Page:A Source Book in Mathematics.djvu/658

 These points are illustrated in the following selections from two articles that were published in the Acta Eruditorum.

The following extract is from “‘A new method for maxima and minima...’’ by Gottfried Wilhelm von Leibniz.

Let there be an axis $$AX$$ and several curves, as $$VV$$, $$WW$$, $$YY$$, $$ZZ$$, whose ordinates $$VX$$, $$WX$$, $$YX$$, $$ZX$$, normal to the axis, are called respectively, $$v$$, $$w$$, $$y$$, $$z$$; and the $$AX$$, cut off from the axis, is called $$x$$. The tangents are $$VB$$, $$WC$$, $$YD$$, $$ZE$$, meeting the axis in the points $$B$$, $$C$$, $$D$$, $$E$$, respectively. Now some straight line chosen arbitrarily is called $$dx$$, and the straight [line] which is to $$dx$$ as $$v$$ (or $$w$$, or $$y$$, or $$z$$) is to $$VB$$ (or $$WC$$, or $$YD$$, or $$ZE$$), is called $$dv$$ (or $$dw$$, or $$dy$$, or $$dz$$) or the difference of the $$v$$’s (or the $$w$$’s, or the $$y$$’s, or the $$z$$’s). These things assumed, the rules of the calculus are as follows:

$$da$$ = 0,

and

$$d\overline{ax}=adx$$

if

$y=v$,

(or [if] any ordinate whatsoever of the curve $$YY$$ [is] equal to any corresponding ordinate of the curve $$VV$$),

$dy = dv$.

Now, addition and subtraction:

if

$z-y+w+x=v$,

$d\overline{z-y+w+x}=dv$,

or

$=dz-dy+dw+dx$.