"Johannes Kepler... imbibed Copernican principles while at the University of Tubingen. His pursuit of science was repeatedly interrupted by war, religious persecution, pecuniary embarrassments, frequent changes of residence, and family troubles. In 1600 he became for one year assistant to... ... His first attempt to explain the solar system was made in 1596, when he thought he had discovered a curious relation between the five regular solids and the number and distance of the planets. The publication of this pseudo-discovery brought him much fame. At one time he tried to represent the orbit of Mars by the oval curve which we now write in polar coördinates, \rho = 2r cos^3\theta. Maturer reflection and intercourse with Tycho Brahe and Galileo led him to investigations and results worthy of his genius—"Kepler's laws." He enriched pure mathematics as well as astronomy. It is not strange that he was interested in the mathematical science which had done him so much service; for "if the Greeks had not cultivated s, Kepler could not have superseded Ptolemy." The Greeks never dreamed that these curves would ever be of practical use; Aristaeus and Apollonius studied them merely to satisfy their intellectual cravings after the ideal; yet the conic sections assisted Kepler in tracing the march of the planets in their elliptic orbits. Kepler made also extended use of logarithms and decimal fractions, and was enthusiastic in diffusing a knowledge of them. At one time, while purchasing wine, he was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum [Vinariorum] in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized 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 , 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 assertion that the circumference of an ellipse, whose axes are 2a and 2b, is nearly π (a + b); (2) a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear; (3) the assumption of the principle of continuity (which differentiates modern from ancient geometry), when he shows that a has a focus at infinity, that lines radiating from this "cæcus focus" are parallel and have no other point at infinity. The Stereometria led Cavalieri... to the consideration of infinitely small quantities."