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1843, while walking with his wife one evening, along the Royal Canal in Dublin, the discovery of quaternions flashed upon him, and he then engraved with his knife on a stone in Brougham Bridge the fundamental formula $\scriptstyle {i^{2}=j^{2}+k^{2}=ijk=-1}$. At the general meeting of the Irish Academy, a month later, he made the first communication on quaternions. An account of the discovery was given the following year in the Philosophical Magazine. Hamilton displayed wonderful fertility in their development. His Lectures on Quaternions, delivered in Dublin, were printed in 1852. His Elements of Quaternions appeared in 1866. Quaternions were greatly admired in England from the start, but on the Continent they received less attenttion. P. G. Tait's Elementary Treatise helped powerfully to spread a knowledge of them in England. Cayley, Clifford, and Tait advanced the subject somewhat by original contributions. But there has been little progress in recent years, except that made by Sylvester in the solution of quaternion equations, nor has the application of quaternions to physics been as extended as was predicted. The change in notation made in France by Hoüel and by Laisant has been considered in England as a wrong step, but the true cause for the lack of progress is perhaps more deep-seated. There is indeed great doubt as to whether the quaternionic product can claim a necessary and fundamental place in a system of vector analysis. Physicists claim that there is a loss of naturalness in taking the square of a vector to be negative. In order to meet more adequately their wants, J. W. Gibbs of Yale University and A. Macfarlane of the University of Texas, have each suggested an algebra of vectors with a new notation. Each gives a definition of his own for the product of two vectors, but in such a way that the square of a vector is positive. A third system of vector analysis has been used by Oliver Heaviside in his electrical researches.