A

THURSDAY, DECEMBER 25, 1873

QUATERNIONS

MATHEMATICIAN is one who endeavours to secure the greatest possible consistency in his thoughts and statements, by guiding the process of his reasoning into those well-worn tracks by which we pass from one relation among quantities to an equivalent relation. He who has kept his mind always in those paths which have never led him or anyone else to an inconsistent result, and has traversed them so often that the act of passage has become rather automatic than voluntary, is, and knows himself to be, an accomplished mathematician. The very important part played by calculation in modern mathematics and physics has led to the development of the popular idea of a mathematician as a calculator, far more expert, indeed, than any banker's clerk, but of course immeasurably inferior, both in resources and in accuracy, to what the "analytical engine" will be, if the late Mr. Babbage's design should ever be carried into execution.

But though much of the routine work of a mathematitician is calculation, his proper work—that which constitutes him a mathematician-is the invention of methods. He is always inventing methods, some of them of no great value except for some purpose of his own; others, which shorten the labour of calculation, are eagerly adopted by all calculators. But the methods on which the mathematician is content to hang his reputation are generally those which he fancies will save him and all who come after him the labour of thinking about what has cost himself so much thought.

Now Quaternions, or the doctrine of Vectors, is a mathematical method, but it is a method of thinking, and not, at least for the present generation, a method of saving thought. It does not, like some more popular mathematical methods, encourage the hope that mathematicians may give their minds a holiday, by transferring all their work to their pens. It calls upon us at every step to form a mental image of the geometrical features represented by the symbols, so that in studying geometry by this method we have our minds engaged with geometrical ideas, and are not permitted to fancy ourselves geometers when we are only arithmeticians.

This demand for thought-for the continued construction of mental representations-is enough to account for the slow progress of the method among adult mathematicians. Two courses, however, are open to the cultivators of Quaternions they may show how easily the principles of the method are acquired by those whose minds are still fresh, and in so doing they may prepare the way for the triumph of Quaternions in the next generation; or they may apply the method to those problems which the science of the day presents to us, and show how easily it arrives at those solutions which have been already expressed in ordinary mathematical language, and how it brings within our reach other problems, which the ordinary methods have hitherto abstained from attacking.

Sir W. R. Hamilton, when treating of the elements of the subject, was apt to become so fascinated by the metaphysical aspects of the method, that the mind of his disciple became impressed with the profundity, rather

VOL. IX.-No. 217

than the simplicity of his doctrines. Professors Kelland and Tait in the opening chapter (II.) of their recently published work * have, we think, successfully avoided this element of discouragement. They tell us at once what a vector is, and how to add vectors, and they do this in a way which is quite as intelligible to those who are just beginning to learn geometry as to the most expert mathematician.

The subject, like all other subjects, becomes more intricate as the student advances in it; but at the same time his ideas are becoming clearer and more firmly established as he works out the numerous examples and exercises which are placed before him.

The technical terms of the method-Scalar, Vector, Tensor, Versor-are introduced in their proper places, and their meaning is sufficiently illustrated to the beginner by the examples which he is expected to work out. The pride of the accomplished mathematician, however (for whom this book is not written), might have been somewhat mollified if somewhere in the book a few pages had been devoted to explaining to him the differences between the Quaternion methods and those which he has spent his life in mastering, and of which he has now become the slave. He is apt to be startled by finding that when one vector is multiplied into another at right angles to it, the product is still a vector, but at right angles to both. His only idea of a vector had been that of a line, and he had expected that when one vector was multiplied into another the result would be something of a different kind from a line, such, for instance, as a surface. Now if it had been pointed out to him in the chapter on vector multiplication that a surface is a vector, he would be saved from a painful mental shock, for a mathematician is as sensitive about "dimensions" as an English schoolboy is about "quantities."

The fact is, that even in the purely geometrical applications of the Quaternion method we meet with three different kinds of directed quantities: the vector proper, which represents transference from A to B; the area or "aperture," which is always understood to have a positive and a negative aspect, according to the direction in which it is swept out by the generating vector; and the versor, which represents turning round an axis.

The Quaternion ideas of these three quantities differ from the old ideas of the line, the surface, and the angle only by giving more prominence to the fact that each of them has a determinate direction as well as a determinate magnitude. When Euclid tells us to draw the line A B, he supposes it to be done by the motion of a point from A to B or from B to A. But when the line is once generated he makes no distinction between the results of these two operations, which, on Hamilton's system, are each the opposite of the other.

Surfaces also, according to Euclid, are generated by the motion of lines, so that the idea of motion is an old one, and we have only to take special note of the direction of the motion in order to raise Euclid's idea to the level of Hamilton's.

With respect to angles, Euclid appears to treat them as if they arose from the fortuitous concourse of right lines;

* "Introduction to Quaternions, with numerous Examples." By P. Kelland, F. R.S., formerly Fellow of Queen's College, Cambridge; and P. G. Tait, formerly Fellow of St. Peter's College, Cambridge; Professors in the Department of Mathematics in the University of Edinburgh. (Macmillan, 1873.

I

but the unsatisfactory nature of this mode of treatment is shown by the fact that in all modern books on trigonometry an angle is represented as generated by motion round an axis in a definite direction.

There are thus three geometrical quantities having direction, and the more than magical power of the method of Quaternions resides in the spell by which these three orders of quantities are brought under the sway of the same system of operators.

The secret of this spell is twofold, and is symbolised by the vine-tendril and the mason's rule and square. The tendril of the vine teaches us the relation which must be maintained between the positive direction of translation along a line and the positive direction of rotation about that line. When we have not a vine-tendril to guide us, a corkscrew will do as well, or we may use a hop-tendril, provided we look at it not directly, but by reflexion in a mirror.

The mason's rule teaches us that the symbol, as written on paper, is not a real line, but a mere injunction, commanding us to measure out in a certain direction a vector of a length so many times that of the rule. Without the rule the symbol would have no definite meaning. Thus the rule is the unit of the Quaternion system, while the square reminds us that the right angle is the unit

versor.

The doctrine of the unit is a necessary part of every exact science, but in Quaternions the application of the same operators to versors, vectors, and areas is utterly unintelligible without a clear understanding of the function of the unit in the science of measurement.

Whether, however, it is better to insinuate the true doctrine into the mind of the student by a graduated series of exercises, or to inculcate it upon him at once by dogmatic statements, is a question which can only be determined by the experience of a new generation, who shall have been born with the extraspatial unit ever present to their consciousness, and whose thoughts, guided by the vine-tendril along the Quaternion path, shall turn always to the right hand, and never to the left. Prof. Kelland tells us in the preface to the work to which we have alluded that, whereas Sir W. R. Hamilton and Prof. Tait have written treatises on Quaternions for mathematicians, the time has come when it behoves some one to write for those who desire to become mathematicians. Whatever, therefore, advanced mathematicians may think of this book, they ought to reserve their judgment as to its difficulty till they have ascertained how it is assimilated by those for whom it is written-those in whom the desire to become mathematicians has not yet become alloyed with the consciousness that they are mathematicians. For while Prof. Kelland-as he has elsewhere told us-finds but little difficulty in teaching the elements of the doctrine of Vectors to his junior classes, Hamilton himself, the great master of the spell, when addressing mathematicians of established reputation, found, for his Quaternions, but few to praise and fewer still to love.

Prof. Kelland, by the clearness and orderliness of his statements, and by boldly getting rid of everything which is unnecessarily abstruse, has done more than any other man towards rendering the subject easy to the student, and reconciling even the case-hardened mathematician to

the new method, as applied to geometrical questions of old-established truth.

The other aspect of Quaternions, as a method which every mathematician must learn in order to deal with the questions which the progress of physics brings every day into greater prominence, is hinted at by Prof. Tait in the last chapter of the book. He there introduces us to the linear and vector function of the first degree under its kinematical aspect of a homogeneous strain. The importance of functions of this kind may be gathered from the fact that a knowledge of their properties supplies the key to the theory of the stresses as well as the strains in solid bodies, and to that of the conduction of heat and electricity in bodies whose properties are different in different directions, to the phenomena exhibited by crystals in the magnetic field, to the thermo-electric properties of crystals, and to other sets of natural phenomena, one or more of which the scientific progress of every year brings before us.

But as we believe that Prof. Tait is about to bring out a new edition of his treatise on Quaternions, in which this higher aspect of the subject will be brought more prominently forward, we reserve our remarks on Quaternions as an instrument of physical research till we have the subject presented to us by Prof. Tait in a form which adequately represents its latest developments.

MARKHAM'S "UNKNOWN REGION”

The Threshold of the Unknown Region. By Clements R. Markham, C.B., F.R.S., Secretary of the Royal Geographical Society, formerly of H.M. Arctic ship Assistance. (London: Sampson Low and Co., 1873).

H

E must be a sorry story-teller who manages to make a traveller's tale uninteresting, especially if the traveller be a voyager, and still more if his voyages have led him into unknown regions. Of all forms of narrative we think it will be generally acknowledged that narratives of discovery are by far the most popular, as is testified by the abundance of this kind of literature, historical and fictitious, provided for the delectation of the young. Νο doubt this may be largely accounted for by the fact that a discoverer of new lands is continually unveiling the unknown to those who listen to his tale, thereby appealing to one of the strongest and most fruitful characteristics of the human mind, that of curiosity. Every step taken by a discoverer, every knot sailed by his "good ship," we know will lead him among fresh wonders. Once upon a time the Unknown Region-that is, the region unknown to those peoples who have had a thirst for knowledge to any fruitful extent-was in sooth wide enough, when first our Aryan forefathers left their eastern home, and had "all the world before them where to choose." Even four centuries ago the greater part of the earth waited the coming of the European descendants of those primitive discoverers who first turned their faces cagerly and inquisitively to the unknown west. But ever since then the boundary of the Unknown Region has been gradually pushed farther and farther back, until now there remains comparatively little to be found out in order to enable geographers to complete the configuration of the lands of the globe. The extent of our dwelling-place is now pretty well known, though there is yet abundance of

that lie between Hall's farthest north point and the Pole.

True, there are a few unhealthy croakers, as there always have been, and will be, we fear, for many generations to come, who ask What is the good of incurring so much danger and expense, for the mere gratification of curiosity, or, at best, to satisfy the wishes of a few men of science? But we feel confident that the great body of the English people will ask no such questions, but would hail with enthusiasm the decision of the Government to crown the glory which England has hitherto gained in Arctic exploration by sending out one more expedition whose task it would be to return with the long-sought-for secret in its keeping. It is beginning to dawn upon the ordinary English mind that, after all, the apparently unpractical researches of scientific men are frequently pregnant with results of the most important practical bearing on the welfare of the country and the race.

As for the element of danger, Mr. Markham convincingly shows by unimpeachable statistics, that the loss of men by the Smith Sound route, from causes connected with the climate and the peculiarities of the service, is almost incredibly small. One of the most distinguished medical officers who has served in the Arctic regions declares, that "of all seas visited by men-of-war the Arctic have proved the most healthy. . . . The risk by climate and disease which is run in a voyage to the Arctic seas-such as a Royal Expedition necessitates-is not greater than that which a ship like the Challenger will incur in her voyage of discovery." The dangers, or rather difficulties, which have to be faced are only such as brave men are eager to confront, and the service is one which our naval officers and men glory in in time of peace, and is certainly an infinitely better use to put them to than to keep them idling at home or on foreign stations. As to the question of expense, the article in a recent number will show that the less said by Government on this score the

better.

All these and many other points in connection with Arctic exploration will be found fully and clearly discussed in Mr. Markham's volume, in which the invaluable results, scientific and practical, in nearly all departments of Science to be obtained from a Government Expedition are set forth with great fulness, clearness, and force. The volume concludes with an account of the interview that took place last year between the Arctic deputation and Mr. Lowe, the result of which was such as to give good grounds for expecting that this year Government will feel bound to organise an adequate expedition to leave our shores next spring to find its way to the Pole by the Smith Sound

route.

The numerous maps by which the volume is illustrated are beautifully drawn, and are of the greatest assistance in enabling the reader to understand the interesting story of Arctic discovery so well told by Mr. Markham. As a mere story the work is a masterly one; and if anyone wants to know within short space what has already been done in the discovery of the Arctic regions, what still remains to be done, and what results are to be expected from further exploration, he could not do better than ead Mr. Markham's "Threshold of the Unknown Region."

OUR BOOK SHELF

Annual Record of Science and Industry for 1872; edited by Spencer F. Baird, with the assistance of eminent men of Science. (New York: Harper and Brothers, 1873.) THE praise which we were able to bestow on the first of Prof. Baird's Annual Records, that for 1871, can be fully repeated with regard to its successor. The only method of "reviewing" a work of this kind, is to refer in general terms to its scope, and to the degree to which the compiler appears to have fulfilled the promises of his programme. On these points we can speak in the most favourable terms. As far as a cursory glance through the pages of the volume enables us to speak, we believe that purchasers of the book will find it a most useful addition to their library shelves. The paragraphs refer to the most noteworthy additions to scientific knowledge piled with commendable terseness and perspicuity from or observation made during the year, and have been coma large range of English, American, and Continental sources. A carefully raisonné table of contents, and an alphabetical index, will enable the student to turn withfreedom from errors, typical and otherwise, can hardly out difficulty to any desired subject. Although absolute be expected in a work with so large a scope, the American "Record" contrasts most favourably in this respect with some similar volumes published in this country. We do not know where to find a more complete record of the science of the year; and we shall hope to see a long series of these useful volumes. The Borderland of Science. By Richard A. Proctor. (London: Smith, Elder, and Co. 1873.)

zine.

99 66

and Rankine. The lecture on Energy is followed by one on the Dynamical Theory; which embraces to some extent the relations of different forces, and the “varied modality" of chemical, thermical, and electrical action. The next lecture relates to Molecular Dynamics. Then in succession:-Electromotive force produced by Chemical Action, by Heat, by mechanical means, and by Induction. Mutual action of Currents and Magnets, Terrestrial Magnetism, Polar Aurora; Atmospheric Electricity; Diamagnetism; Rhumkorff's Coil; Winds; Marine Currents; the Sun; the Doctrine of La Place; the Doctrine of Lyell; Thermogenesis; Atmolysis and Osmosis; Capillarity; the Doctrine of Mayer. The second course treats of electricity, undulations, sonority, musical timbre, echoes, photometry, dispersion and the spectroscope, chromatism, vision, luminous undulations, diffraction, polarisation, radiant heat, action of electricity on organic bodies, the muscular current, electrical nervous phenomena, electrical fishes.

The arrangement is really wonderful. What can possibly warrant the following order for lectures ---diamagnetism, Rhumkorffs coil, winds, marine currents; or again-thermogenesis, atmolysis, capillarity? One lecture ends with "Che cosí mirabilmente si svolgono dall' evoluzione Darwinianæ ;" and the next commences "E impossible proseguire un corso di Fisica e piu ancora quella parte, che tratta delle azioni senza prima definire le parole, atomo, molecula." The Prof. Pozzo can scarcely be expected to lecture on all science: to pass from the sun to an atom, from Darwinism to electro-dynamics, from geology to elliptical polarisation. If he is, the system is a bad one, and his students may get a smattering of many things, and know nothing well. Mechanical philosophy seems to be almost ignored. The book is devoid of mathematics, and without woodcuts; and we imagine the youth of Perugia must yawn over it; and, if the lectures are as dry as the book, spend much of the time which ought to be given to physics in saying "felicissima notte" to each other.

LETTERS TO THE EDITOR

G. F. R.

[The Editor does not hold himself responsible for opinions expressed by his correspondents. No notice is taken of anonymous communications.]

Proposed Alterations in the Medical Curriculum THE remarks made in your number of December 18 by my friend Prof. Balfour are founded on the mistake he has made in supposing that it is proposed to abolish the regulation requiring attendance on the courses of lectures on Botany and Zoology. There is no question raised between mere examining bards and teaching institutions, between compulsory and optional attendance on professors' lectures. It is simply that the candidate for medical degrees be allowed to take the examination in Zoology and Botany earlier than is at present permitted. At present the examination in these subjects in Edinburgh University is fixed by ordinance at the end of the second of the four years of medical study, and in this University, while the Botany comes at that time, the Zoology is actually not till the end of the third year, so that our case is even worse than that of Edinburgh. Prof. Balfour says, "The student might be encouraged to take his science examination at an early period of his curriculum, say at the end of his first year of study." That is exactly the result❘ practically aimed at here, and I am quite at one with him on the subject. But why prevent the student from taking the examination in Botany and Zoology before entering on his medical curriculum proper, if he has attended the professor's class and is ready for it? Very few would at present do so, as it would imply a preliminary year of attendance at the Universities to obtain the courses of Zoology and Botany. But is it not a very desirable thing, from every point of view, to encourage this? So far from lowering the standard in these subjects, or proIt would moting cramming, it would do exactly the reverse. enable real study to take the place of the cramming which is inevitable when these subjects are left over to be mixed up with medical studies proper.

For some time there has been a strong feeling here that the examination in Zoology and Botany should take place not later than the end of the first year, and the Lord Rector of our University in taking this matter up, instead of tinkering as to particular dates, has announced the sound general principle that the student should be encouraged to take the subjects of Botany and Zoology before beginning his medical curriculum proper, with the view both of promoting a more real study of these sciences, and of clearing the subsequent medical curriculum for a more real study of the subjects which belong to it. I see nothing in the resolutions which our distinguished Lord Rector has laid before the University Court either suggesting or implying abolition of compulsory attendance on the professors' courses of Zoology and Botany, and Prof. Balfour might well have taken it for granted that the mere fact of the proposal emanating from Prof. Huxley is security enough that the object could not possibly be to lower the position of the natural sciences or to promote cramming instead of real study. Our Lord Rector has as yet only intimated his resolutions, but when the oracle speaks we shall no doubt hear such good reasons for them that even so enthusiastic a botanist as Prof. Balfour will have his alarm turned into joy.

Will any of those who are so strong on the point of compulsory attendance on courses of Zoology and Botany tell us why they do not say a word for Natural Philosophy? Including such subjects as heat, light, electricity, hydrostatics, pneumatics, optics, acoustics, it is surely of more importance than either of the other two, whether regarded educationally or in its bearing on modern medicine. Yet in the Scotch Universities there is no compulsion to attend a course of lectures on Natural Philosophy, and it is relegated to the preliminary examination in general education. lectures, but it is surely not too much to say that the student The day is past for laying on additional compulsory courses of might be allowed to profess and be examined in Natural Philosophy instead of one of the other two. Aberdeen University, Dec. 20

JOHN STRUTHERS

The Distribution of Volcanoes

SOME of the correspondence in your paper has latterly been so caustic, that timid people may be pardoned for shrinking from writing letters which bring down upon them the hammers of scorn and contempt o vigorously.

Notwithstanding this, the discussion between Mr. Mallet and Dr. Forbes about volcanoes tempts me to write to you on a side issue of that controversy in which I have been interested for some time. What I have to say may not be new, although I believe it to be so. At all events it is not commented upon in the books accessible to me. I will premise that, caring little for laurels of any kind but a good deal for instruction, that if it be discovered that what I say is stale and old, I hope I may be treated as an ignorant scholar, willing to learn, and not as a rival to be crushed, and further, that my results having been obtained independently, they support and make more sure the position of my predecessors.

You were good enough, some months ago, to print some letters of mine on the current elevation of the circumpolar regions of the earth. I have since accumulated much new matter on this subject, which will be shortly published in part in the Journal of the Geographical Society. The general result of my inquiry is, that all the large land surfaces of the earth, the large continental and insular surfaces, are more or less in process of gradual or rapid elevation. There are a few small areas of depression on the outskirts and borders of the great land-masses, but these are very local and unimportant. And with this slight exception the continents of North and South America, Asia, Europe, Africa, and Australia, are all more or less rising. This rise of the land-surfaces necessitates a corresponding sinking, either an absolute or a relative sinking, in the surfaces covered with water. It is comparatively easy to test where a land surface is gradually protruding from the water. is not such a simple matter always to know whether this rise is relative or absolute, for the same effect may be produced by the sinking of the sea-floor as by the actual rising of the land. One thing only we know, that when our measure is water, there must be a corresponding sinking either relative or absolute where there is a rising elsewhere. evidence of the sinking of the sea-bottom is not very easy to find, but such does exist. Students are familiar with the facts collected by Darwin and others, showing from the growth of coral islands, &c., that the Pacific is an area of depression; other evidence consists in the disappearance of well-known rocks, the

It

Direct