Arthur Cayley

Arthur Cayley (August 16 1821 - January 26 1895) was a British mathematician. He helped found the modern British school of pure mathematics.

Quaternions

To the third edition of Tait's Elementary Treatise on Quaternions, Cayley contributed a chapter entitled "Sketch of the analytical theory of quaternions." In it the sqrt{-1} reappears in all its glory, and in entire, so it is said, independence of i, j, k. The remarkable thing is that Hamilton started with a quaternion theory of analysis, and that Cayley should present instead an analytical theory of quaternions. I daresay that Prof. Tait was sorry that he allowed the chapter

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to enter his book, for in 1894 there arose a brisk discussion between himself and Cayley on "Coordinates versus Quaternions,"

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the record of which is printed in the Proceedings of the Royal Society of Edinburgh. Cayley maintained the position that while

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coordinates are applicable to the whole science of geometry and are the natural and appropriate basis and method in the science,

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quaternions seemed a particular and very artificial method for treating such parts of the science of three-dimensional geometry

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as are most naturally discussed by means of the rectangular coordinates x, y, z. In the course of his paper Cayley says:

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:I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications. As another illustration, I compare a quaternion formula to a pocket-map---a capital thing to put in one's pocket, but which for use must be unfolded: the formula, to be understood, must be translated into coordinates.

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He goes on to say,

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:I remark that the imaginary of ordinary algebra---for distinction call this heta---has no relation whatever to the quaternion symbols i, j, k; in fact, in the general point of view, all the quantities which present themselves, are, or may be, complex values a + heta b, or in other words, say that a scalar quantity is in general of the form a + heta b. Thus quaternions do not properly present themselves in plane or two-dimensional geometry at all; but they belong essentially to solid or three-dimensional geometry, and they are most naturally applicable to the class of problems which in coordinates are dealt with by means of the three rectangular coordinates x, y, z.

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To the pocketbook illustration it may be replied that a set of coordinates is an immense wall map, which you cannot carry about,

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even though you should roll it up, and therefore is useless for many important purposes. In reply to the arguments, it may be

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said, first, sqrt{-1} has a relation to the symbols i, j, k for each of these can be analyzed into a unit axis multiplied by sqrt{-1}; second, as regards plane geometry, the ordinary form of complex quantity is a degraded form of the quaternion in which the constant axis of the plane is left unspecified. Cayley took his illustrations from his experience as a traveller. Tait brought forward an illustration from which you might imagine he had visited the Bethlehem Iron Works, and hunted tigers in India. He says,

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:A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. Quaternions, on the other hand, are like the elephant's trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere---alike in the trackless jungle and in the barrack square---directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one.

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The reply which Tait makes, so far as it is an argument, is: There are two systems of quaternions, the i, j, k one, and another one which Hamilton developed from it; Cayley knows the first only, he himself knows the second; the former is an intensely artificial system of imaginaries, the latter is the natural organ of expression for quantities in space. Should a fourth edition of his Elementary Treatise be called for i, j, k will disappear from it, excepting in Cayley's chapter, should it be retained. Tait thus describes the first system:

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:Hamilton's extraordinary Preface to his first great book shows how from Double Algebras, through Triplets, Triads, and Sets, he finally reached Quaternions. This was the genesis of the Quaternions of the forties, and the creature thus produced is still essentially the Quaternion of Prof. Cayley. It is a magnificent analytical conception; but it is nothing more than the full development of the system of imaginaries i, j, k; defined by the equations, i^{2} = j^{2} = k^{2} = ijk = -1 , with the associative, but not the commutative, law for the factors. The novel and splendid points in it were the treatment of all directions in space as essentially alike in character, and the recognition of the unit vector's claim to rank also as a quadrantal versor. These were indeed inventions of the first magnitude, and of vast importance. And here I thoroughly agree with Prof. Cayley in his admiration. Considered as an analytical system, based throughout on pure imaginaries, the Quaternion method is elegant in the extreme. But, unless it had been also something more, something very different and much higher in the scale of development, I should have been content to admire it;---and to pass it by.

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From "the most intensely artificial of systems, arose, as if by magic, an absolutely natural one" which Tait thus further describes. "To me Quaternions are primarily a Mode of Representation:---immensely superior to, but of essentially the same kind of usefulness as, a diagram or a model. They are, virtually, the thing represented; and are thus antecedent to, and independent of, coordinates; giving, in general, all the main relations, in the problem to which they are applied, without the necessity of appealing to coordinates at all. Coordinates may, however, easily be read into them:---when anything (such as metrical or numerical detail) is to be gained thereby.

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Quaternions, in a word, exist in space, and we have only to recognize them:---but we have to invent or imagine coordinates of all kinds."

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To meet the objection why Hamilton did not throw i, j, k overboard, and expound the developed system, Tait says:

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:Most unfortunately, alike for himself and for his grand conception, Hamilton's nerve failed him in the composition of his first great volume. Had he then renounced, for ever, all dealings with i, j, k, his triumph would have been complete. He spared Agog, and the best of the sheep, and did not utterly destroy them. He had a paternal fondness for i, j, k ; perhaps also a not unnatural liking for a meretricious title such as the mysterious word Quaternion; and, above all, he had an earnest desire to make the utmost return in his power for the liberality shown him by the authorities of Trinity College, Dublin. He had fully recognized, and proved to others, that his i, j, k, were mere excrescences and blots on his improved method:---but he unfortunately considered that their continued (if only partial) recognition was indispensable to the reception of his method by a world steeped in---Cartesianism! Through the whole compass of each of his tremendous volumes one can find traces of his desire to avoid even an allusion to i, j, k, and along with them, his sorrowful conviction that, should he do so, he would be left without a single reader.

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