veral of the same kind; then subtract the least of these sums from the greatest, and to the difference prefix the sign of the greater, annexing the common letter or letters, as before. CASE III. When the Quantities are unlike; or some like and others unlike. RULE. Collect all the like quantities together, by taking their sums or differences, as in the foregoing cases, and set down those that are unlike, one after another, with their proper signs. EXAMPLES FOR PRACTICE. 1. Required the sum of (a+b) and (a+b) 2. Add 5x-3a + b + 7 and ther. 4a 3x+2b-9 toge 3. Add 2a+3b - 4c 9 and ba-36 + 2c · 10 toge 5 Add x3 + ax2 + bx +2 and x3+cx2+dx-1 together. SUBTRACTION. SUBTRACTION is the taking of one quantity from another; or the method of finding the difference between any two quantities of the same kind; which is performed as follows (b) : RULE. Change all the signs (+ and -) of the lower line, or quantities that are to be subtracted, into the contrary signs, or rather conceive them to be so changed, and then collect the terms together, as in the several cases of addition. (b) The term subtraction, used for this rule, is liable to the same objection as that for addition; the operations to be performed, being frequently of a mixed nature, like those of the former. 1. Find the difference of (a+b) and (a-b). MULTIPLICATION. MULTIPLICATION, or the finding of the product of two or more quantities, is performed in the same manner as in arithmetic; except that it is usual, in this case, to begin the operation at the left hand, and to proceed towards the right, or contrary to the way of multiplying numbers. The rule is commonly divided into three cases; in each of which, it is necessary to observe, that like signs, in multiplying, produce +, and unlike signs, -. It is likewise to be remarked, that powers, or roots of C the same quantity, are multiplied together by adding their indices thus, : a×a3, or a1 Xa2=a3 ; a3×a3=a3 ; aa×a3=a&; and am Xan=amtn ̧ The multiplication of compound quantities, is also, sometimes, barely denoted by writing them down, with their proper signs, under a vinculum, without performing the whole operation, as 3ab (a-b), or 2a/a2+b2. Which method is often preferable to that of executing the entire process, particularly when the product of two or more factors is to be divided by some other quantity, because, in this case, any quantity that is common to both the divisor and dividend, may be more readily suppressed; as will be evident from various instances in the following part of the work. (c) (c) The above rule for the signs may be proved thus: If в, b, be any two quantities, of which B is the greater, and B-b is to be multiplied by a, it is plain that the product, in this case, must be less than aв, because в- b is less than в; and, consequently, when each of the terms of the former are multiplied by a, as above, the result will be ⚫ (B-b) Xbaв - ab. For if it were aв ab, the product would be greater than aв, which is absurd. B Also, if в be greater than 6, and a greater than a, and it is required to multiply B-b by Aa, the result will be (B —¿)X(A—α)=AB➡aB➡ba+ab B For the product of в➡b by a is A (B—b), or AB—Ab, and that of B bbya, which is to be taken from the former, is ~6(8—b), as has been already shown; whence B-b being less than B, it is evident that the part which is to be taken away must be less than aв; and consequently since the first part of this product is aв, the second part must be ab; for if it were—ab, a greater part than aв would be to be taken from A(B-6), which is absurd. |