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 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 +23 2x2=2a2+4az+2z2 -8x=-8a-8z

Then

=90.

And by rejecting the terms z3+3az2+2z,(Art.569), as being small in comparison with z, we shall have a3+2a2-8a+3x2 z+4az-3z=24;

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

Again, if 3.09 be substituted for a, in the last equa-,

tion, we shall have z=

24—a3 — 2a2 +8a

3a2+4α-8

24-29.503629–19.0962+24.72

28.6443+12.36-8

.00364; and consequently r=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 3x+4x3-5x=140, to find the value of x by approximation. Ans. x=2.07264. Ex. 3. Given x -9x+8x2 -3x+4-0), to find the value of x by approximation.

Ans. 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 3+x2+x=90.

Ex. 6. Given 3+6.75

Ans. x=4.10283. +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 under stood than any other part of the elements of Geome try. 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 ob scure 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

previous knowledge of algebra to render them intelligible.

The algebraic signs commonly used to indicate greater, equal, less, are >, =, <: thus the three expressions ab, c=d, e<f, signify that a is greater than b, that c is equal to d, and that e is less than f. The expression c=d is called an equation or equality; the others ab, eƒ, are called inequalities.

Also when four quantities are proportionals, we shall express this relation in the usual mode by points; thus,

A: B::C: D

is to be read, A is to B as C is to D; or, A has the same ratio to B that C has to D.

THE ELEMENTS OF EUCLID, BOOK V.

Definitions.
I.

A less magnitude is said to be a part of a greater, when the less measures the greater, that is, when the less is contained a certain number of times exactly in the greater.

II.

A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

III.

Ratio is a mutual relation of two magnitudes of the same kind to one another in respect to quantity. IV.

Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

V.

The ratio of the magnitude A to the magnitde B is the number showing how often A contains B; or, which is the same thing, it is the quotient when A is numerically divided by B, whether this quotient be integral, fractional, or surd.

Explication.

This fifth definition, with its corollaries, is used in the present essay instead of Euclid's 5th and 7th definitions: the following examples will sufficiently illustrate the definition. Let A=20, and B=5, A 20 then the ratio of A to B, or of 20 to 5, is or, or

B

4, so that the ratio of 20 to 5 is 4. Again let A=5, A 5 1

and B 20, then

1

and therefore the ratio

of 5 to 20 is 20 is Lastly let A=12/2, and B=4,

4

12/2 to 4 is 3/2.

COROLLARY 1. If four magnitudes A, B, C, D, be

EO related that

C

A
B D

it is evident the ratio of A to

B is the same with the ratio of C to D.

COR. II. Any four magnitudes whatever, so related that the ratio of the first to the second is the same with the ratio of the third to the fourth, may be expressed by

rA, A, rB, B;

the first of the four being rA, the second A, the third rB, and the fourth B; the magnitudes A and B being any whatever, and the letter r denoting each of the two equal ratios or quotients when the first rA is divided by the second A, and the third rB divided by the fourth B.

COR. III. When four magnitudes A, B, C, D, are

that the ratio of A to B is greater than the ratio of C to D; or that the ratio of C to D is less than the ratio of A to B.

The Fifth Definition according to Euclid.

The first of four magnitudes is said to have the same ratio to the second which the third has to the