If a1, …, an and b1, …, bn are two sets of real numbers, then (a1b1 + … + anbb)2 ≤ (a12 + … + an2)(b12 + … + bn2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b1 = … = bn = 0. Write a = (a1, …, an) and b = (b1, …, bn). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x1...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.

In England the sceptre of Mathematics passed from the hands of Wallis to the man who was to raise the exact sciences to a height hitherto unattained—Isaac Newton (1642-1727), that sovereign genius before whom the world of science will ever bow with the profoundest reverence.

The familiar fact that we may cut any triangle into three polygonal pieces which can be rearranged to form a rectangle is one of the most elementary problems of polygonal dissection. This fact is shown in two ways in Figure 1 where both sides of the rectangle are predetermined after setting up the triangle: one side of the triangle and half of the altitude to this side, or one altitude and half of the side at right angles to it.

Some years ago an interesting article appeared in the Gazette [1] about a young student J. E. Cubbon who had found a theorem on the number of odd coefficients appearing in a row of Pascal’s triangle. Whilst trying to prove Cubbon’s conjecture in my own way, I found an unexpected relation between Pascal’s triangle and the construction of regular polygons.

It was at St. Barnabas’ School at Ravenshoe in North Queensland that I met the boy with the stockwhip. I forget his name but it does not matter. His whip was some ten or twelve feet long, and he demonstrated for me a little trick that he used to do. An assistant, the volunteer from the audience, would stand facing him with arms in front of face in case of accidents, the boy would swing the stockwhip towards his assistant with what in lawn tennis is called a forehand drive, the lash at the end would crack behind the assistant’s back, and then curl painlessly round his waist. Presumably the cracking destroyed a lot of kinetic energy.

Charles M. Whish (of the Madras establishment of the East India Company Civil Service) visited Kerala, in southern India, during the early part of the nineteenth century, and collected some antiquities. The information he could gather about these antiquities was summarised and the curious looking items were displayed in his “showroom”, an article entitled “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, Tantra Sangraham, Yukti-Bhasa, Karana-Paddhati and Sadratnamala.” This paper was read at the Royal Asiatic Society in 1832 and published in 1835 in [12]. Sightseers who passed by Whish's “showroom” were unaware of the worth of its exhibits, and the explorative and organisational labours of Whish remained unrecognised for over a century. By the 1940s though, an occasional enthusiast would wander in and the specimens kept mere amazed them and roused their interest and enthusiasm. Gradually more people were drawn to the antiquities, until by the 1980s Whish's small showroom had become die nucleus of a substantial institution. In this article mere will be a tour of the showroom, and some of the information on me exhibits will be brought up to date.

In their book Mathematical Models (second edition, 1960), Cundy and Rollett describe a dissection model to illustrate the result

It is the purpose of this note to describe similar models for the equalities

and

In addition, a dissection for the sum of the cubes of the first n integers is discussed.

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown that

has an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as