CHAPTER VI.

On the Involution and Evolution of Numbers,

and of Algebraic Quantities.

Page

SECT.

I. Involution of algebraic quantities,

304

II. Evolution of algebraic quantities,

315

III. Investigation of the rules for the extraction of

the square and cube roots of numbers,

326

SECT.

CHAPTER VII.

On Irrational and Imaginary Quantities.

I. Theory of irrational quantities,

II. Reduction of radical quantities or surds,

334

341

III. Application of the fundamental rules of arith-

metic to surd quantities,

346

IV. Method of reducing a fraction whose denomina-

tor is a simple or a binomial surd, to another

that shall have a rational denominator,

353

V. Method of extracting the square root of bino-

mial surds,

360

I. Solution of pure equations of the first degree

by involution,

II. Solution of pure equations of the second, and

other higher degrees, by evolution,

III. Examples in which the preceding rules are

applied in the solution of pure equations,

I. Solution of adfected quadratic equations, in-

volving only one unknown quantity,

II. Solution of adfected quadratic equations, in-

volving two unknown quantities,

CHAPTER XI.

On the Solution of Problems producing Quad-

ratic Equations.

SECT.

I. Solution of problems producing quadratic equa-

tions, involving only one unknown quantity,

II. Solution of problems producing quadratic equa-

tions involving more than one unknown

quantity,

CHAPTER XII.

Page

382

389

395

404

413

419

On the Binomial Theorem,

426

SECT.

I. Investigation of the binomial theorem,

427

II. Application of the binomial theorem to the ex-

pansion of series,

431

SECT.

CHAPTER XIII.

On Proportion and Progression.

I. Arithmetical proportion and progression,

II. Geometrical proportion and progression,

III. Harmonical proportion and progression,

IV. Problems in proportion and progression,

CHAPTER XIV.

On Logarithms, - -

SECT.

I. Theory of logarithms,

exponential equations

II. Application of logarithms to the solution of

SECT.

CHAPTER XV.

On the Resolution of Equations of the third and

higher degrees.

Page

438

442

448

450

452

453

460

I. Theory and transformation of equations,

II. Resolution of cubic equations by the rule of

CARDAN, or of SCIPIO FERREO,

462

470

III. Resolution of biquadratic equations by the

method of DES CARTES,

IV. Resolution of numeral equations by the method

of Divisors,

479

482

V. Resolution of numeral equations, by NEWTON'S

method of approximation,

APPENDIX.

Algebraic method of demonstrating the propo-

sitions in the fifth book of Euclid's Elements,

according to the text and arrangement in Sim-

son's edition,

Errata,

487

513

AN

ELEMENTARY TREATISE

ON

ALGEBRA.

INTRODUCTION.

EXPLANATION OF THE

ALGEBRAIC METHOD OF NOTATION :DEFINITIONS AND AXIOMS.

1. Algebra is a general method of computation, in which abstract quantities and their several relations are made the subject of calculation, by means of alphabetical letters and other signs.

2. The letters of the alphabet may be employed at pleasure for denoting any quantities, as algebraical symbols or abbreviations; but, in general, quantities whose values are known or determined, are expressed by the first letters, a, b, c, &c.; and unknown or undetermined quantities are denoted by the last or final ones, u, v, w, x, &c.

3. Quantities are equal when they are of the same magnitude. The abbreviation a=b implies that the quantity denoted by a is equal to the quantity denoted by b, and is read a equal to b; a>b or a greater than b, that the quantity a is greater than the quantity b; and ab or a less than b, that the quantity a is less than the quantity b.

4. Addition is the joining of magnitudes into one sum. The sign of addition is an erect cross; thus, a+b implies the sum of a and b, and is called a plus b, if a represent 8 and b, 4; then, a+b represents 12, or a+b=8+4=12.