The notation adopted in this paper for the triangle ABC is:

G the centroid.

I the centre of the inscribed circle.

I1, I2, I3 the centres of the escribed circles within angles A, B, C.

O the centre of the circumscribed circle.

D, E, F; D1, E1, Fl; D2, E2, F2; D3, E3, F3 the points of contact of the inscribed and escribed circles with BC, CA, AB. The Ds lie all on BC, the Es on CA, and the Fs on AB.

H, K, L the mid points of BC, CA, AB.

X, Y, Z the feet of the perpendiculars from A, B, C, on BC, CA, AB.

A′, B′, C′ the vertices, opposite to A, B, C, of the triangle formed by drawing through A, B, C parallels to BC, CA, AB.

The object of the paper was to draw attention to a few important and well known cases of the harmonic section of a straight line, and to show their application to one or two problems of interest, more especially the method of drawing tangents to a conic by the ruler only. The effort throughout was to secure clearness, brevity, and freshness of proof, coupled with purely geometrical treatment.

The theorem is:—If a1, a2, a3, …, a22 be the middle points of the sides of any convex polygon A1A2A3…A2n then as regards areas

The following proof depends only on the theorem that the line joining the points of bisection of two sides of a triangle cuts off a triangle equal in area to a quarter of the original triangle.

The object of this paper is to bring together the principal properties of figures described on the surface of the sphere that can be established without the use of Solid Geometry or of Trigonometry.

The following properties of the spherical surface, which correspond to the definitions and axioms in Plane Geometry, are assumed. They may be considered as derived from one's general notion of the sphere.

The continuant referred to is that in which the elements of the main diagonal are all equal (to x, say), the elements of the one minor diagonal all equal (to b, say), and the elements of the other minor diagonal all equal (to c, say). It may be denoted by F(b, x, c, n) when it is of the nth order. Professor Wolstenholme has recently given two elegant theorems regarding the condensation of F(1, x, 1, n). I wish to establish the analogous theorems for F(b, x, c, n).

In 1640, when only 18 years of age, Pascal published a tract of a few pages with the above title. It contains only a few enunciations, and concludes with the statement that the author has several other theorems and problems, but that his inexperience, and the distrust he has of his own powers, do not allow him to publish them till they have been examined by competent judges. He afterwards wrote a complete work (opus completum) on the Conics, which was submitted to Leibnitz by M. Périer, Pascal's brother-in-law. Leibnitz recommended that it should be published; but this was not done, and we know its contents only from the analysis which Leibnitz sent back to M. Périer.

This paper was prepared at the suggestion of the committee as the first of a series on the teaching of elementary mathematics, in the belief that an occasional paper of this nature, with discussions, would be useful.

In the introduction it was suggested that, as secondary education in this country was apparently on the eve of considerable changes, the present was an opportune time for discussing the whole subject of school mathematics; and also that the Society should be prepared to form a scheme of a mathematical course for both teaching and examination purposes.

Suppose that we have c candidates, e electors, s seats, v votes.

There are at least three different kinds of voting to consider: Simple, Combinational, and Cumulative.

I. Simple voting. By simple voting I mean any case in which an elector has only one vote. Denote the candidates by A, B, C, D.

Arago observed in 1811 that if a plane polarised ray of light be passed vertically through a plate of quartz cut at right angles to the crystallographic axis, the ray emerges plane polarised but with its plane of polarisation inclined at an angle to the original plane of polarisation, this angle (or the amount of rotation of the plane of polarisation about the direction of the ray) being proportional to the thickness of the plate. Biot further showed that in some quartz crystals this rotation is in one sense, in others in the opposite. In 1821 Herschel proved that the sense of this rotation was connected with the inclination of the so-called plagiedral faces to the faces of the prism. In 1830 Naumann gave a very complete account of the crystallography of quartz, showed that the two kinds of crystals are mirror images of each other, and gave to this relation the name of Enantiomorphism.

If the physical state of a substance is completely defined when the simultaneous values of three of its properties are given, then, by measuring off along three rectangular axes, from any point chosen as origin, lengths proportional to these values, we determine a point which represents completely the physical state of the substance. And, evidently, each point lies on a surface, the equation to which is determined by the three co-ordinate properties. If, in the equation to the surface, we give one of the variables a definite value, we get the equation to a contour-line of the surface which represents the necessary relation subsisting between the remaining two properties when the other is constant.

GivenAB=DE, AC=DF, ∠A=∠D.

Suppose you start from B, and walk along BA a certain distance a to A; then at A you turn at a certain angle into another road AC; then you walk along AC a certain distance b to C. Again you start from E, walk a distance a along ED; turn off at D into DF at the same angle as before; then walk the distance b along DF to F. Since you have gone through the same set of movements in the two cases, and since the same cause always produces the same result, the results in the two cases must be the same, that is, you will arrive in both cases, at the same distance from the starting point. Hence BC = EF.