ALGEBRA. 



problems ; but serves universally for the 

 investigation or invention of theorems, as 

 well as the solution and demonstration of 

 all kinds of problems, both arithmetical 

 and geometrical. The letters used in 

 algebra do each of them, separately, re- 

 present either lines or numbers, sis the 

 problem is either arithmetical or geome- 

 trical ; and, together, they represent 

 planes, solids, and powers, more or less 

 high, as the letters are in a greater or less 

 number. For instance, if there be two 

 letters, a I), they represent a rectangle, 

 whose two sides are expressed, one by 

 the letter a, and the other by b ,- so that 

 by their mutual multiplication they pro- 

 duce the plane a b. Where the same let- 

 ter is repeated twice, as a a, they denote 

 a square. Three letters, a b c, represent 

 a solid, or a rectangular parallelepiped, 

 whose three dimensions are expressed by 

 the three letters a b c ; the length by a, 

 the breadth by 6, and the depth by c ; so 

 that by their mutual multiplication they 

 produce the solid a b c. As the multipli- 

 cation of dimensions is expressed by the 

 multiplication of letters, and as the num- 

 ber of these may be so great as to become 

 incommodious, the method is only to 

 write down the root, and on the right 

 hand to write the index of the power, 

 that is, the number of letters of which 

 the quantity to be expressed consists; as 

 a-, a-, a-*, &c. the last of which signifies 

 as much as a multiplied four times into 

 itself; and so of the rest. But as it is 

 necessary, before any progress can be 

 made hi the science of algebra, to under- 

 stand the method of notation, we shall 

 here give a general view of it. In alge- 

 bra, as we have already stated, every 

 quantity, whether it be known or given, 

 or unknown or required, is usually repre- 

 sented by some letter of the alphabet ; 

 and the given quantities are commonly 

 denoted by the initial letters, , b, c, d, 

 &c. and the unknown ones by the final 

 letters, u, w, x, y, z. These quantities 

 are connected together by certain signs or 

 symbols, which serve to shew their mu- 

 tual relation, and at the same time to 

 simplify the science, and to reduce its 

 operations into a less compass. Accord- 

 ingly the sign -f-, plus, or more, signi- 

 fies that the quantity to which it is prefix- 

 ed is to be added, and it is called a posi- 

 tive or affirmative quantity. Tims, a-\-b, 

 expresses the sum of the two quantities 

 and b, so that if a were 5, and b 3, 

 ti-\-b would be 5+3, or 8. If a quantity 

 have no sign, -J-, plus, is understood, and 

 the quantity is affirmative or positive. 



The sign , minus, or less, denotes that 

 the quantity which it precedes is to be 

 subtracted, and it is called a negative 

 quantity. Thus a b expresses the dif- 

 ference of o and b; so that a being 5, and 

 b 3, a b, or 5 3, would be equal to 2. 

 If more quantities than two were con- 

 nected by these signs, the sum of those 

 with the sign must be subtracted from 

 the sum of those with the sign -)-. Thus 

 a -f- b c d represents the quantity 

 which would remain, when c and d are 

 taken from a and b. So that if a were 7, 

 b 6, c 5, and d 3, a -f- b c d, or 7 -f- 6 

 5 3, or 13 8, would be equal to 5. 

 If two quantities are connected by the 

 sign 03, as a as b, this mode of expres- 

 sion represents the difference of a and b, 

 when it is not known which of them is 

 the greatest. The sign X signifies that 

 the quantities between which it stands 

 are to be multiplied together, or it repre- 

 sents their product. Thus, a X b ex- 

 presses the product of a and b,- a y b X c 

 denotes the product of a, b, and c; (a-\-b) 

 X c denotes the product of the compound 

 quantity -f- b by the simple quantity c , 

 and(+6-j-c) X (ab + c} X (a+b) 

 represents the product of the three com- 

 pound quantities, multiplied continually 

 into one another ; so that if a were 5, b 

 4, and c 3, then would (a -f- b + c) X 

 (a b + c) X (a-f-c) be 12 X 4 X 8, or 

 384. The parenthesis used in the forego- 

 ing expressions indicate that the whole 

 compound quantities are affected by the 

 sign, and not simply the single terms be- 

 tween which it is placed. Quantities that 

 are joined together without any interme- 

 diote sign form a product; thus a b is the 

 same with a X b, and a b c the same with 

 a X b X c. When a quantity is multi- 

 plied into itself, or raised to any power, 

 the usual mode of expression is to draw a 

 line over the quantity, and to place the 

 number denoting the power at the end of 

 it, which number is called the index or 

 exponent. Thus, ( -)- 6) J denotes the 

 same as (a -f- 6) X (u -f- b) or second 

 power, or square, of a -j- 6 considered as 

 one quantity ; and (a -f- 6)3 denotes the 

 swrne as ( '-f b) X (a -f A) X (a + 6), 

 or the third power, or cube, of a+b. In 

 expressing the powers of quantities re- 

 presented by single letters, the line over 

 the top is usually omitted : thus, a- is the 

 same as a a or a X > and b> the same as 

 bbborbxbxb, and a 1 63, the same 

 as aa X b bb or a X a X b X b X b. 

 The full point . and the word into, are 

 sometimes used instead of X as the sign 

 of multiplication. Thus, (a-f-6) . (fl-f-c), 



