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Call No. ^

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Accession No.

Part

One

PLANE ANALYTIC

GEOMETRY

Chapter 1.

Coordinates on a Straight Line

and

in

a Plane

11

1.

An Axis and Segments of an Axis

11

2.

Coordinates on a Line. The Number

Axis

14

3.

Rectangular

Cartesian Coordinates

in

a

Plane. A Note

on Ob-

lique

Cartesian Coordinates

16

4.

Polar

Coordinates

20

Chapter 2.

Elementary

Problems of Plane

Analytic Geometry

23

5.

Projection of

a Line Segment. Distance Between Two Points . .

23

6.

Calculation of the Area of a Triangle

28

7.

Division of a Line Segment in a Given Ratio

of Axes

30

8.

Transformation of

Cartesian Coordinates

by Translation

35

9.

Transformation of

Rectangular Cartesian

Coordinates by Rotation

of

Axes

36

10.

Transformation of

Rectangular

Cartesian

Coordinates

by

Change

of Origin and Rotation of Axes

38

Chapter 3. The Equation of a Curve

.

41

11.

The Concept of the Equation of

a

Curve. Examples

of

Curves

Represented by Equations

41

12.

Examples of Deriving the Equation of a Given Curve

48

13.

The Problem of the Intersection

of

Two Curves

50

14. Parametric Equations of a Curve

51

15.

Algebraic Curves

- .

52

Chapter 4. Curves of the First Order

55

16. The Slope of a Straight Line

55

17.

The Slope-intercept Equation of a Straight Line

56

18.

Calculation of the Angle Between Two Straight Lines. Conditions

for the Parallelism and Perpendicularity of Two Straight

Lines .

59

19.

The Straight Line As the Curve

of

the First

Order. The

General

Equation of the Straight Line

Degree. The

61

20.

Incomplete Equations of the First

Intercept Equation

63

of

a

Straight Line

21. Discussion of a System of Equations Representing Two Straight

Lines

.65

22.

The Normal Equation of a Straight

Line. The Problem

of Cal-

68

culating the Distance of a Point

from

a

Straight Line

23. The Equation of a Pencil of Lines

71

Chapter 5. Geometric Properties of Curves

of

the

Second Order

75

24.

The Ellipse. Definition of the Ellipse

and Derivation of

Its

Canon-

ical

Equation

75

6

Contents

25.

Discussion of the Shape of the Ellipse

79

26.

The Eccentricity of the Ellipse

81

27.

Rational Expressions for Focal Radii of the Ellipse

82

28.

Point-by-point

Construction of the

Ellipse.

The Parametric

83

Equations

of

the Ellipse

29.

The Ellipse as the Projection of

a

Circle on a Plane. The

Ellipse

84

as the Section of a Circular Cylinder

by a

Plane

30.

The Hyperbola. Definition of the

Hyperbola

and

Derivation

of

87

Its

Canonical

Equation

31. Discussion of the Shape of the Hyperbola

.91

32. The Eccentricity of the Hyperbola

97

33. Rational Expressions for Focal Radii of the Hyperbola

98

34.

The Directrices

of

the

Ellipse and

Hyperbola

99

35. The Parabola. Derivation of the Canonical Equation of the Pa-

rabola

.103

36.

Discussion of the Shape of the Parabola

and

Parabola

105

37.

The Polar Equation of

the

Ellipse,

Hyperbola

.

.

107

38. Diameters of Curves of the Second Order

109

39.

The Optical

Properties

of

the

Ellipse,

Hyperbola

and

Parabola

114

40. The Ellipse, Hyperbola and Parabola as Conic Sections

.... 115

Chapter 6. Transformation of Equations by Change of Coordinates

.... 116

41.

Examples

of

Reducing

the General Equation of a Second-Order

Curve to

Canonical

Form

116

42.

The Hyperbola

as

the

Inverse

Proportionality

Graph.

The

Pa-

rabola as

the Graph of a Quadratic Function

125

Part

Two

SOLID

ANALYTIC GEOMETRY

Chapter 7. Some Elementary Problems of Solid Analytic Geometry

. .

. .

131

43.

Rectangular

Cartesian

Coordinates

in

Space

131

44. The Concept

of

a Free

Vector. The

Projection

of

a

Vector on an

134

Axis

45. The Projections of a Vector on the Coordinate Axes

138

46.

Direction

Cosines

Two

.141

47.

Distance

between

Points.

Division of

a

Line Segment

in

a

Given

Ratio

142

Chapter 8. Linear Operations on Vectors

143

48. Definitions of Linear Operations

143

49. Basic Properties of Linear Operations

144

50.

The Vector Difference

147

51.

Fundamental Theorems

on

Projections

. .

149

52. Resolution of Vectors into Components

152

Chapter 9. The Scalar Product of Vectors

157

53. The Scalar Product and Its Basic Properties

157

54.

Representation

of

the

Scalar Product in Terms of the

Coordi-

nates of the Vector Factors

160

Chapter 10. The Vector and Triple Scalar Products of

Vectors

163

55.

The Vector Product and Its Basic

Properties

163

56.

Representation

of the Vector Product in

Terms of

the Coordinates

of

the Vector

Factors

170