Mathematics is very much older than History, which begins in + 1066, as is well known ; for the first mathematician of any note was a Greek named Zeno, who was born in – 494, just 1560 years earlier. Zeno is memorable for proving three theorems: (i) that motion is impossible; (ii) that Achilles can never catch the tortoise (he failed to notice that this follows from his first theorem); and (iii) that half the time may be equal to double the time. This was not considered a very good start by the other Greeks, so they turned their attention to Geometry.

The fact that a mechanical system so well known to so many in its practical aspect as the swing has been thoroughly neglected in textbooks and problem papers may justify a few remarks on it here.

The only reference to the swing which I have been able to find is the question requiring it to be shown that the amplitude of the swing may be increased by internal action only, if, when the swing passes through its lowest position, the distance of the centre of gravity from the point of suspension is (suddenly) shortened from l to (l – h), the full distance l being restored at the end of the swing.

Twice in the course of some work on chemical kinetics it became necessary to determine the type of the solution to a generalised set of any number of first order simultaneous differential equations. The problem reduced in each case to a question of the reality and sign of the roots of the auxiliary equation, which was, of course, determinantal in form. The required information could be deduced from long-established theorems on the roots of symmetric and skew-symmetric determinantal equations, but the following proofs, which are on a much simpler level, may be of interest.

I will illustrate the kind of problem I am going to discuss by showing how it applies to the case of addition. Suppose we add a series of numbers, each correct to the nearest unit ; this may be tenths, hundredths or any other unit. The maximum error in the sum of n such numbers is ½n units. The argument, almost universally applied, is that as the maximum error is ½n units, the answer is unreliable to that extent. This is bad logic. Let us take an analogy from electricity. When a current is switched on the maximum current is reached only after a time which is infinity. Actually, after a very short time, the difference of the current from the maximum is negligible. Likewise to argue that the sum of n items is unreliable to the extent of ½n units is not merely bad logic, it is completely wide of the truth.