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progress may, indeed, at first, be a little retarded; but the real knowledge he thence acquires will abundantly compensate his trouble, and enable him to proceed, afterwards, with certainty and success, in matters of greater difficulty, where authors, and their rules, can yield him no assistance, and he has nothing to depend upon but his own observation and judgment.

This second edition has many advantages over the former, as well with respect to a number of new subjects and improvements, interspersed throughout the whole, as in the order and disposition of the elementary parts in which, particular regard has been had to the capacities of young beginners. The work, as it now stands, will, the author flatters himself, be found equally plain and comprehensive, so as to answer, alike, the purpose of the lower, and of the more experienced class of readers.

P. S. The great reputation of Mr. SIMPSON'S TREATISE of ALGEBRA, and the favourable reception it has universally met with since its first publication, and which testifies it to be the best elementary work upon the subject, has induced the proprietor to have this edition carefully revised and corrected throughout by a very eminent mathematician: he therefore trusts it will be found as worthy the approbation of the public as any former edition, or as if revised by the author himself.

2. The Extermination of unknown Quantities, or the

reduction of two or more equations to a single one 63

VERAL DIMENSIONS

1. Of the origin and composition of Equations

2. How to know whether some, or all the roots of an

Equation be rational, and, if so, what they are

131

ibid.

134

3. Another way of discovering the same thing, by

means of Sir Isaac Newton's method of divisors ;

with the grounds and explanation of that method 136

4. Of the solution of cubic Equations according to

Cardan

-

143

5. The same method extended to other, higher Equa-

tions

145

6. Of the solution of biquadratic Equations according

to Des Cartes

148

7. The solution of biquadratics by a new method with-

out the trouble of exterminating the second term

8. Cases of biquadratic Equations that may be reduced

to quadratic ones

150

153

9. The resolution of literal Equations, wherein the

given, and the unknown quantity are alike affect-

ed

156

10. The resolution of Equations by the common method

of converging series

158

14. A method of solving high Equations, when two, or

more unknown quantities are concerned in each

177

OF FIGURATE NUMBERS

1. The Sums of Series, consisting of the reciprocals of

figurate numbers, with others of the like nature

2. The sums of compound Progressions, arising from

a series of powers drawn into the terms of a geo-

metrical progression

3. The combinations of Quantities

4. A demonstration of Sir Isaac Newton's Binomial

theorem

213

215

219

225

227

SECTION XVI.

OF INTEREST AND ANNUITIES

1. Annuities and Pensions in Arrear, computed at sim-

2. The investigation of Theorems for the solution of

the various cases in compound interest and an-

nuities

SECTION XVII.

229

231

234

OF PLANE TRIGONOMETRY

SECTION XVIII.

THE APPLICATION OF ALGEBRA TO THE

SOLUTION OF GEOMETRICAL PRO-

1. An easy wag of constructing, or finding the roots of

a quadratic equation, geometrically.

241

254

267

2. A demonstration why a problem is impossible when

the square root of a negative quantity is concerned 272

3. A method for discovering whether the root of a ra-

dical quantity can be extracted

4. The manner of taking away radical quantities from

the denominator of a fraction, and transferring

them to the numerator

5. A method for determining the roots of certain high

Equations, by means of the section of an angle

The geometrical construction of a large variety of

linear, and plane Problems; with the manner of

resolving the same numerically

APPENDIX

284

288

301

$15

401