148 A History ofMathematics
one afterwards adopted. Perhaps for this reason, you will find no extracts in Fauvel and Gray which
show how he worked. Here, then, is one in its orginal form. (The ‘standard edition’ ofThe Analytic
Artis a good example of the loss involved when an author’s notation is updated; although it can be
read to get an overall idea of Viète’s project, the changes in terminology, such as ‘BE’ for ‘BinE’
make it both more readable and less interesting; one cannot see what innovations are specifically
Viète’s own.)Book II Zetetic XVII. Given the difference between the roots and the difference between their cubes, to find the roots.
[Try to read through this text to see what it means, if possible, before consulting the notes below.]
LetBbe the difference between the roots andDsolid the difference between the cubes. The roots are to be found.
Let the sum of the roots beE. ThereforeE+Bwill be twice the greater root andE−Btwice the smaller. [Why?] The
difference between the cubes of these isBinEsquared 6+Bcubed 2 which is consequently equal toDsolid 8.Therefore
Dsolid 4
−Bcube
B 3
equalsEsquaredThe squares being given, the root is given, and the difference between the roots and their sum being given, the roots
are given.
Accordingly the difference of the cubes quadrupled, minus the cube of the difference of the sides, being divided by
the difference of the sides tripled, there results the square of the sum of the sides.
IfBis 6,Dsolid is 504, the sum of the sides 1N,1Qequals 100.
Notes. A ‘Zetetic’ is Viète’s word for a method of finding out. In his notation the ‘roots’ are lines, so
the sum of their cubes is a ‘solid’, which is why he calls it ‘Dsolid’; his rule is that (as the Greeks
prescribed) you must always keep track of the dimensions of quantities and not set lines equal to
solids. For example,BandDsolid are denoted by consonants, because they are known; whileEis
a vowel, because it is unknown. Numbers comeafterthe letters, so that ‘Esquared 6’ means what
we would call 6E^2.
What comes out of this, and many other examples like it in theAnalytic Art, is not an outstand-
ingly difficult result. Itisa systematic treatment of algebra in which the objects being manipulated
areletters, which stand not for natural numbers (as in Euclid’s arithmetic), but for quantities, and
in which the proof is not by geometry. In Viète’s example,Bis 6 andDsolid is 504, so thatE^2 is
100, andEis indeed a whole number 10. (Check this; and find the two roots.) But it is clear that
a different choice (e.g.Dsolid=2,B=1) would lead to an ‘irrational’ answer, and that nothing
in the method restricts answers to being whole numbers—or (to anticipate) to being numbers at
all. It is this which leads Klein in particular to give Viète such a high value:But above all—and it is this which gives him his tremendous role in the history of the origins of modern science—
he was the first to assign to ‘algebra’, to this ‘ars magna’,a fundamental place in the system of knowledge in general.
From now on the fundamentalontologicalscience of the ancients is replaced by asymbolicdiscipline whose ontological
presuppositions are left unclarified. (Klein 1968, p. 184)Here, then, (if Klein is right) is the germ of Russell’s ‘Mathematics is the science in which
we do not know what we are talking about’; and its extension to physics via the definition of
‘occult’ quantities, from Newtonian force to atomic spin, whose importance is not that they can
be measured but that they can enter into equations. This is a great deal to ascribe to the work of
a lawyer whose introduction of letters, if we are to believe his English interpreter Thomas Harriot,