A fundamental result in the theory of conics is the following:

Two opposite vertices of a complete quadrilateral are conjugate points for any conic which passes through the four remaining vertices.

Let AB be any chord passing through the fixed point P. Draw the diameter AOC, join CP and let it meet the circumference at D Then in triangle ACP, AC2 = AP2 + CP2 together with either 2AP. PB or 2 CP. PD according as CP is projected on AP or AP on CP. Hence AP.PB = CP.PD. Now draw the diameter DOE, join EP and let it meet the circumference at F. Then the theorem is true for chords CD. EF.

§ 1. The present note gives two methods by which vulgar fractions may often be converted into decimals with great rapidity. It is not pretended, however, that they are more than mere curiosities or that they are of practical use. The first method is not new, but I have never seen the second mentioned anywhere. The proofs are almost self-evident.

The following argument is simple and instructive, but I do not remember seeing it before.

I. Prove sin θ= −sin(− θ), cos θ(− θ)

Let X′OX, Y′OY be rectangular axes, and let OP, OQ, starting from the position OX, describe angles θ, –θ. Then OP, OQ are in every case symmetrically placed with respect to OX. (Illustrate by taking positive and negative values of θ). Hence xP = xQ, yP = –yQ, and the results follow from the definitions of sine and cosine.

We may group the terms of the A. P. a, a + d, a + 2d, …… as follows: the first p terms, the next p + t terms, the next p + 2t terms, and so on. The sum of the terms comprising the nth group is

which is a rational integral function of n of the third degree.

The correct law of refraction, sin i = μ. sin r, is usually assumed to have been first discovered by Snell in 1621, although he did not express it in this form. The astronomer Kepler laboured hard to discover it, but in vain, and Whewell in the History of the Inductive, Sciences says that it is strange that he should have failed, when the law is so simple. There is, however, a good reason for his want of success.