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Address by Professor P. G. TAIT, M.A., F.R.S.E., President of the Section. In opening the proceedings of this Section my immediate predecessors have exercised their ingenuity in presenting its widely differing component subjects from their several points of view, and in endeavouring to coordinate them. What they were obliged to leave unfinished, it would be absurd in me to attempt to complete. It would be impossible, also, in the limits of a brief address to give a detailed account of the recent progress of physical and mathematical knowledge. Such a work can only be produced
by separate instalments, each written by a specialist, such as the admirable “ Reports” which form from time to time the most valuable portions of our annual volume.
I shall therefore confine my remarks in the main to those two subjects, one in the mathematical, the other in the purely physical division of our work, which are comparatively familiar to myself. I wish, if possible, to induce, ere it be too late, native mathematicians to pay much more attention than they have yet paid to Hamilton's magnificent Calculus of Quaternions, and to call the particular notice of physicists to our President's grand Principle of Dissipation of Energy. I think that these are, at this moment, the most important because the most promising parts of our field.
If nothing more could be said for Quaternions than that they enable us to exhibit in a singularly compact and elegant form, whose meaning is obvious at a glance on account of the utter inartificiality of the method, results which in the ordinary their use would be furnished. But it would be unjust to Quaternions to be content with such a statement; for we are fully entitled to say that in all cases, even in those to which the Cartesian methods seem specially adapted, they give as simple an expression as any other method ; while in the great majority of cases they give a vastly simpler one. In the common methods a judicious choice of coordinates is often of immense importance in simplifying an investigation; in Quaternions there is usually no choice, for (except when they degrade to mere scalars) they are in general utterly independent of any particular directions in space, and select of themselves the most natural reference lines for each particular problem. This is easily illustrated by the most elementary instances, such as the following :The general equation of Cones involves merely the direction of the vector of a point, while that of Surfaces of Revolution is a relation between the lengths of that vector and of its resolved part parallel to the axis; and Quaternions enable us by a mero 1871.
mark to separate the ideas of length and direction without introducing the cumbrous and clumsy square roots of sums of squares which are otherwise necessary.
But, as it seems to me that mathematical methods should be specially valued in this Section as regards their fitness for physical applications, what can possibly from that point of view be more important than Hamilton's p? Physical analogies have often been invoked to make intelligible various mathematical processes. Witness the case of Statical Electricity, wherein Thomson has, by the analogy of Heat-conduction, explained the meaning of various important theorems due to Green, Gauss, and others; and wherein Clerk-Maxwell has employed the properties of an imaginary incompressible liquid (devoid of inertia) to illustrate not merely these theorems, but even Thomson's Electrical Images. [In fact he has gone much further, having applied his analogy to the puzzling combinations presented by Electrodynamics.). There can be little doubt that these comparisons owe their birth to the small intelligibility, per se, of what has been called Laplace's Operator, 2.2+
dy2 * dz2
which appears alike in all theories of attraction at a distance, in the steady flow of heat in a conductor, and in the steady motion of incompressible fluids. But when we are taught to understand the operator itself we are able to dispense with these analogies, which, however valuable and beautiful, have certainly to be used with extreme caution, as tending very often to confuse and mislead. Now Laplace's operator is merely the negative of the square of Hamil, ton's V, which is perfectly intelligible in itself and in all its combinations; and can be defined as giving the vector-rate of most rapid increase of any scalar function to which it is applied-giving, for instance, the vector-force from a potential, the heat-flux from a distribution of temperature, &c. Very simple functions of the same operator give the rate of increase of a quantity in any assigned direction, the condensation and elementary rotation produced by given displacements of the parts of a system, &c. For instance, a very elementary application of v to the theory of attraction enables us to put one of its fundamental principles in the following extremely suggestive form :-If the displacement or velocity of each particle of a medium represent in magnitude and direction the electric force at that particle, the corresponding statical distribution of electricity is proportional everywhere to the condensation thus produced. Again, Green's celebrated theorem is at once seen to be merely the well-known equation of continuity expressed for a heterogeneous fluid, whose density at every po nt is proportional to one electric potential, and its displacement or velocity proportional to and in the direction of the electric force due to another potential. But this is not the time to pursue such an inquiry, for it would lead me at once to discussions as to the possible nature of electric phenomena and of gravitation. I believe myself to be fully justified in saying that, were the theory of this operator thoroughly
developed and generally known, the whole mathematical treatment of such physical questions as those just mentioned would undergo an immediate and enormous simplification; and this, in its tum, would be at once followed by a proportionately large extension of our knowledge
* The following extracts from letters of Sir W. R. Hamilton have a perfectly general application, so that I do not hesitate to publish them :-" De Morgan was the very first person to notice the Quaternions in print ; namely in a Paper on Triple Algebra, in the “ Camb. Phil. Trans. of 1844. It was, I think, about that time, or not very long after"wards, that he wrote to me, nearly as follows :- I suspect, Hamilton, that you bare "caught the right sow by the ear!' Between us, dear Mr. Tait, I think that we shall begin
the SHEARING of it !!" “You might without offence to me, consider that I abused the license of hope, which may be indulged to an inventor, if I were to confess that I expect " the Quaternions to supply, bereafter, not merely mathematical methods, but also phy"sical suggestions. And, in particular, you are quite welcome to smile if I that it "does not seem extravagant to me to suppose that a full possession of those à priori prin
ciples of mine, about the multiplication of vectors (including the Law of the Four Scales " and the conception of the Extra-spatial Unit), which have as yet been not much more " than hinted to the public, might have led (I do not at all mean that in my hands they "ever would have done so) to an Anticipation of the great discovery of OERSTED."
"It appears to me that one, and not the least, of the services which quaternions may be expected to do to mathematical analysis generally, is that their introduction will compel
And this is but one of the claims of Quaternions to the attention of physicists. When we come to the important questions of stress and strain in an elastic solid, we find again that all the elaborate and puzzling machinery of coordinates commonly employed can be at once comprehended and kept out of sight in a mere single symbol-a linear and vector function, which is self-conjugate if the strain be pure. This is simply, it appears to me, a proof either that the elaborate machinery ought never to have been introduced, or that its use was an indication of a comparatively savage state of mathematical civilization. In the motion of a rigid solid about a fixed point, a quaternion, represented by a single symbol which is a function of the time, gives us the operator which could bring the body by a single rotation from its initial position to its position at any assigned instant.
In short, whenever with our usual means a result can be obtained in, or after much labour reduced to, a simple form, Quaternions will give it at once in that form ; so that nothing is ever lost in point of simplicity. On the other hand, in numberless cases the Quaternion result is immeasurably simpler and more intelligible than any which can be obtained or even expressed by the usual methods. And it is not to be supposed that the modern Higher Algebra, which has done so much to simplify and extend the ordinary Cartesian methods, would be ignored by the general employment of Quaternions ; on the contrary, Determinants, Invariants, &c. present themselves in almost every Quaternion solution, and in forms which have received the full benefit of that simplification which Quaternions generally produce. Comparing a Quaternion investigation, no matter in what department, with the equivalent Cartesian one, even when the latter has availed itself to the utmost of the improvenients suggested by Higher Algebra, one can hardly help making the remark that they contrast even more strongly than the decimal notation with the binary scale or with the old Greek Arithmetic, or than the well-ordered subdivisions of the metrical system with the preposterous no-systems of Great Britain, a mere fragment of which (in the form of Tables of Weights and Measures) forms perhaps the most effective, if not the most ingenious, of the many instruments of torture employed in our elementary teaching.
It is true that, in the eyes of the pure mathematician, Quaternions have one grand and fatal defect. They cannot be applied to space of n dimensions, they are contented to deal with those poor three dimensions in which mere mortals are doomed to dwell, but which cannot bound the limitless aspirations of a Cayley or a Sylvester. From the physical point of view this, instead of a defect, is to be regarded as the greatest possible recommendation. It shows, in fact, Quaternions to be a special instrument so constructed for application to the Actual as to have thrown overboard everything which is not absolutely necessary, without the slightest consideration whether or no it was thereby being rendered useless for applications to the Inconceivable.
The late Sir John Herschel was one of the first to perceive the value of Quaternions; and there may be present some who remember him, at a British Association Meeting not long after their invention, characterizing them as a
“ Cornucopia from which, turn it how you will
, something valuable is sure to fall.” Is it not strange, to use no barsher word, that such a harvest has hitherto been left almost entirely to Hamilton himself? If but half a dozen tolerably good mathematicians, such as exist in scores in this country, were seriously to work at it, instead of spending (or rather wasting) their time, as so many who have the requisite leisure now do, in going over again what has been already done, or in working out mere details where a grand theory has been sketched, a very great immediate advance would be certain. From the majority of the papers in our few mathematical journals one would almost be led to fancy that British mathematicians have too much pride to use a simple method while an unnecessarily complex one can be
" those who adopt them (or even who admit that they may be reasonably adopted by other “persons) to consider, or to admit that others may usefully inquire, what common grounds " can be established for conclusions common to quaternions and to older branches of ma"thematics."
Could any thing be simpler or more satisfactory? Don't you feel, as well as think, " that we are on a right track, and shall be thanked hereafter? Never mind when.”
had. No more telling example of this could be wished for than the insane delusion under which they permit Euclid to be employed in our elementary teaching. They seem voluntarily to weight alike themselves and their pupils for the race; and a cynic might, perhaps without much injustice, say they do so that they may have mere self-imposed and avoidable difficulties to face instead of the new, real, and dreaded ones (belonging to regions hitherto unpenetrated) with which Quaternions would too soon enable them to come into contact. But this game will certainly end in disaster. As surely as Mathematics came to a relative stand-still in this country for nearly a century after Newton, so surely will it do so again if we leave our eager and watchful rivals abroad to take the initiative in developing the grand method of Hamilton. And it is not alone French and Germans whom we have now to dread, Russia, America, regenerated Italy, and other nations are all fairly entered for the contest.
The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulæ than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting, while even the most abstruse branches of physics, as yet totally incapable of being popularized, attract the attention of the uninitiated. The reason may perhaps be sought in the fact that, while perhaps the only successful attempt to invest mathematical reasoning with a halo of glory—that made in this Section by Prof. Sylvester—is known to a comparative few, several of the highest problems of physics are connected with those simple observations which are possible to the many. The smell of lightning has been observed for thousands of years, it required the sagacity of Schönbein to trace it to the formation of Ozone. Not to speak of the (probably fabulous) apple of Newton, what enormous consequences did he obtain by passing light through a mere wedge of glass, and by simply laying a lens on a flat plate! The patching of a trumpery model led Watt to his magnificent inventions. As children at the sea-shore playing with a “roaring buckie," or in later life lazily puffing out rings of tobacco-smoke, we are illustrating two of the splendid researches of Helmholtz. And our President, by the bold, because simple, use of reaction instead of action, has eclipsed even his former services to the Submarine Telegraph, and given it powers which but a few years ago would have been deemed unattainable.
In experimental Physics our case is not hopeless, perhaps not as yet even alarming. Still something of the same kind may be said in this as in pure Mathematics. If Thomson's Theory of Dissipation, for instance, be not speedily developed in this country, we shall soon learn its consequences from abroad. The grand test of our science, the proof of its being a reality and not a mere inventing of new terms and squabbling as to what they shall mean, is that it is ever advancing. There is no standing still; there is no running round and round as in a beaten donkey-track, coming back at the end of a century or so into the old positions, and fighting the self-same battles under slightly different banners, which is merely another form of stagnation (Kinetic Stability in fact). “A little folding of the hands to sleep,” in chuckling satisfaction at what has been achieved of late years by our great experimenters, and we shall be left hopelessly behind. The sad fate of Newton's successors ought ever to be a warning to us. Trusting to what he had done, they allowed mathematical science almost to die out in this country, at least as compared with its immense progress in Germany and France. It required the united exertions of the late Sir J. Herschel and many others to render possible in these islands a Boole and a Hamilton. If the successors of Davy and Faraday pause to ponder even on their achievements, we shall soon be again in the same state of ignominious inferiority. Who will then step in to save us ?
Even as it is, though we have among us many names quite as justly great as any that our rivals can produce, we have also (even in our educated classes) such an immense amount of ignorance and consequent credulity, that it seems matter for surprise that true science is able to exist. Spiritualists, Circle-squarers, Perpetualmotionists, Believers that the earth is flat and that the moon has no rotation, swarm about us. They certainly multiply much faster than do genuine men of science. This is characteristic of all inferior races, but it is consolatory to remember that in spite of it these soon become extinct. Your quack has his little
day, and disappears except to the antiquary. But in science nothing of value can ever be lost; it is certain to become a stepping-stone on the way to further truth. Still, when our stepping-stones are laid, we should not wait till others employ them. “Gentlemen of the Guard be kind enough to fire first” is a courtesy entirely out of date; with the weapons of the present day it would be simply suicide.
There is another point which should not be omitted in an address like this. For obvious reasons I must speak of the general question only, not venturing on examples, though I could give many telling ones. Even among our greatest men of science in this country there is comparatively little knowledge of what has been already achieved, except of course in the one or more special departments cultivated by each individual. There can be little doubt that one cause at least of this is to be sought in the extremely meagre interest which our statesmen, as a rule, take in scientific progress. While abroad we find half a dozen professors teaching parts of the same subject in one University (each having therefore reasonable leisure), with us one man has to do the whole, and to endeavour as he best can to make something out of his very few spare moments. Along with this, and in great part due to it, there is often found a proneness to believe that what seems evident to the thinker cannot but have been long known to others. Thus the credit of many valuable discoveries is lost to Britain because her philosophers, having no time to spare, do not know that they are discoveries. The scientitic men of other nations are, as a rule, better informed | certainly far better encouraged and less over-worked), and perhaps likewise are not so much given to self-depreciation. Until something resembling the ‘Fortschritte der Physik, but in an improved form, and published at smaller intervals and with much less delay, is established in this country, there is little hope of improvement in this respect. Why should science be imperfectly summarized in little haphazard scraps here and there, when mere property has its elaborate series of Money-articles and exact Broker's Share-lists ? Such a work would be very easy of accomplishment: we have only to begin boldly; we do not need to go back, for in every year good work is being done at almost every part of the boundary between, as it were, the cultivated land and the still unpenetrated forest-enough at all events to show with all necessary accuracy whereabouts that boundary lies.
There is no need of entering here on the question of Conservation of Energy; it is thoroughly accepted by scientific men, and has revolutionized the greater part of Physics. The facts as to its history also are generally agreed upon, but difterences of a formidable kind exist as to the deductions to be drawn from them. These are matters, however, which will be more easily disposed of thirty years hence than now. The Transformation of Energy is also generally accepted, and, in fact, under various unsatisfactory names was almost popularly known before the Conservation of Energy was known in its entirety to more than a very few. But the Dissipation of Energy is by no means well known, and many of the results of its legitimate application have been received with doubt, sometimes even with attempted ridicule. Yet it appears to be at the present moment by far the most promising and fertile portion of Natural Philosophy, having obvious applications of which as yet only a small percentage appear to have been made. Some, indeed, were made before the enunciation of the Principle, and have since been recognized as instances of it. Of such we have good examples in Fourier's great work on Heat-conduction, in the optical theorem that an image can never be brighter than the object, in Gauss's mode of investigating electrical distribution, and in some of Thomson's theorems as to the energy of an electromagnetic field. But its discoverer has, so far as I know, as yet confined himself in its explicit application to questions of Heat-conduction and Restoration of Energy, Geological Time, the Earth’s Rotation, and such like. Unfortunately his long-expected Rede Lecture has not yet been published, and its contents (save to those who were fortunate enough to hear it) are still almost entirely unknown.
But there can be little question that the Principle contains implicitly the whole theory of Thermo-electricity, of Chemical Combination, of Allotropy, of Fluorescence, &c., and perhaps even of matters of a higher order than common physics and chemistry. In Astronomy it leads us to the grand question of the age, or