Plane Analytical Geometry, and elementary Trigonometry, are the beginnings of the systematic application of mathematical analysis to Geometry, i.e. of systematic expression in terms of “number”—which is essentially “real” number.

Complete application of the generality of the algebra (within the restriction to real number) is, of course, the power of the method; and, in particular, sign of the (real) numbers is significant throughout.

Before the end of the first year of the H.S.C. course, a boy has dealt with the elementary theory of indices and logarithms and has been introduced to the exponential and logarithmic functions in Calculus. This introduction, the writer has suggested in a previous Note (1896), may well be intuitive and may be based upon the fact that if y = ax, then dy/dx = ky, the fact being established by reading off gradients and ordinates from a suitable graph. It is now suggested that when the H.S.C. course in Calculus is completed, a mathematically-minded boy may learn a great deal of mathematical technique from a treatment of the exponential and logarithmic functions of the kind outlined below.

There are many methods of solving algebraic equations whose coefficients are real and have given numerical values. In practice it very often happens that the coefficients are not given numerically, but are expressed in terms of certain parameters, and that what is wanted is not the roots of the equation for special values of the parameters, but an indication of the way in which the roots behave as these parameters vary between certain limits. For example, if the equation is the period equation of a mechanical or electrical system, we may wish to know how its roots vary as certain components of the system are changed, not merely the values of the roots for definite values of the components. The method sketched below has been found very useful for studying cubics from this point of view and applies quite well to quartics. It also provides interesting elementary examples in the theory of equations.

Figure 1 is described in detail in my article in the Mathematical Gazette, XIX, No. 236. it is based on a tetrahedron ABCD and an arbitrary point D2. These five points determine all the other points of the figure, and if the tetrahedron ABCD is taken as the tetrahedron of reference and the coordinates of D2 are (α0, β0, γ0, δ0), the coordinates of A1, B1, C1, D1 are (- α0, β0, γ0, δ0), (α0,β0,-γ0, δ0), and so on. in what follows the tetrahedron A1B1C1D1 is called the associated tetrahedron.

Starting with any triangle Δ ≡ ABC, let Δ1 ≡ A1B1C1 be its pedal, Δ2≡A2B2C2 the pedal of Δ1 and so on indefinitely Since the circumradius Rn of Δn is equal to 1/2Rn-1, we may expect a steady contraction towards some definite limiting point. It is proposed to inquire into the position of this point.

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality

Suppose that a particle of mass m is attached to a fixed point O, by means of a light inextensible string of length a, and when hanging at rest is suddenly-given a velocity u0 at right angles to the string. The earlier phases of the motion are dealt with in standard textbooks, but I do not recall having seen a complete analysis of the later stages in the case when the particle leaves its circular path. It is well known that if u02 ≤2ag, the particle executes oscillations like a simple pendulum (this will be referred to later as the case of simple oscillations), while if u02≥5ag, it makes complete revolutions in a circle about O. If 2ag<02< 5ag, the particle leaves its circular path and then travels in a parabolic path until the string becomes taut again. The textbooks do not follow the motion beyond this point, perhaps because it is realised that no real string fulfils the condition of being inextensible. If, however, we assume that the ideal string exists, the resulting motion has some rather surprising features. Owing to the obvious loss of energy, the particle when it first passes through the lowest point again will have a smaller velocity that it had initially, and so will either execute simple oscillations or will leave its circular path at a lower point than previously and lose further energy when the string tightens a second time. It would at first appear that ultimately the energy must decrease to such a value that the particle will finally be moving with simple oscillations. This is not always true, as the following analysis will show.

Dans son “Mémoire sur le tétraèdre (Bulletin de l’Académie royale de Belgique, 1884), J Neuberg a donné des propriétés du point L dont les coordonnées normales sont proportionnelles aux rayons des cercles circonscrits aux triangles des faces d’un tétraèdre isodynamique. Nous avons étendu ces propriétés au point L défini de la même façon dans un tétraèdre quelconque T≡ABCD (Annales de la Société Scientifique de Bruxelles, 1922, pp. 173-177). P Delens lui a consacré aussi des développements mais par des méthodes assez difficiles à lire (Mathesis, 1937 et 1938).

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.