On William K Clifford

 "It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment. - judgment, that is, in matters not admitting of certainty - balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind." (Shiing-Shen Chern, "On Riemannian manifolds of four dimensions". Bulletin of the American Mathematical Society 51 (12), 1945)

"Hermann Grassmann made substantial contributions to algebra and geometry during the 19th century. His ideas were so advanced that many of his colleagues failed to recognize their merit, but later generations quickly gravitated toward Grassmann’s highly abstract and beautiful work. [...] Grassmann also invented the concept of an exterior algebra - another algebra with a special product called the exterior product. This abstract structure was related to the quaternions of Sir William Rowan Hamilton, and was later developed by William Clifford into a tool that has been quite useful in quantum mechanics. The exterior algebra is an important object of study in modern differential geometry. Grassmann’s ideas were quite advanced for his time, and they were accepted slowly [...]" (Tucker McElroy, "A to Z of Mathematicians", 2005)

"The idea of Clifford algebra is basic for Dirac’s theory of the relativistic electron, and hence it is crucial for the fundamental fermions in the Standard Model in particle physics." (Eberhard Zeidler, "Quantum Field Theory I: Gauge Theory", 2006)

"Geometric algebra combines the algebraic structure of Clifford algebra with the explicit geometric meaning of its mathematical elements at its foundation. So, formally, it is Clifford algebra endowed with geometrical information of and physical interpretation to all mathematical elements of the algebra. This intrusion of geometric consideration into the abstract system of Clifford algebra has enriched geometric algebra as a powerful mathematical theory." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)