"The ancients drew tangents to the conic sections, and to the other geometrical curves of their invention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tangents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geometrical curves, which was a great step: but it was previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any; and this operation sometimes required immense, if not absolutely impracticable calculations. The tangent of the , a modern mechanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general method, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1684. The ever memorable paper that contained it is entitled: 'A New Method for Maxima and Minima, and likewise for Tangents, which is affected neither by Fractions nor irrational Quantities; and a peculiar Kind of Calculus for them.' In this we find the method of differencing all kinds of quantities, rational, fractional, or radical, and the application of these calculi to a very complicated case, which points out the mode for all cases. The author afterward resolves a problem de maximis et minimis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly he applies his new calculus to a problem, which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. ...Leibnitz showed in a couple of lines the required curve to be ...the common logarithmic curve."