Understanding the‘ScientificRevolution’ 149
was inspired by the similar language of legal case studies.^12 It might be more reasonable to say
that, following the rediscovery of Diophantus, such a transformation was ‘in the air’.
Stevin by contrast appears as more practical, as can be seen from his biography (Dijksterhuis
1970). He was also more self-consciously innovatory. The quotation above shows his disregard
for the Greeks, and his belief in a ‘lost’ programme of science from an earlier wise age. This was
not completely eccentric, and was shared by a number of his contemporaries. Among the most
important inheritances of the wise age, in his view, was the decimal system of writing numbers, and
his role in promoting decimal fractions is undoubtedly related to that. When, inLa Disme,hegives
the result of a division by three as a decimal with (effectively) as many 3s after the point as you like,
he has finally grasped a fact which seems to have eluded the Babylonians: the existence of repeating
decimals and their necessity.^13 In hisArithmétique, he set out a deliberately ‘controversial’ view on
numbers. The orthodoxy, transmitted in a confused way by the medieval schools from the Greeks,
was that numbers (2, 3, 4,...) were not magnitudes, that fractions or parts of a number were not
numbers, and that ‘one’ was not a number since it was the origin of number. How widely these
statements were believed in practice is uncertain, but Stevin enjoyed demolishing them, suggesting
that those who denied that parts of a unit were numbers were ‘denying that a piece of bread is
bread’. He concludes by a statement of theses: one is a number (thesis I); there are no absurd,
irrational, inexplicable, or surd numbers (thesis IV); and so on. Both Viète’s and Stevin’s viewpoint
can be seen as contributing to the way that mathematics shapes our view of the world today; if
we think of the lawE=mc^2 as an essential equation irrespective of the values ofE,m, andc,
we are following Viète, while if we consider its use in telling us what happens when we substitute
particular (computed) values ofmandc, we are following Stevin.
Exercise 6.ProveViète’s formula for the difference of cubes—it is of course easiest to modernize at least
partly in your working—and deduce the formula for E squared.
8 Descartes
I have constructed a method which, I think, enables me gradually to increase my knowledge and to raise it little by little
to the highest point which the mediocrity of my mind and the short span of my life will allow it to reach. (Descartes
1968b, p. 28)
I have spent some time describing the ways in which a ‘modern’ outlook on numbers can be traced
back to the late sixteenth century. The texts in which the work is done do notlookmodern, because
they are written in a language which is in transition between that of the medieval world and our
own. Descartes’sGeometry, on the other hand, looks modern and is relatively easy to read—for
us; his contemporaries found it difficult, because new. This is because he had the good fortune to
invent the common notation of modern algebra (x,yfor unknowns,a,bfor constants; and 4xy,
for example, instead of Viète’s ‘AinE4’.) Of course, this could be looked at another way: if his
terminology has stayed with us, it is because he had the intelligence to devise one which was
clear and easy to use. As a result of this, and more specifically of his ‘coordinate’ representation for
geometric curves, in the eighteenth century, historians of mathematics (French ones, in particular)
- So, until recently, P. for plaintiff and D. for defendant—see any law book.
- The Babylonians would have had to do more work, since^17 , the first Babylonian repeating decimal, repeats after six sexagesimal
places, not after one.