Using sinh and cosh to solve the cubic

Abstract

We perhaps take for granted the use of the hyperbolic functions sinh x: and cosh x and assume that they always were defined in terms of the exponential function as 1/2(exp (x) ? exp ( – x)). In fact these very useful functions, so prominent in the study of growth and decay, were first motivated by the geometrical analysis of the unit hyperbola compared to the unit circle. The first one to use them, Vincentius Riccati, constructed a kind of logarithm for reducing multiplication to addition. Although he was aware of the exponential function exp(x) he built his ‘analogous logarithm’ structure on ordinary geometry and proceded to prove the theorems we are familiar with today. Among other things he provided a geometrical structure for finding the roots of the cubic x 3 — 3ax — b = 0 according to the method of Cardan. This paper presents the geometry that Riccati used.