science of quantity, but rather as the science of order of progression." Time appeared to him as the picture of such a progression. Hence his definition of algebra as "the science of pure time." It was the subject of years' meditation for him to determine what he should regard as the product of each pair of a system of perpendicular directed lines. At last, on the 16th of October,
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 . 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 attention. 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.
Hermann Grassmann (1809–1877) was born at Stettin,
attended a gymnasium at his native place (where his father was teacher of mathematics and physics), and studied theology in Berlin for three years. In 1834 he succeeded Steiner as
teacher of mathematics in an industrial school in Berlin, but returned to Stettin in 1836 to assume the duties of teacher of mathematics, the sciences, and of religion in a school there.71 Up to this time his knowledge of mathematics was pretty much confined to what he had learned from his father, who had written two books on "Raumlehre" and "Grössenlehre." But now he made his acquaintance with the works of Lacroix,
Lagrange, and Laplace. He noticed that Laplace's results