Substitute coin the first terms of the quantities X, X1, X2, ... Xn; and let the signs of the results be written down in order, and let the number of changes in these signs, from and from to +, be denoted by m; and when is similarly substituted, let the number of variations in the signs ber; then mr is the number of real roots; and n·(mr) the number of imaginary roots. + to EXAMPLE. 1. Given the equation a2a3x2+8 x −12 = 0, to find the number of its real and imaginary roots. -12, and performing on these quantities the process of finding their greatest common measure, the first remainder is 10x2. -2x-88; hence the next divisor is 5x2+x+44; and the next remainder is found to be - 896 x · 1296, so that the next divisor is 56 x +81; and the last remainder is a positive number; hence X = x1 — 2 x3 — x2 + 8 x — 12 X1=423 6x2 2x+8 - X2=5x2+x+44 X3=56x+81 The last remainder being a positive number when its sign is changed, it is = X4; and as it does not alter its value by the substitution of any quantity for a, it is unnecessary to write the number, as only the signs of the results are required. When co and co are substituted in the first terms of these five quantities, the signs of the results are in order in X X1 X2 X3 X4 co, ++ + + -... 3 variations 1 variation In the former series of signs the number of variations or changes from + to and -> to +, is three; and in the latter only one; hence m3, and r1, and m―r=3 1=2= number of real roots. Hence the number of imaginary roots is n (mr)=4—2=2. (688.) To find the number of roots that lie between any two numbers, substitute these numbers for x in all the terms of the quantities X, X1, X2, ... and write down in a row merely the signs of the results, and the difference between the numbers of variations of signs in these two rows will be the number of roots." To find the number of roots of the preceding equation that lie between 1 and 3. Substitute these numbers for x, and the signs of the results are The difference of variations —2—1— 1; or only one root lies between 1 and 3. This root will be found on trial to be 2. EXERCISES. 1. Find the number of real roots of the equation 8x3-6x-1=0. Ans. The three roots are real; there are two roots between 0 and 1, and one between 0 and 1. 2. Find the number of real and imaginary roots of the equation a3. -5 x2+8x-1=0. Ans. There are two imaginary roots, and one real root between 0 and 1. 3. Find the number of real roots of the equation x3-2x-5=0. Ans. It has two imaginary roots, and one real root, between 2 and 3. 4. Find the number of real roots of the equation x42x3-7 x2+10x + 10 = 0. Ans. It has four real roots; two positive ones between 2 and 3; and two negative roots, one between O and 1, and another between 2 and -3. 467 NOTES. NOTE A. It was found in the subject of Equations that their solutions were sometimes negative (pages 187, 253); and the explanation given of these insulated negative quantities was, that they are to be understood in a sense directly the opposite of that of positive solutions. This explanation is that given by some eminent mathematicians; and the chief controverted point connected with this subject is, whether this interpretation is a matter of convention, or a necessary deduction from the definition of the negative sign. While the meaning of the negative sign is restricted to the original one of subtraction, it can properly be prefixed only to a quantity which is to be subtracted from some greater quantity of the same kind. When the negative sign is prefixed to the greater of two quantities that are connected by this sign, the operation indicated by the sign cannot be performed in respect to the whole of the greater quantity, but only in reference to that part of it which is equal to the other quantity; and the difference between the two quantities in this case, without any sign prefixed to it, merely shows how much the quantity proposed to be subtracted exceeds the other; but there is no reason for prefixing the negative sign to this difference more than the positive. It will be necessary to take a particular instance in order to perceive precisely the proper meaning of the result in this last case. Suppose that a person who has only £10 is ordered to pay away £8; then 10-8-2, and he has evidently 2 remaining. But if he is ordered to pay £12, then 12—10—2, and the sum to be paid exceeds the money in his possession by two pounds, so that the transaction is impossible. Now, although 12-10-2, or 10~12=2, it does not follow, from the original meaning of the negative sign, that 10-12 is -2, for this result is not found by subtracting 12 from 10, but the reverse, by subtracting 10 from 12, and then prefixing the sign of the greater, which is an anticipation of the rule of algebraic addition in article (55), and requires to be previously demonstrated by the aid of the conventional meaning stated in art. (44.) Although the remainder 2, therefore, cannot in this case, for any proper reason, have the negative sign prefixed to it, so long as only the original definition of this sign is adhered to, yet it may be prefixed, if by previous agreement or convention a new meaning be assigned to it. If, for example, the 10 pounds be called property, and the 12 pounds debt, then, as the debt exceeds the property by 2, it is manifest that both together amount only to 2 pounds of debt, that is, + 10 and 12 are equivalent to -2, the sign prefixed to the 2 now meaning debt. With this new conventional meaning, therefore, the result-2 is perfectly intelligible, and logically deduced from the premises. When the result - 2 is considered as property, it may be called negative property, or 2 pounds less than nothing, which is a perfectly correct expression when understood in the conventional sense, namely, the opposite of positive property, that is, debt. Hence the greatest of two numbers considered in an absolute sense is the least when understood in the above sense; that is, if - 5 and 12 are insulated negative numbers, then though 125, yet -5 is —— 12. - -- The rules for calculating with such quantities have been demonstrated under the subjects of Addition, Subtraction, &c., which has not hitherto been satisfactorily done. Bourdon accordingly remarks in his Algebre, No. 61, “Les demonstrations qu'ils en donnent n'ont que l'apparence de l'exactitude." Algebraical investigations could be conducted independently of insulated negative quantities, and therefore they might be rejected. Were this done, however, the solutions of questions would not be so general, for the negative solution is actually the direct solution of another case of the question, which, unless these quantities were admitted, would require a new solution. As it was necessary to modify the conditions of an equation, in order to adapt it to the negative solution, so in the application of algebra to geometry, similar modifications must be introduced, in order that the negative solution may be applicable. Thus, the length of a perpendicular drawn from a given point (x', y',) to a given line whose equation is y = ax + b, being y' - ax' L = ± √(1+a2) b if the positive value be taken in reference to the preceding line, then the negative solution, modified by changing b intob, becomes applicable to the line whose equation is y = ax―b, which may be considered as the reverse case of the former, as this line is parallel to the former, and has the same situation relatively to the third and fourth quadrants that the preceding has relatively to the first and second. The perpendiculars therefore are +: y' — ax' - b - √(1+ a2) y' — ax' + b The meaning of negative exponents must be explained in a similar manner, that is, by means of a new definition, for their meaning is not to be deduced from the original definition of the negative sign, as they are arrived at by applying a rule to a case to which it is not properly applicable, not having been demonstrated for it. The negative solutions of physical questions also admit of a consistent interpretation on the same principle. In order to adapt the circumstances to these solutions, it is necessary to change motion in one direction to that in an opposite direction; to change gravity to buoyancy, and so on according to the nature of the question. As negative solutions indicate the modifications necessary to be made, in order that the solution may be directly applicable, so imaginary solutions are indications of some impossibility in the question, and they even show the amount of alteration that must be made in order that the conditions may be compatible, and may lead to real results. Imaginary quantities are indications, not results, of an operation to be performed, to accomplish which, however, is impossible. NOTE B. The subject of Proportion belongs properly to algebra, and not to geometry. A fundamental assumption in analytical |