words, of De Beaune's problem. Two years later (1686) Leibniz published in the Acta Eruditorum a paper containing the rudiments of the integral calculus. The quantities d x and d y are there treated as infinitely small. He showed that by the use of his notation, the properties of curves could be fully expressed by equations. Thus the equation y = 2 x − x 2 + ∫ d x 2 x − x 2 characterises the cycloid.38
The great invention of Leibniz, now made public by his articles in the Leipzig Acts, made little impression upon the mass of mathematicians. In Germany no one comprehended
the new calculus except Tschirnhaus, who remained indifferent to it. The author's statements were too short and succinct to make the calculus generally understood. The first to recognise its importance and to take up the study of it were two foreigners,–-the Scotchman John Craig, and
the Swiss James Bernoulli. The latter wrote Leibniz a
letter in 1687, wishing to be initiated into the mysteries of the new analysis. Leibniz was then travelling abroad, so that this letter remained unanswered till 1690. James Bernoulli succeeded, meanwhile, by close application, in uncovering the secrets of the differential calculus without assistance. He and his brother John proved to be mathematicians of exceptional power. They applied themselves to the new science with a success and to an extent which made Leibniz declare that it was as much theirs as his. Leibniz carried on an extensive correspondence with them, as well as with other mathematicians. In a letter to John Bernoulli he suggests, among other things, that the integral calculus be improved by reducing integrals back to certain fundamental irreducible forms. The integration of logarithmic expressions was then studied. The writings of Leibniz contain many innovations, and anticipations of since prominent methods. Thus he made use of variable parameters, laid the foundation of analysis in situ, introduced the first notion of determinants in his effort
to simplify the expression arising in the elimination of the unknown quantities from a set of linear equations. He resorted to the device of breaking up certain fractions into the sum of other fractions for the purpose of easier integration; he explicitly assumed the principle of continuity; he gave the
first instance of a "singular solution," and laid the foundation to the theory of envelopes in two papers, one of which contains for the first time the terms co-ordinate and axes of co-ordinates.
He wrote on osculating curves, but his paper contained the
error (pointed out by John Bernoulli, but not admitted by him) that an osculating circle will necessarily cut a curve in four consecutive points. Well known is his theorem on the th differential coefficient of the product of two functions of a
variable. Of his many papers on mechanics, some are valuable,
while others contain grave errors.
Before tracing the further development of the calculus we shall sketch the history of that long and bitter controversy between English and Continental mathematicians on the invention of the calculus. The question was, did Leibniz invent it independently of Newton, or was he a plagiarist?