Page:A History of Mathematics (1893).djvu/260

 graphical methods, epicycloids, conchoids, and on magic squares. Michel Rolle (1652–1719) is the author of a theorem named after him.

Of Italian mathematicians, Riccati and Fagnano must not remain unmentioned. Jacopo Francesco, Count Riccati (1676–1754) is best known in connection with his problem, called Riccati's equation, published in the Acta Eruditorum in 1724. He succeeded in integrating this differential equation for some special cases. A geometrician of remarkable power was Giulio Carlo, Count de Fagnano (1682–1766). He discovered the following formula, $\scriptstyle{\pi=2i\log\frac{1-i}{1+i}}$,|undefined in which he anticipated Euler in the use of imaginary exponents and logarithms. His studies on the rectification of the ellipse and hyperbola are the starting-points of the theory of elliptic functions. He showed, for instance, that two arcs of an ellipse can be found in an indefinite number of ways, whose difference is expressible by a right line.

In Germany the only noted contemporary of Leibniz is Ehrenfried Walter Tschirnhausen, who discovered the caustic of reflection, experimented on metallic reflectors and large burning-glasses, and gave us a method of transforming equations named after him. Believing that the most simple methods (like those of the ancients) are the most correct, he concluded that in the researches relating to the properties of curves the calculus might as well be dispensed with.

After the death of Leibniz there was in Germany not a single mathematician of note. Christian Wolf (1679–1754), professor at Halle, was ambitious to figure as successor of Leibniz, but he "forced the ingenious ideas of Leibniz into a pedantic scholasticism, and had the unenviable reputation of having presented the elements of the arithmetic, algebra, and