the cube of b; and the last term is a fraction, whose binomial numerator is the difference between a and b, and whose trinomial denominator is the sum of the cubes of a and b and the fourth power of c. All this is expressed, in one line of algebraic writing, thus; Let a=4, then the value of this quantity is, CHAP. I. ON THE ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF ALGEBRAIC QUANTITIES. 14. PREVIOUS to the application of the fundamental rules of Arithmetic to Algebraic quantities, it may be proper to observe, that the symbols + and - are distinguished from all the other signs or symbols, by giving a kind of quality or affection to the quantities to which they are annexed. As all those terms which have the sign + prefixed to them are to be added, and those which have the sign - prefixed to them are to be subtracted, from the terms which precede them, the former have a tendency to increase, and the latter to diminish, the quantities with which they are combined. A compound quantity, x-a for instance, will therefore be positive or negative, according to the effect which it produces upon some third quantity c. Thus, if x be greater than a, the c+x-a (since x is added and a subtracted) is greater than c; if x be less than a, then c+x-a is less than c; i.e. "if x be greater than a, x-a is positive; and if x be less than a, then x-a is negative." In the same manner it might be shewn that the expression a-b+c-d is positive or negative, according as a+c is greater or less than b+d; and so of all compound quantities whatever. ADDITION. 8 III. ADDITION. From the division of algebraic quantities into positive and negative, like and unlike, there arise three cases of Addition. CASE I. To add like quantities with like signs. 15. In this case, the rule is "To add the coefficients of "the several quantities together, and to the result annex the common sign, and the common letter or letters;" for it is evident, from the common principles of Arithmetic, if +2a, +3a, and +5 a be added together, their sum must be +10a; and if -3, -4 b', and -8b3 be added together, their sum must be 156. In these Examples it may be observed that some of the quantities have no coefficient. In this case, unity or 1 is always understood. Thus, in adding up the first column of Ex. 2. we say, 1+1+11+9+7=29; in the third, 2+1+4+7+5=19; and so of the rest. CASE CASE II. To add like quantities with unlike signs. 16. Since (by Art. 14) the compound quantity a+b−c +d-e &c. is positive or negative, according as the sum of the positive terms is greater or less than the sum of the negative ones, the aggregate or sum of the quantities 2a-4a+7a-3a will be +2a, and of the quantities. 7 b2-5 b2+ 2 b2-8b will be -4b; for in the former case, the excess of the sum of the positive terms above the negative ones is 2a, and in the latter 46. Hence this general rule for the addition of like quantities with unlike signs, "Collect "the coefficients of the positive terms into one sum, and also "of the negative; subtract the lesser of these sums from "the greater; to this difference, annex the sign of the greater together with the common letter or letters, and the "result will be the sum required." and con If the aggregate of the positive terms be equal to that of the negative ones, then this difference is equal to 0; sequently the sum of the quantities will be equal to the second column of Ex. 2. following. O, as in CASE III. 17. There now only remains the case where unlike quantities are to be added together, which must be done by collecting them together into one line, and annexing their proper signs; thus the sum of 3x,-2a, +5l, -4y, is 3x-2a+5b-4y; except when like and unlike quantities are mixed together, as in the following examples, where the expressions may be simplified, by collecting together such quantities as will coalesce into one sum. Ex. 1. 3ab + x-y 4y+x-2y Collecting together like quantities, and beginning with 3ab, we have 3ab+5ab8ab; +x +x=+2x; -y-2y+4y2y=―y; 4c-3c=+c; besides which there are the two quantities +d and +x, which do not coalesce with any of the others; the sum required therefore is 8ab+2x−y+c+d+x3. 18. If it were required to subtract 5-2 (i.e. 3) from 9, it is evident that the remainder would be greater by 2, than if 5 only were subtracted. For the same reason, if b-c were |