sion, &c. of numbers, even in the application thereof: for, when we reason upon the quantities themselves, and not upon the numbers expressing the measures of them, the process becomes purely geometrical, whatever symbols may be used therein, from the algebraic notation; which can be of no other use here than to abbreviate the work. However, after all, it may be necessary to show upon what kind of evidence the multiplication of negative and imaginary quantities is grounded, as these sometimes occur, in the resolution of problems: in order to which it will be requisite to observe, that, as all our reasoning regards real, positive quantities, so the algebraic expressions, whereby such quantities are exhibited, must likewise be real and positive. But, when the problem is brought to an equation, the case may indeed be otherwise; for, in ordering the equation, so much may be taken away from both sides thereof, as to leave the remaining quantities negative; and then it is, chiefly, that the multiplication by quantities absolutely negative takes place. Thus if there were given the equation a- =c b (in order to find x;) then by subtracting the quantity a from each side thereof, we shall have - = C= a; b which multiplied by b, according to the general rule, x will give gives X=- cb+ab; that is 7 by -- b will + x; c by-b, -cb; and -a by -b, ab; which appear to be true; because the products being thus expressed, the same conclusion is derived, as if both sides of the original equation had been first increased by Б c, and then multiplied by b; where both the multiplier and multiplicand are real, affirmative quantities, and where the whole operation is, therefore, capable of a clear and strict demonstration: but then it is not in consequence of any reasoning I am capable of forming independently, that I can be certain that their product ought to be expressed in that manner. So likewise, if there were given the equation a = c; by transposing a and taking the square root on both sides we shall have x2 - a; and this multiplied by ✔, will give ✔2 (or x) ✔cb+ab: which also appears to be true, because the result, this way, comes out exactly the same, as if the operations, for finding x, had been performed altogether by real quantities: but notwithstanding this, it is not from any reasoning that I can form, about the multiplication of the imaginary quantities x2 b and b, &c. considered independently, that I can prove their product ought to be so expressed; for it would be very absurd to pretend to demonstrate what the product of two expressions must be, which are impossible in themselves, and of whose values we can form no idea. It indeed seems reasonable, that the known rules for the signs, as they are proved to hold in all cases whatever, where it is possible to form a demonstration, should also answer here: but the strongest evidence we can have of the truth and certainty of conclusions derived by means of negative and imaginary qnantities, is, the exact and constant agreement of such conclusions with those determined from more demonstrable methods wherein no such quantities have place. In the foregoing considerations, the negative quantities b, c, &c. have been represented, in some cases, as a kind of imaginary or impossible quantities; it may not, therefore, be improper to remark here, that such imaginary quantities serve, many times, as much to discover the impossibility of a problem, as imaginary surd quantities: for it is plain, that, in all questions relating to abstract numbers, or such wherein magnitude only is regarded, and where no consideration of position, or contrary values, can have place; I say, in all such cases, it is plain that the solution will be altogether as impossible, when the conclusion comes out a negative quantity, as if it were actually affected with an imaginary surd; since, in the one case, it is required that a number should be actually less than nothing; and in the other, that the double rectangle of two numbers should be greater than the sum of their squares; both which are equally impossible: but, as an instance of the impossibility of some sort of questions, when the conclusion comes out negative, let there be given, in a right-angled triangle, the sum of the hypothenuse and perpendicular = a, and the base = b, to find the perpendicular; then (by what shall hereafter be shown in its proper place) the answer a2_b2 and is possible, or impossible, ac2a will come out cording as the quantity as_b2 is affirmative or negative, or as a is greater or less than b; which will manifestly appear from a bare contemplation of the problem: and the same thing might be instanced in a variety of other examples. SECTION V. of Division. DIVISION in species, as in numbers, is the converse of multiplication, and is comprehended in the seven following cases. 1o. When one simple quantity is to be divided by another, and all the factors of the divisor are also found in the dividend, let those factors be all cast off or expunged, then the remaining factors of the dividend, joined together, will express the quotient sought. But it is to be observed, that, both here and in the succeeding cases, the same rule is to be regarded in relation to the signs, as in multiplication, viz. that like signs give +, and unlike -. It may also be proper to observe, that, when any quantity is to be divided by itself, or an equal quantity, the quotient will be expressed by an unit, or 1. Thus aa, gives 1; and 2ab÷2ab gives 1; moreover 3abcd÷ac, gives 3bd; and 16bc8b, gives 2c: for the dividend here, by resolving its co-efficient into two factors, becomes 2 × 8 xbx c; from whence casting off 8 and b, those common to the divisor, we have 2×c, or 2c. In the same manner, by resolving or dividing the co-efficient of the dividend by that of the divisor, the quotient will be had in other cases: Thus, 20abc divided by 4c, gives 5ab; and - 51ab ✔ xy × √ xx + yy, divided by — 17a ✓ xy, gives + 3b xx + yy. - 2° But if all the factors of the divisor are not to be found in the dividend, cast off those (if any such there "be) that are common to both, and write down the remaining factors of the divisor, joined together, as a denominator to those of the dividend; so shall the fraction thus arising express the quotient sought. But if, by proceeding thus, all the factors in the dividend should happen to go off, or vanish, then an unit will be the numerator of the fraction required. Thus, abc divided by bcd, gives a And 16a ba3 divided by 8abcx2, gives 2ax The first rule, given above, being exactly the converse of rule 1° in the preceding section, requires no other demonstration than is there given. The second rule (as well as those that follow hereafter upon fractions) depend on this principle, that, as many times as any one proposed quantity is contained in another, just so many times is the half, third, fourth, or any other assigned part of the former, contained in the half, third, fourth, or other corresponding part of the latter; and Likewise 27 ab✔xy divided by 9a2 ✔xy, gives 1 And 8ab ay divided by 16a2b✔ay, gives 2a 36 a 3o One fraction is divided by another, by multiplying the denominator of the divisor into the numerator of the dividend for a new numerator, and the numerator of the divisor into the denominator of the dividend for a new de nominator. just so many times likewise is the double, triple, quadruple, or any other assigned multiple of the former contained in the double, triple, quadruple, or other corresponding multiple of the latter. The demonstration of this principle (though it may be thought too obvious to need one) may be thus: let A and B represent any two proposed quantities, and AC and BC their equimultiples (or, let AC and BC be the two quantities and A and B their like parts:) I say, then, that AC A' = BC B For the multiple of AC B(vid. p. 14 and 15) AC: therefore, seeing the equimultiples of the two proposed quantities are the same, the quantities themselves must necessarily be equal. The second rule, given above, is nothing more than a bare application of the principle here demonstrated; |