Newton to Euler

The Method of Fluxions, translated by J. Colson from Newton's

Latin, was first published in 1736, or sixty-five years after it was written. In it he explains first the expansion into series of fractional and irrational quantities,–-a subject which, in his first years of study, received the most careful attention. He then proceeds to the solution of the two following mechanical problems, which constitute the pillars, so to speak, of the abstract calculus:–-

"I. The length of the space described being continually (i.e. at all times) given; to find the velocity of the motion at any time proposed.

"II. The velocity of the motion being continually given; to find the length of the space described at any time proposed."

Preparatory to the solution, Newton says: "Thus, in the equation $y=x^2$, if $y$ represents the length of the space at any time described, which (time) another space $x$, by increasing with an uniform celerity $\dot{x}$, measures and exhibits as described: then $2x\dot{x}$ will represent the celerity by which the space $y$,

at the same moment of time, proceeds to be described; and contrarywise."

"But whereas we need not consider the time here, any farther than it is expounded and measured by an equable local motion; and besides, whereas only quantities of the same kind can be compared together, and also their velocities of increase and decrease; therefore, in what follows I shall have no regard to time formally considered, but I shall suppose some one of the quantities proposed, being of the same kind, to be increased by an equable fluxion, to which the rest may be referred, as it were to time; and, therefore, by way of analogy, it may not improperly receive the name of time." In this statement of Newton there is contained a satisfactory answer to the objection which has been raised against his method, that it introduces into analysis the foreign idea of motion. A quantity thus increasing by uniform fluxion, is what we now call an independent variable.

Newton continues: "Now those quantities which I consider as gradually and indefinitely increasing, I shall hereafter call fluents, or flowing quantities, and shall represent them by the

final letters of the alphabet, $v$, $x$, $y$, and $z$; and the velocities by which every fluent is increased by its generating motion (which I may call fluxions, or simply velocities, or celerities), I shall represent by the same letters pointed, thus, $\dot{v}$, $\dot{x}$, $\dot{y}$, $\dot{z}$. That is, for the celerity of the quantity $v$ I shall put $\dot{v}$, and so for the celerities of the other quantities $x$, $y$, and $z$, I shall put $\dot{x}$, $\dot{y}$, and $\dot{z}$, respectively." It must here be observed that Newton does not take the fluxions themselves infinitely small. The "moments of fluxions," a term introduced further on, are

infinitely small quantities. These "moments," as defined and used in the Method of Fluxions, are substantially the differentials of Leibniz. De Morgan points out that no small amount of

confusion has arisen from the use of the word fluxion and the

notation $\dot{x}$ by all the English writers previous to 1704, excepting Newton and Cheyne, in the sense of an infinitely small increment.35

Strange to say, even in the Commercium Epistolicum