and greater No. = 2 + x = 3, 1, 2 + (-25), or 2 — (— 25), = where it must be observed that if 3 be taken for the value of the greater number, 1 must be taken for that of the less, and so on for the three remaining corresponding values. As a second example, suppose the equations b b ===n, dividing by a and putting n, to avoid fractions a .. z2+2z+1+1+ + +1+1+ 29 = n2 - 2nz s; USE OF THE DOUBLE SIGN. x2 - 2mz + m2 = m2 − 1 = (m + 1) (m −1), 55 Scholium IV. From notions derived from the practice of arithmetic, the student, on first being introduced to problems capable of double solution, is frequently inclined to question the propriety of the second value of x; and, consequently, to reject or neglect the second or minus sign, in taking the square root of a quadratic equation. Such neglect must never be allowed; for the double sign (+) is necessary-and, so far from being any thing like an excrescence or deformity in algebraical language, is a symbol of the greatest utility, as will appear hereafter when we come to apply the foregoing principles to Geometry. The following problem, illustrative of the use of the double sign, will suffice for the present. Required the point in the line joining the centres of the earth and moon where their attractions are equal. Fig. 1. M C Let E be the earth, M the moon and C the point of equal attraction, and let the quantity of matter in the earth be represented by e and that in the moon by m, the radius of the moon's orbit, or distance from the earth, by r, the distance CM by u and consequently, C E by r — u. Now it is a principle of physics that the attraction of a sphere is proportional to the quantity of matter directly and to the square of the distance to its centre inversely, or But the mass of the moon, inferred from her action in raising the tides, is of that of the earth ;* or the distance of the point of equal attraction from the moon, is nearly of the moon's radius, or of moon's radius, where it remains to interpret the sign in the second answer. If we subtract from u (that is, from C lay off a line in the direction of M) a quantity less than u, the point of equal attraction will approach M, and the more, the greater the line subtracted, so that, when the line subtracted is = CM, u will = 0, and when > CM, as CC', u will become, being = MC-CC' = MC — (CM+MC')=-MC'. Therefore if u =+ • r, gives the point C on the left or this side, -⚫r corresponds to the point C' on the right or beyond the moon. Indeed, had we supposed the point of equal attraction at C' instead of at C, there would have resulted The two values of u are therefore perfectly applicable, and, in fact, the problem would not be completely solved without them. The second value has revealed a truth, viz.: that there is a second point of equal attraction beyond the moon-which, though not contemplated in the statement of the problem, is yet a direct consequence of the equation drawn from it; and which, when once discovered by the aid of algebraical symbols, recommends itself to the understanding without them, since the attraction of the earth is greater than that of the moon. Suppose, for a simple illustration, the masses of E and M were as 9 and 4, and their distances to C' as 3 9 4 and and con32 22' and 2; then their attractions at C' would be as sequently equal. As a last example in which a single equation is equivalent to two independent equations, we give the following important theorem: *Poisson, Traité de Mécanique, p. 258. CONSTANTS AND VARIABLES. 57 PROPOSITION III. Whenever a problem gives an equality between constant (63) and variable terms, the variables being capable of indefinite diminution so as to become less than any assignable quantity; two independent equations will be formed, one between the constants, the other between the variables. represent any equation in which A, B, are the sums of the constant, those of the variable terms in each member, X and Y, being capable of continually decreasing and becoming less than any assignable quantity, that is less than any assignable part of A or B. Then will A = B, and X = Y; for, if possible, let A be greater or less than B; first suppose A > B by some quantity D, or or A = B+ D ; Y = D+ X, whence it appears that Y can never be less than D, even should X become actually = 0; therefore Y is greater than, or at least as great as, an assignable quantity D, which is contrary to the hypothesis, and this contradiction holds as long as D has any value, or as long as A is > B. Again suppose or AB, or A+D=B; X = D+Y; whence X is always greater than, or at least as great as D, and this is also contrary to the hypothesis, according to which X, as well as Y, is to be capable of indefinite diminution. Therefore it is absurd to suppose A either greater or less than B, or otherwise than equal to B; A = B, and consequently X = Y, which was to be proved. The method of proof here employed is called that of "reductio ad absurdum," leading to an absurdity; where, instead of showing directly that the proposition must be true, we show that it can not be otherwise than true. As an example, let it be required to find the sum of the repeating decimal fraction 333 &c. = '3'03+003+... [ad infinitum]. But (1) may be made less than any assignable quantity by taking n sufficiently great, which may be done since the number of terms is infinite. Thus, if we take 1, 2, 3, &c., terms, or make successively n = 1, 2, 3, 4, (1)" '1, 01, 001, 0001, 00001, ...; and X becoming < any assignable quantity, we have (63) = 23 '99 = 23 99 And generally, if C any circulate containing c digits, we shall have 3c A-X= C(1) + C(1) + C(1) + ... [n]; We observe that a quantity may increase constantly by an infinite number of additions, and yet never exceed a determinate finite quantity or limit; as 333, &c. constantly approaches to which it can never surpass. This gives us our first notion of limits. Def. "When a variable magnitude A - X can be made to (64) approach another A which is fixed in such a way as to render their difference X less than any given magnitude, without, however, their being able ever to become rigorously equal, the second A is called a limit of the first A-X."* A direct consequence of (63) and further illustrating (56), is the following important principle : * Francœur, Mathematiques pures, p. 166. 2 |