The subject of the definition of irrational numbers is so intimately connected with questions about the existence of a limit of a sequence, which has recently formed the subject of much discussion in this Gazette, that a fuller consideration of the subject, and, in particular, of how irrational numbers are to be introduced into a course introductory to ‘higher’ mathematics, may not be undesirable.

In the standard text-books on trigonometry the object of the authors is to reduce geometry to algebra, to give geometrical proofs of a few fundamental formulae and afterwards to develop the subject on algebraic lines. When alternative geometrical proofs are given they are not often designed to illustrate the steps in the algebra. De Morgan’s Double Algebra and Hayward’s Vector Algebra may be mentioned as works in which free use is made of geometry, but the emphasis laid on the interpretation of a vector as a complex quantity makes their method appear rather advanced. The slides which I shewed to the annual meeting were intended to illustrate the

The following presentation of the fundamental theorem of The Integral Calculus has been evolved as the result of attempts to put § 31 of Whittaker’s “Analysis” before a University class. It is to be regarded as the first application of the Infinite Sequence Theory which has been the subject of articles in recent numbers of The Mathematical Gazette. It is in the details, making for stricter logical accuracy, that what is new will be found.

In a letter having the title “Classification and Mathematics,” which appeared in Nature of May 28th, it was suggested that models of spaces and correspondences might usefully be employed in elementary mathematics. The present paper is restricted to the consideration of some simple one-toone correspondences.

It will be taken tor granted that, in order to arrive at any functional dependence or one-to-one correspondence of things, a classification and crossclassification is necessary. For example, the correspondence of English and French words tabulated in an English-French dictionary might be given as an illustration of a function (cf. Oliver Lodge, Easy Mathematics, 1905, p. 164). Here the words are divided into two classes, English and French, and are crossclassified by meaning.

In the twelve slides described in this paper the Picture Plane is that determined by the two thickened vertical lines which appear in each diagram. It will be called the P. PI.

Similarly the Ground Plane is that determined by the two thickened horizontal lines which appear in each diagram. It will be called the G. PI.

The possibility of solving the problem of the elementary division of a circumference into equal parts was known to the ancients for the cases

n = 2p, n = 3, n = 5, and their products.

Gauss in his Disquisitiones Arithmeticae proved the possibility of solving the problem by quadratics, and therefore geometrically with ruler and the compass, for all the prime numbers of the series

n = 22p + 1.