5) 8 THE POPULAR SCIENCE MONTHLY. 



then we would have a times a, or a a ; and in the same way we might 

 have a times a a, or a a a, etc. But why write all the 's ? Put one 

 and a number above to right to tell how many, call the number b, and 

 we have a h . 



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 New- 

 ton'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 illus- 

 trating. 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 develojied sufficiently to appreciate the definition of an alge- 

 bra, 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 de- 

 fined 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 



