"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)
"His optical and dynamical investigations were prophetic and foreshadowed the quantum theory of our days. His quaternions foreshadowed the space-time world of relativity. The quaternion algebra was the first example of a noncommutative algebra, which released an avalanche of literature in all parts of the world. Indeed, his professional life was fruitful beyond measure." (Cornelius Lanczos, "William Rowan Hamilton - an appreciation", American Scientist 2, 1967)
"[...] the consensus now is that the quaternion system is but one of many comparable mathematical systems, and though it is interesting as a rather special system, it offers little value for application." (Michael J Crowe, "A History of Vector Analysis", 1967)
"[...] gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. Of these, negative and complex numbers were most troublesome. It was because these two types of numbers had no 'reality' in nature that they were still suspect at the beginning of the nineteenth century, even though freely utilized by then. The geometrical representation of negative numbers as points or vectors in the complex plane, which, as Gauss remarked of the latter, gave them intuitive meaning and so made them admissible, may have delayed the realization that mathematics deals with man-made concepts. But then the introduction of quaternions, non-Euclidean geometry, complex elements in geometry, n-dimensional geometry, bizarre functions, and transfinite numbers forced the recognition of the artificiality of mathematics." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
"[...] the quaternions [...] were not quite what the physicists wanted. They sought a concept that was not divorced from but more closely associated with Cartesian coordinates than quaternions were. (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
"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)
"While translations are well animated by using vectors, rotation animation can be improved by using the progenitor of vectors, quaternions. [...] By an odd quirk of mathematics, only systems of two, four, or eight components will multiply as Hamilton desired; triples had been his stumbling block." (Ken Shoemake, "Animating Rotation with Quaternion Curves", ACM SIGGRAPH Computer Graphics Vol. 19 (3), 1985)
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