Newton to Euler

of velocity or fluxion, used the infinitely small increment as a means of determining it, while with Leibniz the relation of the infinitely small increments is itself the object of determination. The difference between the two rests mainly upon a difference in the mode of generating quantities.35

We give Newton's statement of the method of fluxions or rates, as given in the introduction to his Quadrature of Curves. "I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux: and so on in other quantities. These geneses really take place in the nature of things, and are daily seen in the motion of bodies.

"Fluxions are, as near as we please (quam proxime), as the increments of fluents generated in times, equal and as small as possible, and to speak accurately, they are in the prime ratio of nascent increments; yet they can be expressed by any lines whatever, which are proportional to them."

Newton exemplifies this last assertion by the problem of tangency: Let $AB$ be the abscissa, $BC$ the ordinate, $VCH$ the tangent, $Ec$ the increment of the ordinate, which produced meets $VH$ at $T$, and $Cc$ the increment of the curve. The right line $Cc$ being produced to $K$, there are formed three small triangles, the rectilinear $CEc$, the mixtilinear $CEc$, and the rectilinear $CET$. Of these, the first is evidently the smallest, and the last the greatest. Now suppose the ordinate $bc$ to move into the place $BC$, so that the point $c$ exactly coincides with

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the point $C$; $CK$, and therefore the curve $Cc$, is coincident with the tangent $CH$, $Ec$ is absolutely equal to $ET$, and the mixtilinear evanescent triangle $CEc$ is, in the last form, similar to the triangle $CET$; and its evanescent sides $CE$, $Ec$, $Cc$, will be proportional to $CE$, $ET$, and $CT$, the sides of the triangle $CET$. Hence it follows that the fluxions of the lines $AB$, $BC$, $AC$, being in the last ratio of their evanescent increments, are proportional to the sides of the triangle $CET$, or, which is all one, of the triangle $VBC$ similar thereunto. As long as the points $C$ and $c$ are distant from each other by an interval, however small, the line $CK$ will stand apart by a small angle from the tangent $CH$. But when $CK$ coincides with $CH$, and the lines $CE$, $Ec$, $cC$ reach their ultimate ratios, then the points $C$ and $c$ accurately coincide and are one and the same. Newton then adds that "in mathematics the minutest errors are not to be neglected." This is plainly a rejection of the postulates of Leibniz. The doctrine of infinitely small quantities

is here renounced in a manner which would lead one to suppose that Newton had never held it himself. Thus it appears that Newton's doctrine was different in different periods. Though, in the above reasoning, the Charybdis of infinitesimals is safely avoided, the dangers of a Scylla stare

us in the face. We are required to believe that a point may be considered a triangle, or that a triangle can be inscribed in a point; nay, that three dissimilar triangles become similar and equal when they have reached their ultimate form in one and the same point.

In the introduction to the Quadrature of Curves the fluxion of $x^n$ is determined as follows:–-

"In the same time that $x$, by flowing, becomes $x+0$, the power $x^n$ becomes $(x+0{)}^n$, i.e. by the method of infinite series $x^n+n0x^{n-1}+\frac{n^2-n}{2}{0}^2{x}^{n-2}+\text{etc.},$ and the increments $0\text{ and }n0x^{n-1}+\frac{n^2-n}{2}{0}^2{x}^{n-2}+\text{etc.},$ are to one another as $1\text{ to }n{x}^{n-1}+\frac{n^2-n}{2}0x^{n-2}+\text{etc.}$