The first half of Temple's Inaugural Lecture is here.
We now continue with the second half of the lecture:-
The first half of Temple's Inaugural Lecture is here.
We now continue with the second half of the lecture:-
Fashions change in natural philosophy and dust begins to gather on the classical tomes of dynamics, elasticity, hydrodynamics, and acoustics. The romantic themes of relativity, plasticity, and gas dynamics have captured the imagination of the young mathematical physicist. The last of these, gas dynamics, admirably exemplifies the modern romantic movement in applied mathematics. Classical dynamics appeared to be based on a few principles as clear and almost as compelling as Euclid's axioms. Gas dynamics does not really start with a few self-evident principles but with a wild welter of experimental facts which are still in the process of being organized. We can indeed discern afar off the shining ramparts built by Euler, Navier, and Stokes; but these remote headquarters are of little avail in the daily adventuring among the perils of shock waves, turbulence, and boundary effects. The present struggle is to establish advanced bases which can dominate the strange territory of the motion of real fluids. Here all equations are nonlinear. Here each physical quality, such as thermal conductivity, viscosity, and compressibility, reacts with every other quality and refuses to be studied in abstract isolation. Here the world of supersonics presents us with a concrete realization of Alice's adventures through the Looking Glass. To make significant advances in this field requires the keenest physical intuition, the most powerful methods of mathematical analysis, and often the infinite capacity for arithmetic of the modem digital computing machine. Saintsbury's characterization of the romantic fits fluid dynamics like a glove - energy, freedom, fancy, caprice - these are the authentic notes of this branch of natural philosophy.
The extraordinary appeal of modern fluid dynamics is no doubt partly due to the fact that, so far from being just a topic in applied mathematics, it combines within itself some of the most exciting subjects in physics, chemistry, astronomy, and engineering, and demands some of the most recent techniques in pure mathematics.
In physics, the special theory of relativity is justly famous for its power of co-ordinating a vast array of puzzling phenomena, but many a physicist would confess to a feeling of personal loss now that relativity seems to have eliminated the aether and to have offered in its place only the principle of the inaccessibility of the velocity of light. In gas dynamics we are concerned not with waves of light or heat propagated through an imaginary aether, but with waves of pressure propagated through air or water. The critical velocity is not the velocity of light but the velocity of sound; and the crucial fact is that this acoustic speed, unlike the speed of light, offers no barrier at all to the motion of aeroplanes or missiles. The only barrier is erected by our own imaginations which find it difficult to envisage the strange effects produced under supersonic conditions, when aircraft or rockets travel faster than sound. For the physicist it will be sufficient to say that these effects are analogous to the electromagnetic disturbances which would be produced by an electron moving faster than light. For the biologist it may be a help to compare the supersonic flow of a gas through a converging-diverging duct with the motion of a crowd along a passage which first narrows and then expands. In both cases the speed of the gas or of the crowd first falls and then rises, in direct contradiction to expectations based on experiences of incompressible liquids, such as the high speed of a river in a narrow gorge. The analogy is exact, for we can show that the crowd is moving with supersonic speed. All that is necessary is to initiate a disturbance by abruptly halting one passenger. The disturbances produced by the consequent collisions do not spread out in all directions, but are confined to a wedge-shaped region, just as predicted by Ernst Mach for pressure waves in supersonic flow. To inquire the most exciting part of chemistry may be to raise a controversial question, but there will be general acknowledgement that a special interest attaches to the chemistry of high temperature and low pressures. Under these conditions equilibrium is brought about by binary collisions between molecules under the dominant influence of radiation. The study of such conditions in a gas is forced upon the fluid dynamicist in three very different regimes - in the atmosphere at altitudes above the tropopause, in shock tubes, and in colliding star clouds in intergalactic space. At heights of sixty miles above the surface of the earth, the tenuous air is heated by unfiltered solar radiation to a temperature at which molecular ionization becomes an important factor in the thermodynamics of the air. This is the cruising height of intercontinental rockets. Such conditions can be simulated in a terrestrial laboratory when powerful shock waves are produced in long tubes by an initial explosion. Behind the shock the gas becomes hot, ionized, and luminous. On the cosmic scale astronomers see similar phenomena when shock waves are produced by the head-on collision of star clouds.
Great as we may admit the intellectual appeal of gas dynamics to be in the speculative order, the practical developments of this subject have their own special attractions. The interaction between theory, experiment, and full-scale flight has been intimate, continuous, and fruitful. The demands of the aircraft designer, the specifications insisted upon for civil and military purposes, the need for speed and manoeuvrability in military aircraft, the requirement of safety and regularity in civil aircraft, and the universal demand for high pay-loads - all these have pressed upon the aerodynamicist and have stimulated him to some of the most remarkable designs and projects. As typical examples of these we may cite the use of flaps to give high lift, sweptback wings to avoid shock waves and shock stall, and boundary layer suction to reduce drag. When mathematics is applied to such ends, the standard Of 70 per cent achievement - often reckoned as a first-class performance - is intolerably slovenly. Complete accuracy is the minimum requirement: physical intuition almost equivalent to inspiration is the maximum. Practical needs have infused a healthy atmosphere of urgency and responsibility into a branch of mathematics once conceived as leisurely and fantastic.
According to Courant and the New York school of applied mathematicians the two indispensable prerequisites for a study of theoretical physics are the theory of linear algebra and the theory of distributions. In gas dynamics Schwartz's theory of distributions seems destined to play an important part, but there are other techniques which cannot be neglected, such as the theory of non-linear differential equations and the calculus of variations. Even in the simplest problems of aerodynamics singular integral equations occur with embarrassing frequency. By a strange paradox the advent of automatic or electronic computing machines has not eased problems for the theoretical aerodynamicist, but has created an urgent demand for still more powerful and profound analytical methods. Perhaps it is not the least of the attractive features of gas dynamics that the pace of development is so rapid that the best of textbooks is already a few years out of date, and that the printed word must be supplemented by oral tradition.
As a final and instructive case study in the distinction of the classic and romantic, let me quote from Henri Poincaré's lectures on 'électricite et Optique', [Paris, Carré et Naud, 1901] in which the great French mathematician gives us his full, free, and frank opinion of the writings of Clerk Maxwell, whom we in Great Britain regard as the creator of electromagnetic theory. I quote, or rather I paraphrase:
'The first time that a French reader opens Maxwell's book, a feeling of uneasiness, and often even of mistrust mingles at first with his admiration. It is only after a prolonged acquaintance and at the price of much effort that this feeling disappears. Some eminent minds retain it always.
'Why is it that the ideas of the learned Englishman have such difficulty in getting acclimatised among us? It is doubtless because the education received by most enlightened Frenchmen predisposes them to relish precision and logic before every other quality.
'In this respect the older theories of mathematical physics give us complete satisfaction. All our masters, from Laplace to Cauchy, have proceeded in the same way. Starting from sharply formulated hypotheses, they have deduced all the consequences with mathematical rigour and have then compared them with experiment. It seems to be their wish to give to every branch of Physics the same precision as to Celestial Mechanics....
'Maxwell does not give a mechanical explanation of electricity and magnetism; he limits himself to showing that such an explanation is possible....
'When the reader has agreed to set this limit to his expectations, he will encounter other difficulties; the learned Englishman does not endeavour to construct a unique edifice, definitive and well proportioned; it seems rather that he builds a great number of provisional and independent constructions, between which communications are difficult and sometimes impossible.
'Take for example the chapter which explains electrostatic attractions by the pressures and tensions which exist in the dielectric medium. This chapter could be suppressed without the remainder of the volume becoming less clear or less complete, and on the other hand it contains a theory which is self sufficient and which we can understand without having read a single one of the lines which precede it or follow. But it is not only independent of the remainder of the work; it is difficult to reconcile it with the fundamental ideas of the book ...; Maxwell himself did not attempt this reconciliation, he contents himself with saying: "I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric."
'[In these lectures] I must not therefore flatter myself that I have avoided every contradiction; but I must reconcile myself to the inevitable. In fact, if we do not mix them up, and if we do not seek to get to the bottom of things, two contradictory theories can both be useful instruments of research, and perhaps Maxwell's book would be less suggestive if it had not opened to us so many divergent paths.'
This little-known criticism of Maxwell's great treatise most admirably expresses the difference between the classic and romantic in theoretical physics. To do Maxwell justice I add one more quotation, attributed by Gertrude Stein to Picasso. [Gertrude Stein, The Autobiography of Alice B Toklas, 1933, p. 24-]
Gertrude Stein had inquired of Miss Toklas if she had noticed any differences between the pictures exhibited by Picasso and by the other painters. To which Miss Toklas replied with devastating frankness, 'Picasso's pictures were awful and the others were not.' 'Sure,' said Miss Stein, 'as Pablo (Picasso) once remarked, when you make a thing, it is so complicated making it that it is bound to be ugly, but those that do it after you they don't have to worry about making it and they can make it pretty, and so everybody can like it when others make it.' This is perhaps one of the most lucid sentences in Gertrude Stein's writings, and it contains the whole philosophy of the romantic in art, literature, or mathematical physics.
I have now carried out my programme of literary criticism by exhibiting a few topics and books of natural philosophy in the light of the classic and the romantic. Our investigations, however, are not merely speculative; they have a practical application with which I shall conclude this lecture.
The long extract which I have ventured to quote from Poincaré very properly displays the disadvantages of the romantic style in treatises on mathematical physics - the lack of unity, order, and proportion, the exasperating changes of theme and of method, the tantalizing inconclusiveness and incompleteness. To suffer under these defects is the price we must pay for the privilege of assisting at 'work in progress' and for the experience of intellectual cooperation in a masterpiece of original research.
The disadvantages of the classic style are not so openly admitted, but they are equally, if not more, fatal to the enthusiasm of the earnest student. What are, perhaps, the two main disadvantages may be briefly expressed by saying, first, that although there have been romantic movements, there cannot in the nature of things be a classic movement; and second, that classics may be models, but they are not guides.
A classic movement is almost a contradiction in terms, for a classic is essentially static and derives its static character from its very perfection and completeness. Often a classic marks the end of an epoch: it summarizes compendiously all that has been done before in its own department. But there are minds who find this faultless perfection and this all-inclusive completeness discouraging for further work, and repellent for present study. They sink under the weight of those monumental tomes, and would fain exclaim, in the metre of Herrick's poem on 'The Poetry of Dress', the sentiments of the following imitation, for which I offer all due apologies:
The Poetry of a Scientific Address
A sweet disorder in th' address
Gives lecture notes a wantonness:
Mystery about the subject thrown into a fine distraction,
An erring sign, which here and there
Enthrals the staid compositor,
A theme unfinished, and thereby
Corollaries flow confusedly,
A winning guess, upon whose stem
Flowers the tempestuous theorem,
A careless knot, within whose tie
I see a whole topology,
Do more bewitch me, than when art
Is too precise in every part.
It is almost part of the definition of a classic that it should be approved as a model and taken as a standard. But the carefully finished product gives no hint of the skills and crafts which went to make it a work of art, and therefore gives no help to the novice just 'commencing author'. All the artifice is successfully concealed, all the false starts have been eliminated, all the abortive efforts quietly forgotten, all the imperfections removed. But it is the presence of these imperfections which give such value to the romantic style in natural philosophy for the student and the research worker.
If Newton had only told us how he discovered the law of central attraction for a sphere, instead of concealing this crucial discovery in a geometric language as dead and as difficult as Etruscan! If Eddington had only told us of the wild guesses and conjectures, the mistakes and rejected constructions, which passed through his mind as he wrote his Fundamental Theory, we might be nearer to solving the outstanding riddle of that great and puzzling book.
We shrink, however, from exposing ourselves in print, and therefore give the world an expurgated edition of our mental struggles. Arthur Symons has remarked that 'the occasional notes and sayings of such men as Blake and Rossetti are often of more essential quality than their more ordered and elaborate comments. The essence they contain is undiluted. They are what is remembered over from a state of inspiration; and they are to be received as reports are received from eye witnesses, whose honesty has already proved itself in authentic deeds'. [Introduction to Everyman edition of Coleridge's Biographia Literaria.] Alas, natural philosophers are shy of publishing their 'occasional notes and sayings', although Newton's reputation rests as much upon his Queries as upon his Principia.
In our lectures, however, it is possible to recapture something of that 'first, fine, careless rapture' with which we first saw in a sudden flash the truth of some theorem or the power of some technique. There is a tradition at the City and Guilds (Engineering) College that Henrici would lecture, year after year, on the differentiation of xn in such a manner as to convey the impression that he was sharing with the class a discovery made only the night before. This I should call the romantic style in lecturing. The classic style is much easier . . . for the lecturer; but it is useless for the student until he knows all about the subject. Then it provides an elegant medium for revision. Indeed, scarcely anything pleases a body of experts more than to be told something they know already, but told it decently and in order with the calm perfection of classic exposition.
These views on the appropriateness of the romantic and classic styles to lectures and textbooks are admirably expressed by Professor J E Littlewood, [The Elements of the Theory of Real Functions, Cambridge, W Heffer & Sons Ltd., 1926.] in the Introduction to his lectures on the theory of real functions. There he defends the lecture as a medium of instruction as follows:
I am one of those who believe that lectures can have great value, and particularly at a certain moderately advanced stage of a mathematical education. The modern standard of conciseness and lucidity in original papers and advanced textbooks is on the whole a high one, but the style is one for the expert only. We may demand two things of an original paper, a complete and accurate exposition on the one hand, and on the other that it should convey what is the real 'point' of the subject matter. For various reasons, among which a sufficient one nowadays is sheer lack of space, the second demand is inevitably sacrificed to the first. A lecture, however, more particularly when it is supported by a complete exposition in print, is the very place for the provisional nonsense that the second generally calls for. This would appear ridiculous if enshrined permanently in print, and its real function is to disappear when it has served its turn.... The infinitely greater flexibility of speech enables me to do here without a blush what I shrink from doing in print.
Classic perfection should be reserved for the monograph: the successful lecture is almost inevitably a romantic adventure. It is at once the grandeur and misery of a scientific classic that it says the last word: it is the charm of a scientific romance that it utters the first word, and thus opens the windows on a new world.
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