On William R Hamilton

"The discoveries of Newton had done more for England and for the race, than has been done by the whole dynasties of British monarchs; and we doubt not that in the great mathematical birth of 1843, the Quaternions of Hamilton, there is as much real promise of benefit to mankind as in any event of Victoria’s reign." (Thomas Hill, "Review of Sir William Rowan Hamilton's Lectures on quaternions", North American Review 85, 1857)

"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 a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the […] development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

"While vector analysis was being created and afterward there was much controversy between the proponents of quaternions and the proponents of vectors as to which was more useful. The quaternionists were fanatical about the value of quaternions but the proponents of vector analysis were equally partisan. On one side were aligned the leading supporters of quaternions such as Tait and, on the other, Gibbs and Heaviside. Apropos of the controversy, Heaviside remarked sarcastically that for the treatment of quaternions, quaternions are the best instrument. On the other hand Tait described Heaviside’s vector algebra as 'a sort of hermaphrodite monster, compounded of the notations of Grassmann and Hamilton'. [...] The issue was finally decided in favor of vectors. Engineers welcomed Gibbs’s and Heaviside’s vector analysis, though the mathematicians did not. By the beginning of the present century the physicists too were quite convinced that vector analysis was what they wanted. Textbooks on the subject soon appeared in all countries and are now standard. The mathematicians finally followed suit and introduced vector methods in analytic and differential geometry." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Hamilton’s discovery of the quaternions can be appreciated better today than in his own time because we now see that they are 'nearly impossible'. The quaternions are an amazing rarity (and from the ndimensional point of view, so are the real and complex numbers)." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"Hamilton’s dream of three-dimensional numbers was indeed impossible, but the reality turned out to be more interesting. The known systems of numbers (real and complex) are exceptional structures, existing only in one and two dimensions, and the system of quaternions is even more exceptional. It is the only n-dimensional structure satisfying all the laws of algebra except commutative multiplication. This was first proved by the German mathematician Georg Frobenius in 1878, but unfortunately Hamilton did not live long enough to see it." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)