This essay-review of Jeffreys’ very welcome and valuable Tract with the above title has been written at the editor’s request. Many readers of the Gazette must have heard of Heaviside’s operational method of solving the equations of dynamics and mathematical physics. If they have tried to learn about them from Heaviside’s own works, they have attempted a difficult task. Nothing more obscure than his mathematical writings is known to me. A Cambridge Tract is now at their disposal. Prom it much may be learned; but the air of mystery still—at least in part—remains.

The theory of the orthocentre is usually considered in conjunction with ordinary triangles only, that is to say, with triangles whose three vertices or sides are distinct. The treatment for the case when two or more sides or vertices coincide is generally overlooked, but it leads to many interesting results and applications. An exposition of some of these is the object of this note.

On previous occasions in the Gazette (and in other places) I have repeatedly expressed the opinion that the most natural line of approach to the logarithmic function is the historical one (followed both by Napier and by Newton) in which the logarithm is effectively defined as an integral. From conversation with boys (from eleven to fourteen) I am still convinced that the idea of an area is much easier to grapple with than the idea of a limit. In spite, then, of other suggestions (which appear in the Gazette three or four times a year on the average), I still accept this method as the best: more details will be found in Appendix II. of my book on Infinite Series (both editions).

The Editor of the Mathematical Gazette has kindly allowed me to see an advance proof of Prof. Carslaw’s paper, and I take the opportunity of making some comments both on that paper and on Dr. Jeffreys’ Tract.

p, q, r are lengths of perpendiculars from A, B, C on line, and have same or opposite signs according as A, B, C are on same or opposite sides of the line.

It is worthy of note that the imaginary was introduced into mathematics in the course of strictly non-utilitarian researches on the theory of quadratic equations, being for a long time rejected by the mathematicians as an entirely useless conception. This formal condemnation of the imaginary seems less extraordinary when we recollect that at the time of its inception, and indeed for long afterwards, serious scruples were experienced by some mathematicians even regarding the admission of negative quantities in their results. The last writers who objected to the use of negative quantities were Baron Maseres, a Fellow of Clare College, Cambridge, and William Frend, a second wrangler. Both these men had written extensively in condemnation of the use of negative quantities; their attitude in this matter is clearly evinced by the following extract from the preface to Frend’s Principles of Algebra, published in 1796, at which period, strangely enough, the notion of the imaginary was firmly eatablished.

It is sometimes fascinating to return to the old geometry and try to find new properties of the triangle and quadrilateral. One of the most interesting ways of discovering such properties is to start with some well-known result and generalise it. The following notes are the result of an attempt to find circles in a triangle which satisfy the same kind of conditions as the Brocard circle.

This is a clumsily long title, but it might have been much more lengthy if it had gone on to suggest how the problem in probability arose. That would involve us in many questions, so we may say at once that the problem is equivalent to the following: If from a bag containing M different balls N drawings are made, where M, N are very large numbers (the ball being replaced after each drawing), what is approximately the chance that exactly r balls should fail to appear?

It is generally recognised that the proofs given in the school text-books on Geometry are incomplete and that many theorems are assumed because they are intuitively ‘obvious,’ and one of my aims this afternoon is to show wherein this incompleteness lies and to state the ‘obvious’ assumptions.

We start, as Euclid did, with the vague notion that, if A and B are two magnitudes of the same kind, the ratio of A to B is some relation between them in respect of magnitude, i.e. in respect of relative magnitude, independent of the actual magnitude of each.

When I began the preparation of this paper, I turned up the bulky report of 1912 published by the Board of Education. In the article by Prof. G. H. Hardy I found this passage :

In view of the popular appeal of Lawn Tennis, and of the fact that a large number of professional mathematicians depend upon this game for much of their recreation, it seems strange that no one has pointed out that a good deal may be learned (as to suitable tactics) by applying a few quite simple principles of dynamics and geometry to the game.

In an equation with real coefficients, if the terms are vectors, they must be codirectional if the equation is possible with all the quantities real. Thus to obtain a geometrical solution of the equation a cos θ + b sin θ =c, a and b must be drawn at right angles to each other since a cos θ makes an angle θ with a, and b sin θ makes an angle 90° − θ with b. The solution is obtained by the intersections of the circles r=a cos θ + b sin θ, r=c.

The object of the first conversation (Math. Gazette, March 1928) was to suggest the advantages of using a definition of right angle different from that of Euclid for the study of “spaces” of which the dimensions can be distinguished from each other, a very common type of space. For brevity we will refer to this study as “Dimensional Geometry,” and call the right angles “dimensional right angles.” The second conversation is intended to suggest that “dimensional spaces” facilitate the study of Non-Euclidean Geometry.

If a rod of fixed length a slides with its ends on two fixed lines at right angles, its envelope is an astroid or hypocycloid of four cusps whose equation referred to the two fixed lines as axes is .

If the two lines are not at right angles but inclined at an angle 2a (a < 46°), the envelope of the rod is a curve with four eusps in shape like an astroid elongated along one of the bisectors of the axes. This curve can be shown to be an involute of the astroid.