I Want to urge on the Association the importance of including the geometrical solution of quadratic equations in the school curriculum.

We admit a certain amount of algebraic reasoning as it is, and there seems no reason against its extension, and there are many problems which admit of ready solution if this method is known.

Descartes at the beginning of his work on Geometry (1637), which would now be called a treatise on Algebraical Geometry, in order to shew the relation between Algebra and Geometry considers four types of quadratic equations. It will be noticed that these are special forms, the constant term of a quadratic equation being as a rule, not a square, but the product of two lengths, and necessitating the use of Euc. VI, 13 to find the equivalent square.

Many teachers know that Euclid’s proofs by superposition have been much criticised, but to most of us, perhaps, even after the publication of the Report on Geometry, the matter has remained somewhat obscure. As for purposes of my own I have had to try to clear my mind on the subject, possibly what I have so learnt may be of use to others.

Interest in the trick that we all know was not discovered by Cardan is greatly enhanced if we think less of a cubic equation and more of a three-one relation between complex variables. It is convenient to take the relation in the form