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 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, quae 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 dx and dy is not made clear. It has been inferred from this that Leibniz himself had no definite and settled ideas on this subject. Are dy and dx 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 dx, then we designate the line which is to dx as y is to the sub-tangent, by dy, 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 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 dx and dy 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,

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