In like manner, by taking m and n each =1, p=b, and q=0, there will arise, by transposition, and rejecting b in the two consequents, Each of which proportions may be easily verified by making the product of the extremes equal to that of the means, and observing that ad=bc. Lastly, taking any number of equations of the form before used, for expressing proportions, as which, according to the common method, are called a series of equal ratios, and are usually denoted by a : b :: c d :: e: ƒ :: g: h :: &c. we shall necessarily have, from the fractions being all equal, i=4, 2=q;=q, {=q, &c. And by multiplying q by each of the denomi nators a=bq; c=dq, e=fq, g=hq, &c. Whence, equating the sum of all the terms on the left hand side of these equations, with those on the right, we have a+c+e+g+&c.={b+d+f+h+&c.)q. And consequently, by division, and the properties of proportionals before shown, a + c + e + g + &c. a =-= a + c + e b b+d b + d + f Which result shows, that, in a series of equal ratios, the sum of any number of the antecedents is to that of their consequents, as one, or more of the antecedents, is to one, or the same number, of consequents. Having thus treated of proportion, we may now proceed to progression; which consists of a series of terms so formed, that the quotient arising from dividing either of them by that which precedes or follows it is always the same. Thus, designing the first term of such a series by a, and the common factor or multiplier by q, the several terms of the progression, taken in order, will be a, aq', aq, aq', aqt, aq", &c. Where, supposing to be the general term of the series, or that whose rank is n, its value will evidently be Also, if s be made to denote the sum of n terms of the series, including the first, we shall have And multiplying each side of this equation by q, Whence, subtracting the first of these equations. from the second, and observing that all the terms except a and aq" destroy each other, we shall have Or, substituting 1 for the last term aq"-', as above found, this expression will become From which two equations, if any three of the quantities a, q, n, l, s, be given, the rest may be found. When the common factor q, in the above series, is a whole number, the terms a, aq, aq, aq3, . . . aq"-1, form an increasing progression; in which case n may be so taken, that the value of s shall be greater than any assignable quantity. But if q be a proper fraction, as, the series will then be a decreasing one, and the above expression, by substituting this value for q, will be 1 Where it is plain, that the term will be indefinitely small when n is indefinitely great; and consequently, by prolonging the series, s may be ar r-1 made to differ from by less than any quantity that can be named. Whence, by supposing the series to be of the and to be continued ad infinitum, or without end, we shall, in that case, have Which is what some writers call the radix, and others the limit of the series; as being of such a value, that the sum of no number of its terms can ever exceed it, and yet may be made to approach nearer to it than by any given difference. DOCTRINE OF EQUATIONS./ (N) THE doctrine of equations being that branch of algebra upon which the solutions of all kinds of analytical problems ultimately depend, I shall here. lay down snch of the theoretical principles, relating to this part of the science, as appear to be most necessary for the elucidation of the following articles, leaving the rest to be treated of as they occur. In the first place, then, it may be observed, that the term root, as applied to equations, has a more extensive signification than that which it commonly bears in arithmetic; being here used to denote such a number, or quantity, whether real or imaginary, as, when substituted for the unknown quantity, will make both sides of the equation vanish, or become equal to each other, 1. This being premised, it may now be readily shown, that if a be a root of any equation, Xm+Axm-1+ Bxm-2 + Cxm-3+. + Tx+y=0(i), (2) The highest term of the equation, in every case of this kind, is always supposed to be positive; and if it has a coefficient prefixed to it, this must be taken away, by dividing all the rest of the terms by it; when it will be reduced to the form of that above given; which is the state that every equation is supposed to be taken in, throughout the following articles. It is also to be observed, that the coefficients A, B, C, &c. of the proposed equation, as well as the root a, may be taken either in + or in ~; and that the proposition will be equally true whether the equation be complete or has some of its terms wanting. the left hand member of that equation will be exactly divisible by x-a. For, substituting a for x, agreeably to the above definition, we shall necessarily have am +▲am-1 + Bam-2 + cam−3+ + Ta+v=0. And consequently, by taking each of the terms, except the last, to the other side of the equation Whence, if this expression be substituted for v in the first equation, we shall have Or, by uniting the corresponding terms, and placing them all in a line, 2-1 Where, since m is a whole positive number, each of the quantities (-aTM), (xTM-1 — aTM-1); (xTM-9 — aTM-2), &c. are, by Art. (E), Part 1, divisible by x-a. And, therefore, the whole polynomial m-1 m-2 m (xTM — aTM) + A(xTM-1 — aTM-1) + B(xTM-2 — aTM-o) &c. which is equivalent to the first member of the proposed equation, is also divisible by x-a; as was to be shown. But if a be a quantity greater or less than the root, this conclusion will not take place, because, in that case, we shall not have |