As will be seen by the reader of the former Preface (reprinted below) the point of view which I have, from the first, adopted presents Quaternions as a Calculus uniquely adapted to Euclidian space, and therefore specially useful in several of the most important branches of Physical Science. After giving the necessary geometrical and other preliminaries, I have endeavoured to develope it entirely from this point of view ; and, though one can scarcely avoid meeting with elegant and often valuable novelties to whatever branch of science he applies such a method, my chief contributions are still those contained in the fifth and the two last Chapters. When, twenty years ago, I published my paper on the application of ∇ to Greens and other Allied Theorems, I was under the impression that something similar must have been contemplated, perhaps even mentally worked out, by Hamilton as the subject matter of the (unwritten but promised) concluding section of his Elements. It now appears from his Life (Vol. iii. p. 194) that such was not the case, and thus that I was not in any way anticipated in this application (from any point of view by far the most important yet made) of the Calculus. But a bias in such a special direction of course led to an incomplete because one sided presentation of the subject. Hence the peculiar importance of the contribution from an Analyst like Prof. Cayley.
T.Q.I. b p.viIt is disappointing to find how little progress has recently been made with the development of Quaternions. One cause, which has been specially active in France, is that workers at the subject have been more intent on modifying the notation, or the mode of presentation of the fundamental principles, than on extending the applications of the Calculus. The earliest offender of this class was the late M. Hoüel who, while availing himself of my permission to reproduce, in his Théorie des Quantités Complexes, large parts of this volume, made his pages absolutely repulsive by introducing fancied improvements in the notation. I should not now have referred to this matter (about which I had remonstrated with M. Hoüel) but for a remark made by his friend, M. Laisant, which peremptorily calls for an answer. He says:—“M. Tait…trouve que M. Hoüel a altéré l’œuvre du maître. Perfectionner n’est pas détruire.” This appears to be a parody of the saying attributed to Louis XIV.:—“Pardonner n’est pas oublier”: but M. Laisant should have recollected the more important maxim “Le mieux est l’ennemi du bien.” A line of Shakespeare might help him:—
“…with taper-light
To seek the beauteous eye of heaven to garnish,
Is wasteful and ridiculous excess.”
Even Prof. Willard Gibbs must be ranked as one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector Analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann.
Àpropos of Grassmann, I may advert for a moment to some comparatively recent German statements as to his anticipations &c. of Quaternions. I have given in the last edition of the Encyc. Brit. (Art. Quaternions; to which I refer the reader) all that is necessary to shew the absolute baselessness of these statements. The essential points are as follows. Hamilton not only published his theory complete, the year before the first (and extremely imperfect) sketch of the Ausdehnungslehre appeared; but had given ten years before, in his protracted study of Sets, the very processes of external and internal multiplication (corresponding to the Vector and Scalar parts of a product of two vectors) which have been put forward as specially the property of Grassmann. The scrupulous care with which Hamilton drew up his account of p.vii the work of previous writers (Lectures, Preface) is minutely detailed in his correspondence with De Morgan (Hamilton’s Life, Vol. iii.).
Another cause of the slow head-way recently made by Quaternions is undoubtedly to be ascribed to failure in catching the “spirit” of the method:—especially as regards the utter absence of artifice, and the perfect naturalness of every step. To try to patch up a quaternion investigation by having recourse to quasi-Cartesian processes is fatal to progress. A quaternion student loses his self-respect, so to speak, when he thus violates the principles of his Order. Tannhäuser has his representatives in Science as well as in Chivalry ! One most insidious and dangerous form of temptation to this dabbling in the unclean thing is pointed out in § 500 below. All who work at the subject should keep before them Hamilton’s warning words (Lectures, § 513):—
“I regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto [1853] been unfolded, whenever it becomes, or seems to become, necessary to have recourse……to the resources of ordinary algebra, for the solution of equations in quaternions.”
As soon as my occupation with teaching and with experimental work perforce ceases to engross the greater part of my time, I hope to attempt, at least, the full quaternion development of several of the theories briefly sketched in the last chapter of this book ; provided, of course, that no one have done it in the meantime. From occasional glimpses, hitherto undeveloped, I feel myself warranted in asserting that, immense as are the simplifications introduced by the use of quaternions in the elementary parts of such subjects as Hydrokinetics and Electrodynamics, they are absolutely as nothing compared with those which are to be effected in the higher and (from the ordinary point of view) vastly more complex branches of these fascinating subjects. Complexity is no feature of quaternions themselves, and in presence of their attack (when properly directed) it vanishes from the subject also :—provided, of course, that what we now call complexity depends only upon those space-relations (really simple if rightly approached) which we are in the habit of making all but incomprehensible, by surrounding them with our elaborate scaffolding of non-natural cöordinates.
p.viiiMr Wilkinson has again kindly assisted me in the revision of the proofs; and they have also been read and annotated by Dr Plarr, the able French Translator of the second edition. Thanks to their valuable aid, I may confidently predict that the present edition will be found comparatively accurate.
With regard to the future of Quaternions, I will merely quote a few words of a letter I received long ago from Hamilton :— “Could anything 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.”
The special form of thanks which would have been most grateful to a man like Hamilton is to be shewn by practical developments of his magnificent Idea. The award of this form of thanks will, I hope, not be long delayed.
| Chap. I. | 31 (k), (m), 40, 43. |
| II. | 51, 89. |
| III. | 105, 108, 116, 119–122, 133–4. |
| IV. | 140 (8)–(12), 145–149. |
| V. | 174, 187, 193–4, 196, 199. |
| VI. | The whole. |
| VIII. | 247, 250, 250*. |
| IX. | 285, 286, 287. |
| X. | 326, 336. |
| XI. | 357–8, 382, 384–6. |
| XII. | 390, 393–403, 407, 458, 473 (a)–(l), 480, 489, 493, 499, 500, 503, 508–511, 512–13. |
There are large additions to the number of Examples, some in fact to nearly every Chapter. Several of these are of considerable importance ; as they contain, or suggest, processes and results not given in the text.
p.ix
To the first edition of this work, published in 1867, the following was prefixed :—
‘The present work was commenced in 1859, while I was a Professor of Mathematics, and far more ready at Quaternion analysis than I can now pretend to be. Had it been then completed I should have had means of testing its teaching capabilities, and of improving it, before publication, where found deficient in that respect.
‘The duties of another Chair, and Sir W. Hamilton’s wish that my volume should not appear till after the publication of his Elements, interrupted my already extensive preparations. I had worked out nearly all the examples of Analytical Geometry in Todhunter’s Collection, and I had made various physical applications of the Calculus, especially to Crystallography, to Geometrical Optics, and to the Induction of Currents, in addition to those on Kinematics, Electrodynamics, Fresnel’s Wave Surface, &c., which are reprinted in the present work from the Quarterly Mathematical Journal and the Proceedings of the Royal Society of Edinburgh.
‘Sir W. Hamilton, when I saw him but a few days before his death, urged me to prepare my work as soon as possible, his being almost ready for publication. He then expressed, more strongly perhaps than he had ever done before, his profound conviction of the importance of Quaternions to the progress of physical science ; and his desire that a really elementary treatise on the subject should soon be published.
‘I regret that I have so imperfectly fulfilled this last request p.x of my revered friend. When it was made I was already engaged, along with Sir W. Thomson, in the laborious work of preparing a large Treatise on Natural Philosophy. The present volume has thus been written under very disadvantageous circumstances, especially as I have not found time to work up the mass of materials which I had originally collected for it, but which I had not put into a fit state for publication. I hope, however, that I have to some extent succeeded in producing a thoroughly elementary work, intelligible to any ordinary student ; and that the numerous examples I have given, though not specially chosen so as to display the full merits of Quaternions, will yet sufficiently shew their admirable simplicity and naturalness to induce the reader to attack the Lectures and the Elements; where he will find, in profusion, stores of valuable results, and of elegant yet powerful analytical investigations, such as are contained in the writings of but a very few of the greatest mathematicians. For a succinct account of the steps by which Hamilton was led to the invention of Quaternions, and for other interesting information regarding that remarkable genius, I may refer to a slight sketch of his life and works in the North British Review for September 1866.
‘It will be found that I have not servilely followed even so great a master, although dealing with a subject which is entirely his own. I cannot, of course, tell in every case what I have gathered from his published papers, or from his voluminous correspondence, and what I may have made out for myself. Some theorems and processes which I have given, though wholly my own, in the sense of having been made out for myself before the publication of the Elements, I have since found there. Others also may be, for I have not yet read that tremendous volume completely, since much of it bears on developments unconnected with Physics. But I have endeavoured throughout to point out to the reader all the more important parts of the work which I know to be wholly due to Hamilton. A great part, indeed, may be said to be obvious to any one who has mastered the preliminaries ; still I think that, in the two last Chapters especially, a good deal of original matter will be found.
p.xi‘The volume is essentially a working one, and, particularly in the later Chapters, is rather a collection of examples than a detailed treatise on a mathematical method. I have constantly aimed at avoiding too great extension ; and in pursuance of this object have omitted many valuable elementary portions of the subject. One of these, the treatment of Quaternion logarithms and exponentials, I greatly regret not having given. But if I had printed all that seemed to me of use or interest to the student, I might easily have rivalled the bulk of one of Hamilton’s volumes. The beginner is recommended merely to read the first five Chapters, then to work at Chapters VI, VII, VIII* (to which numerous easy Examples are appended). After this he may work at the first five, with their (more difficult) Examples ; and the remainder of the book should then present no difficulty.
‘Keeping always in view, as the great end of every mathematical method, the physical applications, I have endeavoured to treat the subject as much as possible from a geometrical instead of an analytical point of view. Of course, if we premise the properties of i, j, k merely, it is possible to construct from them the whole system†; just as we deal with the imaginary of Algebra, or, to take a closer analogy, just as Hamilton himself dealt with Couples, Triads, and Sets. This may be interesting to the pure analyst, but it is repulsive to the physical student, who should be led to look upon i, j, k, from the very first as geometric realities, not as algebraic imaginaries. The most striking peculiarity of the Calculus is that multiplication is not generally commutative, i.e. that qr is in general different from rq, r, and q, being quaternions. Still it is to be remarked that something similar is true, in the ordinary cöordinate methods, of operators and functions : and therefore
p.xii the student is not wholly unprepared to meet it. No one is puzzled by the fact that log. cos. x is not equal to cos. log. x, or that √(dy/dx) is not equal to (d/dx)√(y). Sometimes, indeed, this rule is most absurdly violated, for it is usual to take cos2x as equal to (cos x)2 , while cos−1x is not equal to (cos x)−1. No such incongruities appear in Quaternions ; but what is true of operators and functions in other methods, that they are not generally commutative, is in Quaternions true in the multiplication of (vector) cöordinates.
‘It will be observed by those who are acquainted with the Calculus that I have, in many cases, not given the shortest or simplest proof of an important proposition. This has been done with the view of including, in moderate compass, as great a variety of methods as possible. With the same object I have endeavoured to supply, by means of the Examples appended to each Chapter, hints (which will not be lost to the intelligent student) of farther developments of the Calculus. Many of these are due to Hamilton, who, in spite of his great originality, was one of the most excellent examiners any University can boast of.
‘It must always be remembered that Cartesian methods are mere particular cases of Quaternions, where most of the distinctive features have disappeared ; and that when, in the treatment of any particular question, scalars have to be adopted, the Quaternion solution becomes identical with the Cartesian one. Nothing therefore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to choice of particular cöordinate axes.
‘There is one very desirable object which such a work as this may possibly fulfil. The University of Cambridge, while seeking to supply a real want (the deficiency of subjects of examination for mathematical honours, and the consequent frequent introduction of the wildest extravagance in the shape of data for “Problems”), p.xiii is in danger of making too much of such elegant trifles as Trilinear Cöordinates, while gigantic systems like Invariants (which, by the way, are as easily introduced into Quaternions as into Cartesian methods) are quite beyond the amount of mathematics which even the best students can master in three years’ reading. One grand step to the supply of this want is, of course, the introduction into the scheme of examination of such branches of mathematical physics as the Theories of Heat and Electricity. But it appears to me that the study of a mathematical method like Quaternions, which, while of immense power and comprehensiveness, is of extraordinary simplicity, and yet requires constant thought in its applications, would also be of great benefit. With it there can be no “shut your eyes, and write down your equations,” for mere mechanical dexterity of analysis is certain to lead at once to error on account of the novelty of the processes employed.
‘The Table of Contents has been drawn up so as to give the student a short and simple summary of the chief fundamental formulae of the Calculus itself, and is therefore confined to an analysis of the first five [and the two last] chapters.
‘In conclusion, I have only to say that I shall be much obliged to any one, student or teacher, who will point out portions of the work where a difficulty has been found; along with any inaccuracies which may be detected. As I have had no assistance in the revision of the proof-sheets, and have composed the work at irregular intervals, and while otherwise laboriously occupied, I fear it may contain many slips and even errors. Should it reach another edition there is no doubt that it will be improved in many important particulars.’
To this I have now to add that I have been equally surprised and delighted by so speedy a demand for a second edition—and the more especially as I have had many pleasing proofs that the work has had considerable circulation in America. There seems now at last to be a reasonable hope that Hamilton’s grand invention will soon find its way into the working world of science, to which it is certain to render enormous services, and not be laid T. Q. I. c p.xiv aside to be unearthed some centuries hence by some grubbing antiquary.
It can hardly be expected that one whose time is mainly engrossed by physical science, should devote much attention to the purely analytical and geometrical applications of a subject like this; and I am conscious that in many parts of the earlier chapters I have not fully exhibited the simplicity of Quaternions. I hope, however, that the corrections and extensions now made, especially in the later chapters, will render the work more useful for my chief object, the Physical Applications of Quaternions, than it could have been in its first crude form.
I have to thank various correspondents, some anonymous, for suggestions as well as for the detection of misprints and slips of the pen. The only absolute error which has been pointed out to me is a comparatively slight one which had escaped my own notice : a very grave blunder, which I have now corrected, seems not to have been detected by any of my correspondents, so that I cannot be quite confident that others may not exist.
I regret that I have not been able to spare time enough to rewrite the work ; and that, in consequence of this, and of the large additions which have been made (especially to the later chapters), the whole will now present even a more miscellaneously jumbled appearance than at first.
It is well to remember, however, that it is quite possible to make a book too easy reading, in the sense that the student may read it through several times without feeling those difficulties which (except perhaps in the case of some rare genius) must attend the acquisition of really useful knowledge. It is better to have a rough climb (even cutting one’s own steps here and there) than to ascend the dreary monotony of a marble staircase or a well-made ladder. Royal roads to knowledge reach only the particular locality aimed at—and there are no views by the way. It is not on them that pioneers are trained for the exploration of unknown regions.
But I am happy to say that the possible repulsiveness of my early chapters cannot long be advanced as a reason for not attacking this fascinating subject. A still more elementary work than p.xv the present will soon appear, mainly from the pen of my colleague Professor Kelland. In it I give an investigation of the properties of the linear and vector function, based directly upon the Kinematics of Homogeneous Strain, and therefore so different in method from that employed in this work that it may prove of interest to even the advanced student.
Since the appearance of the first edition I have managed (at least partially) to effect the application of Quaternions to line, surface, and volume integrals, such as occur in Hydrokinetics, Electricity, and Potentials generally. I was first attracted to the study of Quaternions by their promise of usefulness in such applications, and, though I have not yet advanced far in this new track, I have got far enough to see that it is certain in time to be of incalculable value to physical science. I have given towards the end of the work all that is necessary to put the student on this track, which will, I hope, soon be followed to some purpose.
One remark more is necessary. I have employed, as the positive direction of rotation, that of the earth about its axis, or about the sun, as seen in our northern latitudes, i.e. that opposite to the direction of motion of the hands of a watch. In Sir W. Hamilton’s great works the opposite is employed. The student will find no difficulty in passing from the one to the other ; but, without previous warning, he is liable to be much perplexed.
With regard to notation, I have retained as nearly as possible that of Hamilton, and where new notation was necessary I have tried to make it as simple and, as little incongruous with Hamilton’s as possible. This is a part of the work in which great care is absolutely necessary; for, as the subject gains development, fresh notation is inevitably required ; and our object must be to make each step such as to defer as long as possible the revolution which must ultimately come.
Many abbreviations are possible, and sometimes very useful in private work ; but, as a rule, they are unsuited for print. Every analyst, like every short-hand writer, has his own special contractions ; but, when he comes to publish his results, he ought invariably to put such devices aside. If all did not use a common p.xvi mode of public expression, but each were to print as he is in the habit of writing for his own use, the confusion would be utterly intolerable.
Finally, I must express my great obligations to my friend M. M. U. Wilkinson of Trinity College, Cambridge, for the care with which he has read my proofs, and for many valuable suggestions.
College, Edinburgh,
October 1873.