Lunar Theory from the 1740s to the 1870s – A Sketch

Abstract

The attempt to cope with Newton’s three-body problem not geometrically as Newton had done but algebraically, using the calculus in the form elaborated by Leibniz, got under way in the 1740s. That this attempt had not been made earlier appears to have been due to lack of an appreciation, among Continental mathematicians, of the importance of trigonometric functions for the solution of certain differential equations; they failed to develop systematically the differential and integral calculus of these functions. Newton had used derivatives and anti-derivatives of sines and cosines, but had not explained these operations to his readers. Roger Cotes, in his posthumous Harmonia mensurarum of 1722, articulated some of the rules of this application of the calculus. But Euler, in 1739, was the first to provide a systematic account of it. In the process he introduced the modern notation for the trigonometric functions, and made evident their role qua functions. Thus sines and cosines having as argument a linear function of the time, t, could now be differentiated and integrated by means of the chain rule. Differential equations giving the gravitational forces acting on a body could be formulated and solved – though only by approximation.