Ladies and Gentlemen,

There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.

The pitch of a musical note is defined by its frequency, measured in cycles per second. The frequencies of pure musical tones form an infinite set of real numbers lying between the lower and upper limits of audibility. The notes of the pianoforte form a finite subset of this infinite ‘spectrum’ containing usually 88 members. Other instruments provide different subsets of available notes, these subsets being infinite in the case of the strings and the trombone, whereas in the case of most other instruments, we have finite subsets of discrete musical tones.

Everyone connected with the teaching or practice of mathematics is aware that in recent years there has been a great increase in the number and variety of opportunities for the employment of mathematicians. In particular, industry, and especially engineering industry, has found a need to recruit mathematicians in large numbers with a view to employing them to do mathematics as a full-time occupation. This is quite a new development; although the value of the mathematically trained mind has long been recognised in many places, it is only quite recently that mathematics, as opposed to mathematicians, has been extensively employed by industry This change has, of course, largely coincided with the advent of the electronic digital computer, and the computer has certainly had a great influence on it; new mathematical disciplines have grown up as a result of the existence of computers, and it is now practicable to solve problems whose solution would be inconceivable without the use of a computer. As a result of this it has become fashionable to attribute the spread of mathematics to the computer. I think it more likely that the computer has accelerated rather than created the demand for mathematicians, and that the development of new industries and the introduction of new techniques to old ones would have brought about a demand for mathematicians even in the absence of computers. However there is no doubt that this demand exists, and it has certainly been stimulated by the existence of the computer.

The older sixth form masters have seen a fundamental change from a unified programme guided by the form master, to an à la carte curriculum where the form master may see some members of his form only once or twice a week. The old liberal character of form work must now be transformed into certain elusive elements in the subjects themselves and in the way they are studied. In this respect sixth form teachers are in the same vulnerable position as the editors of newspapers: they must be resolute in resisting the monopolists, whether inside or outside the school.

The Computer Education Group of the British Computer Society lists as one of its aims, “To actively further the spread of the use of computers in education at all levels.” I cannot agree that this is desirable even if it is financially possible.

The following extends the result given by the author for the Nth power of a square matrix to the case of negative and fractional values of N. See The Mathematical Gazette, 1965, XLIX, p. 421.

In a recent note (The Mathematical Gazette, 1965, XLIX, 388) the late Prof. D. Markovitch of Belgrade states that the method published by me (The Mathematical Gazette, 1960, XLIV, 241) is not new as he had described the same method in the period 1954 to 1956. By the courtesy of the Editor I have now had an opportunity of studying the full manuscript submitted by Prof. Markovitch together with copies of the published papers.

Last week I felt that one of my sixth form applied mathematics sets was finding its subject too straightforward. Imaginations had been stuffed with smooth spheres, perfectly elastic and moving as in a vacuum. Methods relied almost exclusively on the less complicated aspects of Newton’s work, with even solid rigid bodies seeming to behave like the ubiquitous point masses. Advanced level text books contain a good store of examples and, needless to say, the class had worked solidly through its own book. “Solidly” was the right word! I wondered how to fan a flagging interest, whether to bring out my own constructional toys for another lesson and to set up a pulley system or project particles from one side of the room to the other, whether to take over the physics laboratory and let the pupils use some proper apparatus. Alas, my pupils are more sophisticated than I and therefore eschew the use of toys, while regrettably the physics staff arrange their physics experiments so much more ably than I could. Besides, it would be entertaining to see the sixth formers investigating a situation which could not be explained with the help of Advanced level mathematics. Let them flounder, as perhaps our forefathers floundered whilst discovering such trivia as Hooke’s Law or Newton’s Experimental Law!

Textbooks dealing with this topic are usually content to solve approximately the equations of motion for a particle moving under constant gravity g, using the fact that the angular velocity w of the Earth is small. (See A. S. Ramsey—Dynamics, Part II (§§4.6-4.74) or L. A. Pars—A Treatise on Analytical Dynamics § 10.10.) These equations of motion may be solved exactly, of course, without placing any restriction on w. The modified equations of motion for the particle moving in a medium offering resistance kv per unit mass, v being the velocity of the particle, may also be solved exactly.