Background

With the growth of international trade and warfare, there was a need for many to understand arithmetic and geometry. New methods for Algebra were invented in Italy, (Pacioli, 1500, Del Ferro, 1520, Tartaglia, 1530, Cardano 1540 and Bombelli, 1550). These included solutions by radicals of cubic and quartic polynomials. In France (Chuquet, 1480, manipulation of exponents), Germany (Rudolff, Stifel 1520, cos algebra) and England (Recorde, 1540).

 Later there were two outstanding mathematicians Viète (1580) in France, who unified algebraic and geometric methods and intoduced the use of literal coefficients in algebra, and Stevin in Holland.

Simon Stevin (1548--1620) was born in Flanders (Belgium) but spent his adult life in the service of the Republic of the Netherlands. He was a military engineer and tutor in mathematics and adviser in finance and navigation to the rulers William of Orange and Maurice of Nassau. He organized the school of engineering at the University of Leiden and wrote textbooks. His most important work was "De Thiende", translated as " Art of Tenths". 

For an image of the French manuscript of 'De Thiende", click here.

 Apart from the introduction of new notation and algorithms, the fundamental idea was to erase the Greek distinction between number and magnitude.

 Although decimal fractions were re-invented several times between 1000 and 1600, they were not in widespread use. Common fractions and sexigesimal fractions were used as appropriate, but were regarded as difficult. Also, it was not easy to recognize approximations to eg root 2 or p .

 Stevin states in his Introduction that the purpose of "De Thiende" is to teach the easy performance of all reckonings, computations and accounts without common fractions. A commencement is a whole number followed by the symbol (O) , [O with a circle round it] representing units. Each tenth of a unit, called a prime is written as digits, possibly 0, followed by (1) , and so on, for example 364 (0) 9 (1) 5 (2) 7 (3) means 364 957/1000. Stevin calls these decimal numbers and the symbols (1) etc signs. He describes how to add, multiply, subtract and divide decimal numbers. Naturally the main problem is to decide what are the signs of the digits in a product or quotient.

 The connection between this notation and the basic concept of number in Stevin is his declaration that Number is that which explains the quantity of each thing . In particular, 1 is a number . For example root 8 is just a number, not the side of a square of area 8. 

Simon Stevin's "De Thiende" 

 The Preface of Simon Steuin.

 To Astronomers, Land-meaters, Measurers of Tapistry, Gaudgers, Stereometers in general, Money-Masters, and to all Marchants, Simon Steuin wisheth health.

 Many, seeing the smallnes of this Book, and considering your worthynes, to who{m} 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 inuentio{n}, for as the countryman by chance sometime findeth a great treasure, without any vse of skill or cunning, so hath it happened herein. Therefore, if any will thinke that I vaunt my selfe of my knowledge, because of the explicatio{n} of these vtilities, out of doubt, he sheweth himselfe to haue neyther iudgeme{n}t, vnderstanding, nor knowledge, to discerne 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 definition following) is number, the vse and effects of which your selues 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 means of the declination of the Sunne, to describe the true Longitudes, Latitudes, situatio{n}s & distances of places, &c.) a Paradise, abou{n}ding 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 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 damage 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 man's busines, in such sort, as that the foure Principles of Arithmetick namely, Addition, Substractio{n}, Multiplication, & Devisio{n}, by whole numbers, may satisfie these effects, affording the like facility vnto those that vse Counters. 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 therein, 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 practised, effect nothing of worth, as it often hapneth to the searchers 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 Disme hath two parts, that is, Definitions & Operations: by the 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 Diuision of Disme numbers....

The first Definition.

 DIsme is a kind of Arithmeticke, invented by the tenth progression, consisting in Characters of Cyphers; whereby a certain number is described, and by which also all accounts which happen to 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 conuenie{n}t that the things whereof we would speake, haue names, and that this maner of 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 Trealtie 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 the vnity of the Prime, we call the Second, whose signe is (2), and so of the 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 1/10 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 understand, that in this Disme we vse no 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 beforegoing, are generally called Disme numbers. 

The end of the Definitions.

 The second part of the Disme.

 Of the Operation or Practize.

 The first proposition of Addition.

 Disme numbers being given 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:

    (0) (1) (2) (3)

    27  8   4    7

   37  8   7    5

 875  7   8   2

 The summe (by the 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 27(0) 8(1) 4(2) 7(3) fiuen, make by the 3 Definitions before 27 8/10 4/100 7/1000, together 27 847/1000, and by the same reason, the 37(0) 8(1) 7(2) shall make 37 875/1000, and the 875(0) 7(1) 8(2) 4(3) [Note 6] will make 875 782/1000, which three 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 natural order, the place of the defectant shal be filled. As forexample, 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. 

(0) (1) (2) 
 8   5    6 
 5   0    7 
---------- 
13  6   3 

This aduertisement shall also serue in the three following propositions, wherein the order of the defayling figures must be supplied, as was done in the former example..... 

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 nu{m}bers, in this maner, 

           (0) (1) (2)
           32   5    7
           89   4    6

Now 29137122 to know how much the value, ioyne (0) (1) (2) (3) (4) the two last signes together as the one (2) and the other (2) also, which together make (4), and say the 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 us take 2/10 and 3/100 which (by the third Definition of this Disme) are 2(1) 3(2) their product is 6/10000 [Note 10] 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.