567. In order to investigate the above-mentioned method, let there be taken the following general equation,

xm+pxm-1+qxm--2+rxm--3 + .. sx2+tx+u≈0. (1). Then, supposing a to be a near value of x, found by trial, and z to be the remaining part of the root, we shall have x=a+z; and, consequently, by substituting this value for x in the given equation, there will arise

(a+z)+p(a+z)m-1 + ..s(a + 2)2+t(a+z)+n=0; which last expression, by involving its terms, and taking the result in an inverse order, may be transformed into the equation

P+Qz+Rz2+Sz3+ ·· +z"=0 (2), where P, Q, R, &c. are polynomials, composed of certain functions of the known quantities, a, m, p, q, r, &c. which are derived from each other, according to a regular law.

568. Thus, by actually performing the operations above indicated, or by referring to (Art. 539), it will be found that

l'=am+pam-1+7am-2+

...

sa2 + ta+u; which value is obtained by barely substituting a for x in the equation first proposed.

And, by collecting the several terms of the coeffi cients of z, it will likewise appear, that

...

Q=mam--1+m(m−1)pum2 + +2sa+t; which last value is found by multiplying each of the terms of the former by the index of a in that term, and diminishing the same index by unity.

569. Hence, since z in equation (2), is, by hypothesis, a proper fraction, if the terms that involve its several powers z2, z3, z1, &c. which are all, successively, less than z, be neglected in the transformed equation, we shall have

a""+pa-1+

P+Qz=0, or z m--1

+ta+u.

ma + (m − 1) pam-2 + ..+t And, consequently, if the numeral value of this expression be calculated to one or two places of deci

mals, and put equal to b, the first approximate part of the root will be z=b, or x=a+b=a'.

Whence also, if this value of x, which is nearer its true value than the assumed number a, be substituted in the place of a in the above formula, it will be.

come

am+pa'm-1+

2 'mam--1+ (m—1)pam--2+.+t

which expression being now calculated to three or four places of decimals, and put equal to c, we shall have, for a second approximation towards the unknown part of the root

z=c, or x=a'+c=a".

And, by proceeding in this manner, the approximation may be carried on to any assigned degree of exactness; observing to take the assumed root a in defect or excess, according as it approaches nearest to the root sought, and adding or subtracting the corrections b, c, &c. as the case may require.

570. A negative root of any equation may also be found in the same manner, by first changing the signs of all the alternate terms, (Art. 541), and then taking the positive root of this equation, when determined as above, for the negative root of the proposed equation.

571. In the practical application of this rule we must endeavour to find two whole numbers, between which some one root of the given equation lies; and by substituting each of them for x in the given equation, and then observing which of them gives a result most nearly equal to 0, we shall ascertain the whole number to which x most nearly approaches; we must then assume a equal to one of the whole numbers thus found, or to some decimal number which lies between them, according to the circumstances of the

case.

х

572. Since any quantity, which from positive becomes negative, passes through 0 (Art. 496), if any two whole numbers, n and n'; one of which, when

substituted for x in the proposed equation, gives a positive, and the other a negative result; one root of the equation will, therefore, lie between n and n'. This, of course, goes upon the supposition that the equation contains at least one real root.

! 573. It is necessary to observe, that, when a is a much nearer approximation to one root of the given equation than to any other, then the foregoing method of approximation can only be applied with any degree of accuracy. To this we also farther add, that, when some of the roots are nearly equal, or differ from each other by less than unity, they may be passed over without being perceived, and by that means render the process illusory; which circumstance has been particularly noticed by LAGRANGE, who has given a new and improved method of approximation, in his Traité de la Resolution des Equations Numériques. See, for farther particulars relating to this, and other methods, BONNYCASTLE'S Algebra, or BRIDGE's Equations.

Ex. 1. Given x3+2x2-8x=24, to find the value of x by approximation.

Here by substituting 0, 1, 2, 3, 4, successively for x in the given equation, we find that one root of the equation lies between 3 and 4, and is evidently very nearly equal to 3. Therefore let a=3, and x=a+z. x2=a3+3a2 z+3az2+z3 2x22a2+4az+223

Then

-8x= =- -8a-8z

=90.

And by rejecting the terms z3+3az2 +2z2, (Art.569), as being small in comparison with z, we shall have a3+2a2-8a +3x2z+4az-8z=24;

and consequently x=a+2=3.09, nearly.

Again, if 3.09 be substituted for a, in the last equation, we shall have z=

24-a3-2a2+8a_24-29.503629-19.0962+24.72

3a2+4α-8

28.6443+12.36-8

=

.00364; and consequently a=a+z=3.09+.00364 =3.09364, for a second approximation.

And, if the first four figures, 3.093, of this number, be substituted for a in the same equation an approximate value of x will be obtained to six or seven places of decimals. And by proceeding in the same manner the root may be found still more correctly. Ex. 2. Given 3x5 +4x3-5x=140, to find the value of x by approximation. Ans. x 2.07264. Ex. 3. Given x 9x3+8x2-3x+4-0, to find the value of x by approximation.

Ans. x=1.114789. Ex. 4. Given x3+23.3x2 -39x-93.3=0, to find the values of x by approximation.

Ans. x 2.782; or-1.36; or-24.72; very nearly. Ex. 5. Find an approximate value of one root of the equation + x2+x=90. Ans. x=4.10283. Ex. 6. Given 3+6.75x2+4.5x-10.25=0, to find the values of x by approximation.

Ans. x.90018; or-2.023; or-5.627; very nearly.

END OF THE TREATISE ON ALGEBRA.

APPENDIX.

Algebraic Method of demonstrating the Propositions in the fifth book of Euclid's Elements, according to the text and arrangement in Simson's edition.

SIMSON'S Euclid is undoubtedly a work of great merit, and is in very general use among mathematicians; but notwithstanding all the efforts of that able commentator, the fifth book still presents great difficulties to learners, and is in general less understood than any other part of the elements of Geometry. The present essay is intended to remove these difficulties, and consequently to enable learners to understand in a sufficient degree the doctrine of proportion, previously to their entering on the sixth book of Euclid, in which that doctrine is indispensable.

I have omitted the demonstrations of several propositions, which are used by Euclid merely as lemmata, but are of no consequence in the present method of demonstration.

Instead of Euclid's definition of proportion, as given in his 5th definition of the 5th book, I make use of the common algebraic definition; but I have shown the perfect equivalence of these two definitions. This perfect reciprocity between the two definitions is a matter of great importance in the doctrine of proportion, and has not (as far as I can learn) been discussed by any preceding mathematician.

With respect to compound ratio, I have also given another definition of it instead of that given by Dr. Simson; as his definition is found exceedingly obscure by beginners, and is in my judgment one of the most objectionable things in his edition of Euclid's Elements.

The literal operations made use of in the present paper are extremely simple, and require very little