To his researches on series we owe the creation of the theory of definite integrals by the development of the so-called Eulerian integrals. He warns his readers occasionally against the use of divergent series, but is nevertheless very careless himself. The rigid treatment to which infinite series are subjected now was then undreamed of. No clear notions existed as to what constitutes a convergent series. Neither Leibniz nor Jacob
and John Bernoulli had entertained any serious doubt of the
correctness of the expression . Guido Grandi went so far as to conclude from this that $12 = 0 + 0 +
0 +$. 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
these added give zero. Euler has no hesitation to write , 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 , 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
chapters of his Institutiones calculi differentialis, and then deduced the differential calculus from it. He established a theorem on homogeneous functions, known by his name, and
contributed largely to the theory of differential equations, a
subject which had received the attention of Newton, Leibniz,
and the Bernoullis, but was still undeveloped. Clairaut,
Fontaine, and Euler about the same time observed criteria of
integrability, but Euler in addition showed how to employ them to determine integrating factors. The principles on which the criteria rested involved some degree of obscurity. The celebrated addition-theorem for elliptic integrals was first