I\

A FIRST COURSE IN PROJECTIVE GEOMETRY

MACMILLAN AND CO., Limited

LONDON • BOMBAY • CALCUTTA

MELBOURNE

THE MACMILLAN COMPANY

NEW YORK • BOSTON • CHICAGO

DALLAS • SAN FRANCISCO

THE MACMILLAN CO. OF CANADA, Ltd

TORONTO

THE CONIC SECTIONS.

1. Hyperbola. 2. Parabola. 3. Ellipsk.

[In each of the above photographs the horizontal and vertical threads define a

plane through the source of light parallel to the screen on which the light is received.

Tlie latter thread, Ijeinf^ the common section of this j)lane ^%•ith the plane of tha

circle, re^jresents the position of the vanishing line.

A FIRST COURSE

IN

PROJECTIVE GEOMETRY

BY

E. HOWARD SMART, M.A.

HEAD OF THE MATHEMATICAL DEPARTMENT, BIRKBECK COLLEGE, LONDON

MACMILLAN AND CO., LIMITED

ST. MARTIN'S STREET, LONDON

1913

COPYRIGHT

PREFACE

This work is intended for the use of students who have read

the substance of Euclid, Books I. -XL, and who desire some

introduction to the properties of the conic before proceeding

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to the study of the more advanced works on modern pure

geometry.

The subject of Geometrical Conies, through the medium of

which such an introduction is usually acquired, often proves

repulsive to the average student.

In my opinion this is due to two causes : first, the demands

which it makes upon the memory owing to its lack of coher-

ence as commonly treated ; and, secondly, to the very slight

extension of outlook as regards method which it affords.

In the presentation here adopted, which, I venture to think,

is in some respects original, I have endeavoured to overcome

as far as possible these defects, while at the same time giving

a slight sketch of the method of projection, and of the great

principles of homography and duality upon which the further

development of pure geometry so greatly depends.

. No systematic treatment of imaginary elements or of the

theory of involution is attempted, as these subjects are, in my

judgment, unsuitable for a first course.

On the other hand, considerable use has been made of con-

crete illustrations in the form of examples involving practical

drawing and calculation, as my teaching experience has con-

Otttt 1 "^ I

vi PREFACE

vinced me of their value in bringing home to the beginner

the extent and variety of application of the principles involved.

With a view to further stimulating interest, a short historical

note has bet'ii a})pended to most of the chapters.

In conclusion, I desire to express my hearty thanks to the

Senate of the University of London for permission to make

use of examples extracted from the degree examination papers

of that University ; to my friend, Mr. W. H. Salmon, Lecturer

in Mathematics at the Northampton Technical Institute,

Clerkenwell, for his great kindness in reading the work both

in manuscript and proof, and for his helpful criticisms and

suggestions ; and to Mr. H. H. Smart for his valuable assist-

ance with the photographs reproduced in the frontispiece.

E. HOWARD SMART.

Orpington,

May 8th, 1913.

CONTENTS

CHAPTER I

INTRODUCTORY.

SECT. PAGE

1. The Geometric Elements 1

2. Fundamental Axioms - 1

3. 4. Elements at Infinity 2, 4

5. Examples 4

6. Direction in Space 5

7. Notation 5

8. The Prime Geometric Forms 5

Historical Note .. - . - - 6

Examples _ - '. - - - 8

CHAPTEE n.

PROJECTION AND DUALITY.

1. The Fundamental Method of Projection ... 9

2. Vertex. Axis. Plane of Projection. Projectors - - 10

3. Orthogonal and Parallel Projection. Object of Projec-

tive Geometry - - - - 10

4. Projection of Point and Straight Line. Projection of

Triangle 10

5. 6. Projection of Range and Pencil. Projective Figures - 12, 13

viii CONTENTS

6KCT. PAOK

7. Any three Collinear Points are projective with any

other three -.. - -_. 14

8. Geometrical Properties — Metrical and Descriptive - 15

9. Point to Point Correspondence 15

10. Duality of Figures. Complete Quadrangle and Quadri-

lateral .. - -. - - 16

11. Duality in Space. The Fundamental Operations (Pro-

jecting and Cutting) - - - - - - - 19

Historical Note 20

Examples 20

CHAPTER III.

METRICAL RELATIONS. PERSPECTIVE FIGURES.

1. Metrical Relations. Sign of a Segment- - - - 22

2. BC.AD + CA.BD + AB.CD = 23

3. Ceva's Theorem and its Converse 23

4. Menelaus' Theorem and its Converse - - - - 25

5. Desargues' Theorem and its Converse - . - - - 27

6. Perspective Figures. Definition 28

Historical Note 29

Examples 30

CHAPTER IV.

HARMONIC FORMS AND THEIR ELEMENTARY PROPERTIES.

1. Definition of Harmonic Range 32

2. Given A, B, C to find D so that ABCD is a harmonic range 32

3. Harmonic Conjugates 33

4. Properties of the Harmonic Range - - - - 33

(1) AB, AC, AD in h.p. (2) 0B.0D = 0C2

CONTENTS ix

SECT. PAGE

5. Harmonic Pencils — Fundamental Property of Section - 34

Harmonic Ranges and Pencils project into Harmonic

Ranges and Pencils 35

6. Three Equidistant Collinear Points and Point at Infinity

on same line form a Harmonic Range - - - 35

7. If a pair of Conjugate Rays at Right Angles, they are the

Internal and External Bisectors of the Angles between

the other pair 36

8. Harmonic Properties of the Complete Quadrangle and

Complete Quadrilateral 37

9. Construction by use of the Ruler only for the Fourth

Harmonic of three Collinear Points - - - - 39

„ „ of three Concurrent Rays - - - - 39

Historical Note - 40

Examples - - ....-40

CHAPTER V.

INVERSION. SIMILITUDE. COAXAL CIRCLES.

1. Introductory 42

2. Inversion. Inversion with respect to a Circle - - 42

3. Any Circle through a pair of Inverse Points cuts the

given Circle orthogonally 43

4. The Operation of Inversion. Centre and Radius of In-

version - - - -• 43

5. Inverse of a Circle with respect to a Point on it - - 43

6. „ „ „ any Point - - 44

7. If two Curves cut, their Inverses cut at the same Angle 45

8. Similitude. Centres of Similitude of two Circles - - 46

X CONTENTS

SK(T. PAGE

9. If a Variable Circle touches two Fixed Circles, the join

of the Points of Contact passes through one or other

of two Fixed Points 47

10. Similar Curves, nomothetic Figures - - - - 48

11. nomothetic Centre. Direct Similarity - - - - 49

12. Symmetry about a Point 50

13. „ „ Line 50

13 (a). Inverse Similarity 51

14. Coaxal Circles. The Radical Axis of two Circles - - 52

15. The Radical Axes of any three Circles are concurrent - 53

16. Construction of the Radical Axis for Non -intersecting

Circles. Examples - - - - 53

17. Circles of a Coaxal System. Construction of such a

System 55

18. Through any Point one and only one Circle can be drawn

coaxal with given Circles 56

19. Limiting Points. Inverse Points for every Circle of the

System. Examples - - - - 56

20. TP2-TP'2 = 2CC'.TN. Two Important Corollaries - 57

Historical Note 58

Examples -59

CHAPTER VI.

POLES AND POLARS. HARMONIC PROPERTIES OF THE

CIRCLE.

1. Locus of Intersection of Tangents at the ends of Chords

of a Circle which pass through a Fixed Point is a

Straight Line 61

2. Definition of Pole and Polar. Case of Point outside the

Circle. Tangent is the Polar of its Point of Contact.

Construction of Polar. Polar of the Centre - - 62

CONTENTS xi

SECT. PAGE

3. Reciprocal Property of Pole and Polar - - - - 63

4. Salmon's Theorem 63

5. Definition of Conjugate Points and Lines.

An infinite number of pairs of Conjugate

/Points on a given Straight Line \

(.Straight Lines through a given Point/

can be found 64

6. Polars of a pair of Conjugate Points are a pair of Conju-

gate Lines 65

7. Self-Conjugate or Self-Polar Triangles. Construction of

such. Centre of Circle is the Orthocentre of such

Triangles 65

8. Polars of a Harmonic Range form a Harmonic Pencil - 66

9. The Fundamental Harmonic Property of Pole and Polar

for the Circle 67

10. Any pair of Conjugate Lines, together with the Tangents

from their Point of Intersection, form a Harmonic

Pencil, and conversely - - - 68

IL If a Quadrangle be inscribed in a Circle, its Diagonal

Points form a Self-Conjugate Triangle - - - 69

12. If a Quadrilateral be circumscribed to a Circle, its

Diagonal Triangle is Self -Con jugate with respect to

the Circle - 70

13. The Angular Points of the Diagonal Triangle of the

Circumscribed Quadrilateral coincide with the Diagonal

Points of the Quadrangle formed by the Points of

Contact - - - - 71

14. Principle of Duality. Reciprocation - - - - 72

15. Same continued. Summary 74

Examples - 75

xii CONTENTS

CHAPTER VII.

PROJECTION {continued).

SECT. PAOK

1. Introductory 77

2. The Vanishing Lines 77

3. The Projection of an Angle. To project Concurrent

Lines into Parallels - 78

4. To project two Angles into given A.ngles and a given

Straight Line to Infinity 79

5. To project a Triangle into a given Triangle and a given

Straight Line into Infinity . . . . _ 80

6. To project a Quadrilateral into a Square - - - 81

7. Proof by Projection of the Harmonic Property of the

Conaplete Quadrilateral 82

8. The Drawing of Actual Forms of Projected Figures.

Rabatment .-.-. - - 82

9. Projection of a Point and a Straight Line. Examples - 84

10. Actual Projection of a Quadrilateral into a Square - 85

11. Parallel and Orthogonal Projection - - - - 87

12. Examples on Orthogonal Projection - - - - 89

13. Orthogonal Projection of an Area 91

Examples - 93

CHAPTER VIII.

THE CONIC.

1. Definition of Order and Class of Curve - - - - 95

2. Circle is Curve of Second Order and Second Class - - 95

3. Preservation of Tangency, Order and Class after Pro-

jection - - - 96

4. Definition of a Conic 96

CONTENTS xiii

SECT. PAOE

5. Species of Conies Eealisation by Model - - - - 97

6. General Features of the Conies 98

1. Hyperbola.

2. Parabola.

3. Ellipse.

7. General Properties 99

I. Line euts in two Points. Two Tangents ean be drawn.

II. Pole and Polar Property for the Conic.

III. Reciprocal Property of Pole and Polar.

IV. Conjugate Points and Lines.

V. Self -Con jugate Triangles.

VI. Fundamental Harmonic Property of Pole and Polar.

VII. Polars of Harmonic Range form a Harmonic Pencil.

VIII. Construction of Polar of P by use of Ruler only.

IX. Duality and Reciprocation for Conic. Base Conic.

8. Symmetry about (1) a Point, (2) a Line, recapitulated. 104

9. Pole of Line at Infinity is Centre. Centre of Parabola.

Appropriateness of term Centre. All Chords of

Conic through the Centre bisected there - - - 105

10. Definition of Diameters. Locus of Middle Points of

System of Parallel Chords is a Diameter - - - 105

Cor. 1. Tangents at ends of Diameter are Parallel.

CoR. 2. Tangents at ends of double Ordinate intersect

on Diameter.

Cor. 3, CV.CT = CP2.

11. Diameters of a Parabola. TP=PV . . . . 107

12. Def. Conjugate Diameters. Each bisects all Chords

parallel to the other 108

CoR. Chord and the Diameter through its Middle Point

give the Directions of a pair of Conjugate Diameters.

13. An infinite number of pairs of Conjugate Diameters.

Only one pair at Right Angles. The Axes of a Conic 109

xiv CONTENTS

SKCT. PAGE

14. Symmetry of Conic. Lengths of Axes. Vertices - - 112

Tangents at Vertices.

Polar of Point on Axis perpendicular to Axis,

One pair of Conjugate Lines at Point on an Axis are

Orthogonal.

15. Asymptotes defined 113

16. Some Properties of Asymptotes 113

I. Form with any pair of Conjugate Diameters a Har-

monic Pencil.

Cor. Axes bisect the Angles between the Asymptotes.-

II, Intercepts on a Secant between a Hyperbola and its

Asymptotes are equal.

Cor. Tangent bisected at its Point of Contact.

17. Given Asymptotes and one Point to find any number of

Points 114

18. Particular and Degenerate Cases of the Conic - - 115

19. 20. Drawing Exercises 115, 118

Examples 120

CHAPTER IX.

CARNOT'S THEOREM.

1. Carnot's Theorem 123

2. Newton's Theorem. Cor. - 125

3. Application to the Central Conic. QV- : PV .VP'== const. 126

Pseudo-Conjugate Diameter for the Hyperbola.

QV2/PV = const, for the Parabola.

4. Forms of latter results for Principal Axes - - - 129

Cartesian Equation of Central Conic and Parabola,

Equation of a Conic referred to a pair of Conjugate

Diameters as Axes.

CONTENTS

XV

SECT. PAOE

5. Kesumption from Chapter VIII. of Asymptotic Pro-

perties 131

III. LP = CD.

IV. Asymptotes are the DiagODals of the Parallelogram

whose Median Lines are a pair of Conjugate

Diameters.

Def. Conjugate Hyperbola.

V. Area of Triangle cut off from the Asymptotes by

any Tangent is constant.

Another Proof.

6. Properties of a Diameter and its Pseudo-Conjugate - 135

1. Area of Parallelogram in (IV.) above is constant.

-CD.PF = «6.

2. CP2 - CD2 = const. =a? - lP- for the Hyperbola.

7. Ellipse is obtainable as the Orthogonal Projection of a

Circle 136

8. CPH CD2 = const. = a2 + 62 for the Ellipse - - - 138

9. Area of Parallelogram circumscribing an Ellipse at the

ends of Conjugate Diameters is constant - - - 139

10. Def. Auxiliary Circle. Corresponding Figure for the

Parabola 139

Case of the Hyperbola and Rectangular Hyperbola.

Historical Note 140

Examples 142

CHAPTER X.

THE FOCI.

1. Conjugate Lines at Right Angles - - - - - 144

Lemma. If PA, PA' meet the Polar of S (a Point on the

Axis inside the Curve) at K, K', SK, SK' are Conjugate

Lines.

xvi CONTENTS

SECT. PAGE

Prop. There is only one position of S through which

more than one pair of Perpendicular Conjugate Lines

can be drawn, and every pair of Conjugate Lines

through this Point is Orthogonal.

2. Def. Focus and Directrix 146

Prop. In any Central Conic there are two Real Foci on

the Major or Transverse Axis, but none on tlie Con-

jugate Axis. Also the join of two Foci riiust be an

Axis.

CS2 = AC2-BC2 (Ellipse).

CS- = AC- + BC2 (Hyperbola).

3. Latus Rectum. SR.AC=^BC2 - - - 148

4. Case of Parabola. One Finite Focus - - - - 148

5. The Fundamental Focus and Directrix Property - - 150

6. Eccentricity in terms of the Semi- Axes. 6^ = 1^1)^10^

(Ellipse, Hyperbola), e = l (Parabola)- ... 151

7. The Bifocal Property. Examples 152

8. Mechanical Construction of the Conic - - - - 154

9. Semi-latus Rectum is a Harmonic Mean between the Seg-

ments of any Focal Chord 155

10. The Foci, Directrices and Eccentricity in the Particular

and Degenerate Cases of the Conic - - - - 156

11. Foci of Sections of a Right Circular Cone - - - 156

Historical Note 158

Examples - 159

CHAPTER XL

FOCAL, TANGENT AND NORMAL PROPERTIES.

1. Tangents subtend Equal or Supplementary Angles at a

Focus 161

CONTENTS xvii

SECT. PAGE

Portion of the Tangent between the Point of Contact

and the Directrix subtends a Right Angle at the

Focus.

2. Tangent is equally inclined to the Focal Distances of its

Point of Contact 162

Cor. (1) SG = SP for the Parabola.

(2) Particular Case of this.

(3) Confocal Ellipses and Hyperbolas cut at Right

Angles.

3. (1) Auxiliary Circle as Pedal of the Focus - - - 164

(2) SY.SY' = BC2.

(3) PE = PE' = AC.

Cor. (1) Particular Case for the Parabola.

(2) One Conic only, given Foci and any Line as

Tangent.

(3) Envelope of Perpendiculars to Radial Lines at

their Intersections with a Fixed Circle. Examples.

4. Relative Positions of Asymptotes, Directrices, Foci, and

Auxiliary Circle in the Hyperbola - - - - 166

5. Tangents from an External Point are equally inclined

to the Focal Distances 167

Notes and Example.

6. {a)CG = e'^CH,(b)PG = byp 169

CoRl. P^ = a2/p. Cor. 2. PG/CD = 6/a.

7. Given a pair of Conjugate Diameters to construct the

Axes 170

8. Parabola Properties. (1) NG = 2a, (2) AT = AN, (3) SUP,

SUP' similar Triangles, UP, UP' being Tangents, (4)

Circumcircle of Triangle of Tangents passes through

the Focus - - - 171

9. Director Circle. Case of Parabola 173

10. Gaskin's Theorem 175

11. Steiner's Theorem 176

b

xviii CONTENTS

SECT, PAGE

12. Curvature, Comparison of two Curves. Curvature of

Circle 177

13. Curvatureof any Curve. Circle and Radius of Curvature 179

14. Chord of Curvature of a Curve in any given direction - 180

15. Chord of Curvature through the Centre, and Radius of

Curvature at any Point of a Central Conic - - 181

16. Chord of Curvature through the Focus of a Parabola - 182

Historical Note 184

Examples 184

CHAPTER XII.

CROSS-RATIOS.

1. Definition of Cross-Ratio. Notation - - - - 187

2. Prop, The Cross-Ratio of a Range of four Points is

equal to that of the Points in any order obtained by

the Simultaneous Interchange of pairs of Letters - 188

3. Prop, The 24 Cross-Ratios of four Points reduce to six 188

4. Cross-Ratio unaltered by Projection - - - - 189

5. Definition of Cross-Ratio of Pencil, Notation - - 191

Cross-Ratio of Pencil can be expressed in terms of the

Mutual Inclinations of the Rays only.

6. Discussion of Sign Conventions in the Expression for the

Cross-Ratio of a Pencil 191

7. Dualistic Representation of Properties of a Range or

Pencil 192

8. Expression of a Cross-Ratio as the Ratio of two Lengths 193

9. If three of four Elements of a Range or Pencil be given,

it is possible to place the fourth in one way only, so

that the Cross-Ratio of the four Elements is given - 194

10, Construction of a Range or Pencil having a given Cross-

Ratio 194

CONTENTS xix

SECT.

11. The six Cross-Eatios of four Points can all be expressed

in terms of the Trigonometrical Functions of an Angle 195

12. Cross-Eatio of a Pencil unaltered by Projection - - 196

Historical Note 196

Examples 197

CHAPTER Xni.

HOMOGRAPHIC RANGES AND PENCILS.

1. Definition of nomographic Ranges and Pencils - - 200

2. If two nomographic Forms at Different Bases have a

Common Element their remaining Elements are

Incident - - - - 200

3. Ranges and Pencils in Perspective. Definition - - 202

4. Construction of Homographic Ranges on two Intersect-

ing Lines. Construction of Homographic Pencils at

two Vertices - - - - 203

5. Notation. Cross -Axis and Cross-Centre - - - 205

6. Projective Ranges and Pencils are Homographic and

vice versa - - - - - 207

7. Vanishing Points of two Homographic Ranges - - 207

8. Homographic Forms at the same Base - - - - 209

9. Similar Homographic Forms 209

Historical Note 209

Examples 210

CHAPTER XIV.

PROJECTIVE PROPERTIES OF THE CONIC.

1. Cross-Ratio of four Collinear Points equal to that of

their four Polars 212

CoR. Diameters and Conjugates.

XX CONTENTS

SECT. PAGE

2. (a) Four Fixed Points subtend at any fifth Point a

Pencil of Constant Cross-Ratio - - - 213

(6) Four Fixed Tangents determine on any fifth Tan-

gent a Range of Constant Cross -Ratio ■ • - 213

3. Deductions from these 214

(1) Harmonic Property of Pole and Polar.

(2) Constant Area cut off' from Asymptotes by any

Tangent.

(3) Property of Tangents to a Parabola.

4. Pappus' Theorem and Chasles' Theorem - - - - 218

5. (a) Locus of Intersections of Corresponding Rays of two

nomographic Pencils at Different Vertices, but not

in Perspective, is a Conic through their Vertices - 219

(6) Envelope of joins of Corresponding Points of two

nomographic Ranges on Diff'erent Lines, and not in

Perspective, is a Conic touching the Lines - - - 219

CoRs. 1 and 2. Line-Pair and Point-Pair.

6. {a) Through any five Points can be drawn one and only

one Conic - - - 222

(6) Touching any five Lines can be drawn one and only

one Conic 222

CoRS. 1 and 2. Note.

7. (rt) Locus of Point P which moves so that P'ABCDf is

constant is a Conic, A, B, 0, D being Fixed Points - 223

(6) Envelope of Line p which moves so tha^t p{abcd} is

constant is a Conic, «, 6, c, d being Fixed Lines - 223

Example: Locus of Centres of Conies circumscribing a

given Quadrangle.

8. The Projection of a Conic is a Conic - . - . 225

Historical Note - - - 225

Examples - - 226

CONTENTS xxi

CHAPTER XV.

PASCAL'S AND BRIANCHON'S THEOREMS.

SECT. PAGE

1. Pascal's and Brianchon's Theorems. First Proof - - 228

2. „ „ „ Second Proof - 230

3. The Converses of these Theorems - - - - - 231

4. Deductions from these Theorems - - - 231

5. Given five Points (Tangents) of a Conic to find Tangent

at (Point of Contact of) any one of them - - - 232

6. Tangents at Opposite Points of Inscribed Quadrangle

meet on Third Diagonal (and Dual) - - - - 233

7. Note on Opposite Elements - - - 233

8. I. If the Vertices of two Tiiangles are on a Conic their

six sides touch a Conic, and conversely - - - 234

II. If two Triangles are Self -Con jugate for a Conic, the

six Vertices lie on a Conic and the six Sides touch a

Conic . - -. - .- 235

Historical Note 235

Examples 236

CHAPTER XVI.

SELF-CORRESPONDING ELEMENTS.

1. nomographic Ranges on a Conic - - - - 237

2. Self-Corresponding Points 238

3. nomographic Sets of Tangents and Self-Corresponding

Lines - - - - - 238

4. Determination of the Self-Corresponding Points of two

nomographic Ranges on the same Straight Line.

Problems of the First and Second Order and their

Algebraic Equivalent - - - 239

xxii CONTENTS

SKCT. PAGE

5. Honiograpliic Pencils at the same or different Vertices.

Determination of Common or Parallel Rays - - 241

6. Application of these results to the Conic - - - 241

7. Envelope of joins of Corresponding Points of two Homo-

graphic Eanges on a Conic is a Conic having Double

Contact with the given Conic 243

Examples 245

^245^

CHAPTER XVn.

CONSTRUCTION OF A CONIC FROM GIVEN CONDITIONS;— \

1. Five Conditions necessary

2. Given five Points (Tangents) to construct the Conic by

Points (Tangents). Maclaurin's Construction - - 247

3. Use of Pascal's and Brianchon's Theorems to solve the

same Problems - - - - 249

4. Examples. Construct a Conic - - - - - 250

1. (a) Given four Points and a Tangent at one of them.

1. (6) Given four Tangents and a Point on one of them.

2. (a) Given three Points and Tangents at two of them.

2. (6) Given three Tangents and Points of Contact of

two of them.

5. Directions of Asymptote, Axis of Parabola, etc. - - 251

6. Given Directions of both Asymptotes and three Points

on the Curve. Examples - - - - - -251

7. Parabola given three Tangents and Direction of Axis - 253

8. Applications involving the Determination of Common

Elements of two nomographic Forms - - - 255

(a) Points in which a given Line meets a Five-Point

Conic.

(b) Tangents from a given Point to a Five-Tangent

Conic.

CONTENTS xxiii

SECT. . I"A(iK

9. Centre of Five-Point Conic or Five-Tangent Conic.

Asymptotes of latter 255

10. Parabola through four Points 256

Historical Note 257

Examples - 258

CHAPTER XVni.

RECIPROCATION.

1. Recapitulation and Examples 260

2. Identity of Reciprocal of Conic C, whether considered as

(1) Envelope of Polars of Points on C with respect to

base X, or

(2) Locus of Poles of Tangents to C with respect to

baseX 261

3. Reciprocal of a Conic, for a Conic C is a Conic C - - 262

4. Pole and Polar for C reciprocate into Polar and Pole

forC - - - - 262

5. Reciprocal of Circle with respect to Circle, Centre S, is

an Ellipse, Parabola or Hyperbola, according as S is

inside, on, or outside the given Circle, and conversely 263

6. Reciprocals of the Asymptotes. Centre of the Reciprocal

Conic. Examples - - - - 265

7. Reciprocal of the Angle between two Lines - - - 265

8. Two Examples of Reciprocation 266

9. Coaxal Circles reciprocate into Confocal Conies. Examples. 267

10. Species of Reciprocal Conic determined by Position of

Centre of Base Conic - - - - 268

11. Fregier's Theorem 268

12. Examples of Reciprocation - 269

Examples - - - - - - - - - 271

CHAPTER I.

INTRODUCTOEY.

§1. The Geometric Elements.

Points, straight lines, and planes are called the geometric

elements.

§ 2. The following statements concerning their mutual

relations are assumed as fundamental :

(1) Through two points can be drawn one and only one

straight line.

(2) Two coplanar straight lines intersect in one point only,

or are parallel.

Two non-coplanar straight lines do not intersect.

(3) Three non-collinear points,

onr a point and a straight line not passing through it,

or two intersecting straight lines, determine a plane.

(4) A straight line either intersects a plane in one point

only or is parallel to it.

(5) Two planes either intersect one another in one straight

line only, or are parallel.

(6) Through a given point only one straight line can be

drawn parallel to a given straight line.

By the term 'parallel' in the above we mean that the

elements concerned do not meet, however far produced.