even though he resorted to solutions in form of infinite series. Newton's third case comes now under the solution of partial differential equations. He took the equation and succeeded in finding a particular integral of it.
The rest of the treatise is devoted to the determination of maxima and minima, the radius of curvature of curves, and other geometrical applications of his fluxionary calculus. All this was done previous to the year 1672.
It must be observed that in the Method of Fluxions (as well as in his De Analysi and all earlier papers) the method employed by Newton is strictly infinitesimal, and in substance like that of Leibniz. Thus, the original conception of the calculus in England, as well as on the Continent, was based on infinitesimals. The fundamental principles of the fluxionary calculus were first given to the world in the Principia; but its peculiar notation did not appear until published in the second volume of Wallis' Algebra in 1693. The exposition given in the Algebra was substantially a contribution of Newton; it rests on infinitesimals. In the first edition of the Principia (1687) the description of fluxions is likewise founded on infinitesimals, but in the second (1713) the foundation is somewhat altered. In Book II. Lemma II. of the first edition we read: "Cave tamen intellexeris particulas finitas. Momenta quam primum finitæ sunt magnitudinis, desinunt esse momenta. Finiri enim repitgnat aliquatenus perpetuo eorum incremento vel decremento. Intelligenda sunt principia jamjam nascentia finitorum magnitudinum." In the second edition the two sentences which we print in italics are replaced by the following: "Particulæ finitæ non sunt momenta sed quantitates ipsæ ex momentis genitæ." Through the difficulty of the phrases in both extracts, this much distinctly appears, that in the first, moments are infinitely small quantities. What else they are in the second is not clear.[35] In the
