Although some years have elapsed since the first appearance of Professor Halsted’s translations of Lobatchewsky’s Theory of Parallels and John Bolyai’s Science Absolute of Space, they are not so widely known and studied as they deserve to be. There is nothing in the mathematics of these small volumes of a kind to deter any one from reading them; they are short and elementary treatises, disclosing a system of geometry quite distinct from Euclid, by rejecting the axiom concerning parallels. The Science Absolute of Space is a most interesting exposition of one branch of non-Euclidean geometry; but it is unfortunately extremely condensed, and occasionally obscure. It would have been a great advantage if explanatory notes had been added to the English translation.

In working problems on Collision or Impact, the following method is often useful:

Imagine a rigid plane moving with velocity equal to that of the centre of gravity of two bodies, so that during collision it touches and separates the surfaces of contact. (We may either suppose that the mass of the plane is infinite, or that it is constrained to move as stated.) Then the motions of the two bodies due to their mutual impact will obviously be the same as if each separately had impinged on this plane. Thus the problem of two bodies is reduced to two simpler problems of one body impinging on a body whose motion is fixed. And we may by giving to each of the bodies in question, in addition to its own motion, a motion equal and opposite to that of the centre of gravity of the system, bring the plane of impact to rest, so that the motion of each body relative to this plane is the same as that of a body impinging on a fixed body.

This begins with an “Epistle Dedicatory” by Edm. Wingate and the Preface of John Kersey. The Table of Contents divides the whole book into three parts; one half of the first portion was Wingate’s and the other half was Kersey’s improvement upon Wingate, while nearly the whole of the second part and all the Appendix are comprised in “chapters that have been entirely composed by the said J. K.”