MATHEMATICS 23 



i.e., for indefinite quantities of all kinds, known and 

 unknown. 



Algebra is a further extension of that process of 

 abstraction which is employed in arithmetic. In arith- 

 metic we use symbols to denote definite quantities of 

 undefined things. Thus, we use 7, 9, and 12 to denote 

 such definite quantities of any kind of things whatever. 

 In algebra we use symbols to denote undefined quan- 

 tities of undefined things. An algebraic statement 

 e.g., a + a=2a applies to any possible quantities or any 

 possible or impossible things. That economy of human 

 effort which is effected by arithmetic is, as before said, 

 carried to enormously greater extent by algebra. 



Such indefinite quantities as are treated of in algebra 

 are represented by letters. It is usual in elementary 

 algebra to represent definite and constant quantities by 

 the first letters of the alphabet, a, b, c, d, &c., and to 

 represent quantities which are variable, are under in- 

 vestigation, and have to be determined, by the last 

 letters of the alphabet, z, y, x, w, &c. 



Capital letters, Greek letters, and various other 

 symbols, are used to denote quantities according to cir- 

 cumstances. As to symbols denoting relations between 

 quantities, in addition to those lately referred to as of 

 special use in algebra, the following may be added out 

 of a variety of other ones : The sign > between two 

 quantities signifies that the quantity expressed on the 

 left hand of the sign is greater than that on its right, as 

 a > b means that a is greater than b. 



Similarly, a < b means that the right-hand quantity 

 (here a) is less than that on the left hand. 



When letters representing quantities are enclosed in a 

 bracket, or have a line drawn over them, each of these 

 symbols signifies that such quantities are to be taken 



