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 the Rules of this Association, there is a declaration that its object shall be

Having regard to such a policy in action, having regard also to the professional duties of what probably is the majority of the members, I must appeal for a measure of your indulgence when my address, in retiring from office as your President, is devoted to some reflections concerning the present range of elementary Applied Mathematics.

Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .