Chapter III: Introduction: Page

GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE 17

THE OBJECT OF MATHEMATICS 18
Measuring Magnitudes 18
Difficulties 19
General Method 20
Illustrations 21
1. Falling Bodies 21
2. Inaccessible Distances 23
3. Astronomical Facts 24

TRUE DEFINITION OF MATHEMATICS 25
A Science, not an Art 25

ITS TWO FUNDAMENTAL DIVISIONS 26
Their different Objects 27
Their different Natures 29
_Concrete Mathematics_ 31
Geometry and Mechanics 32
_Abstract Mathematics_ 33
The Calculus, or Analysis 33

EXTENT OF ITS FIELD 35
Its Universality 36
Its Limitations 37

BOOK I.

ANALYSIS.

CHAPTER I.

Page

GENERAL VIEW OF MATHEMATICAL ANALYSIS 45

THE TRUE IDEA OF AN EQUATION 46
Division of Functions into Abstract and
Concrete 47
Enumeration of Abstract Functions 50

DIVISIONS OF THE CALCULUS 53
_The Calculus of Values, or Arithmetic_ 57
Its Extent 57
Its true Nature 59
_The Calculus of Functions_ 61
Two Modes of obtaining Equations 61
1. By the Relations between the given
Quantities 61
2. By the Relations between auxiliary
Quantities 64
Corresponding Divisions of the Calculus of
Functions 67

CHAPTER II.

ORDINARY ANALYSIS; OR, ALGEBRA. 69

Its Object 69
Classification of Equations 70

ALGEBRAIC EQUATIONS 71
Their Classification 71

ALGEBRAIC RESOLUTION OF EQUATIONS 72
Its Limits 72
General Solution 72
What we know in Algebra 74

NUMERICAL RESOLUTION OF EQUATIONS 75
Its limited Usefulness 76
Different Divisions of the two Systems 78

THE THEORY OF EQUATIONS 79

THE METHOD OF INDETERMINATE COEFFICIENTS 80

IMAGINARY QUANTITIES 81

NEGATIVE QUANTITIES 81

THE PRINCIPLE OF HOMOGENEITY 84