On Quaternions ( - 1874)

"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, [letter to William R Hamilton] 1843)

"There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]" (Sir William R Hamilton, "On Quaternions; or on a new System of Imaginaries in Algebra", 1844)

"The algebraically real part may receive [...] all values contained on the one scale of progression of number from negative to positive infinity; we shall call it therefore the scalar part, or simply the scalar of the quaternion, and shall form its symbol by prefixing, to the symbol of the quaternion, the characteristic Scal., or simply S., where no confusion seems likely to arise from using this last abbreviation. On the other hand, the algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion; andmay be denoted by prefixing the characteristic Vect., or V." (William R Hamilton, 1846)

"The quaternion [was] born, as a curious offspring of a quaternion of parents, say of geometry, algebra, metaphysics, and poetry. [...] I have never been able to give a clearer statement of their nature and their aim than I have done in two lines of a sonnet addressed to Sir John Herschel: 'And how the one of Time, of Space the Three,/Might in the Chain of Symbols girdled be'" (William R Hamilton, [letter to  Rev. Townsend] 1855)

"Every man is ready to join in the approval or condemnation of a philosopher or a statesman, a poet or an orator, an artist or an architect. But who can judge of a mathematician? Who will write a review of Hamilton’s Quaternions, and show us wherein it is superior to Newton’s Fluxions?" (Thomas Hill, 'Imagination in Mathematics', North American Review 85, 1857)

"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language. It would defy all the power of Mercury himself to explain to a person ignorant of the science what is meant by the single phrase “functional exponent.” How much more impossible, if we may say so, would it be to explain a whole treatise like Hamilton’s Quaternions, in such a wise as to make it possible to judge of its value! But to one who has learned this language, it is the most precise and clear of all modes of expression. It discloses the thought exactly as conceived by the writer, with more or less beauty of form, but never with obscurity. It may be prolix, as it often is among French writers; may delight in mere verbal metamorphoses, as in the Cambridge University of England; or adopt the briefest and clearest forms, as under the pens of the geometers of our Cambridge; but it always reveals to us precisely the writer’s thought." (Thomas Hill, North American Review 85, 1857)

"The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions." (Peter G Tait, "Note on a Quaternion Transformation", [communication read] 1863) 

"If nothing more could be said of Quaternions than that they enable us to exhibit in a singularly compact and elegant form, whose meaning is obvious at a glance on account of the utter inartificiality of the method, results which in the ordinary Cartesian co-ordinates are of the utmost complexity, a very powerful argument for their use would be furnished. But it would be unjust to Quaternions to be content with such a statement; for we are fully entitled to say that in all cases, even in those to which the Cartesian methods seem specially adapted, they give as simple an expression as any other method; while in the great majority of cases they give a vastly simpler one. In the common methods a judicious choice of co-ordinates is often of immense importance in simplifying an investigation; in Quaternions there is usually no choice, for (except when they degrade to mere scalars) they are in general utterly independent of any particular directions in space, and select of themselves the most natural reference lines for each particular problem." (Peter G Tait, Nature Vol. 4, [address] 1871)