The purpose of these notes is to suggest diagrams that may illustrate the setting up and ordering of equivalence classes, their addition and multiplication.

My attention was first called to this attractive curve by a pupil, M. J. Burleigh; so, lacking any other name, we called it “Burleigh’s Oval”. It is mentioned in Salmon’s Higher Plane Curves (p. 107), where a cartesian equation is given; and in Hilton’s Plane Algebraic Curves (p. 64), where it is described, with a diagram, as the reciprocal of the Limaçon. It also occurs in a very interesting general treatment of negative pedals by A. Ameseder, (Archiv der Mathematik und Physic, LXIV, 1879).

My main thesis is that men have too long allowed themselves to be bemused by the magic of numbers, though the kinds of spell that numbers have cast over us have changed with the ages. In seeking to understand why it is difficult for us to break the spell, I have been led to suspect that schoolmasters (and perhaps especially schoolmistresses) are largely to blame because of the way in which they use school marks. And a special responsibility in this matter rests upon the mathematical staffs of schools, not because they are more “mark-minded” than their colleagues, but because marks appear to be numbers or measures of some kind and the non-mathematicians must look mainly to the mathematician for guidance on the manipulation and interpretation of numbers and measures. Any lead that may be needed to reform our school marking systems must be given, if it is to be given at all, by the mathematicians. That is my justification for addressing my comments on School Marks to the Mathematical Association.