Page:A History of Mathematics (1893).djvu/270

and John Bernoulli had entertained any serious doubt of the correctness of the expression $\scriptstyle {{\frac {1}{2}}=1-1+1-1+\cdots }$. Guido Grandi went so far as to conclude from this that $\scriptstyle {{\frac {1}{2}}=0+0+0+\cdots }$. In the treatment of series Leibniz advanced a metaphysical method of proof which held sway over the minds of the elder Bernoullis, and even of Euler.^[46] The tendency of that reasoning was to justify results which seem to us now highly absurd. The looseness of treatment can best be seen from examples. The very paper in which Euler cautions against divergent series contains the proof that

$\scriptstyle {\cdots {\frac {1}{n^{2}}}+{\frac {1}{n}}+1+n+n^{2}+\cdots =0}$ as follows:

$\scriptstyle {n+n^{2}+\cdots ={\frac {n}{1-n}},~1+{\frac {1}{n}}+{\frac {1}{n^{2}}}+\cdots ={\frac {n}{n-1}};}$

these added give zero. Euler has no hesitation to write $\scriptstyle {1-3+5-7+\cdots =0}$, and no one objected to such results excepting Nicolaus Bernoulli, the nephew of John and Jacob. Strange to say, Euler finally succeeded in converting Nicolaus Bernoulli to his own erroneous views. At the present time it is difficult to believe that Euler should have confidently written $\scriptstyle {\sin \phi -2\sin 2\phi +3\sin 3\phi -4\sin 4\phi +\cdots =0}$, but such examples afford striking illustrations of the want of scientific basis of certain parts of analysis at that time. Euler's proof of the binomial formula for negative and fractional exponents, which has been reproduced in elementary text-books of even recent years, is faulty. A remarkable development, due to Euler, is what he named the hypergeometric series, the summation of which he observed to be dependent upon the integration of a linear differential equation of the second order, but it remained for Gauss to point out that for special values of its letters, this series represented nearly all functions then known.

Euler developed the calculus of finite differences in the first