The Classic & Romantic in Natural Philosophy
by G Temple
by G Temple
The precept that 'practice makes perfect' is rarely applicable to inaugural lectures, for few professors lead lives of such vicissitude and change that they are called upon to inaugurate more than twice or thrice. The only guiding principles are that the inaugural lecture, like the farewell speech, is delivered once for all; and, like the farewell, should have a theme not entirely irrelevant to the special interests and concerns of the lecturer, or of his hearers. With this wide freedom the inaugurator may well look to history to provide him with some models to imitate. History, indeed, does furnish specimens of at least three types of inaugural lecture-the sublime, the prophetic, and the familiar.
The supreme example of the sublime is the inaugural lecture delivered by St Thomas Aquinas in the University of Paris in 1256. Here the Angelic Doctor meditates on the sublimity of his subject; on the dignity required of its teachers; on the humility, sense, and responsiveness needed by its hearers; and on the conditions for effective communication from teacher to hearer. It would not be difficult to transpose these themes into the mode appropriate to secular studies, but I will resist the temptation and content myself with quoting one phrase from the Brief Beginning of St Thomas, which I gladly make my own, as I embark upon the venture of my own inaugural address:
Sed quamvis aliquis per se, ex se ipso, non sit sufficiens ad tantum ministerium, sufficientiam tamen potest a Deo sperare.In the prophetic style the inaugural lecture becomes the formal announcement of a scholastic policy, sketching the outlines of a projected course of study, teaching, or research. But few English professors take themselves or their subjects so seriously as to make this the occasion to formulate the grand strategy of their future work.
The inaugural lecture delivered by Professor William Thomson, afterwards Lord Kelvin, on his appointment to the University of Glasgow in 1846, was a straightforward account of the characteristics of natural philosophy and of the principal divisions of the subject. It formed in fact a full syllabus of the courses of study which he proposed to teach. I must confess that I derived much consolation from Kelvin's own feelings after the inauguration ceremony. According to his biographer, Silvanus P Thompson, 'Maxwell used to declare that when Thomson read his lecture its delivery took less than an hour, and that the lecturer was greatly downhearted at its conclusion.' However, the manuscript of the lecture shows signs of revisions made in later years, and at different times, so that Kelvin clearly made good use of his inaugural on later occasions.
For my part I find it natural and congenial to adopt the more modest and friendly manner of the familiar style. I should like to seize this opportunity of making my subject better known and more widely understood. And I shall therefore endeavour to describe in the vernacular some of the more translatable features of that esoteric subject - applied mathematics.
The branch of learning prescribed for the Sedleian Professor is described 'Natural Philosophy'; and any difficulty which we might expect in determining the precise significance of this description is removed by the statutes of this University where 'Natural Philosophy' is interpreted as 'Mathematical Physics'. For our present purposes Natural Philosophy may therefore be taken to mean the organization of our observational and experimental knowledge of the physical world by means of mathematical principles and methods. This is indeed the historical sense in which the phrase is employed in Newton's Philosophiae Naturalis Principia Mathematica, and as sanctioned by the authority of Thomson and Tait in their incomplete Treatise on Natural Philosophy.
There is a proverbial difficulty in making known the achievements of mathematical physics to those who are innocent of the exacting disciplines of mathematics and physics; and those who attempt such a work of 'vulgarization' may well exclaim 'Pius labor, sed periculosa praesumptio.' [St Jerome, Epistula ad Damasum.]
The root of the difficulty is that the proper medium for the accurate expression of physical principles, methods, and conclusions is mathematics: and that physical mathematics, regarded as a language, has an almost untranslatable vocabulary. It is scarcely translatable because it has been created for the direct expression of physical truths, which can be done into English only partially, imperfectly, and inaccurately. In these circumstances we are almost obliged to take refuge in analogies; and, for this reason, it seemed to me that perhaps some light might be thrown on the concepts and methods of mathematical physics by employing the similitudes of literary and artistic criticism. Perhaps, also, some idea can be formed of the high interest and intellectual appeal of the topics, literature, and personalities of mathematical physics by considering them under the categories of 'classicism and romanticism'. This is the purpose of my inaugural lecture today; and I must therefore crave the indulgence of teachers of humane studies for venturing to appropriate some of their noble concepts and terminology.
The classification of works of art as classic or romantic is neither exhaustive nor exclusive, and this may be one reason why even the resources of the great critics or of the Oxford English Dictionary fail to provide more than suggestive hints. Goethe, as is well known, identified the classic with what is healthy, energetic, and fresh, and the romantic with what is ailing, feeble, and stale. For George Saintsbury [Encyclopaedia Britannica 11th edition, article 'Romance'] classicism implies method, order, lucidity, and proportion; while romanticism implies energy, freedom, fancy, and caprice. For Sainte-Beuve [Profils et jugements littéraires, VoI. i, p. 215: 'Qu'est-ce qu'un classique?'] a true classic is an author who has enriched the spirit of men by shedding some new light on old truths in a style which combines grandeur, a nice sensitiveness, health, and beauty.... It is manifest that there is no agreement between these great critics and no sharp distinction in their characterizations of the classic and romantic, which at times appear to be reduced to terms of praise or blame.
It is, perhaps, presumptuous for a mathematician to attempt a more definite description of the classic and romantic; and I offer the following attempt at an objective criterion with diffidence, and, it must be confessed, with an eye to later applications to Natural Philosophy.
It will, I think, be generally agreed that the fundamental note of a classic is unity, a unity of purpose and achievement, a unity of form which is perfectly compatible with a great diversity of matter. The unity of a great classic manifests itself in two ways, which may be aptly described by two algebraic terms, as completeness and irreducibility. By these terms I mean that nothing can be added to or removed from a classic without impairing its unity. Coleridge expresses this forcefully by saying [S T Coleridge, Biographia Literaria, p. 12, Everyman edition.] '... it would be scarcely more difficult to push a stone out of the Pyramids with the bare hand, than to alter a word, or the position of a word, in Milton or Shakespeare ... without making the poet say something else, or something worse than he does say'. The Pyramids, in their ideal or mathematical conception, are indeed a most appropriate image for the great classics which lack nothing and have nothing superfluous. And the testimony of Coleridge is all the more convincing, when we reflect that scarcely any other English author is so diverse and multitudinous, so incomplete, and so full of redundancies and superfluities.
It is by no means so easy to distil the essence of the romantic. The obvious trap is to define the romantic by what theologians call the via negationis, as lacking all that constitutes a classic. If we resist this temptation and seek for some positive characterization of the romantic, we may, perhaps, best find it in the spirit of seeking and accepting whatever may happen or occur to the spirit of a writer committed to some prescribed matter. Quite literally it is the spirit of adventure. A romance can therefore scarcely be complete, for it is always open to fresh calls: and redundancy is no disqualification in a romantic writer. A romance can always be continued; it is essentially advancing. In a romance neither the end nor the incidents can be foreseen, as in the predestined doom of a classic tragedy. Hence the air of novelty which even familiarity cannot take from a true romance.
You will remember that when Sir Patrick O'Prism [Thomas Love Peacock, Headlong Hall, chap. iv.] was displaying the grounds of Headlong Hall to his guests, Mr Gall ventured to say, 'I distinguish the picturesque and the beautiful, and I add to them, in the laying out of grounds, a third and distinct character, which I call unexpectedness.'
'Pray sir,' said Mr Milestone, 'by what name do you distinguish this character, when a person walks round the grounds for a second time?'
I suggest that it is just this quality of continual surprise and renascent unexpectedness which characterizes the romantic.
To describe mathematics merely as a language may seem to pure mathematicians a sad impoverishment of a great and noble discipline. But I hope I can extend an olive branch to my geometrical, algebraical, and analytical colleagues by discussing mathematics as much more than a language-as a literature. And I shall endeavour to arouse a more widespread appreciation of that great literature, both outside and inside the Faculty of Physical Sciences, by reference to the rich concepts of the classic and romantic.
In pure mathematics the great works seem for the most part to conform to the classic style. In its ideal form a treatise on a branch of pure mathematics has a severe and closely cemented structure of precise definitions and conclusive arguments, building up a great edifice of powerful theorems. The foundations are a small number of undefined entities and unproved propositions, which are demonstrated, where possible, to be consistent, independent, and categorical. This logical structure imposes a formal unity on the whole architecture. The rigorous canons of mathematical style eliminate all superfluities and excesses; and the passion for complete generality ensures that there are no omissions of any significance. Thus the requirements of unity, completeness, and irreducibility are satisfied almost by the very nature of pure mathematics, rather than by any conscious literary effort on the part of the pure mathematician.
It is, however, applied mathematics which is the proper subject of my lecture. Here the division into classic and romantic is far from complete or exclusive. Regarded as a literature, applied mathematics embraces almost every variety of style, from the rather dreary didactic prose of elementary dynamics to the epic heights of Sir Edmund Whittaker's History of the Theories of Aether and Electricity. The cosmological works of our late friend and colleague, Professor E A Milne, frequently rise to the heights of the lyric. The still unsolved riddle of Sir Arthur Eddington's great speculations seems to entitle his work to be labelled as prophetic. And if I may venture to refer to my distinguished predecessor, Professor Sydney Chapman, I feel that the gigantic character of the cosmic problems with which he grappled, and the great victories which he achieved, justify us in classifying his style as the heroic.
Tempting though it is to continue in this vein, I must return to my main theme-the classic and romantic in natural philosophy. But first of all I must remove a possible misconception of the character of natural philosophy, a misconception fostered by the exigencies of elementary manuals and textbooks. Many of these primers of natural philosophy give the impression that the various divisions of this great subject are mainly and essentially deductive systems, each solidly based on a few general principles, which themselves are almost immediate inferences from a few simple and unequivocal experiments or observations. These experimental prolegomena are rapidly summarized and dismissed in an introductory chapter. Thereafter the textbook almost loses sight of the real, physical world, and seems to operate in a realm of mathematical abstractions. Occasionally we arrive at some detailed deduction which makes contact with physical reality, but the issue of a comparison of theory with experiment is generally regarded as a foregone conclusion.
In reality natural philosophy is far more complex and far more interesting. There are in fact two great movements in natural philosophy - one leading from experiment to general principles and the other returning from general principles to experiment. These movements do not take place along ready-made highways of thought, but along tracks which have to be cleared in a pathless jungle. The discovery of first principles by inference, induction, and abstraction is as much an integral part of the subject as the prediction of new experimental results by deduction, analysis, and computation.
The relative amount of traffic in these two directions varies from subject to subject, and from one period of time to another. When the fundamental principles seem to be firmly established, and are readily accepted without dispute, then the main intellectual movement is in the direction of deduction and prediction. But when the basic laws are still unknown, then scientific effort is mainly directed to making a pathway back towards the base. In Newton's Principia the basic laws of dynamics are set down as accepted and undisputed principles, and the main business of that great natural philosopher was to determine the consequences of the basic laws. In the theory of nuclear structure the basic laws are unknown and the whole effort of nuclear physicists is to cut a path through the chaotic mass of experimental data to some position which can be taken as an intellectual origin.
If this account of the flux and reflux of thought in natural philosophy is substantially correct, it would seem natural and appropriate to find there both the classic and the romantic. The description of the deductive movement in a branch of natural philosophy easily adopts the classic style. The unquestioned first principles dominate the whole work and give it the necessary unity, while the endeavours to achieve logical rigour and the widest generality in deduction eliminate the superfluous and enlarge the field of application to the utmost, so that all our canons of the classic style are observed.
On the other hand an account of the inductive movement seems to be ineluctably romantic. The search for general laws or first principles in a confused maze of apparently contradictory facts has the typically adventurous spirit of a knightly quest in Arthurian legend. The object of the search is known and yet unknown; it is known because it will be recognized when found by its general character as a group of principles unifying and organizing a mass of experimental data, it is unknown inasmuch as its specific form still eludes our mental vision. The incidents of the search are full of the unexpected: surprise and novelty are its natural notes. Even the history of achieved research can stir again the same emotions in the reader.
Consider for example the study of the optical properties of media which are in motion with respect to the observer. There were the strange and puzzling experimental results of Arago, Fresnel, and Michelson; difficulty after difficulty in the theories of a universal aetherial medium for the transmission of light and electromagnetic action; and finally the sudden discovery by Einstein of the Special Theory of Relativity in the light of which all the optical phenomena form a harmonious whole. Age cannot wither nor custom stale the appeal of this story, which exemplifies all the elements which we have identified in the concept of the romantic.
It is an entertaining exercise in the study of applied mathematics as a literature to examine the classic or romantic features in the great treatises. The following examples are selected only because of their personal interest for the lecturer, and they are far from forming a systematic survey.
The branch of applied mathematics upon which a student first embarks is Dynamics, the study of the local motion of rigid bodies under the action of gravitation and of contact forces. There was a happy time of innocence, before the age of Einstein, when Newton's Laws of Motion reigned supreme, and a treatise on Dynamics could fail to be classic only by being formless and amorphic. Of this period there are two examples of the classic style which have always had a special appeal for me - Klein and Sommerfeld's Theorie des Kreisels [F Klein and A Sommerfeld, Über die Theorie des Kreisels. Leipzig, B G Teubner, 1897.] and Whittaker's Analytical Theory of Dynamics. [E T Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, 4th edition, 1937.]
The great treatise of Klein and Sommerfeld, of nearly a thousand pages, is a complete account of gyroscopic theory, giving, in the words of Sir George Greenhill, 'the full analytical development, where no mathematical difficulty is passed over or ignored'. Klein and Sommerfeld begin with a complete and leisurely survey of the kinematics and kinetics of tops and gyroscopes, and then develop the grand theory of the top in steady motion, employing the full resources of the theory of elliptic functions. Their account of the perturbation theory of the spinning top takes in a cosmic range of astronomical and geophysical phenomena; and the monograph concludes with a detailed survey of all the technical applications known at the date of writing, such as gyroscopic compasses, gyroscopic stabilizers for ships, and the gyroscopic monorail. From the abstract algebra of quaternions and rotation groups to the theory of the bicycle this great volume pursues its majestic way.
The encyclopaedic work of Sir Edmund Whittaker is nothing less than a complete and unified survey of the whole field of analytical dynamics. To give an adequate idea of the range of the topics discussed in both pure and applied mathematics would require a whole lecture by itself, but the classic quality of the exposition can be clearly seen in the titles of chapters IV and VI, which read as follows: 'The Soluble Problem of Particle Dynamics' and 'The Soluble Problem of Rigid Dynamics'. There is a sublime and magisterial character about these titles, which subsume in some ninety pages all that has been achieved, or that can be achieved, in these two subjects.
In the mathematical theory of continuous media there are three standard works which are indispensable to the advanced student - The Theory of Elasticity by my distinguished predecessor, A E H Love, Hydrodynamics, by H Lamb, and The Theory of Sound by Lord Rayleigh. Broadly speaking, these volumes are devoted, respectively, to the dynamics of deformable solids, of liquids, and of gases.
Love's classic volume derives its unity from its basic principle that stresses and strains in an elastic solid are connected by linear relations. This generalization of Hooke's Law determines inevitably and ineluctably the character of the analytical investigations, and the nature of the problems which are passed under review.
Of Lamb's Hydrodynamics it has been somewhat unfairly said that it discusses the physical problems of fluid motion as if they were propositions in pure mathematics. A more judicious appreciation would recognize that this invaluable work, like Love's Elasticity, acquires its character from the limitations within which it operates. This classic style is most clearly visible in the earlier editions, written in those far off, happy times when all liquids were incompressible and all gases were perfect. In the later editions it seems as if some Gothic wings had been added to a Greek temple. Viscosity and compressibility are discussed more as real physical properties rather than as entertaining mathematical speculations. The classic shades of the Mathematical Tripos begin to be lit by some fitful gleams of the romantic.
From classical elasticity and hydrodynamics it is a far cry to the romantic acoustics of Rayleigh's Sound. This amazing treatise is still almost an unworked mine of treasures of natural philosophy. It lacks almost any formal unity, and such coherence as it possesses is derived from its subject-matter - vibrations in general, from the aerial to the electric. It contains a wealth of original analysis and of experimental data. Each chapter contains new surprises and opens up new perspectives. The reader feels that he himself is assisting at a series of scientific discoveries, and that a world of new possibilities lies before him. Throughout there is a direct contact with physical reality, and the mathematical theories can be studied in what old-fashioned chemists used to call the 'nascent state'.
The second half of Temple's Inaugural Lecture is here.