Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744) (1744) - Page 15

COROLLARY III.

19. However, to make it more clearly evident how the differentials differentials: the infinitesimal changes in a value, which in modern calculus are represented as derivatives like dy/dx of each degree of $y$ vanish original: "evanescant"; in this context, it means they are eliminated or replaced by simpler variables through substitution through these substitutions, it will be helpful to add the following table.


$dy$	$= p dx$
$ddy$	$= dp dx$	$= q dx^2$
$d^3 y$	$= dq dx^2$	$= r dx^3$
$d^4 y$	$= dr dx^3$	$= s dx^4$
$d^5 y$	$= ds dx^4$	$= t dx^5$
&c.	&c.	&c.

COROLLARY IV.

20. But if the arc of the curve original: "arcus curvae"; this refers to the actual length along the curved path rather than a straight line corresponding to the abscissa $x$ should also occur, along with its differentials of any degree; all these can be expressed through these letters in such a way that no other differentials besides $dx$ are present. For if we set the arc $= w$, it will be:

$w = \int \sqrt{(dx^2 + dy^2)} = \int dx \sqrt{(1 + pp)}$

$dw = dx \sqrt{(1 + pp)}$

$ddw = \frac{p q dx^2}{\sqrt{(1 + pp)}}$

$d^3 w = \frac{p r dx^3}{\sqrt{(1 + pp)}} + \frac{q q dx^3}{(1 + pp)^{3:2}}$

&c.

COROLLARY V.

21. In a similar way, from these [substitutions], the radius of curvature radius of curvature: the radius of a circle that perfectly fits the curve's bend at a specific point; Euler uses the Latin "radius osculi" (kissing radius) or the radius of the curve's bending at any location can be expressed through quantities that appear, at least in form, to be finite. For since, assuming the element $dx$ is constant, the length of the radius of curvature $= \frac{— d w^3}{dx ddy}$ Euler uses a negative sign here to indicate the direction of the radius relative to the curve's concavity; it will become $= \frac{— (1 + pp)^{3:2}}{q}$.