fractions or broken numbers.
Explication.
LEt the certaine number be one thousand, one hundred and eleuen, described by the Characters of Cyphers thus 1111, in which it apeareth that ech 1 is the 10th part of his precedent character 1: likewise in 2378, each vnity of 8 is the tenth of each vnity of 7, and so of all the others: But because it is conueniēt that the things whereof we would speake, haue names, and that this maner of [Page]computation is found by the consideration of such tenth or disme progression; that is, that it consisteth therein entirely, as shall hereafter appeare: Wee call this Treatise fitly by the name of Disme, whereby all accounts hapning in the affayres of man, may be wrought and effected without fractions or broken numbers, as hereafter appeareth.
The second Definition.
EVery number propounded, is called Comencement, wose signe is thus ( 0).
Explication.
BY example, a certaine number is propounded of three hundred sixty foure: we call the 364 Comencements, described thus 364 ( 0) and so of all other like.
The third Definition.
ANd each tenth part of the vnity of the Comencement, wee call the Prime, whose signe is thus ( 1), and each tenth part of y e vnity of the Prime, we call the Second, whose signe is ( 2), and so of y e other: each tenth part of the vnity of the precedent signe, alwayes in order, one further.
Explication.
AS 3 ( 1) 7 ( 2) 5 ( 3) 9 ( 4) that is to say, 3 Primes, 7 Seconds, 5 Thirds, 9 Fourths, and so proceeding infinitly: but to speake of their valew, you may note, that according to this definition, the sayd numbers are 3/10 7/100 5/1000 9/10000, together 3759/10000 and likewise 8 ( 0) 9 ( 1) 3 ( 2) 7 ( 3) are worth 8 9/10 3/100 7/1000 together 8 937/1000 and so of other like. Also you may vnderstand, that in this Disme we vse no [Page]fractions, and that the multitude of signes, except ( 0) neuer exceede 9: as for example, not 7 ( 1) 12 ( 2) but in their place 8 ( 1) 2 ( 2), for they valew as much.
The fourth Definition.
THe numbers of the second and third Definitions before going, are generally called Disme numbers.
The second part of the Disme. Of the Operation or Practize.
The first proposition of Addition.
DIsme numbers being giuen how to adde them to find their summe.
The explication propounded; there are 3 orders of Disme numbers giuen, of which the first 27 ( 0), 8 ( 1), 4 ( 2), 7 ( 3), the second 37 ( 0), 8 ( 1), 7 ( 2), 5 ( 3), the third 875 ( 0), 7 ( 1) 8 ( 2), 2 ( 3). The explication required, we must find their totall summe.
Construction.
The numbers giuen, must be placed in order as here adioyning, [...] adding them in the vulgar maner of adding of whole numbers in this maner: The summe (by y e first Probleme of Arithmetick following) is 941504, which are (that which the signes aboue the numbers do shew) 941 ( 0) 5 ( 1) 0 ( 2) 4 ( 3). I say, they are the summe required. Demonstration: the [Page]27 ( 0) 8 ( 1) 4 ( 2) 7 ( 3) giuen, make by the 3 Definition before 27 8/10 4/100 7/1000, together 27 847/1000, and by the same reason, the 37 ( 0) 8 ( 1) 7 ( 2) 5 ( 3) shall make 37 875/1000, and the 875 ( 0) 7 ( 1) 8 ( 2) 4 ( 3) will make 875 782/1000, which thrée numbers make by common addition of vulgar Arithmeticke 941 304/1000. But so much is the summe 941 ( 0) 5 ( 1) 0 ( 2) 4 ( 3): therefore it is the true summe to be demonstrated. Conclusion: Then Disme numbers being giuen to bee added, wee haue found their summe, which is the thing required.
Note, that if in the number giuen, there want some signes of their naturall order, the place of the defectant shal be filled. As for example, let the numbers giuen bee 8 ( 0) 5 ( 1) 6 ( 2) and 5 ( 0) 7 ( 2): [...] in which, the latter wanted the signe of ( 1), in the place thereof shall 0 ( 1) bee put, take then for that latter number giuen 5 ( 0) 0 ( 1) 7 ( 2) adding them in this sort.
This aduertisement shall also serue in the thrée following propositions, wherein the order of the defayling figures must be supplied, as was done in the former example.
The second Proposition. Of Substraction.
A Disme number being giuen to substract: another lesse Disme number giuen out of the same to finde their rest.
[Page] Explication propounded: be the numbers giuen 237 ( 0) 5 ( 1) 7 ( 2) 8 ( 3) & 59 ( 0) 7 ( 1) 3 ( 2) 9 ( 3) The Explicatiō required; to find their rest.
Construction: the numbers giuen shalbe placed in this sort, substracting according to vulgar maner of substractiō of whole nūbers, thus [...]
The rest is 177839 which valueth as the signes ouer them do denote 177 ( 0) 8 ( 1) 3 ( 2) 9 ( 3), I affirme y e same to be the rest required.
Demonstration: the 237 ( 0) 5 ( 1) 7 ( 2) 8 ( 3) make by the third Definition of this Disme, 237 5/10 7/100 8/1000 together 237, 578/1000 and by the same reason, the 59 ( 0) 7 ( 1) 4 ( 2) 9 ( 3) value 59 749/1000 which substracted from 237 578/1000 there resteth 177 839/1000 but so much doth 177 ( 0) 8 ( 1) 3 ( 2) 9 ( 3) value: that is then the true rest which should be made manifest. Conclusion: a Disme being giuen, to substract it out of another Disme number, and to know the rest, which we haue found.
The third Proposition: of Multiplication.
A Disme number being giuen to be multiplied, and a multiplicator giuen to find their product:
The Explication propounded: be the number to be multiplied 32 ( 0) 5 ( 1) 7 ( 2), and the multiplicator 89 ( 0) 4 ( 1) 6 ( 2)
The Explication required: to find the product. Construction: the giuen numbers are to be placed as here is shewed, [...] multiplying according to the vulgar maner of multiplication by whole nūbers, in this maner, [Page]giuing y e product, 29137122: [...] Now to know how much they value, ioyne the two last signes together as the one ( 2) and the other ( 2) also, which together make ( 4), and say y t the last signe of the product shall be ( 4) which being knowne, all the rest are also knowne by their continued order. So that the product required, is 2913 ( 0) 7 ( 1) 1 ( 2) 2 ( 3) 2 ( 4).
Demonstration: The number giuen to be multiplyed, 32 ( 0) 5 ( 1) 7 ( 2) (as appeareth by the third Definition of this Disme) 32 5/10 7/100 together 32 57/100: and by the same reason the multiplicator 89 ( 0) 4 ( 1) 6 ( 2) value 89 46/100 by the same, the said 32 57/100 multiplied, giueth the product 2913, 7122/10000 But it valueth 2913 ( 0) 7 ( 1) 1 ( 2) 2 ( 3) 2 ( 4). It is then the true product which we were to demonstrate. But to shew why ( 2) multiplied by ( 2) giueth the product ( 4) which is the summe of their numbers, also why ( 4) by ( 5) produceth ( 9), and why ( 0) by ( 3) produceth ( 3) &c. Let vs take 2/10 and 3/100 which (by the third Definition of this Disme) are 2 ( 1) 3 ( 2) their product is 6/10000 which value by the said third Definition 6 ( 3), multiplying then ( 1) by ( 2) the product is ( 3) namely a signe compounded of the summe of the numbers of the signes giuen.
Conclusion.
A Disme number to multiply, and to be multiplyed, being giuen, we haue found the product, as we ought.
Note.
IF the latter signe of the number to bee multiplied, bee vnequall to the latter signe of the multiplicator, as for example, the one 3 ( 4) 7 ( 5) 8 ( 6), the [Page]other 5 ( 1) 4 ( 2), they shal he handled as aforesayd, and the disposition thereof shalbe thus. [...]
The fourth Proposition: of Diuision.
A Disme number for the diuident, and diuisor, being giuen to find the Quotient.
Explication proposed: let the number for the diuident be 3 ( 0) 4 ( 1) 4 ( 2) 3 ( 3) 5 ( 4) 2 ( 5) and the diuisor 9 ( 1) 6 ( 2). Explication required: to find their Quotient.
COnstruction: the numbers giuen diuided (omitting the signes) according to the vulgar maner of diuiding of whole numbers, giueth the Quotient, 3587; now to know what they value; the latter signe of the diuisor ( 2) must be substracted from the latter signe of the diuident which is ( 5), resteth ( 3) for the latter signe of the latter Character of y e Quotient, which being so knowne, all y e rest are also manifest by their continued order, thus 3 ( 0) 5 ( 1) 8 ( 2) 7 ( 3) are the Quotient required.
DEmonstration: the number diuident giuen 3 ( 0) 4 ( 1) 4 ( 2) 3 ( 3) 5 ( 4) 2 ( 5) maketh (by the third Definition of this Disme) 3 4/10 4/100 3/1000 5/10000 2/100000 together 3 44352/100000 and by y e same reason, the diuisor 9 ( 1) 6 ( 2) valueth 96/100, by which 3 44352/100000 being diuided, giueth the Quotient 3 587/1000; but the sayd Quotient valueth 3 ( 0) 5 ( 1) 8 ( 2) 7 ( 3): therefore it is the true Quotient to be demonstrated.
Conclusion: a Disme number being giuen for the diuident [Page]and diuisor, we haue found the Quotient required.
Note, if the diuisors signes be higher then the signes of the diuident, there may be as many such Cyphers 0 ioyned to the diuident as you will, or many as shalbe necessary: as for example, [...] 7 ( 2) are to be diuided by 4 ( 5), I place after the 7 certaine 0 thus 7000, diuiding them as aforesayd, & in this sort it giueth for the Quotient 1750 ( 7).
It hapneth also sometimes, that the Quotient cannot be expressed by whole numbers, as 4 ( 1) diuided by 3 ( 2) in this sort, [...] whereby appeareth, that there will infinitly come from the 3 the rest of ⅓ and in such an accident you may come so neere as the thing requireth, omitting the remaynder, it is true, that 13 ( 0) 3 ( 1) 3⅓ ( 2) &c. shalbe the perfect Quotient required: but our intention in this Disme is to worke all by whole numbers: for seing that in any affayres, men reckon not of the thousandth part of a mile, grayne, &c. as the like is also vsed of the principall Geometricians, and Astronomers, in cō putacions of great consequence, as Ptolome & Iohannes Monta-regio haue not described their Tables of Arches, Chords, or Sines, in extreme perfection (as possibly they might haue done by Multinomall numbers,) because that imperfection (considering the scope and end of those Tables) is more conuenient then such perfection.
Note 2. the extraction of all kinds of Roots may also be made by these Disme numbers: as for example, To extract the square roote of 5 ( 2) 2 ( 3) 9 ( 4), which is performed in the vulgar maner of extraction in this sort, [...] and the root shalbe 2 ( 1) 3 ( 2), for the moitye or [Page]halfe of the latter signe of the numbers giuen, is alwayes the latter signe of the roote: wherefore if the latter signe giuen were of a number imper: the signe of the next following shalbe added, and then it shalbe a number per; and then extract the Root as afore. Likewise in the extraction of the Cubique Roote, the third part of the latter signe giuen shalbe alwayes the signe of the Roote: and so of all other kind of Roots.
The Appendix.
The Preface.
SEing that we haue already described the Disme, we will now come to the vse thereof, shewing by vi. Articles, how all computations which can happen in any mans busines, may be easily performed thereby: beginning first to shew how they are to be put in practize, in the casting vp of the content or quantity of Land measured as followeth.
The first Article, of the Computations of Land-meating.
CAll the Pearch or Rood also Comencement, which is 1 ( 0), diuiding that into 10 equall parts, whereof each one shalbe 1 ( 1); thē diuide each prime againe [Page]into 10 equall parts, each of which shalbe 1 ( 2); and againe each of them into 10 equall parts, and each of them shalbe 1 ( 3); proceeding further so, if neede be; but in Land-meating, diuisions of seconde wilbe small enough: yet for such things as require more exactnes, as Fathome of the Lead, Bodyes &c. there may be thirds vsed: and for as much as the greater number of Land-meaters vse not the Pole, but a chayne line of three, foure or fiue Perch long marking vpon the yard of their crosse staffe certaine feete 5 or 6 with fingers, palmes &c. the like may be done here: for in the place of their fiue or sixe feete with their fingers, they may put 5 or 6 primes with their seconds.
THis being so prepared, these shalbe vsed in measuring, without regarding the feete and fingers of the Pole, according to the Custome of the place: & that which must be added, substracted, multiplied or diuided according to this measure, shalbe performed according to the doctrine of the precedent examples.
AS for example, we are to adde 4. tryangles or surfaces of Land, whereof the first 345 ( 0) 7 ( 1) 2 ( 2), y e second 872 ( 0) 5 ( 1) 3 ( 2), the third 615 ( 0) 4 ( 1) 8 ( 2) y e fourth 956 ( 0) 8 ( 1) 6 ( 2); [...]
THese being added according to the manner declared in the first Proposition of this Disme in this sort, their summe wil be 2790 ( 0) or Perches 5 ( 1) 9 ( 2), the sayd Roods or Perches, diuided according to the custome of the place; (for euery Acre contayneth certaine Perches) by the number of perches you shall haue the Acres sought. BVt if one would know how many feete and fingers are in the 5 ( 1) 9 ( 2) (that which Land-meater shall need to doe but once, and that at the end of the casting vp of the proprietaries, although most men esteeme it vnnecessary to make any mention of feete and fingers) it will appeare [Page]vpon the Pole how many feete and fingers (which are marked, ioyning the tenth part vpon another side of the Rood) accord with themselues.
In the second, out of 57 ( 0) 3 ( 1) 2 ( 2) substracted 32 ( 0) 5 ( 1) 7 ( 2) it may be effected according to the second proposition of this Disme, in this maner: [...]
In the third (for multiplication of the sides of certaine Triangles and Quadrangles) multiply 8 ( 0) 7 ( 1) 3 ( 2), by 7 ( 0) 5 ( 1) 4 ( 2) 2 this may be performed according to the third proposition of this Disme, in this manner: [...]
And giueth for the product or superfices 65 ( 0) 8 ( 1) &c.
In the fourth let A, B, C, D, be a certaine Quadrangle Rectangular (from which we must cut 367 ( 0) 6 ( 1) and the side A D: maketh 26 ( 0) 3 ( 1):
The question is, how much we shall measure from A, towards B, to cut off, (I meane by a line paralell to A D,) the said 367 ( 0) 3 ( 1)
Deuide 367 ( 0) 6 ( 1) by 26 ( 0) 3 ( 1) according to the fourth proposition of this Disme: so the Quotient giueth from A, towards B, 13 ( 0) 9 ( 1) 7 ( 2), which is A E.
And if wee will, wee may come neerer (although it bée needles) by the second note of the fourth Proposition, [Page]the demonstrations of all these examples are alreadie made in their propositions. [...]
The II. Article: of the Computations of the measures of Tapistry, or Cloth.
THe Ell of the Measurer of Tapistrie or cloth, shall be to him 1 ( 0), the which he shall deuide (vpon the side whereon the partitions, which are according to the ordinance of y e Towne, is not set out) as is done aboue on y e Pole of y e Land meater, namely into 10 equall parts, whereof each shall be 1 ( 0), then each 1 ( 1) into 10 equall partes, of which each shall be 1 ( 2) &c. And for the practise seeing that these examples doe altogether accord with those of the first Article of Landmeating, it is thereby sufficiently manifest, so as we need not here make any mention againe of them.
The III. Article: of the Computations, seruing to Gaudging, and the measures of all Liquor vessels.
ONe Ame (which maketh 100 pots Antwerp) shalbe 1 ( 0), the same shall be deuided in length and deepnes, into 10 equall parts (namely, equall to respect of the wine, not of the Rod; of which the parts of the pro [...]ditie shalbe vnequall) & each part shalbe 1 ( 1) containing 10 pots, then again each 1 ( 1) into 10, parts equal as afore, and each will make 1 ( 2) worth 1 pot, then each 1 ( 2) into 10. equal parts making each 1 ( 3).
[Page]Now the Ro [...] being so deuided, to know the content of the Tunne, multiply and worke as in the precedent first Article, of which (being sufficiently manifest) we will not speake here any farther.
But seing that this tenth diuision of the deepnes is not vulgar, wee will explaine the same. Let the rod bee one Ame. A. B. which is 1 ( 0) deuided (according to the custome) into the points of the déepnes of these nine:
C, D, E, F, G, H, I, K, A, making each part 1 ( 1) which shall bee againe each part deuided into 10. thus. Let each 1 ( 1) bee deuided into two so: draw the line, B M. with a right angle vpon A B. and equall to 1 ( 1), B C, then (by the 13 proposition of Euclid his 6: booke) find the meane proportionall betweene B M and his moytie, which is B N: cutting B O: equall to B N: And if N O: bee equall to B C: the operation is good. Then note the length N C: from B towards A, as B P: the which being equall to N C: the operation is good: likewise the length of B N: from B to Q: and so of the rest.
It remaineth yet to deuide each length as B O & O C, &c. into fiue, thus: Seeke the meane proportionall betweene B M: & his 10 part which shalbe B R: cutting B S: equal to B R: Then the length S R. noted from B towards A: as B T: and likewise the length T R: from B to V: & so of the others: & in like sort proceeding to deuide B S: and S T: &c. into ( 3), I say that B S: S T: and T V: &c. [Page]are the desired ( 2) which is thus to bee demonstrated.
For that B N: is the meane proportionall line (by the Hipothesis betwéene B M: and his moytie, the square of B N: (by the 17. proposition of the sixt booke of Euclide) shalbe equall to the Rectangle of B M: & his moytie: But the same Rectangle is the moytie of y e square of B M: the square then of B N: is equall to the moytie of the square of B M: But B O is (by Hipothesis) equall to B N: and B C: to B M: the square then of B O: is equall to the moitie of the square of B C. And in like sort it is to be demonstrated, that the square of B S is equall to the tenth part of the square of B M. Wherefore &c. we haue made the demonstration briefe, because wee write not this to learners, but vnto masters in their science.
The IIII. Article: of Computations of Stereometrie in generall.
TRue it is, that Gaudgerie which we haue before declared, is Stereometrie (y t is to say, the Art of measuring of bodies) but considering the diuers diuisions of the Rod, Yard, or Measure of the one and other, and that and this doe so much differ, as the Genus and the Species: they ought by good reason to be distinguished. For all Stereometrie is not Gaudgerie. To come to the point, the Stereometrian shall vse the measure of the towne or place, as the Yard, Ell &c. with his tenne partitions, as is described in the first and second Articles, the vse and practise thereof, (as is before shewed) is thus: Put case wee haue a Quadrangular, Rectangular Columne to bee measured, the length whereof is 3 ( 1) 2 ( 2), the breadth 2 ( 1) 4 ( 2), the height 2 ( 0) 3 ( 1) 5 ( 2), The question is, how much the substance or matter of that Piller is: Multiply (according to the doctrine of the 4. proposition of this Disme) the length by the bredth, & the product again by the height in [Page]this manner, [...]
And the product appeareth to be 1 ( 1) 8 ( 2) 4 ( 4) 8 ( 5).
NOte, some ignorant (and vnderstanding not that wee speake here) of the Principles of Stereometry, may maruayle wherefore it is sayd, that the greatnes of y e abouesaid colūn is but 1 ( 1) &c. seing that it cōtayneth more then 180 cubes, of which the length of each side is 1 ( 1), he must know that the body of one yard is not a body of 10 ( 1) as a yard in length, but 1000 ( 1) in respect whereof 1 ( 1) maketh 100 Cubes, each of 1 ( 1) as the like is sufficiently manifest amongst Land-meats in surfaces: for when they say 2 Roodes, 3 Feete of Land, it is not barelymeant 2 square Roods, and three square feete, but two Roods (and counting but 12 feete to the Rood) 36 feete square: therefore if the sayd Question had beene how many Cubes each being 1 ( 1) was in the greatnes of the sayd Piller, the solution should haue bene fitted accordingly, considering that each of these 1 ( 1) doth make 100 ( 1) of those; and each 1 ( 2) of these maketh 10 ( 1) of those &c. or otherwise, if the tenth part of the yard be the greatest measure that the Stereometrian proposeth, he may call it 1 ( 0), and so as abouesayd.
The fift Article; of Astronomicall Computations.
THe ancient Astronomers hauing diuided their Circles each into 360 degrees, they saw, that the Astronomicall Computations of them with their parts was too laborious: and therefore they diuided also each degree into certaine parts, and those againe into as many, &c. to the end thereby to worke alwayes by whole numbers, chusing the 60th progression, because that 60 is a number measurable by many whole measures, namely, 1, 2, 3, 4, 5, 6 10, 12, 15, 20, 30: but if experience may be credited (we say with reuerence to the venerable antiquity, and moued with the common vtility) the 60th progression was not the most conuenient, (at least) amongst those that in nature consist potentially, but the tenth which is thus: we call the 360 degrees also Comencements, expressing them so 360 ( 0), and each of them a degree 12 1 ( 0) to be diuided into 10 equall parts, of which each shall make 1 ( 1), and againe each 1 ( 1) into 10 ( 2) and so of the rest, as the like hath already bene often done.
Now this diuision being vnderstood, we may describe more easily that we promised in Addition, Substraction, Multiplication, and Diuision; but because there is no difference betweene the operation of these, and the foure former propositions of this booke, it would but be losse of time, and therefore they shall serue for examples of this Article: yet adding thus much, that we will vse this maner of partition in all the Tables & computations which happen in Astronomy, such as we kepe to diuulge in our vulgar Germane Language, which to the most rich adorned and perfect Tongue of all other, & of the most singularity, of which we attend a more abundant demonstration, then Peter and Iohn haue made thereof in the Bewysconst and Dialectique, lately diuulged, and haue in the lease following placed a necessary Table, for the reducing of the [Page]minutes, seconds, &c. of the 60, progression, into primes, seconds, &c. of the tenth progression: the vse whereof followeth.
The vse of this Table.
VVHen any number of minutes, seconds, thirds, fourths &c. of the 60th progressiō, are giuen to be reduced into the primes, seconds, thirds, &c. of the tenth progression, seeke the giuen number in this Table, or if the number be not there to be found, take the neerest: if none be there great enough, take halfe or one quarter of the giuen: if there be none small enough, double, treble, or quadruple, the giuen, and then as aforesayd seeke the neerest number thereunto in the Table, and the two numbers in whose common Angle the giuen number is found, or neerest found, shall shew you the quantity and quality of the subdiuisions of the ten progressions proper to that giuen number, namely, the number standing in the toppe or front of the table directly ouer it, shall shew the quantity, and the number directly against it in the first Columne toward the left hand, shall denote the quality; as for example, be the pronumber giuen [...], seeke it in the Table, and you shall find to stand in the front directly ouer it the figure 7, and in y e first Columne directly against it toward the left hand ( 5): therefore according to the rule aboue mentioned, I conclude, that [...] of the 60 progression valueth iust 7 ( 5) of the tenth progression &c. This example I thinke sufficient to enlighten the ingenious practizer: onely this, that if there be no number to be found in the Table, iust or neere the number giuen, you may take two, three or more of those that will come neerest, and so worke as before: as for example also, be the number giuen [...] of the 60 progression; you shall find them all by taking 4 of the numbers of y e colūne vnder 3, to be [...] of the tenth progression: and so with a small diligence may any other number of the one progression be reduced into the other, which I omit to speake any further of at this time.
The sixt Article; of the Computations of Money-masters, Marchants, and of all estates in generall.
TO the end we speake in generall and briefly of the sūme and contents of this Article, it must be alwayes vnderstood, y t all measures (be they of length, liquors, of mony &c.) be parted by the tenth progression, and each notable species of them, shalbe called Comencement: as a Marke, comencement of weight, by the which Siluer and Gold are wayed, Poūd of other cōmon weights, Liuers-degros in Flanders, Pound sterling in England, Ducat in Spaine &c. Comencement of Money: the highest signe of the Marke shalbe ( 4), for 1 ( 4) shall weigh about the halfe of one Es of Antwerp, the ( 3) shall serue for the highest signe of the Liure de gros, seing that 1 ( 3) maketh lesse then the quarter of one DS.
The subdiuisions of weight to weigh al things, shalbe (in place of the halfe pound, quarter, halfe quarter, ounce, halfe ounce, esterlin, graine, Es, &c. of each signe, 5, 3, 2, 1, that is to say, that after the pound or 1 ( 0) shall follow the halfe pound or 5 ( 1), then the 3 ( 1) then the 2 ( 1) then the 1 ( 1), and the like subdiuisions haue also the 1 ( 1) and the other following.
VVE thinke it necessary, that each subdiuision, what matter soeuer the subiect be of, be called Prime, Second, Third, &c. and that because it is notable vnto vs, y t the Second, being multiplied by the Third, giueth in y e product the Fifth (because two and three make fiue, as is sayd before) also the Third diuided by the Secōd, giueth y e Quotient Prime &c. that which so properly cannot be done by any other names: but when it shalbe named for distinction of the matters (as to say, halfe an Ell, half a pound, halfe a pynte &c.) we may call them Prime of Marc, Second of Marc, Second of Pound, Second of Ell, &c.
But to the'nd wee may giue example, suppose 1 Mark [Page]of Gold value 36 lib. 5 ( 1) 3 ( 2) the Question what valueth 8 Marks 3 ( 1) 5 ( 2) 4 ( 3): multiply 3653 by 8354 giuing the product by the fourth Proposition (which is also the solution required) 395 lib. 1 ( 1) 7 ( 2) 1 ( 3); as for the 6 ( 4) and 2 ( 5) they are here of no estimation.
SVppose againe, y t 2 Ells and 3 ( 1) cost 3 lib. 2 ( 1) 5 ( 2) y e question is, what shall 7 Ells 5 ( 1) 3 ( 2) cost: multiply according to the custome the last terme giuen by the second, and diuide the product by the first, that is to say, 753 by 325 maketh 244725, which diuided by 23, giueth the Quotient and Solution 10 lib. 6 ( 1) 4 ( 2).
VVE could also more amply demonstrate by easie examples of broken numbers, y e comparison and great difference of the facility of this more then that, but we will passe them ouer for breuity sake.
LAstly, it may be sayd, that there is some difference betweene this last sixt Article, and the 5 precedent Articles, which is, that each one may exercise for them selues the tenth partition of the said precedent 5 Articles, thoughst be not giuen by the Magistrate of the place as a generall order, but it is not so in this latter: for the examples hereof, are vulgar computations, which do almost continually happen to euery man, to whom it were necessary that the solution so found, were of each accepted for good and lawfull: Therefore considering the so great vse, it would be a commendable thing, if some of those who expect the greatest commodity, would solicit to put y e same in execution to effect, namely, that ioyning the vulgar partitions that are now in weight, measures, and moneyes (continuing still each Capitall measure, waight and Coyne in all places vnaltred) that the same tenth progression might be lawfully ordained by the superiors, for euery one that would vse the same it might also do well, if the values of Moneys, principally the new Coynes, might be valued and reckned vpon certayne Primes, Seconds, Thirds &c [Page]But if all this be not put in practize so soone as we could wish, yet it will first content vs, y t it wil be beneficiall to our successors, if future men shal hereafter be of such nature as our predecessors, who were neuer negligent of so great aduantage. Secondly, that it is not vnnecessary for each in particular, for so much as concerneth him, for that they may all deliuer them selues when they will, from so much and so great labour. And lastly, although the effects of the sixt Article appeare not immediatly, yet it may be; and in the meane time may each one exercise himselfe in the fiue precedent, such as shalbe most conuenient for them; as some of them haue already practized.