The last section of Arithmetica Infinitorum is devoted to the ingenious argument by which Wallis suceeded in reaching the first modern expression of π as an infinite series.

We begin with an argument which, although in Wallis’s hands it proved abortive as an attempt to determine π, is yet useful for the comprehension of his more successful procedure, and has (as we shall see) great historical importance.

Dr. John Wallis, who lived through the greater part of the seventeenth century (1616-1703), was a typical child of that energetic age. Son of a Kentish parson, he began his academic career at Cambridge, but ended as Savilian professor at Oxford. He was a scholar who apparently read mathematics in Arabic as easily as he wrote on them in Latin; a founder of the Royal Society; a grammarian; and (needless to say) a vigorous theologian. Into the middle of his famous Algebra he “spatch-cocked” (with apologies for its irrelevance) one of the earliest modern inquiries in experimental psychology : in the course of a sleepless night he had amused himself by extracting mentally the square root of 3 to 20 places and, a month later, showed in the presence of a distinguished foreigner that he could remember the result correctly. Nor did these accomplishments exhaust his versatility For as a supporter of the Parliamentarians he confounded the Royalists by his astonishing facility in unravelling their cipher messages. Those who are offended by his adherence to the cause of the Parliament may be reconciled by the knowledge that he was one of the remonstrants against the execution of the King—a defection which, happily, did not blind the authorities to his great merits when the Savilian professorship fell vacant in the following year.

To establish the order of the locus we must note that besides the above mentioned n intersections with the circumference, the point D also belongs to the curve, so that the latter cuts the circle in n+1 points. The order of the curve is therefore Let n be even.

The ratio is not an integer, so that a complete rotation of the straight line BC must take place before the two generating straight lines can be in their initial mutual position; the line revolving round D will be, after a semi-rotation of BC, parallel to BC in DE, and we may say that all the loci for n = 2p are asymptotic in the direction BC, on account of the initial position of BC and DA.

That some advantages have been derived from the freedom now enjoyed in the teaching of Geometry will be denied by no one; but there are disadvantages equally obvious. The, confusion both of order and method, the neglect of the theoretical and deductive work as compared with the practical and experimental, and the failure to grasp the value of Euclid’s work as an educational discipline may be mentioned. Perhaps it is not too late to express the hope that one of our great mathematicians—or a representative group chosen from among them—may yet produce a Textbook of Geometry which will be to English-speaking people what Legendre’s Eléments has been, during more than 100 years, to so large a part of the Continent of Europe.

Professor Bryan’s paradox on Fourier’s theorem (Gazette, vol. iv. p. 390) seems to call for a few words of explanation.

In the first place a function f(θ) is constructed of the form

and the function is arranged so as to have the same values as an arbitrary function F(θ) at the places θ = 0, a, 2a, …, (n-1)a, where α=2Π/n. It will be observed here that we have only n data from which to determine (2n-1) coefficients, and consequently the form of f(θ) is largely at our disposal ; but the particular form selected by Professor Bryan leads to the formulae for the coefficients,

A Meeting of the North Wales Branch was held on June 18th at Llanberis. The discussion on examinations which had been begun at the previous meeting was reopened by Miss Prior, Girls’ County School, Bangor, who dealt in turn with the points raised in Prof. Bryan’s paper, from the point of view of the school-girl.