Page:A History of Mathematics (1893).djvu/168

 of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him. Detailed investigations on the famous old problem of the section of an angle into an odd number of equal parts, led Vieta to the discovery of a trigonometrical solution of Cardan's irreducible case in cubics. He applied the equation $$\scriptstyle{(2 \cos \frac{1}{3} \phi)^3-3(2 \cos \frac{1}{3} \phi)=2 \cos \phi}$$ to the solution of $$\scriptstyle{x^3-3a^2x=a^2b}$$, when $$\scriptstyle{a>\frac{1}{2}b}$$, by placing $$\scriptstyle{x=2a \cos \frac{1}{3} \phi}$$, and determining $$\scriptstyle{\phi}$$ from $$\scriptstyle{b=2 a \cos \phi}$$.

The main principle employed by him in the solution of equations is that of reduction. He solves the quadratic by making a suitable substitution which will remove the term containing x to the first degree. Like Cardan, he reduces the general expression of the cubic to the form $$\scriptstyle{x^3+mx+n=0}$$; then, assuming $$\scriptstyle{x=(\frac{1}{3}a-z^2) \div z}$$ and substituting, he gets $$\scriptstyle{z^6-bz^3-\frac{1}{27}a^3=0}$$. Putting $$\scriptstyle{z^3=y}$$, he has a quadratic. In the solution of bi-quadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration. In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question.

The most epoch-making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but