In these days, when payment for goods received is so commonly made in instalments spread over a period, it is not surprising that examination questions on Commercial Arithmetic, or on Arithmetic with a Commercial bias, not infrequently take the form

In a previous paper, which appeared in The Mathematical Gazette for December 1938, the writer showed that, assuming the concept of a rigid rod, it is possible to derive the fundamental formulae of special Relativity from the consideration of photographs taken by two observers in relative uniform motion.

The extension of this method of approach to the consideration of the relativity of time measures appears to furnish some interesting results.

Some time ago a Hungarian post-graduate, Dr. P. Erdös, studying under Prof. L. J Mordell at Manchester University, enunciated an interesting theorem connected with a triangle, for which he asked that a proof should be given.

Triangles with integral sides and circumradius.

A general formula giving such triangles was stated by Gauss, we propose to find a general formula giving prime solutions—that is, a formula giving integral sides and circumradius, a, b, c, R, which have highest common factor unity

If α β γ are the angles of an integral triangle, then sin α, sin β, sin δ and cos α, cos β, cos γ are rational.

It is economical in the use of symbols to express the sides of a triangle fractionally, by dividing the expressions for a, b, c given in my previous Note (Gazette, XXIII, October, 1939, Note 1398, p. 380) by qy. Thus writing h for x/y and K for p/q, we have

a = hk + h − k + 1,

b = hk − h + k + 1,

c = 2(h + k),

and n = hk − 1,

where l, m, n are the medians bisecting a, b, c.

I am no artist, but I believe my artistic friends when they tell me that among the first requisites for good art are simplicity and truthfulness. Architecture is also outside my range; but I have architectural friends too, and they tell me of the need for simplicity and truthfulness in Architecture. After taking my degree in Mathematics I went for nearly a year to a Theological College, and I very soon found out for myself what were the qualities which a Theological College most urgently requires.

Rouse Ball in his Mathematical Recreations tells us that there are no rules for the construction of bordered squares ; I have made the following rules, which are now published for the first time as far as I am aware.

It is remarkable that the rules for odd bordered squares are apparently not known as they are so simple. We use two diagrams, one a skeleton to make the method clear, and the other completed to show the full square.

Rule I. Place the letter M at the (n +1 )th square as shown in Fig. 1, where n denotes the order of the square (here n = 9).

Rule II. Now, commencing at the second column, write the numbers 1, 2, 3, … up to and under M. Then write the next number in the diagonal at the bottom right-hand corner ; the next number in the same column and in the second row, and continue until we arrive at the square above M.

We have all heard the scheme recommended to Headmasters for their guidance in making the first three reports on a new pupil. The first should be devastating in order to impress the parent with the toughness of the job that the school is up against in remedying the defects in the earlier training of the pupil. The second should be glowing in order to show the progress made when the boy gets to the right school. The third should be very guarded in order to moderate any undue optimism created by the second, and to make it clear that any failure that may lie in the future will be due solely to lack of aptitude or application on the part of the pupil. I fear that some of my colleagues in the universities are prone to adopt a similar line of defence. Certainly one not infrequently hears in college common rooms of the shockingly incompetent teaching given in the schools in this subject or in that. One may sometimes suspect that this is a professional particularisation of the proposition that “all good workmen have bad tools”, though every beginner in logic knows that this may not be simply converted to “all who have bad tools are good workmen” Even if the teachers are as bad as some would have us believe, we professors who have had a hand in the training of so many of them had better keep rather quiet about it.

In two recent Notes (1354 and 1289) formulae have been given for the determination of some, but not all, triangles having integral sides and one integral median. All such triangles can be found, with the aid of a list of primes of the form 4n + 1.

From the equation AB2 + AC2 = 2AD2 + 2BD2 it follows that if AB, AC and AD are to be integral, then 2BD2 is an integer, whence, if BC is to be integral, BD is an integer.

I feel somewhat diffident in addressing you on this subject, because I am only too conscious of the fact that my qualifications for doing so are but slight. Most of my audience will have had some experience of teaching algebra to the higher forms in schools, some will have had considerable experience. I have had none. My only knowledge of the subject on which I address you has been gained as an examiner and, a little, as a university don teaching undergraduates who come from your schools. I am as ready as the next man to admit that the examiner does not see everything; to admit, too, that he does not even see all that he thinks he sees. But when all has been said against the examiner—and more than a little has been said of late—the examiner does see something. And so, with these admittedly slight qualifications, I venture to put before you some few points about algebra.

A little to the north of the Yenching University near Peking stand the ruins of the famous Yuan Ming Yuan. It was a palace built during the reign of Emperor K’ang Hsi of the Ch’ing Dynasty, about 200 years ago, and was given to and repaired and enlarged by Yung Cheng, his son. The Chinese unit of length at the time of building may have been different from the modern unit, and it is therefore interesting, mathematically and archaeologically, to determine the old unit by a mathematical method. Additional interest arises from the fact that the cut stones remaining belong to a section of the palace designed in Western style by Jesuit Fathers who came as missionaries to China. Did they use Chinese or French units in their plans?

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service.

The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.

Introduction. Impulses occur not only in technical work but in everyday existence. There is hammering and door banging which may cause annoyance, handclapping which denotes enthusiasm and enjoyment, motoring on rough roads which causes discomfort (especially after a good meal!), speech, electrical communication in the Morse code, and so on ad infinitum. Impulses have been treated by applied mathematicians with reference to various technical problems, e.g. Heaviside in his work on electromagnetic theory. Moreover, the subject is one which merits attention, so in what follows a brief outline will be given of recent work based upon a particular case of the Mellin inversion theorem.

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.

The theorem of J E. Littlewood stated below includes Maclaurin’s integral test and furnishes the basis for a unified theory of convergence criteria for series of positive terms.

Theorem A. If F(x) is positive monotone decreasing and (Dn) is such that

then converges or diverges with .

The diagram which it is the purpose of these notes to explain is so simple in idea that it is surprising that it is not well known. Its idea is merely that the shape of a triangle depends only on two data and can therefore be represented in various ways by the coordinates of a point. To do this, suppose the scale of any triangle reduced so that its largest side is unity and let the other sides be x and y in descending order, then the point whose coordinates are x and y will represent the triangle. Moreover x and y are limited by the inequalities 1 ≥ x, x ≥ y, x + y ≥ 1, so the point representing any triangle lies in the quarter of the unit square shown in the diagram. Of this the boundary line x = 1 corresponds to isosceles triangles whose greater sides are equal, the line x = y to isosceles triangles whose smaller sides are equal, and the third line x + y = 1 to nugatory triangles with two angles equal to zero. The circle x2 + y2 = 1 corresponds to right-angled triangles, and triangles are obtuse-angled or acute-angled according as their representative points lie inside or outside this circle.

Of all the reasons urged in favour of the teaching of mathematics and the favoured position of mathematics in the secondary school curriculum, the most widespread is the belief that to learn mathematics is valuable as a training in the art of reasoning. In this paper I want to explore the implications of this belief and its bearing upon the teaching of geometry I shall try to avoid a pronouncement on whether the belief in the value of mathematics as a training in reasoning is justified, by adopting the mathematical method of exploring the consequences of an assumption.

In the Gazette No. 249, Vol. XXII, Mr. Gibbins has deduced some interesting formulae connected with the triangle ABC, of which the side BC is given by L = lx + my +n= 0 and AB, AC by the equation

The equations of the circles in § 3 of Mr. Gibbins’ note are very elegant indeed. But, in principle, it does not seem desirable to have recourse to pure geometry in dealing with problems in algebraic geometry. The algebraic analysis is not difficult. It is not necessary to resolve S into factors. Let us assume that S≡vw, where v ≡ l2x + m2y + n2 and w ≡ l3x + m3y + n3.