In the discussion held at the annual meeting of the Mathematical Association in January 1936 on “Rider-Work in Geometry”, various interesting proposals were made and examples put forward to show how school geometry might be given a research aspect by means of the “discovery rider” or the “inductive method” in order to develop the full powers of the mind and to stimulate the creative abilities. Teachers who have tried to put into practice the ideas presented on that occasion will have obtained rather varied, if not actually contradictory, results. However many may have been successful there are always quite a number who come to the false conclusion that the average pupil has no creative abilities and that all efforts in that direction will fail, since only the very few mathematical specialists are found by them by benefit by it.

Several mathematicians have studied the question of symmetry in the knight’s tour, one of the earliest being Euler and one of the latest Dr. Kraitchik of Brussels. Their investigations have been in the nature of random experiments, and they have confined themselves chiefly to square boards, thus missing such elegant results as

Fig. 1, which was published anonymously in the German periodical Denken und Raten in 1931.

1. It is necessary to challenge Mr. F H. V Gulasekharam’s Note 1448 (May, 1940) in which he claims to give simpler methods than usual for dealing with the circle

S ≡ (Σlx)(Σx) − Σa2yz = 0.

Objection must be taken at the very beginning where he says “Suppose the circle cuts the sides of the triangle of reference ABC at D1D2” The arguments of §§ 1 and 2 are sound when these points are real, but it is asking too much of a young student to expect him to understand what is meant by the ratios D1C/BD1 and D2C/BD2 when D1 and D2 are imaginary Moreover the writer considers that the arguments of §§ 3 and 4 are not simple but very difficult.

From four given lengths as sides it is possible in general to construct three distinct cyclic quadrilaterals, whose circumcircles are equal. Any pair of these quadrilaterals has one diagonal-length in common.

An article by Lawrence Crawford in the Gazette for February 1934 (see also Gazette, February 1935) pointed out that the textbooks rarely make any attempt to relate coordinates of foci to the equations of corresponding directrices, etc. The article went on to show how this could be done.

In what follows an entirely different method is adopted. The method leads to actual formulae for all the principal parts of the conic.