CONNECTED WITH HUMAN MOETALITT. 
513 
among the causes of my preference. I call the differential notation furtive, on I think a 
moral ground, and also on the ground of its introducing an interruption and an incon- 
venience in practice. The moral ground is, that it appears to give Leibnitz a greater 
claim to originality, to the prejudice of Newton, than I think he is justly entitled 
to. And the other ground is, it steals from the alphabet a letter — and one which it 
is most convenient to retain, in order to keep up the regular order of notation — to use 
it for a purpose of different intent to that for which it was originally used ; and may 
introduce confusion. And with respect to the superior advantage of the fluxional 
calculus over the differential calculus, I observe that if x and y be rectangular coordi- 
nates of a curve in a plane, and z the length of the curve from a given point in it to 
the point of which x and y are coordinates, the fluxional calculus gives z=\/ x^+y"^^ 
which is strictly true ; and may be proved to be so without the introduction of infinitely 
small quantities: but the differential calculus gives dz='^ dx\ -\-dy\ , true only on con- 
sideration of infinitely small quantities ; and even with that consideration it cannot be 
proved luminously to be true; because only expresses the length of the 
chord of an infinitely small arc, and not of the arc itself, as they have no part common 
with each other, but at the points of intersection: but in the fluxional calculus 
which, I think, is a much neater and a much more commodious expression, 
z, X, y only express finite values ; namely, the velocities in the several directions, at the 
point to which x and y are coordinates, with which the point describing the curve is 
moving in the relative directions parallel to the axes of x and y^ and that of the tangent 
to the curve at the point ; in the same way as if all causes which might incurvate the 
future path of the point were to cease; and similar observations may be made with 
respect to the luminous character of the fluxional analysis, when compared with the 
differential analysis, in the application of it to physical subjects. Thus if / be a force 
acting on a body, v the velocity which is generated by its momentary impulse, that is to 
say its single impulse, by which is meant the finite space the body would describe in 
consequence of it in the finite time i, but not the variable portions of space it would 
describe if that force were considered to be an infinitely small action, as it were continually 
active during infinitely small portions of time ; the fluxional calculus gives and is 
correctly true, however large v and t are. But the differential analysis f .dt=dv^ 
which is correct only if dt and dv are infinitely small, and is then only to be considered 
so in virtue of the hypothesis that infinitely small quantities of the second and higher 
degree may be omitted. 
But whilst I am endeavouring to clear away the shadowing clouds which may obstruct 
the briUiant light of Newton’s lamp from being duly perceived by the scientific eye, I 
am willing to acknowledge that Leibnitz’s differential d has, in many instances, done 
great service in his own hands, in the hands of Eulbe, Lageange, Laplace, and of a host 
of scientific men whose names cannot be pronounced without gratitude and reverence. 
Amd I observe that other letters, such as /, &c., when used as characteristics 
4 A 2 
