Proceed from column to column exactly in the same manner as with the last column, until the whole is constructed; then the last number of every column is the co-efficient of the corresponding term of the transformed equation to that of the original equation, of which the number at the head is the co-efficient.

N.B. The addition which is here spoken of is that of algebraic addition, and is the sign of the number by which the root is to be increased, must be considered negative.

Er. 1. Diminish the root of the cubic equation +5xx2+7x-47 =0 by the number 2.

Whence the equation (x-2)+11(x-2)2+39(x-2)-5=0, has its roots less by 2 than the roots of the proposed equation. Er. 2. Diminish the root of the biquadric equation x-423 +8x316x+20=0 by the number 3.

Whence (x-3)*+8(x—3)3 +26(x−3)3 + 32(x−3)+17=0 is the

equation required.

Er. 3. Diminish the root of the cubic equation x+3x-4x+1 by the number='3.

(x-3)+3′9(x—·3)—193(x-3)+097=0 is the equa

Whence tion required.

Ex. 4. Increase the roots of the cubic equation x3-7x+7=0 by

the number 3.

Whence (x+3)-9(x+3)+20(x+3)+1=0 is the equation required, or substituting v for x+3 the transformed equation will be v3-9v2+20v+1=0.

To ascertain the first digit of every possible root of an equation.

Proposition VII. Problem.

Find such a digit a, either at once or by trial, that when the original equation is diminished separately by a and a+ 1, the transformed equation in x-a-1 may have one or more changes of signs less than those of the equation in x-a; then for every change that disappears so many affirmative roots will be contained between a and a+1; and, if the number of changes that disappear be odd, the equation will at least have one possible root; therefore, if only one change disappears, a will either be the entire root, or the first figure of the root, according as the value of the absolute number of the equation which arises by diminishing the root of the original equation by a is zero, or a real magnitude; but if more changes disappear, a will be the first digit of each root contained between a and a+1.

If we now ascertain every two transformed equations, of which the root of the one is less than that of the other by unity, and of which the changes of signs, in that which has its root diminished, are fewer than in that other, we shall have all the affirmative roots.

In order to discover the negative roots, we may either substitute - for a in the original equation, and proceed as before, or we may increase the roots of the original equation, and consider the permanences of signs that disappear instead of the changes.

N.B. One certain sign of two or some even numbers of impossible roots, is, when any co-efficient becomes zero between two adjacent co-efficients which have like signs, viz. either both + or both ·

Ex. 1. Find the first digit of each of the roots of the cubic equation r-11x+5x—14.

This equation having no permanencies of signs has no negative roots, that is, all its roots are affirmative.

Having now executed the three transformed equations; by coupling the signs of every two adjacent terms, of every succeeding equation as it arises, we have the four series of signs

Where the first series, belonging to the original equation in x, has three changes of signs, the second in x-1 only one change; therefore the equation in x-1 has two roots less than the equation in x: when these two roots are real, the first figure of each will be O in the unit's place.

Again, since the equation in x-11 has no changes of signs, and the equation in x-10 has one change, the first two figures of this root of the original equation are 10.

Er. 2. Find the first digit of each root of the cubic equation x37x+7=0.

Having executed the two transformed equations, we have, by

arranging as in the preceding example, the following series of signs:

Then because the original equation in x, and the transformed equation in -1, have each two changes of signs, they have each two affirmative roots; but in transforming the equation in x-1 to that in r-2, the two roots have disappeared, and since the roots lie between 1 and 2, the first digit of each of these roots is ; and since the signs of the terms of the last transformed equation are all affirmative, the riginal equation in x has only two affirmative roots.

In order to discover the negative root of the equation x3-7x+7.

Again, transforming the equation in z+3, by increasing its root by unity, we have the following operation:

Let us now make trial of two numbers, as 3 and 4, in order to increase the roots of the proposed equation, then these two transformed equations being executed as above, the signs of the terms will stand thus:

The first equation in x+3, has one continuation of the same sign, that is, one permanency, therefore it has one negative root, as in the proposed equation; but in the equation in x+4, the signs of the terms have no permanencies, and, consequently, no more negative roots; therefore the first digit of the negative root is -3.

Proposition VIII. Problem.

To extract Roots by their Limits.

Having found the first digit a of one of the roots, by Prop. VII, transform the equation in x-a, into two others in x-a-b, and -a-b-1, so that the equations in x-a-b and x-a—b—1 may have respectively the same signs as the equations in x-a and x-a-i and b will be the second figure of the root.

Proceed in this manner, from one transformed equation to another, until the root is either complete, or as many figures ascertained in it as may be found necessary.

Proposition IX. Problem.

To extract the root of the general equation

x” +Bx~~1±Cx*-+ Lx=N.

...

Find the first digit of every possible root by the preceding problem; then, in order to extract any one of them, let a be the first digit be longing to that root. Having annexed a cypher to the absolute number of the equation in x-a, divide it by the co-efficient of the single power of the unknown quantity, and let the quotient be b; diminish the root of the equation in x-a by the quantity b; then, if the equation

in x-a-b has the same number of changes of signs as the equation in s-a, i will be the second digit of the root; but if the sign of the absolute number of the equation in x-a-b is contrary to that of the absolute number of the equation in x-a, the operation of finding b must be repeated; supposing that b is properly found, annex a cypher to the absolute number of the equation in x-a-, and divide it by the co-efficient of its second term; diminish the root of the equation in-a-b by c, and the co-efficients and absolute number of the equation in x-a-b-c will be obtained, and thus three digits a+b+c will have been found in the root of the original equation in x.

Proceed in this manner, always dividing the absolute number of the last transformed equation, increased by annexing a cypher, as before, by its co-efficient, then, diminishing the root of that last transformed equation by the quotient, then the quotient is the next figure of the

root.

N.B. The whole of the transformations may be formed in a single opera. tion, as will appear by the following examples.

Er. 1. Find one of the roots of the cubic equation x3—11x3+5x— 14=0.

In this operation, the first digit 10 is found as in example first of Prop. VIII., the proposed equation here being the same as in that example; then, dividing the absolute number 64 of the transformed equation in x-10 by 85, the co-efficient of the single power, the quotient is 7; but 7 being tried, by diminishing the root of the proposed equation in x-10 by 7 is found not to succeed, as the coefficients of the equation in x—10—7, or in x-10.7 become all affirmative, which shows we have passed over the root; but, on trying 6, and diminishing the root of the equation in x-10 by 6, the transformed equation in x-106, has the same number of changes of signs as the equation in x-10; therefore, 6 is the next digit of the

root.

The remaining figures of the root are found at once without trial.