"Quantity is that which is operated with according to fixed mutually consistent laws. Both operator and operand must derive their meaning from the laws of operation. In the case of ordinary algebra these are the three laws already indicated [the commutative, associative, and distributive laws], in the algebra of quaternions the same save the law of commutation for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague; but the reader will do well to reflect that any definition must include the linear algebras of Peirce, the algebra of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quantity enables us to see how operators may be treated as quantities, and thus to understand the rationale of the so called symbolical methods." (George Chrystal, "Mathematics", Encyclopedia Britannica, 1875)
"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)
"Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quantities are coördinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze." (John Stallo, "The Concepts and Theories of Modern Physics", 1881)
"[....] this definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective. Recent investigations, of which quaternions is the most noteworthy instance, make it manifest that the old definition is too restricted. The sphere of mathematics is here extended, in accordance with all the derivation of its name, to all demonstrative research, so as to include all knowledge capable of dogmatic teaching." (Benjamin Peirce, 1881)
"This symbol [v-1] is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power. The immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases. But the strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebras", American Journal of Mathematics Vol. 4, 1881)
"I think the time may come when double algebra will be the beginner’s tool; and quaternions will be where double algebra is now. The Lord only knows what will come above the quaternions." (Augustus De Morgan, "A. Graves’ Life of Hamilton" Vol. 3, 1882-1889)
"The merits or demerits of a pamphlet printed for private distribution a good many years ago do not constitute a subject of any great importance, but the assumptions implied in the sentence quoted are suggestive of certain reflections and inquiries which are of broader interest, and seem not untimely at a period when the methods and results of the various forms of multiple algebra are attracting so much attention. It seems to be assumed that a departure from quaternionic usage in the treatment of vectors is an enormity. If this assumption is true, it is an important truth; if not, it would be unfortunate if it should remain unchallenged, especially when supported by so high an authority. The criticism relates particularly to the notations, but I believe that there is a deeper question of notions underlying that of notations. Indeed, if my offence had been solely in the matter of notations, it would have been less accurate to describe my productions as a monstrosity, than to characterize its dress as uncouth." (Josiah W Gibbs, "The Rôle of Quaternions in the Algebra of Vectors, Nature vol. xliii, 1891)
"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892)
"[...] it is as unfair to call a vector a quaternion as to call a man a quadruped." (Oliver Heaviside, 1892)
"The invention of quaternions must be regarded as a most remarkable feat of human ingenuity. Vector analysis, without quaternions, could have been found by any mathematician [...] but to find out quaternions required genius." (Oliver Heaviside, 1892)
"Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies." (William T Kelvin, [footnote in Letter to George F FitzGerald] 1896)
"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." (Peter G Tait [in Alexander MacFarlane's "Lectures on Ten British Mathematicians", 1916])
"[...] of possible quadruple algebras the one [...] by far the most beautiful and remarkable was practically identical with quaternions, and [...] it [is] most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels." (Benjamin Peirce) [atributed by William E Byerly, former student of Peirce)
"Quaternions came from Hamilton [...] and have been an unmixed evil to those who have touched them in any way. Vector is a useless survival [...] and has never been of the slightest use to any creature." (William T Kelvin)
"There is still something in the system [of quaternions] which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. [...] If with your alchemy you can make three pounds of gold, why should you stop there?" (John T Graves)
"Time is said to have only one dimension, and space to have three dimensions [...] The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." (William R Hamilton [in Robert P Graves's "Life of Sir William Rowan Hamilton", 1882-1889])
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