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.^[1]

The following extract is from “‘A new method for maxima and minima...’’ by Gottfried Wilhelm von Leibniz.^[2]

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:

If $a$ is a given constant,

$da$ = 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

$=dz-dy+dw+dx$.

---

1. The Latin is frequently bad. The translator wishes to acknowledge her indebtedness to Professors Carter and Hahn, both of Hunter College, who kindly made a number of corrections.
2. "Nova methodus pro maximis & minimis, itemque tangentibus, qua nec irrationales quantitates moratur, & singulare pro illis calculi genus, per G.G.L." (his Latin initials) Acta Eruditorum, October, 1684.