of the nature of whole Numbers or Integers, with respect to

Of the properties of Integers with respect to their Divisors.

Of Fractions in general.

CHAPTER VII.

1

3

7

CHAPTER VIII.

Of the properties of Fractions.

24

Of Square Roots, and of Irrational Numbers resulting from

them.

26

29

33

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CHAPTER XIII.

Of Impossible or Imaginary Quantities, which arise from the

Of Cube Roots, and of Irrational Numbers resulting from them. 44

Of the different Methods of Calculation, and of their mutual

Connexion.

SECTION SECOND.

53

60

58

of the Extraction of Roots applied to Compound Quantities.

74

83

86

CHAPTER VIII.

Of the calculation of Irrational Quantities.

90

CHAPTER IX.

Of Cubes, and of the Extraction of Cube Roots.

94

CHAPTER X.

Of the higher Powers of Compound Quantities.

96

CHAPTER XI.

Of the Transposition of the Letters, on which the demonstra-

tion of the preceding Rule is founded.

CHAPTER XII.

CHAPTER XIII.

Of the expression of Irrational Powers by Infinite Series.

Of the resolution of Negative Powers.

SECTION THIRD.

OF RATIOS AND PROPORTIONS.

CHAPTER I.

Of Arithmetical Ratio, or of the difference between two Num-

102

106

109

Of the greatest Common Divisor of two given numbers.

CHAPTER VII.

of Geometrical Proportions.

CHAPTER VIII.

Observations on the Rules of Proportion and their utility.

CHAPTER IX.

of Compound Relations.

128

132

135

140

OF ALGEBRAIC EQUATIONS, AND OF THE RESOLUTION OF THOSE

EQUATIONS.

CHAPTER I.

Of the Solution of Problems in general.

CHAPTER II.

Of the Resolution of Simple Equations, or Equations of the

first degree.

CHAPTER III.

Of the Solution of Questions relating to the preceding chapter.

CHAPTER IV.

158

161

165

Of the Resolution of two or more Equations of the First Degree. 175

CHAPTER V.

Of the Resolution of Pure Quadratic Equations.

CHAPTER VI.

Of the Resolution of Mixed Equations of the Second Degree.

CHAPTER VII.

Of the Nature of Equations of the Second Degree.

184

190

198

Questions for Practice.

204

New demonstrations of the fundamental propositions of the

5th book of Euclid's Elements.

217

INTRODUCTION

TO THE

ELEMENTS OF ALGEBRA.

SECTION 1.

OF THE DIFFERENT METHODS OF CALCULATION APPLIED TO SIMPLE
QUANTITIES.

CHAPTER I.

of Mathematics in general.

ARTICLE I.

WHATEVER is capable of increase or diminution, is called magnitude or quantity.

A sum of money, for instance, is a quantity, since we may increase it or diminish it. The same may be said with respect to any given weight, and other things of this nature.

:

2. From this definition, it is evident, that there must be so many different kinds of magnitude as to render it difficult even to enumerate them and this is the origin of the different branches of mathematics, each being employed on a particular kind of magnitude. Mathematics, in general, is the science of quan- ! tity; or the science which investigates the means of measuring quantity.

3. Now we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known piece of money (as a dollar, a crown, a ducat, or some other coin,) and shew how many of

1