ALGEBRA. 



and a + b into a + c, signify the same 

 thing as (a -f 6) X (a + c), or the pro- 

 duct of a -f- 6 by a + c. The sign -r- 

 is the sign of division, as it denotes that 

 the quantity preceding it is to be divided 

 by the succeeding quantity. Thus, c -f- b 

 signifies that c is to be divided by b , and 

 (a + ) -r- (a + c), that a -f- b is to be 

 divided by a -f- c. The mark ) is some- 

 times used as a note of division ; thus 

 a -{- b) a b denotes that a b is to be divid- 

 ed by a -f- b. But the division of alge- 

 braic quantities is most commonly ex- 

 pressed by placing the divisor under the 

 dividend, with a line between them, like 



a vulgar fraction. Thus, represents the 

 quantity arising by dividing c by b, or 

 the ^quotient, and - represents the 



quotient of a-f-6 divided by a-\-c. Quan- 

 tities thus expressed are called algebraic 

 fractions. 



The sign ^/ expresses the square root 

 of any quantity to which it is prefixed ; 

 thus v/ 25 signifies the square root of 25, 

 or 5, because 5 x 5 is 25 ; and v/ (a 6) 

 denotes the square root of a b , and 



/ ( - - J denotes the square root of 



, or of the quantity arising from 



the division of a b -f b c by d ; but 



v/ (a b H- b c) . . . , 



- - - , winch has the separating 



line drawn under v/, signifies that the 

 square root of a b -\- b c is to be first ta- 

 ken, and afterwards divided by d , so 

 that if a were 2, b 6, c 4, and d 9, 



v/ (a b + be) ... v /36 6 



^ ' - - - -, would be z-fi or-; but 



v/4, which is 2. The sign ^/ with a 

 figure over it is used to express the cubic 

 or biquadratic root, &c. of any quantity ; 

 thus ^/ 64 represents the cube root of 

 64, or 4, because 4 X 4 x 4 is 64 ; and $/ 

 (a b -f- c d} the cube root of a b -{- c d. In 

 like mannerl^/ 16 denotes the biquadratic 

 root of 16, or 2, because 2x2x2x2 is 16, 

 and v/ ( a b-\-c d) denotes the biquadra- 

 tic root of a b + c d; and so of others. 

 Quantities thus expressed are called ra- 

 dical quantities, or surds ; of which those, 

 consisting of one term only, as ^/ a and 

 ^/ (a b) are called simple surds; and 

 those consisting of several terms or num- 

 bers, as v/ (a 1 6>) and $f (a 1 b+b c) 



are denominated compound surd&. 

 ther commodious method of expressing 

 radical quantities is that which denotes 

 the root by a vulgar fraction, placed at 

 the end of a line drawn over the quantity 

 given. In this notation, the square root 

 is expressed by , the cube root by -i, 

 the biquadratic root by , &c. Thus a $ 

 expresses the same quantity with ^/ a y 

 i. e. the square root of a, and (c^-f-a b) * 

 the same as *f (a j + 6), i. e. the cube 



root of c^-fa b ; and~^ f denotes the 

 cube root of the square o f a, or the 

 square of the cube root of a; and (a-J-z)^. 

 the seventh power of the biquadratic root 

 of a + x; and so of others; (a 1 ) \ is a, 

 asj-y is a, &c. Quantities that have no ra- 

 dical sign (vO or index annexed to them, 

 are called rational quantities. The sign 

 =, called the sign of equality, signifies 

 that the quantities between which it oc- 

 curs are equal. Thus 2 + 3 =5, shews 

 that 2 plus 3 is equal to 5 ; and x = a b 

 shews that x is equal to the difference of 

 a and b. The mark : : signifies that the 

 quantities between which it stands are 

 proportional. As a : b : : c : d denotes that 

 a is in the same proportion to b as c is to 

 d,- or that if a be twice, thrice, or four 

 times, 8cc. as great as 6, c will be twice, 

 thrice, or four times, 8cc. as great as d. 

 When any quantity is to be taken more than 

 once, the number which shows how many 

 times it is to be taken must be prefixed; 

 thus 5 a denotes that the quantity a is to 

 be taken 5 times, and 3 b c represents 

 three times b c, and 7 v/ (a'X^ 3 ) denotes 

 that <S (a 1 -f- 6') is to be taken 7 times, 

 &c. The numbers thus prefixed are call- 

 ed co-efficients ; and if a quantity have 

 no co-efficient, unit is understood, and it 

 is to be taken only once. Similar or like 

 quantities are those that are expressed; 

 by the same letters under the same pow- 

 ers, or which differ only in their co-effi- 

 cients ; thus, 3 6 c, 5 b c, and 8 b c, are like 

 quantities, and so are the radicals 



But un- 



like quantities are those which are ex- 

 pressed by different letters, or by the 

 same letters with different powers, as 2 a 

 b, 5 a 6 2 , and 3 a 1 b. When a quantity is 

 expressed by a single letter, or by several 

 single letters multiplied together, without 

 any intervening sign, as a or, 2 a b, it is 

 called a simple quantity. But the quan- 

 tity which consists of two or more such 

 simple quantities, connected by the signs 

 -|- or , is called a compound quantity : 

 thus, a 2a6-r-5a&cis a compound 



