Newton'sPrincipia contains a neglected passage, which is nevertheless the true source of inertial navigation, occurring in De Motu Corporum Liber Primus, toward the end of ‘Sect. X. De Motu Corporum in Superficiebus datis, deg; Funipendulorum Motu reciproco’ as ‘Corol. 2’ to Prop. LIII. Prob. XXXV. Concessis figurarum curvilinearum Quadraturis, invenire vires quibus corpora in datis curvis lineis Oscillationes semper Isochronas peragent’. The corollary reads in the first edition:

‘Corol. 2. Igitur in Horologiis, si vires a Machina in Pendulum ad motum conservandum impressae ita cum vi gravitatis componi possint, ut vis tota deorsum semper sit ut linea quae oritur applicando rectangulum sub arcu TR & radio AR, ad sinum TN, Oscillationes omnes erunt Isochronae’.

This text shows only one or two very minor changes in the second and third editions, but the two extra drawings added in the third edition are helpful. Cajori, in a readily available publication quite adequate for reference here, gives these drawings, and his version of Motte's translation reads:

‘Cor. II. And therefore in clocks, if forces are impressed by some machine upon the pendulum which continues the motion, and so compounded with the force of gravity that the whole force tending downwards will be always as a line which is obtained by dividing the product of the arc TR and the radius AR, by the sine TN, then all the oscillations will become isochronous.’