When we arrive at two quantities which, substituted in the place of the unknown quantity in an equation, lead to two results with contrary signs, we may infer, that one of the roots of the proposed equation lies between these two quantities, and is consequently real. Let there be, for example, the equation

if we substitute, successively, 2 and 20 in the place of x, in the first member, instead of being reduced to zero, this member becomes, in the former case, equal to 31, and in the latter, to +2939; we may therefore conclude, that this equation has a real root between 2 and 20, that is, greater than two and less than 20.

As there will be frequent occasion to express this relation, I shall employ the signs > and <, which algebraists have adopted to denote the inequality of two magnitudes, placing the greater of two quantities opposite the opening of the lines, and the less against the point of meeting. Thus I shall write

x > 2, to denote, that x is greater than 2,

x20, to denote, that x is less than 20.

Now in order to prove what has been laid down above, we may reason in the following manner. Bringing together the positive terms of the proposed equation, and also those which are negative, we have

x37x-(13x2 + 1),

a quantity, which will be negative, if we suppose x = 2, because, upon this supposition,

and which becomes positive, when we make x = 20, because, in this case,

x37x13x+1.

Moreover, it is evident, that the quantities

x2+7x and 13 2+1,

each increase, as greater and greater values are assigned to x, and that, by taking values, which approach each other very nearly, we may make the increments of the proposed quantities as small as we please. But since the first of the above quantities, which was originally less than the second, becomes greater, it is evident, that it increases more rapidly than the other, in consequence of which its deficiency is made up, and it comes at length

we arrive at two results with contrary signs; but putting — y in the place of x, the proposed equation is changed to

and we have

whence

y+2y3-3y2+15y30,

P = y1 + 2y3 + 15 y, N = 3 y2 + 3,

PN, when y = 0,

P>N, when y = 1.

Reasoning as before, we may conclude, that the equation in y has a real root, found between 0 and + 1; whence it follows, that the root of the equation in x lies between 0 and — 1, and, consequently, between + 2 and 1.

As every case the proposition enunciated can present, may be reduced to one or the other of those which have been examined, the truth of this proposition is sufficiently established.

212. Before proceeding further, I shall observe, that whatever be the degree of an equation, and whatever its coefficients, we may always assign a number, which, substituted for the unknown quantity, will render the first term greater than the sum of all the others. The truth of this proposition will be immediately apparent from what has been intimated of the rapidity, with which the several powers of a number greater than unity increase (126); since the highest of these powers exceeds those below it more and more in proportion to the increased magnitude of the number employed, so that there is no limit to the excess of the first above each of the others. Observe, moreover, the method by which we may find a number that fulfils the condition required by the enunciation,

It is evident, that the case most unfavourable to the supposition, is that, in which we make all the coefficients of the equation negative, and each equal to the greatest, that is, when instead of 8cm + Pxm-1 + Qxm- + Tx+U = 0,

....

S representing the greatest of the coefficients, P, Q,.......... T, U. Giving to the first member of this equation the form

xm S (xm−1 + xm-2... + 1),

we may observe, that

xm 1

xm-1+xm-2

...

+1=

(158);

a quantity, which evidently becomes positive, if we make

Now if we divide each member of this equation by M", we have

By substituting therefore for x the greatest of the coefficients found in the equation, augmented by unity, we render the first term greater than the sum of all the others.

A smaller number may be taken for M, if we wish simply to render the positive part of the equation greater than the negative; for to do this, it is only necessary to render the first term greater than the sum arising from all the others, when their coefficients are each equal, not to the greatest among all the coeffi cients, but to the greatest of those which are negative; we have, therefore, merely to take for M this coefficient augmented by unity.*

Hence it follows, that the positive roots of the proposed equation are necessarily comprehended within 0 and S + 1.

In the same way we may discover a limit to the negative roots; for this purpose we must substitute -y for x, in the proposed equation, and render the first term positive, if it becomes negative (178). It is evident, that by a transformation of this kind, the positive values of y answer to the negative values of x, and the reverse. If R be the greatest negative coefficient after this change, R + 1 will form a limit to the positive values of y; consequently R-1 will form that of the negative values of x. Lastly, if we would find for the smallest of the roots a limit approaching as near to zero as possible, we may arrive at it by

* In the Résolution des équations numériques, by Lagrange, there are formulas, which reduce this number to narrower limits, but what has been said above is sufficient to render the fundamental propositions for the resolution of numerical equations independent of the consideration of infinity.

substituting for x in the proposed equation, and preparing the

y

equation in y, which is thus obtained, according to the directions. given in art. 178. As the values of y are the reverse of those of x, the greatest of the first will correspond to the least of the second, and, reciprocally, the greatest of the second to the least of the first. If, therefore, S' + 1 represent the highest limit to the values of y, that is, if

Indeed, it is very evident, that we may, without altering the relative magnitude of two quantities separated by the sign < or >, multiply or divide them by the same quantity, and that we may also add the same quantity to or subtract it from each side of the signs and >, which possess, in this respect, the same properties as the sign of equality.

<

213. It follows from what precedes, that every equation of a degree denoted by an odd number has necessarily a real root affected with a sign contrary to that of its last term; for if we take the number M such, that the sign of the quantity

Mm+PMm-1 + QMm-2 .....+ TM ± U, depends solely on that of its first term Mm, the exponent m being an odd number, the term Mm will have the same sign as the number M (128). This being admitted, if the last term U has the sign, and we make x =— M, we shall arrive at a result affected with a sign contrary to that, which the supposition of x = 0 would give; from which it is evident, that the proposed equation has a root between 0 and M, that is, a negative If the last term U has the sign we make x + M; the result will then have a sign contrary to that given by the supposition of x = 0, and in this case, the root will be found between 0 and +M, that is, it will be positive.

root.

214. When the proposed equation is of a degree denoted by an even number, as the first term Mm remains positive, whatever sign we give to M, we are not, by the preceding observations, furnished with the means of proving the existence of a real root,

0,

if the last term has the sign +, since, whether we make x or x M, we have always a positive result. But when this term is negative, we find, by making

three results, affected respectively with the signs +, -, and +, and, consequently, the proposed equation has, at least, two real roots in this case, the one positive, found between M and 0, the other negative, between 0 and M; therefore, every equation of an even degree, the last term of which is negative, has at least two real roots, the one positive and the other negative.

215. I now proceed to the resolution of equations by approximation; and in order to render what is to be offered on this subject more clear, I shall begin with an example.

Let there be the equation

the greatest negative coefficient found in this equation being — 4, it follows (212), that the greatest positive root will be less than 5. Substituting -y for x, we have

y1 + 4 y3 + 3y + 27 = 0;

and as all the terms of this result are positive, it appears, that y must be negative; whence it follows, that x is necessarily positive, and that the proposed equation can have no negative roots; its real roots are, therefore, found between 0 and + 5.

The first method, which presents itself for reducing the limits, between which the roots are to be sought, is to suppose successively

x = 1, x = 2, x = 3, x=4 ;

and if two of these numbers, substituted in the proposed equation, lead to results with contrary signs, they will form new limits to the roots. Now if we make

x = 1, the first member of the equation becomes + 21,

it is evident, therefore, that this equation has two real roots, the one found between 2 and 3, and the other between 3 and 4. To approximate the first still nearer, we take the number 2,5, which occupies the middle place between 2 and 3 (Arith, 129), the present limits of this root; making then x = 2,5, we arrive at the result