The title of this paper was chosen before I had put my thoughts together, and even before I had considered thoroughly whether it were possible to make any general conclusions at all. I hope I shall not be accused afterwards of making something out of nothing, or of seeing connections where none exist. Perhaps it will be that, instead of having a vague notion that mathematics and morals are strangers or opposites, we may arrive at the end of the evening with a reasoned conviction that they are indeed so. But that the subject is not unprofitable to discuss I am strengthened in believing by noticing that Henri Poincaré, perhaps the greatest mathematician of the last fifty years, in the closing years of his life wrote an essay on the “Relations of Morals to Science,” which is included in the volume of his last thoughts.

Let A B C be three travellers going along a straight road, each at constant speed. Let the following three events happen in succession, the passing of C by A at O (Fig. 1), the passing of B by A at V, and the passing of B by C at W. At the meeting of A and C let A set his watch by C’s, and at the meeting of A and B let B set his watch by A’s. Our question is whether the watches of B and C will agree when they meet at W.

In the traditional text-book presentation and evaluation of logarithmic, exponential, circular and hyperbolic functions there is a lack of uniformity of method, and the demands on the analytical powers of the average student are considerable, if not excessive It has been shown in The Mathematical Gazette of June, 1926, that logarithmic and exponential functions may readily be evaluated by ab initio arithmetical methods. Incidentally it may be noted that the theory of the method is independent of the theory of indices. The definition of loge a is virtually and leads to a simple assignment of meaning to irrational and imaginary powers of a number.

In all general equations in mechanics the fundamental units, length, time, mass, enter homogeneously; the result is that these equations are true whatever the units of length, time, and mass may be. Angles, however, occupy an anomalous position. An equation which involves angles, angular velocities or accelerations, is true only for a certain particular unit of angle, usually the radian. Thus the formula T = 2π/ω, for periodic time with angular velocity ω, is only true when the angular unit is the radian; so also for arc = rθ, or for . The reason is, that these equations are not homogeneous in angle; π, sin θ, and cos θ are pure numbers, arc and r are lengths. In explanation it is commonly said that angle is of zero dimensions in length, time, and mass. So it is, and yet it is not a pure number, but a quantity just as much as length or any other concrete magnitude. If π were the symbol for a certain angle, instead of being a pure number, equations like T — 27π/ω, α+β + γ=π, etc., would become homogeneous, and true for all angular units, π would then resemble the symbol g, which stands for a particular acceleration, 32.2 ft./sec.2 or 981 cm./sec.2, etc.; but π has too long been identified with the number 3.14159… , and is too useful in all sorts of questions not involving angles at all, to be appropriated in this way. If, however, we introduce another symbol, say P, for the angle π radians or the angle of half a revolution, we may make all our equations homogeneous in angles, and restore their generality; it will do equally well, and be generally simpler, if instead of P we use the symbol K, denoting an angle of one radian.