0.1 Preface

From An Elementary Treatise On Quaternions by Peter Guthrie Tait.

IN the present edition this work has been very greatly enlarged; to the extent, in fact, of more than one-third. Had I not determined to keep the book in moderate compass it might easily have been doubled in size. A good deal of re-arrangement also has been thought advisable, especially with reference to the elementary uses of q()q⁻1, and of ∇. Prominent among the additions is an entire Chapter, On the Analytical Aspect of Quater nions, which I owe to the unsolicited kindness of Prof. Cayley.

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 im portant 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 my 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.

It 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. Houel who, while availing himself of my permission to reproduce, in his Theorie des Quantites 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. Houel) but for a remark made by his friend, M. Laisant, which peremptorily calls for an answer. He says: ”M. Tait...trouve que M. Houel a altere l’ceuvre du maitre. Perfectionner n’est pas detruire.” 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.

Apropos 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 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. Tannhauser 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 coordinates.

Mr 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.

P. G. TA1T.