Page:A History of Mathematics (1893).djvu/215

 in the numerators respectively $\scriptstyle{-\frac{1}{8}}$,|undefined $\scriptstyle{+\frac{1}{16}}$,|undefined $\scriptstyle{-\frac{5}{128}}$,|undefined etc.; hence the required area for the circular segment is $\scriptstyle{x-\frac{\frac{1}{2}x^3}{3}-\frac{\frac{1}{8}x^5}{5}-\frac{\frac{1}{16}x^7}{7}-}$etc. Thus he found the interpolated expression to be an infinite series, instead of one having more than one term and less than two, as Wallis believed it must be. This interpolation suggested to Newton a mode of expanding $\scriptstyle{(1-x^2)^\frac{1}{2}}$,|undefined or, more generally, $\scriptstyle{(1-x^2)^m}$, into a series. He observed that he had only to omit from the expression just found the denominators 1, 3, 5, 7, etc., and to lower each power of x by unity, and he had the desired expression. In a letter to Oldenburg (June 13, 1676), Newton states the theorem as follows: The extraction of roots is much shortened by the theorem

where A means the first term, $\scriptstyle{P^\frac{m}{n}}$,|undefined B the second term, C the third term, etc. He verified it by actual multiplication, but gave no regular proof of it. He gave it for any exponent whatever, but made no distinction between the case when the exponent is positive and integral, and the others.

It should here be mentioned that very rude beginnings of the binomial theorem are found very early. The Hindoos and Arabs used the expansions of $$\scriptstyle{(a+b)^2}$$ and $$\scriptstyle{a+b)^3}$$ for extracting roots; Vieta knew the expansion of $\scriptstyle{(a+b)^4}$; but these were the results of simple multiplication without the discovery of any law. The binomial coefficients for positive whole exponents were known to some Arabic and European mathematicians. Pascal derived the coefficients from the method of what is called the "arithmetical triangle." Lucas de Burgo, Stifel, Stevinus, Briggs, and others, all possessed something from which one would think the binomial theorem could have