Page:Calculus Made Easy.pdf/78

work will be done. If we write $w$ for work, then an element of work will be $dw$; and we have

$dw=f\times dy;$;

which may be written

$dw=ma\cdot dy$;

or

$dw=m{\frac {d^{2}y}{dt^{2}}}\cdot dy$;

or

$dw=m{\frac {dv}{dt}}\cdot dy$.

Further, we may transpose the expression and write

${\frac {dw}{dy}}=f$.

This gives us yet a third definition of force; that if it is being used to produce a displacement in any direction, the force (in that direction) is equal to the rate at which work is being done per unit of length in that direction. In this last sentence the word rate is clearly not used in its time-sense, but in its meaning as ratio or proportion.

Sir Isaac Newton, who was (along with Leibniz) an inventor of the methods of the calculus, regarded all quantities that were varying as flowing; and the ratio which we nowadays call the differential coefficient he regarded as the rate of flowing, or the fluxion of the quantity in question. He did not use the notation of the $dy$ and $dx$, and $dt$ (this was due to Leibnitz), but had instead a notation of his own. If $y$ was a quantity that varied, or “flowed,” then his symbol for its rate of variation (or “fluxion”) was