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we obtain
3 Ì © D sec © D ter ©.Although notation still had far to go, from the modern point of view, at least it was
no longer necessary to use a different letter to represent each power of the unknown
in a problem.
Frangois Viete. The French lawyer Frangois Viete (1540-1603), who worked as
tutor in a wealthy family and later became an advisor to Henri de Navarre (the
future king Henri IV), found time to study Diophantus and to introduce his own
ideas into algebra. Viete is credited with several crucial advances in the subject.
In his book Artis analyticae praxis (The Practice of the Analytic Art) he begins by
giving the rules for powers of binomials (in words). For example, he describes the
fifth power of a binomial as "the fifth power of the first [term], plus the product
of the fourth power of the first and five times the second,...." Viete's notation
was slightly different from ours, but is more recognizable to us than that of Stevin.
He would write the equation A^3 + SB A — D, where the vowel A represented the
unknown and the consonants  and D were taken as known, as follows (Zeuthen,
1903, p. 98):
A cubus + Â planum in A3 aequatur D solido.
As this quotation shows, Viete appears to be following the tedious route of
writing everything out in words, and to be adhering to the requirement that all the
terms in an equation be geometrically homogeneous.
This introduction is followed by five books of zetetics (research, from the Greek
word zetein, meaning seek). The mention of "roots" in connection with the bino-
mial expansions was not accidental. Viete studied the relation between roots and
coefficients in general equations. By using vowels to represent unknowns and con-
sonants to represent data for a problem, Viete finally achieved what was lacking in
earlier treatises: a convenient way of talking about general data without having to
give specific examples. His consonants could be thought of as representing num-
bers that would be known in any particular application of a process, but were left
unspecified for purposes of describing the process itself. His first example was the
equation A^2 + AB = Z^2 , in other words, a standard quadratic equation. According
to Viete these three letters are associated with three numbers in direct proportion,
Æ being the middle, Â the difference between the extremes, and A the smallest
number. In our terms, Æ — AT and  = Ar^2 — A. Thus, the general problem
reduces to finding the smallest of three numbers A, Ar, Ar^2 given the middle value
and the difference of the largest and smallest. Viete had already shown how to do
that in his books of zetetics.
This analysis showed Viete the true relation between the coefficients and the
roots. For example, he knew that in the equation x^3 — 6x^2 + 1 lx = 6, the sum and
product of the roots must be 6 and the sum of the products taken two at a time
must be 11. This observation still did not enable him to solve the general cubic
equation, but he did study the problem geometrically and show that any cubic could
be solved provided that one could solve two of the classical problems of antiquity:
constructing two mean proportionals between two given lines and trisecting any
angle. As he concluded at the end of his geometric chapter: "It is very worthwhile
to note this."