Introduction. Many students interest themselves in finding a function which is equal to factorial n for integral values of n. There is a certain fascination about this function : it is so different from the transcendental functions of applied mathematics.

Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.

In this article an attempt has been made to indicate the ordinary methods of integration of the total differential equation

Then the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is

We consider here three geometrical problems, the solutions of which are not hard, but each involves some rather unobvious point. The reader is invited to attempt them before looking at the solutions. The names of the original proposers are noted in brackets.

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.

What is the most symmetric quadrilateral you can think of? Most people would say ‘a square’, but having given this (or any other) answer, can you create your own definition of a ‘quadrilateral’, and of ‘most symmetric’, so that you can justify your answer as a rigorous piece of mathematics? This is a challenging investigation because the answer may not be the obvious one.

Often the main difficulty in solving a problem lies in finding appropriate definitions; here we must first consider the questions ‘what is a quadrilateral?’, and ‘what is a symmetry of a quadrilateral?’.