In all the English text-books on Analytical Trigonometry, so far as I know, the power-series for cos x; is obtained as the limit, when n tends towards infinity, of the finite sum for cos x,

where and the number of terms is equal to the integer next greater than . The power-series for sin x is obtained similarly; the two series and the method of finding them being both due to Euler.

The locus of the points of contact of tangents from A to the conics of the confocal system with foci B, C is the same curve as the locus of the feet of normals from A to conies of the system. For any tangent AP to one of the conics is a normal AP to the orthogonal conic of the system through P. Hence if the coordinates of A referred to BC and the perpendicular bisector of BG be x0y0 the equation of this locus is the eliminant of λ between

The difficulties which arise in teaching the elementary power series appear to form part of a larger question, namely, the proper course of mathematical education after a sufficient knowledge of Elementary Geometry, Algebra and Trigonometry has been acquired. The question of Geometry is hardly germane to the present discussion, and I will only mention it so far as to say that I believe it is the hope of many teachers of mathematics that boys of 15 or 16 may in the future learn far more of Projective and Descriptive Geometry than is the case at present.

§ 1. The projective theory of conies is based chiefly on the fact that a conic is a unieursal curve, and the simplest analytical method of treatment is to take for reference a triangle formed by two tangents and their chord of contact, so that with suitable multiples of distances for coordinates the equation of the conic may be written in the form

y2 — xz=0.

We recognise as decadence the substitution for Euclid’s theorem I. 20, “Any two sides of a triangle are together greater than the third side,” of the objectionable pseudo-axiom, “A straight line is the shortest line from one point to another.”

This phrase, as Hilbert has well said, is of necessity meaningless when the concept “length of a curve” has not been defined. Every book on elementary geometry which has introduced as axiom or definition of a straight line any phrase equivalent to “ A straight line is the shortest distance between two points,” every book which has at its beginning as axiom or definition of a straight line any phrase equivalent to “A straight line is the shortest distance between two points,” has been in error. We believe the advances in regard to the foundations of elementary geometry make timely the rejection of this piece of decay from the body of the elements, and that the time is ripe for the return at this point to the pristine purity of Euclid.