1. Let f(x) be a real function of a real variable x. The meanings of when λ is a positive integer, a negative integer and zero, are well known. In the first case, denotes the λth integral of f(x) with respect to x, with an arbitrary lower limit of integration. In the second case, stands for the (−λ)th differential coefficient of f(x) with respect to x. Lastly, when λ = 0, means f(x).

1. When we proceed to obtain the equation of the join of points on the ellipse x2/a2 + y2/b2 = 1 whose eccentric angles are θ and ϕ, we find that cos ½ (θ − ϕ) is a factor of each term, and similarly a common factor is found to exist for any other case of parameters. We may therefore suspect the existence of a common factor when we find the equation of the secant in the general case. It is proposed to find this common factor and to apply the simplified equation to the solution of a well-known problem.

When the subject of the teaching of differentials was discussed by this Association at its Annual Meeting two years ago, I read a paper on the theoretical aspect of the subject. In that paper I maintained that it is futile to discuss the teaching of differentials until we have agreed upon the theory of differentials which we wish to teach, and I proceeded to summarise the three main theories of differentials. It was obvious that each of these theories was intrinsically more difficult and abstract than the theory of limiting processes which has become the classical method of introducing the calculus, and it was unanimously agreed that each theory was, in practice, quite unteachable.

It has been shown that the evolution of logarithms involves consideration of Napier’s knowledge of Greek; this in turn is connected with his “educational and cultural environment”. He certainly studied abroad for some years but where he spent the time is largely a matter for speculation. Now even speculation must be based on some form of reasoning, therefore let it be taken as a hypothesis that as his mathematical equipment was considerably in advance of that of his contemporaries, this equipment must have been the result of ideas planted by his teachers. Consequently an explanation of Napier’s travels may be made by considering his work in mathematics and by discovering similarities in the works of mathematicians who were in Europe during the years of his residence abroad. It must be remembered also that it was a time of acute religious differences, and that he belonged to a family that was ardently Protestant in its sympathies. His own Protestantism is quite openly expressed in his work on the Apocalypse. Therefore these speculations are based on the “historico-religious” background of the period and on Napier’s mathematical attainments.