A recent report on the teaching of geometry by a committee of the Mathematical Association (The Teaching of Geometry in Schools) suggests that a coherent geometrical system could be erected on a foundation other than that of Euclid’s axioms and postulates. Indeed, a cogent argument is advanced for the replacement of the principle of superposition and the parallel-postulate by the “principles of congruence and similarity” which are regarded as the expressions of more profound beliefs. The success of the recommendation, to substitute the principle of congruence for the principle of superposition, is undoubted, and the comparative reluctance to adopt the twin “principle of similarity” may be due, in part, to an obscurity in the treatment of parallels with this principle as basis.

1 To prove that a line meeting one given conic in points harmonic to its intersections with another given conic, touches a conic.

2. Lemma I. Let two conies intersect in P, Q, R, S as shown in Fig. 1.

Let UV, FG be a pair of common tangents intersecting in T, and let the tangents at P to the two conies cut UF, VG in the points M, N respectively. Then M, N, T are collinear

For: QR, PS, FU, GV all meet in a point X, and the range MFUX is related to the pencil P(PFUS), which is related to F(PFUS), which is related to PQ′XS, where Q′ is the intersection of SP and FG Similarly, NGVX is related to PQ′XS. Hence MFUX and NGVX are related, and thus MN passes through T

For a subject whose death has been several times announced, Geometry is at present in rather a thriving state. During this century considerable progress has been made in two kinds of topology, in general differential geometry and in the theory of ovals, and now we have the new development of “Textile Geometry”, largely due to Graf and Sauer, and to Blaschke and his followers. It is the work of the latter writers that is considered in this review

Whereas the works on Trigonometry, Geometry and Determinants were published over the name of Charles L. Dodgson, Student and Mathematical Lecturer of Christ Church, Oxford, the two small works on Logic—The Game of Logic and Symbolic Logic, Part I — and the two interesting logical notes in Mind appeared over the signature of the creator of Alice. Thus on the theory of double personality which the Rev C. L. Dodgson tried so hard to establish (and so unsuccessfully, since his was a completely integrated though singular personality), the formal works on logic are to be classed with those informal ones which are the delight of the civilised world.

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.

If the elements of a determinant of order n are given by

ars(r, s = 1, 2, ... n),

then the rule for determining the sign to be affixed to any term of the determinant is usually given in a form equivalent to the following arrange the n elements of the term with one set of suffixes in ascending order (i.e. 1, 2, ... n), and count the number of inversions, equal to i, say, in the other set. Then the correct sign is ( − )i.

The teaching of mathematics has been brought to so high a pitch of excellence by the eminent teachers of the past century that it may be thought there is little scope for improvement in their methods.

Since the radical revision of our geometry text-books towards the end of the nineteenth century, when modern methods first superseded the age-old system of Euclid, we find no fundamental change of doctrine until we come to consider the place of trigonometry and of the calculus in present-day curricula.