Glaisher, I believe, once said that no subject loses more than mathematics by any attempt to dissociate it from its history. It is still customary, however, in schools and most universities to regard the history of mathematics as an “extra” outside the ordinary student’s syllabus—indeed it is usually treated as an exotic luxury. The point I would like to emphasise is that while the primary aim of the tyro should be the mastery of technique, yet in most cases this mastery is liable to become a mere mechanical acquirement unless it is accompanied by some appreciation of the cultural significance of the “Queen of the Sciences”.

A difficulty which is frequently met in dealing with examination results is that of bringing to a common standard the marks obtained from two different papers. The most common form of variation occurs when the mark distribution curves are correct in general character but the peaks are at different points of the scale. We describe a machine devised by the author which has been used for the adjustment of marks. Its theory has no statistical justification, but where the distortions are not excessive the results do not differ appreciably from those of the rather lengthy theoretical methods and are usually absorbed in the approximation of a fraction to its nearest unit.

I am afraid that what I am going to place before you to-night will be somewhat incomplete. This is unavoidably so because I have not had sufficient time at my disposal to test my views thoroughly. The evidence which I have to offer, therefore, in support of my case is necessarily scanty, and it would be premature to force unwarranted conclusions from it. Moreover, what little testing I have done has been confined almost wholly to one special problem. On that account there is a danger that my presentation of the topics will appear to lack proportion. I hope, therefore, that you will bear these facts in mind when judging this paper, for otherwise your verdicts could not be altogether fair.

In talking about the relations of mathematics and culture or, rather, about the cultural value of mathematics, I think it advisable that I should first say a little about what I suppose the word “culture” to mean. In its general use it contains, I think, two ingredients, as it were. It refers to information and it refers to sensibility An ignorant man, however refined his natural sensibilities may be, is not regarded as cultured. And a man who has amassed an immense knowledge of facts, but is quite unable to distinguish between what is valuable and what is not valuable in his information, is not regarded as cultured. Of these two elements, I am inclined to think that sensibility is the more important. What the old writers on aesthetics used to call taste is, I think, more important than knowledge for the formation of a cultured outlook.

The discussion on symbolic operators in differential equations in recent numbers of the Gazette shows the need of a different method of attacking the linear equation of the second order with constant coefficients, which is the case most wanted in practice. It seems a pity that no method is given in the English books except the symbolic one, in which pass students get lost owing to their not understanding exactly what they can or cannot do with the operators. And is this to be wondered at when a theorem which gives erroneous results if not interpreted in a particular way has stood in Forsyth’s Treatise for forty years?

1. One of the two curious self-reciprocal formulae stated by Rama-nujan,

where

of which I have recently given a proof, can be established by using a result proved by Stieltjes that, if x > 0, m ≥ 0, 0≤a≤ 1,

The proof only requires the above result in the case in which m = 1, a = 0. While seeking an extension of the result (1. 1), I have obtained an expression for the continued fraction (1. 3) in the form of an infinite series.

One of the chief appeals of Trigonometry to the beginner lies in the practical nature of the subject. The first aim in this respect is to obtain the formulae for the solution of the general triangle, but hitherto there has been much delay in the logical development of the subject up to that stage because of the need

(a) to prove the half-angle formula cos A =cos2 ½A − sin2 ½A and

(b) to show that cos (180° − A) = −cos A and sin(180°−A)= sin A.