Page:A History of Mathematics (1893).djvu/212

 already mentioned elsewhere Wallis's solution of the prize questions on the cycloid, which were proposed by Pascal.

The Arithmetic of Infinites, published in 1655, is his greatest work. By the application of analysis to the Method of Indivisibles, he greatly increased the power of this instrument for effecting quadratures. He advanced beyond Kepler by making more extended use of the "law of continuity" and placing full reliance in it. By this law he was led to regard the denominators of fractions as powers with negative exponents. Thus, the descending geometrical progression $$\scriptstyle{x^3,~x^2,~x^1,~x^0}$$, if continued, gives $$\scriptstyle{x^{-1},~x^{-2},~x^{-3}}$$, etc.; which is the same thing as $$\scriptstyle{\tfrac{1}{x},~\tfrac{1}{x^2},~\tfrac{1}{x^3}}$$. The exponents of this geometric series are in continued arithmetical progression, $$\scriptstyle{3,~2,~1,~0,~-1,~-2,~-3}$$. He also used fractional exponents, which, like the negative, had been invented long before, but had failed to be generally introduced. The symbol $$\scriptstyle{\infin}$$ for infinity is due to him.

Cavalieri and the French geometers had ascertained the formula for squaring the parabola of any degree, $$\scriptstyle{y=x^m}$$, m being a positive integer. By the summation of the powers of the terms of infinite arithmetical series, it was found that the curve $$\scriptstyle{y=x^m}$$ is to the area of the parallelogram having the same base and altitude as 1 is to $$\scriptstyle{m+1}$$. Aided by the law of continuity, Wallis arrived at the result that this formula holds true not only when m is positive and integral, but also when it is fractional or negative. Thus, in the parabola $$\scriptstyle{y=\sqrt{px}}$$, $$\scriptstyle{m=\tfrac{1}{2}}$$; hence the area of the parabolic segment is to that of the circumscribed rectangle as $$\scriptstyle{1:1\tfrac{1}{2}}$$, or as $$\scriptstyle{2:3}$$. Again, suppose that in $$\scriptstyle{y=x^m}$$, $$\scriptstyle{m=-\tfrac{1}{2}}$$; then the curve is a kind of hyperbola referred to its asymptotes, and the hyperbolic space between the curve and its asymptotes is to the corresponding parallelogram as $$\scriptstyle{1:\tfrac{1}{2}}$$. If $$\scriptstyle{m=-1}$$, as in the common equilateral hyperbola $$\scriptstyle{y=x^{-1}}$$ or $$\scriptstyle{xy=1}$$, then this ratio is $$\scriptstyle{1:-1+1}$$, or $$\scriptstyle{1:0}$$, showing that its asymptotic space