I am venturing to call the attention of the Association to a difficulty that I have to contend with, as a private tutor, when assisting boys fresh from school either to continue their mathematical studies or to prepare for some of the public examinations. It is of course impossible for me to assert positively that this difficulty is experienced by tutors generally, but the strong conviction that it is felt by others as well as by myself has led to my bringing the matter forward.

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.

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′.

This product of Queen Anne’s reign consists of five parts: I. Arithmetick, Vulgar and Decimal. II. Algebra or Arithmetick in Species. III. The Elements of Geometry. IV. Conick Sections. V. The Arithmetick of Infinites explained, with its application to Superficial and Solid Geometry. To these is added an Appendix of Practical Gauging, and in the eighth edition is also found the History of Logarithms.