Such was, in brief, the progress in the evolution of the new calculus made by Leibniz during his stay in Paris. Before his departure, in October, 1676, he found himself in possession of the most elementary rules and formulæ of the infinitesimal calculus.
From Paris, Leibniz returned to Hanover by way of London and Amsterdam. In London he met Collins, who showed him
a part of his scientific correspondence. Of this we shall speak later. In Amsterdam he discussed mathematics with Sluze,
and became satisfied that his own method of constructing tangents not only accomplished all that Sluze's did, but even more, since it could be extended to three variables, by which tangent planes to surfaces could be found; and especially, since neither irrationals nor fractions prevented the immediate application of his method.
In a paper of July 11, 1677, Leibniz gave correct rules for the differentiation of sums, products, quotients, powers, and roots. He had given the differentials of a few negative and fractional powers, as early as November, 1676, but had made some mistakes. For he had given the erroneous value , and in another place the value ; for occurs in one place the wrong value, , while a few lines lower is given , its correct value.
In 1682 was founded in Berlin the Acta Eruditorum, a journal usually known by the name of Leipzig Acts. It was a partial imitation of the French Journal des Savans (founded in 1665), and the literary and scientific review published in Germany. Leibniz was a frequent contributor. Tschirnhaus, who had studied mathematics in Paris with Leibniz, and who was familiar with the new analysis of Leibniz, published in the Acta Eruditorum a paper on quadratures, which consists principally of subject-matter communicated by Leibniz to Tschirnhaus during a controversy which they had had on this subject. Fearing that Tschirnhaus might claim as his own and publish the notation and rules of the differential calculus, Leibniz decided, at last, to make public the fruits of his inventions. In 1684, or nine years after the new calculus first dawned upon the mind of Leibniz, and nineteen years after Newton first worked at fluxions,
and three years before the publication of Newton's Principia, Leibniz published, in the Leipzig Acts, his first paper on the differential calculus. He was unwilling to give to the world all his treasures, but chose those parts of his work which were most abstruse and least perspicuous. This epoch-making paper of only six pages bears the title: "Nova methodus pro maximis et minimis, itemque tangentibus, quæ nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus." The rules of calculation are briefly stated without proof, and the meaning of d x and d y is not made clear. It has been inferred from this that Leibniz himself had no definite and settled ideas on this subject. Are d y and d x finite or infinitesimal quantities? At first they appear, indeed, to have been taken as finite, when he says: "We now call any line selected at random d x , then we designate the line which is to d x as y is to the sub-tangent, by d y , which is the difference of y ." Leibniz then ascertains, by his calculus, in what way a ray of light passing through two differently refracting media, can travel easiest from one point to another; and then closes his article by giving his solution, in a few