Page:A History of Mathematics (1893).djvu/188

 solids known to Archimedes and then takes up others. Kepler introduced a new idea into geometry; namely, that of infinitely great and infinitely small quantities. Greek mathematicians always shunned this notion, but with it modern mathematicians have completely revolutionised the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the Method of Exhaustion, which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria.

Other points of mathematical interest in Kepler's works are (1) the statement of the earliest problem of inverse tangents; (2) an investigation which amounts to the evaluation of the definite integral $$\scriptstyle{\int\limits_{0}^{\phi} \sin \phi d \phi = 1 - \cos \phi}$$; (3) the assertion that the circumference of an ellipse, whose axes are $$\scriptstyle{2a}$$ and $$\scriptstyle{2b}$$, is nearly $$\scriptstyle{\pi(a+b)}$$; (4) a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear; (5) the assumption of the principle of continuity (which differentiates modern from ancient geometry), when he shows that a parabola has a focus at