of (A), of which is the first figure; and diminishing the roots of (1) by r', we shall have for the second transformed equation A+A"3+A"¿x''2+A","+N"=0... (2); and the third figure '" of the original root will, in like manner, be the first figure of a root of this. And thus by finding at the outset the leading figurer of the root of (A) sought, then the leading figure r' of the same root, diminished by r in (1), then the leading figure of the same root still further diminished by r" in (2), and so on, we shall obtain the figures of the root sought, one after another, by a series of uniform steps. It must be noted, however, that in speaking of these successive figures, r, r', r'', &c. we have regard not only to the mere numerical symbol but to the place it occupies in the numerical scale; for instance, the 3 in the number 324, is not 3 merely, it is 300; in like manner, the 2, having regard to the place it occupies, is 20. If this number 324 were a root of (A), the root here supposed to be selected, then we should say that its first figurer, is 3 in the hundreds' place, or 300; the second figure r', is 2 in the tens' place, or 20; and the third figure '', is 4 in the units' place, or simply 4: for the determination of the complete root 324, no transformation beyond (2) would be necessary: if the root extend to several decimals, the transformations must be carried on till the required number of decimals are obtained, just as in the common arithmetical operations for the square and cube roots. From what has been shown in Article 91, the learner must perceive that the determination of the successive transformed equations (1), (2), &c. is an operation of pure arithmetic, very simple and easy in its character, and admitting of rapid performance; but we cannot enter upon it till we know how to find the leading figure of a root: the determination of (1) presupposes that is found from (A); the determination of (2) pre-supposes that r' is found from (1), and so on. The problem, therefore, of finding the first figure of a root of an equation, is a problem of the greatest consequence in the practical solution of numerical equations. And this fact is deserving of especial notice: for whatever improvements or facilities be attempted or proposed in reference to the general solution of numerical equations, if they be not exclusively directed to this initial step of the process, they may be confidently predicted to be of but little practical value: the researches of Mr. Horner have conferred upon the other part of the process such a degree of finish and perfection as not only to preclude all further improvement, but even to extinguish all desire for it. From these remarks the learner will be able to foresee that the discovery of the first figure of each of the roots of a numerical equation is the only thing of any difficulty connected with the process of solution. This difficulty is, however, in general, but little felt after the leading step of the work; that is to say, after it has been surmounted with respect to the proposed equation (A): the first figure of the root in (1) is in general much more readily discovered; the first figure of the root in (2) still more readily, and so on, till at a certain stage of the development all difficulty disappears, and the successive figures from that point become determinable one after another with the greatest ease. A little reflection upon the nature of the arithmetical operation for the square or cube root, will at once suggest to the learner an explanation of this: the first step in the extraction of such a root requires extraneous aid, which we must seek in a pre-formed table of roots either actually before us or preserved in the memory; but after this first step we are independent of such aid; the operation assumes a character analogous to that of ordinary division, and a trial divisor, suggestive of the following figure, offers its assistance at every subsequent step; and we know that this trial divisor becomes more and more effective—that is, it approaches nearer and nearer to the true divisor, as the work advances. It is just the same with the process for finding the figures of a root of a numerical equation: the determination of the first figure requires external aid; for the discovery of the second we are furnished with a trial divisor; for the determination of the third, with a trial divisor still more effective; and so on, till, as in the square and cube roots, these trial divisors become unerring guides to the remaining figures of the root. 93. In the search after the first figure of a root of an equation, we are materially assisted by the following principle, namely: if two numbers when separately substituted for in the polynomial which forms the significant side of an equation, give results with opposite signs, a root of the equation must necessarily lie between those numbers: that is, a root of the equation will be greater than one of the numbers substituted, and less than the other. For this is only saying that between two numbers which give results, the one positive and the other negative, there must lie an intermediate number that will give the intermediate result zero; that is, a number that will be a root of the equation: for instance, if I be substituted for x in the first member of the equation x3-3x2-4x-10=0, the result will be minus 10; but if 2 be substituted, the result will be plus 2: we conclude, therefore, that one of the roots of the equation must lie between 1 and 2: in other words, that the first figure of one of the roots must be 1 in the units' place; so that if 1, followed by certain decimals unknown to us at present, were substituted for x, the result would be neither plus nor minus, but the intermediate zero. 94. It would, however, be a very tedious business, in many equations that might be proposed, to prosecute the search after the places of the roots by such chance substitutions as these; and analysts have accordingly directed their efforts, and with considerable success, to the discovery of efficient means of reducing the labour within practicable limits: a full account of their researches in this important field of inquiry will be found in the "Theory of Equations" before referred to. Into these we are of course precluded from entering in the present introductory work; but we shall here offer to the learner certain assistance in this matter that is not afforded to him elsewhere, and which will always prove acceptable whatever other aid may be combined with it. (1) Required the leading figure of a root of the equation x3-3x2-4x-10=0. This equation we shall multiply by 3, for the purpose of making the second term a rational square: that is, we shall replace the equation by 3a+9x-12x-30-0. Decomposing into factors, as already explained (Art. 86), we have for the factors of the first member 3x-3x+12x+30); or 3x—2±√(—3x3+34). Now that value of x which is a root of the equation, must be such as to render one or other of these factors zero; but no real value of x can possibly do this that causes the quantity covered by the radical to become negative: for then the irrational part of each factor would be imaginary and the other part real; and the sum or difference of two quantities, the one real and the other imaginary, can never be zero. But for all positive integral values of x above 2, the quantity -33+34 is negative; hence the first figure, of whatever positive root the equation may have, cannot be greater than 2; that is, the first figure must be either 2 or a smaller number, so that our trial substitutions need not extend beyond the limit 2; putting, therefore, 2 for x in the proposed equation, we find the result of the substitution by the process already explained (pages 199, 200). -4-10[2 1 3 the result, therefore, is plus 2, and as a glance at the coefficients of the proposed equation shows that, for x= =1, the result is negative, we conclude that a root of the equation lies between 1 and 2; and, consequently, that the first figure of that root is 1. (2) Find the first figure of a root of the equation x3-3x2+2x-71=0, or 3x3+9x2+6x-213=0. 3x±√(−3x3—6x+213). Here, in order that the quantity under the radical may be positive, x must be less than 4. As the substitution x=4 gives plus 59, and the substitution x=3 gives minus 11, we conclude that 3 is the first figure of a root. It may be remarked that the multiplication of the proposed equation by 3 might have been dispensed with, as we may write the factors thus: √3.x√(x3-2x+71). (3) Find the first figure of a root of the equation 2x+x-3x-416793-0, x±√(-2x+3x+416793), and as the quantity under the radical remains the same whether be positive or negative, it follows that the roots, both positive and negative, are each numerically less than 30. Hence a root of the equation lies between 20 and 30; the first figure of it is, therefore, 2 in the tens' place. Although in these two examples we have substituted the superior limits of the roots, as inferred from the factors of the polynomial, yet such substitution is never necessary, for the result will always be positive. This is pretty obvious; for the first member of the equation, being the product of its two factors, is the square of the common rational part of each factor, minus the square of the irrational part: when, therefore, this latter square is a negative number, the product is the sum of two squares. (4) Find the first figure of a root of As we know that the substitution of 7 for x will give a positive result, we have to try inferior substitutions till we arrive at a result which is negative. Hence the first figure of a root is 5. (5) Find the first figure of a root of a1—17x2-20x-6=0. that is to say, a negative root lies between 0 and -4. therefore the first figure of this root is —3. In this search after the first figure of a root, the chances which lie in our way of actually decomposing the polynomial into two factors in which x shall be free of the radical, as already exemplified; should not be disregarded: thus, in the present in |