Page:A History of Mathematics (1893).djvu/240

A curve is here considered to be a polygon. The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. It was first employed by Barrow in England, but appears to have been reinvented by Leibniz. From it Leibniz observed the connection existing between the direct and inverse problems of tangents. He saw also that the latter could be carried back to the quadrature of curves. All these results are contained in a manuscript of Leibniz, written in 1673. One mode used by him in effecting quadratures was as follows: The rectangle formed by a sub-tangent p and an element a (i.e. infinitely small part of the abscissa) is equal to the rectangle formed by the ordinate y and the element l of that ordinate; or in symbols, $\scriptstyle {pa=yl}$. But the summation of these rectangles from zero on gives a right triangle equal to half the square of the ordinate. Thus, using Cavalieri's notation, he gets

$\scriptstyle {{\text{omn. }}pa={\text{omn. }}yl={\frac {y^{2}}{2}}}$ (omn. meaning omnia, all).

But $\scriptstyle {y={\text{omn. }}l}$; hence

$\scriptstyle {{\overline {{\text{omn. }}{\overline {{\text{omn. }}l}}{\frac {l}{a}}}}={\overline {\frac {{\text{omn. }}l^{2}}{2a}}}}$.

This equation is especially interesting, since it is here that Leibniz first introduces a new notation. He says: "It will be useful to write $\scriptstyle {\int }$ for omn., as $\scriptstyle {\int l}$ for $\scriptstyle {{\text{omn. }}l}$, that is, the sum of the l's"; he then writes the equation thus:—

$\scriptstyle {{\frac {\int {\overline {l^{2}}}}{2a}}=\int {\overline {\int {\overline {l}}{\frac {l}{a}}}}}$.

From this he deduced the simplest integrals, such as

$\scriptstyle {\int x={\frac {x^{2}}{2}},\quad \int (x+y)=\int x+\int y}$.