SECTION VI.

INTEGERS, THEIR DIVISORS, FORMS, ETC.

Prime and Composite Numbers.

....

Divisors and Multiples-Greatest Common Divisor and Least Common

Multiple....

PAGE

128

Method of finding Prime Numbers

Rule for finding the Greatest Common Divisor

Rule for finding the Least Common Multiple.

SECTION VII.

FRACTIONS.

Reduced to their Lowest Terms...

Mixed Numbers Reduced to Improper Fractions..

Improper Fractions Reduced to Mixed Numbers

Fractions Reduced to a Common Denominator

Addition of Fractions

Subtraction of Fractions

Multiplication of Fractions.

Division of Fractions..

129

135

141

150

157

158

160

161

163

165

167

169

Method of finding the Sum of any Number of Terms of an Arithmetical

Progression....

Method of finding any Term of a Proportion from the other Terms ..

Geometrical Progressions-Method of forming them from their first Terms

and Ratios of Increase or Decrease

203,

205

Method of inserting any Number of Geometrical Means between given

Terms.....

Method of finding the Sum of a Geometrical Series.

Abridged Proportions-Harmonical Proportions and Progressions.

Reduction of Compound to Single Ratios

PAGE

207

209

211

215

220

223

225

Fundamental Principle of Single and Double Position, and of the Differ-

POWERS AND INVOLUTION.

Method of obtaining any Power from its Root.

New Investigation of the Binomial Theorem

Method of finding any Power of a Polynomial

SECTION XI.

EVOLUTION.

269

271

279

Uses of the Surd Sign and Fractional Indices in the Extraction of Roots. 282

The Root of any Exact Algebraic Power Extracted by means of the Great-

Surds of Different Indices Reduced to those having a Common Index ... 317

Method of Freeing any Algebraic Expression from Surds .

Impossible or Imaginary Roots

326

330

SECTION XIII.

OF EQUATIONS.

Algebraic and Transcendental Equations..

338

Rule for Solving any (Simple) Determinate Equation containing but One

Unknown Letter

348

Solution of Two or More separate Simple Equations containing as many

Unknown Letters as there are Equations...

Particular Methods of Solution illustrated by Examples.

PAGE

353

362

388

Appendix to Simple Equations.....

SECTION XIV.

QUADRATIC EQUATIONS.

Simple or Pure Quadratics Solved by Extracting the Square Root....

Affected or Complete Quadratics-Rule for their Solution..

394

396

Development of the Roots of Equations of the nth Degree

439

Equations of the nth Degree shown to have n Roots..

440

Illustration by Examples..

441

Method of Approximating to the Ratio of Two Numbers or Quantities-

Illustrated by Examples

448

Solution of the Reducible Case by Cardan's Formulæ (A), (A')

Solution of the Irreducible Case by Pure Algebra

462

464

469

Solution of Biquadratic Equations

SECTION XVIII..

LIMITS OF THE REAL ROOTS OF AN EQUATION OF THE 7th Degree.

DEVELOPMENT OF THE REAL AND IMAGINARY ROOTS OF AN EQUATION OF THE

7th DEGREE.

Illustrated by Examples..

494

New and General Method of Developing the Roots of Equations.

512

Illustrated by Examples.

514

RESOLUTION OF WHOLE NUMBERS INTO THEIR FACTORS

Illustrated by Examples...

535

537

SECTION XXII.

PERMUTATIONS AND COMBINATIONS.

544