We have seen that for solving problems on the simultaneous impact of more than two bodies, we require in addition to the laws of Abstract Dynamics, not only a generalization of Newton’s Empirical Law as to change of relative velocity, but also the assumption that the impacts are simultaneous in the special sense of having their greatest compression at the same instant. In other words, it is only in a particular set of special cases that elementary methods can lead to a solution.

John Dee, commonly though improperly called Dr. Dee, whose life extended over more than eighty years (1527-1608), was a mathematician and astronomer for more than fifty years, and a magician and raiser of spirits in his old age. It is by these latter practices that he is generally best known, and full accounts of them are given in the Dictionary of National Biography and in the life of Sir Edward Kelley in the recently published book, Twelve Bad Men, edited by Mr. Seccombe.

I was led to the present subject by observing that every one, almost without exception, is interested in astronomy, yet from lack of learning spherical trigonometry very few of those who have passed through an ordinary school and college course of mathematics can understand a book of physical astronomy, or at least any other than a very elementary and popular exposition of this royal science. The ordinary well-educated man cannot only not predict an eclipse, but he has no idea of the method followed in the prediction. His astronomy is all taken on faith or on authority; he knows only results, nothing of processes. The work of Kepler is as much a mystery to him as a theological dogma. If he possess a telescope, he can use it little better than a toy; he has no power of adding anything to the sum of astronomical knowledge. Now this seems a pity: I think it is not necessary, for my experience tells me that spherical trigonometry is not much harder than plane trigonometry.

In problems where more than two bodies are simultaneously in collision, the rules usually stated in the text books are not, in general, sufficient for their solution. It has been suggested to me that I should continue the subject of Collision by a discussion of the method required in such cases.

In what follows I confine myself to the case of smooth spheres impinging on one another, or on fixed bodies, so that rotations do not fall to be considered.

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.

The majority of text-books on Geometrical Conies do not give any proof of the fundamental property that one and only one conic passes through any five given points, no three of which lie on a straight line. Special attention is drawn to this omission by the Rev. Dr. C. Taylor in an article published in the Gazette for May, 1895. The following is a description of three methods which various attempts at obtaining an elementary proof of the theorem have suggested. It is, however, very desirable to have a simpler proof than any of these, if such can be given.

Let PQ, PQ′ (Fig. 1) be tangents to a conic, S, H the foci, Q′Cq a diameter, then Kq the tangent at q is ⅡPQ′. By a construction and a proof identical with those of an article in the second number of this Gazette we can show that rectle PQ.QK = SQ.QH, ∠ SKQ = QHP = PHQ′, ∆s PKS and PHQ′ similar, and ∴ rectle PS . PH = PK. PQ′. Through C draw a straight line ⅡQQ′, meeting PQ, PQ′ in F, F′, and cutting the tangent qK in E. QQ′ being ⅡFF′ and bisected by CP, CF = CF′; also Q′q being a diameter, CE = CF′. Thus E and F are the same point and coincide with K, and ∴ KC is ⅡQQ′.

The square of the tangent from a point P to a circle S, divided by the diameter of S, will, in what follows, be denoted by (PS).

If S open out into a straight line, it is easily seen that (PS) equals in the limit the perpendicular distance of P from this line; accordingly the length of the perpendicular from a point P on a straight line S will be denoted by the same symbol (PS).