Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744) (1744) - Page 16

COROLLARY VI.

22. Furthermore, from these same substitutions, it will be as follows:

Subtangent subtangent: the segment on the x-axis between the x-coordinate of a point and the point where the tangent line intersects the x-axis $= \frac{y dx}{dy} = \frac{y}{p}$

Subnormal subnormal: the segment on the x-axis between the x-coordinate of a point and the point where the normal line (perpendicular to the tangent) intersects the x-axis $= \frac{y dy}{dx} = py$

Tangent tangent: here referring to the length of the segment of the tangent line from the point on the curve to the x-axis $= \frac{y dw}{dy} = \frac{y \sqrt{(1 + pp)}}{p}$

Normal normal: the length of the segment of the line perpendicular to the curve from the point on the curve to the x-axis $= \frac{y dw}{dx} = y \sqrt{(1 + pp)}$

And, in a similar way, all finite quantities pertaining to the curve—unless they involve integrals original: "integralia"; Euler distinguishes between properties that can be calculated at a single point (finite) and those requiring the sum of parts along the curve (integrals)—can be expressed through finite quantities of this kind in such a way that no differentials seem to be present any longer.

DEFINITION IV.

23. The Formula of the Maximum or Minimum, for any problem, shall be for us that quantity which must obtain the maximum or minimum value in the sought-after curve.

COROLLARY I.

24. Since in all problems to which this method is adapted, a curve is sought which—either among all possible curves, or only among countless curves determined in a specific way—enjoys the property of being a maximum or minimum; this very property, which must be the maximum or minimum in the sought-after curve, will be a quantity, and it will be expressed by the Formula, which we here call the Formula of the maximum or minimum.