Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.

Some seven years ago an attempt was made to tell something of the story of Francis Walkingame, the author of The Tutor’s Assistant, which was first issued in 1751 and became the most popular arithmetic text book for over a century afterwards. Recent contributions to the Gazette shows that this author and his famous book are still unknown to many readers. Interest was aroused by Dr. T. M. Flett’s description of Edward Winder’s Exercise Book, which was clearly based on an edition of The Tutor’s Assistant of about 1830. In his description of John Stewart’s Exercise Book, written at Dundee some thirty years earlier, Mr. W More remarked that the rules were the same as Winder used and quoted additionally the Rule of False in very similar words to those used by Walkingame. This example is interesting as showing that Walkingame’s book was used in Scotland at the turn of the century. Richard Hartley’s 1846 letter gives a list of the later sections in Part I (Integers) of Walkingame and shows that he had just started Part II on Vulgar Fractions.

Several books on Group Theory show that the isomorphism between the group of proper rotations of a cube and P4, the group of 24 permutations on four arbitrary symbols, can be established by a consideration of the four diagonals of the cube. To each arrangement of these diagonals corresponds a unique position of the cube and therefore a unique rotation from one position as reference to that position. The extension of this result to other polyhedra does not appear in the books that I know.

As Geometry was developed in recent centuries, it became realised that some of the more interesting results were not metrical in nature, and that even some of the metrical theorems were immediately derivable from more general non-metrical ones. For example, mid-point theorems were often derivable from harmonic theory, and confocal conics from general tangential pencils. It seemed inelegant as well as unnecessary to use the ladder of a metric to achieve non-metrical results and then to kick it away as if it had never existed; a far more satisfactory procedure would be to develop pure geometry from pure non-metrical axioms.

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process. Then certain standard forms are evaluated, the rules for application to differential equations are laid down and a selection of worked examples follows. But such a treatment can mystify a student who wants to know why, in following these rules, he should, in effect, firstly, multiply each term of the equation by e-px, secondly, integrate the resultant products from 0 to ∞, and, thirdly, obtain in this way the complete solution.

Suppose there are n towns every pair of which are connected by a single one-way road (roads meet only at towns). Is it possible to choose the direction of the traffic on all the roads so that if any two towns are named there is always a third from which the two named can be reached directly by road?

If a, b, and c are the sides of a triangle then

We shall prove an inequality in the reverse direction.

We shall also prove that

The first session of this Institute was in the summer of 1950. At the 1949 Vancouver meeting of the Canadian Mathematical Congress a Committee on Research was set up with R. L. Jeffery, who was then Head of the Mathematics Department of Queen’s University, as Chairman. He raised with the National Research Council the question of a central location at which Mathematicians could carry on their programmes of research during the summer. Several reasons for such an arrangement were presented.

It is now a truism that apparent defects in our methods of studying and teaching mathematics are causing anxiety to all those who are concerned with the subject. There is, in fact, a widespread feeling that some positive action is needed—together with some uncertainty about what the action should be and who should do it. The inevitable feeling of frustration that follows is hardly surprising in the light of the technical complexity of the subject and the diffusion of control among the several bodies which are responsible for it.