Disme: the art of tenths, or decimall arithmetike teaching how to perform all computations whatsoeuer, by whole numbers without fractions, by the foure principles of common arithmeticke: namely addition, subtraction, multiplication, and diuision. Inuented by the excellent mathematician, Simon Steuin. Published in English with wholesome additions by Robert Norton, Gent.

The second part of the Disme. Of the Operation or Practize. (Book 2)

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, 〈 math 〉〈 math 〉 adding them in the vulgar maner of adding of whole numbers in this maner: The summe (by y^e first Probleme of Arithme∣tick 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

27 (⁰) 8 (¹) 4 (²) 7 (³) giuen, make by the 3 Definition be∣fore 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 num∣bers 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. Con∣clusion: Then Disme numbers being giuen to bee added, wee haue found their summe, which is the thing requi∣red.

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): 〈 math 〉〈 math 〉 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 follow∣ing 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.

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, substra∣cting according to vulgar maner of substractiō of whole nūbers, thus 〈 math 〉〈 math 〉

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 to∣gether 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 mul∣tiplied 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, 〈 math 〉〈 math 〉 multiplying according to the vulgar maner of multiplicati∣on by whole nūbers, in this maner,

giuing y^e product, 29137122: 〈 math 〉〈 math 〉 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 al∣so 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 pro∣duct 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 de∣monstrate. 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 Defini∣tion 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, be∣ing giuen, we haue found the product, as we ought.

Note.

IF the latter signe of the number to bee multipli∣ed, bee vnequall to the latter signe of the multiplica∣tor, as for example, the one 3 (⁴) 7 (⁵) 8 (⁶), the

other 5 (¹) 4 (²), they shal he handled as aforesayd, and the disposition thereof shalbe thus. 〈 math 〉〈 math 〉

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 (omit∣ting the signes) according to the vulgar maner of di∣uiding of whole numbers, giueth the Quotient, 3587; now to know what they value; the latter signe of the diui∣sor (²) must be substracted from the latter signe of the diui∣dent which is (⁵), resteth (³) for the latter signe of the lat∣ter 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): there∣fore it is the true Quotient to be demonstrated.

Conclusion: a Disme number being giuen for the diui∣dent

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, 〈 math 〉〈 math 〉 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, 〈 math 〉〈 math 〉 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 re∣quireth, 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 num∣bers: 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 ex∣tract the square roote of 5 (²) 2 (³) 9 (⁴), which is perfor∣med in the vulgar maner of extraction in this sort, 〈 math 〉〈 math 〉 and the root shalbe 2 (¹) 3 (²), for the moitye or

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 follo∣wing 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.