then we would have a times a, or aa; and in the same way we might have a times a a, or a a a, etc. But why write all the a's? Put one and a number above to right to tell how many, call the number b, and we have ab.
These are the three direct operations, seemingly mere devices to spare a little trouble. You could hardly believe the conquest of the thought-world was lying dormant in them. Yet their undoing or inversion leads to the four inverse operations, and the seven, together with their working laws, are the algorithm of your algebra. So are they also of Descartes's application of algebra to form, and even Newton's fluxional calculus to a certain extent presupposes them, so that it was looked upon rather as an extension, a generalization, than as a new algebra of infinitesimals formulating its own working algorithm.
Therefore, much as we prefer Newton's character, and believe in his prior invention of the calculus, it is to Leibnitz that we assign the high honor first to have grasped the plural whose growth we are illustrating. After two of the most extraordinary of modern algebras were discovered and published, it was found that the possibility of each had been indicated by Leibnitz more than a century and a half before.
Toward the modern deep study of the formal laws involved in a pure science, Lagrange and Laplace led on also by the conclusion that theorems proved to be true for symbols representing numbers are also true for all symbols subject to the same laws of combination. Hence followed the principle of the separation of symbols of operation from those of quantity, with the "calculus of operations." The world of mind had now developed sufficiently to appreciate the definition of an algebra, though when it was first given I do not know. An algebra is an abstract science or calculus of symbols combining according to defined laws. There may be an indefinitely large number of sets of such defined laws—that is, of distinct, different, and independent algebras.
In the history of science it is a worthy illustration of the rhythmic character of great advance that, as if by an irruption of genius, the same year (1844) published three of the most fundamentally new and interesting modern algebras, and stamped for immortality the names of Rowan Hamilton, Hermann Grassmann, and George Boole.
Among the first men to systematically consider symbols combining according to laws more complicated than those of natural number was Sir Rowan Hamilton. After a struggle of ten years from 1833, his genius enabled him to escape from the rut of common thought by casting away the commutative principle in multiplication, which in numbers formulates the fact that twice three gives precisely the same result as thrice two. So, in 1843, he presented to the Irish Academy the principles of the algebra of quaternions, and published an article on the subject in the "Philosophical Magazine" in 1844. At the same time had appeared in Germany Grassmann's "Ausdehnungslehre," a more extraordinary algebra, which contains quaternions as a special
