The ancient Greeks recognised two branches of arithmetic. One of them, the philosophical study, expounded by Euclid in Books VII-IX, developed as a logical science the properties of whole numbers. It was not so much concerned with addition, multiplication and the manipulation of fractions, in which the notation developed in the middle ages has greatly improved the methods of the ancients, as with prime, odd and even numbers, with digressions to triangular, polygonal and perfect numbers. The Greeks were familiar with the facts (1) that when a prime number is a factor of the product of two whole numbers ab it must be a factor of either a or 6, and (2) that every whole number can be expressed, in only one way, as a product of prime factors. A late offshoot of this branch of the subject, destined to develop into one division of the modern Theory of Numbers, was considered by Diophantos of Alexandria (English edition by Sir T. L. Heath), who was mainly concerned with finding solutions in rational numbers of indeterminate equations such as

yz + a=u2, zx + a = v2, xy + a=w2.

The second branch of arithmetic known to the Greeks (Aoyto-riK?/) was concerned with its applications to measurements, to monetary calculations and to various problems in which the science was used in practical life. No exposition of this is known to have survived the ages.

I found the article Chemically, the same or different? by G. T. Woods [1] to be very interesting. On p. 253 of that article the problem of providing a formula for the number of isomers, as a function of the number n of carbon atoms, of certain molecular structures is suggested.

In opening this discussion on the influence of the university on the school course in geometry I have two dangers to avoid. Many of the things which it would seem natural to say are true and a few of them may be important, but when I wrote them down and took a second look at them, I saw that they were not concerned in any specific way with geometry; we sometimes forget that there is a Section of the British Association which exists to rescue the rest of us from the generalities and platitudes to which teachers are only too prone. The other danger is of entering so far into particulars as to compose a syllabus round which we might be expected to quarrel instead of talking about principles on which we can hope to agree; here I am fortunate in the subjects chosen by the speakers who are to follow me, for where what I have to say is most in need of illustration, their expositions will provide you with examples worked out at the appropriate length.

BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.