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* OF THE

UNIVERSITY OF CALIFORNIA.

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A TEXT-BOOK

OF

EUCLID'S ELEMENTS.

A TEXT -BOOK

OF

EUCLID'S ELEMENTS

FOR THE USE OF SCHOOLS

BOOKS Lâ€” VI. AND XL

BY

IH. S. HALL, M.A.

FORMKRLY SCHOLAR OF CHRISt's COLLEGE, CAMBRIDGE ;

AND

F. H. STEVENS, M.A.

FORMERLY SCHOLAR OP QUEEN's COLLEGE, OXFORD ;

MASTERS OF THE MILITARY AND ENGINEERING SIDE, CLIFTON COLLEGE,

SECOND EDITION REVISED AND ENLARGED.

Hontron: frrWT VT?t? ^T^

MACMILLAN AND COT fj * ^ ^^^ \

AND NEW YOKE. ^^^tlf^"^!^

iÂ«9i ^^

\A It" Rights reserved.]

Richard Clay and Sons, Limited,

london and bungay.

FirÂ»t Edition 18S8.

Second Edition (Book XI. added) 18S9.

Reprinted 1890. 1891.

PEEFACE TO THE FIEST EDITION.

This volume contains the first Six Books of Euclid's

Elements, together with Appendices giving the most im-

portant elementary developments of Euclidean Geometry.

The text has been carefully revised, and special atten-

tion given to those points which experience has shewn to

present difficulties to beginners.

In the course of this revision the Enunciations have

been altered as little as possible; and, except in Book V.,

very few departures have been made from Euclid's proofs:

in each case changes have been adopted only where the old

text has been generally found a cause of difficulty; and

such changes are for the most part in favour of well-recog-

nised alternatives.

For example, the ambiguity has been removed from the

Enunciations of Propositions 18 and 19 of Book I.: the

fact that Propositions 8 and 26 establish the complete

identical equality of the two triangles considered has been

strongly urged; and thus the redundant step has been

removed from Proposition 34. In Book II. Simson's ar-

rangement of Proposition 13 has been abandoned for a

well-known alternative proof. In Book III. Proposition

25 is not given at length, and its place is taken by a

/! /^

VI PREFACE.

simple equivalent. Propositions 35 and 36 have been

treated generally, and it has not been tliought necessary

to do more than call attention in a note to the special

cases. Finally, in Book VI. we have adopted an alterna-

tive proof of Proposition 7, a theorem wliich has been too

much neglected, owing to the cumbrous form in whicli it

has been usually given.

These are the chief deviations from the ordinary text

as regards method and arrangement of proof: they are

points familiar as difficulties to most teachers, and to name

them indicates sufficiently, without further enumeration,

the general principles which have guided our revision.

A few alternative proofs of difficult propositions are

given for the convenience of those teachers who care to

use them.

With regard to Book V. we have established the princi-

pal propositions, both from the algebraical and geometrical

definitions of ratio and proportion, and we have endeavoured

to bring out clearly the distinction between these two modes

of treatment.

In compiling the geometrical section of Book V. we

have followed the system first advocated by the late Pro-

fessor De Morgan ; and here we derived very material

assistance from the exposition of the subject given in the

text-book of the Association for the Improvement of Geo-

metrical Teaching. To this source we are indebted for the

improved and more precise wording of definitions (as given

on pages 286, 288 to 291), as well as for the order and

substance of most of the propositions which appear between

pages 297 and 306. But as we have not (except in the

points above mentioned) adhered verbally to the text of

the Association, we are anxious, while expressing in the

fullest manner our obligation to their work, to exempt the

PREFACE. Vll

Association from all responsibility for our treatment of the

subject.

One purpose of the book is to gradually familiarise the

student with the use of legitimate symbols and abbrevia-

tions; for a geometrical argument may thus be thrown into

a form which is not only more readily seized by an advanced

reader, but is useful as a guide to the way in which Euclid's

propositions may be handled in written work. On the

other hand, we think it very desirable to defer the intro-

duction of symbols until the beginner has learnt that they

can only be properly used in Pure Geometry as abbrevia-

tions for verbal argument: and we hope thus to prevent

the slovenly and inaccurate habits which are very apt to

arise from their employment before this principle is fully

recognised.

Accordingly in Book I. we have used no contractions

or symbols of any kind, though we have introduced verbal

alterations into the text wherever it appeared that con-

ciseness or clearness would be gained.

In Book II. abbreviated forms of constantly recurring

words are used, and the phrases therefore and is 'equal to

are replaced by the usual symbols.

In the Third and following Books, and in additional

matter throughout the whole, we have employed all such

signs and abbreviations as we believe to add to the clear-

ness of the reasoning, care being taken that the symbols

chosen are compatible with a rigorous geometrical method,

and are recognised by the majority of teachers.

It must be understood that our use of symbols, and the

removal of unnecessary verbiage and repetition, by no

means implies a desire to secure brevity at all hazards.

On the contrary, nothing appears to us more mischievous

than an abridgement which is attained by omitting

Vlli PREFACE.

steps, or condensing two or more steps into one. Such

uses spring from the pressure of examinations; but an

examination is not, or ought not to be, a mere race; and

while we wish to indicate generally in the later books how

a geometrical argument may be abbreviated for the pur-

poses of written work, we have not thought well to reduce

the propositions to the bare skeleton so often presented to

an Examiner. Indeed it does not follow that the form most

suitable for the page of a text-book is also best adapted

to examination purposes; for the object to be attained

in each case is entirely different. The text-book should

present the argument in the clearest possible manner to the

mind of a reader to whom it is new : the written proposition

need only convey to the Examiner the assurance that the

proposition has been thoroughly grasped and remembered

by the pupil.

From first to last we have kept in mind the undoubted

fact that a very small proportion of those who study Ele-

mentary Geometry, and study it with profit, are destined

to become mathematicians in any real sense; and that to

a large majority of students, Euclid is intended to serve

not so much as a first lesson in mathematical reasoning,

as the first, and sometimes the only, model of formal and

rigid argument presented in an elementary education.

This consideration has determined not only the full

treatment of the earlier Books, but the retention of the

formal, if somewhat cumbrous, methods of Euclid in many

places where proofs of greater brevity and mathematical

elegance are available.

We hope that the additional matter introduced into

the book will provide sufficient exercise for pupils whose

study of Euclid is preliminary to a mathematical edu-

cation.

PREFACE. IX

The questions distributed tlirough the text follow very

easily from the propositions to which they are attached,

and we think that teachers are likely to find in them

all that is needed for an average pupil reading the subject

for the first time.

The Theorems and Examples at the end of each Book

contain questions of a slightly more difficult type : they

have been very carefully classified and arranged, and brought

into close connection with typical examples worked out

either partially or in full ; and it is hoped that this section

of the book, on which much thought has been expended,

will do something towards removing that extreme want of

freedom in solving deductions that is so commonly found

even among students who have a good knowledge of the

text of Euclid.

In the course of our work we have made ourselves

acquainted with most modern English books on Euclidean

Geometry : among these we have already expressed our

special indebtedness to the text-book recently published by

the Association for the Improvement of Geometrical Teach-

ing; and we must also mention the Edition of Euclid's Ele-

ments prepared by Mr J. S. Mackay, whose historical notes

and frequent references to original authorities have been of

the utmost service to us.

Our treatment of Maxima and Minima on page 239 is

based upon suggestions deriyed from a discussion of the

subject which took place at the annual meeting of the

Geometrical Association in January 1887.

Of the Riders and Deductions some are original; but

the greater part have been drawn from that large store of

floating material which has furnished Examination Papers

for the last 30 years, and must necessarily form the basis

of any elementary collection. Proofs which have been

"X > PREFACE.

found in two or more books without acknowledgement

liave been regarded as common property.

As regards figures, in accordance with a usage not

uncommon in recent editions of Euclid, we have made a

distinction between given lines and lines of construction.

Throughout tlie book we have italicised those deductions

on which we desired to lay special stress as being in them-

selves important geometrical results : this arrangement we

think will be useful to teachers who have little time to

devote to riders, or who wish to sketch out a suitable course

for revision.

We have in conclusion to tender our thanks to many of

our friends for the valuable criticism and advice which we

received from them as the book was passing through the

press, and especially to the Rev. H. C. Watson, of Clifton

College, who added to these services much kind assistance

in the revision of proof-sheets.

H. S. HALL,

F. H. STEVENS.

July, 1888.

PREFACE TO THE SECOND EDITION.

In the Second Edition the text of Books I â€” VI. has

been revised ; and at the request of many teachers we have

added the first twenty-one Propositions of Book XI. together

with a collection of Theorems and Examples illustrating the

elements of Solid Geometry.

September, 1889.

CONTENTS.

BOOK I.

PAGE

Definitions, Postulates, Axioms 1

Section I. Propositions 1 â€” 26 11

Section II. Parallels and Parallelograms.

Propositions 27 â€” 34 50

Section III. The Areas of Parallelograms and Triangles.

Propositions 35â€”48 66

Theorems and Examples on Book I.

Analysis, Synthesis 87

I. On the Identical Equality of Triangles , . . 90

II. On Inequalities 93

III. On Parallels . .95

IV. On Parallelograms 96

V. Miscellaneous Theorems and Examples . . . 100

VI. On the Concurrence of Straight Lines m a Tri-

angle 102

VII. On the Construction of Triangles with given

Parts 107

VIII. On Areas 109

IX. On Loci 114

X. On the Intersection of Loci 117

XI 1 CONTENTS.

BOOK II.

I'AGE

DEriNITIONS, &0 ... 120

Propositions 1 â€” 14 122

Theorems and Examples on Book II 1-14

BOOK m.

Definitions, (fee 149

Propositions 1 â€” 37 153

Note on the Method of Limits as Applied to Tangency . 213

Theorems and Examples on Book III.

I. On the Centre and Chords of a Circle . . 216

n. On the Tangent and the Contact of Circles.

The Common Tangent to Two Circles, Problems on

Tangency, Orthogonal Circles 217

IIL On Angles in Segments, and Angles at the Centres

AND Circumferences op Circles.

The Orthocentre of a Triangle, and properties of the

Pedal Triangle, Loci, Simson's Line . . , 222

IV. On the Circle in Connection with Kectangles.

Further Problems on Tangency 233

V. On Maxima and Minima 239

VI. Harder Miscellaneous Examples 246

BOOK IV.

Definitions, &c. 250

Propositions 1 â€” 16 251

Note on Regular Polygons 274

Theorems and Examples on Book IV

I. On the Triangle and its Circles.

Circumscribed, Inscribed, and Escribed Circles, The

Nine-points Circle 277

II. Miscellaneous Examples 283

CONTENTS. Xlll

BOOK V.

PAGE

Inteoductory 285

Definitions 286

Summary, with Algkbbaical Proofs, op the Principal

Theorems of Book V 292

Proofs of the Propositions derived from the Geometrical

Definition of Proportion 297

BOOK VI.

Definitions 307

Propositions 1 â€” D 308

Theorems and Exaviples on Book VI.

I, On Harmonic Section 359

II. On Centres of Similarity and Similitude . . . 363

III. On Pole and Polar 365

IV. On the Eadical Axis of Two or More Circles . . 371

V. On Transversals 374

VI. Miscellaneous Examples on Book VI 377

BOOK XI.

Definitions 383

Propositions 1â€”21 393

Exercises on Book XI , . . . 418

Theorems and Examples on Book XI. . . . . . 420

- OP THE

IVEB.SITY]

, . OB'

EUCLID'S ELEMENTS.

BOOK I.

Definitions.

1. A point is that which has position, but no mag-

nitude,

2. A line is that which has length without breadth.

The extremities of a line are points, and the intersection of two

lines is a point.

3. A straight line is that which lies evenly between

its extreme points.

Any portion cut off from a straight lineis called a segment of it.

4. A surface is that which has length and breadth,

but no thickness.

The boundaries of a surface are lines.

5. A plane surface is one in which any two points

being taken, the straight line between them lies wholly in

that surface.

A plane surface is frequently referred to simply as a plane.

Note, Euclid regards a point merely as a viark of position, and

he therefore attaches to it no idea of size and shape.

Similarly he considers that the properties of a line arise only from

its length and position, without reference to that minute breadth which

every line must really have if actually draion, even though the most

perfect instruments are used.

The definition of a surface is to be understood in a similar way.

II. ]â€¢:. 1.

2 EUCLID'S KLEMENTS.

6. A plane angle ia the inclination of two straiglit

lines to one another, which meet together, but are not in

the same straight line.

The point at which the straight lines meet is called the vertex of

the angle, and the straight lines themselves the arms of the angle.

When several angles are at one point O, any one

of them is expressed by three letters, of which the

letter that refers to the vertex is put between the

other two. Thus if the straight lines OA, OB, OC

meet at the point O, the angle contained by the

straight lines OA, OB is named the angle AOB or

BOA ; and the angle contained by OA, OC is named

the angle AOC or COA. Similarly the angle con-

tained by OB, 00 is referred to as the angle BOG

or COB. But if there be only one angle at a point,

it may be expressed by a single letter, as the angle

atO.

Of the two straight lines OB, OC shewn in the

adjoining figure, we recognize that OC is rnore in-

clined than OB to the straight line OA : this we

express by saying that the angle AOC is greater

than the angle AOB. Thus an angle must be

regarded as having magnitude. q

It should be observed that the angle AOC is the sum of the

angles AOB and BOC ; and that AOB is the difference of the angles

AOC and BOC.

The beginner is cautioned against supposing that the size of an

angle is altered either by increasing or diminishing the length of its

arms.

[Another view of an angle is recognized in many branches of

mathematics ; and though not employed by Euclid, it is here given

because it furnishes more clearly than any other a conception of what

is meant by the magnitxidc of an angle.

Suppose that the straight line OP in the figure

is capable of revolution about the point O, like the

hand of a watch, but in the opposite direction ; and

suppose that in this way it has passed successively

from the position OA to the positions occupied by

OB and OC.

Such a line must have undergone more turning

in passing from OA to OC, than in passing from OA to OB; and

consequently the angle AOC is said to be greater than the angle AOB.]

DEFINITIONS.

7. When a straight line standing on

another straight line makes the adjacent

angles equal to one another, each of the an-

gles is called a right angle ; and the straight

line which stands on the other is called a

perpendicular to it.

8. An obtuse angle is an angle whicli

is greater than one right angle, but less

than two right angles.

9. An acute angle ,is an angle which is

less than a right angle.

B

[In the adjoining figure the straight line

OB may be supposed to have arrived at

its present position, from the position occu-

pied by OA, by revolution about the point O

m either of the two directions indicated by

the arrows : thus two straight lines drawn

from a point may be considered as forming

tico angles, (marked (i) and (ii) in the figure)

of which the greater (ii) is said to be reflex.

If the arms OA, OB are in the same

straight line, the angle formed by them

on either side is called a straight angle.]

10. Any portion of a plane surface bounded by one

or more lines, straight or curved, is called a plane figure.

The sum of the bounding lines is called the perimeter of the figure.

Two figures are said to be equal in area, when they enclose equal

portions of a plane surface. <,..â€¢

11. A circle is a plane figure contained

by one line, which is called the circum-

ference, and is such that all straight lines

drawn from a certain point within the

figure to the circumference are equal to one

another : this point is called the centre of

the circle.

A radius of a circle is a straight line drawn from the

centre to the circumference.

1â€”2

4 Euclid's elements.

12. A diameter of a circle is a straiglit line drawn

ilirougli the centre, and terminated both ways by the

circumference.

13. A semicircle is tlie figure bounded by a diameter

of a circle and the jmrt of the circumference cut off by the

diameter.

14. A segment of a circle is the figure bounded by

a straight line and the part of the circumference which it

cuts oft".

15. Rectilineal figures are those which are bounded

by straight lines.

16. A triangle is a plane figure ])ounded by three

straight lines.

Any one of the angular points of a triangle may be regarded as its

vertex ; and the opposite side is then called the base.

17. A quadrilateral is a plane figure bounded by

four straight lines.

The straight line which joins opposite angular points in a qiiadri-

lateral is called a diagonal.

18. A polygon is a plane figure bounded by more

than four straight lines.

19. An equilateral triangle is a triangle

whose three sides are equal,

L

20. An isosceles triangle is a triangle two

of whose sides are equal.

21. A scalene triangle is a triangle which

has three unequal sides.

DEFINITIONS.

22. A right-angled triangle is a triangle

which has a right angle.

The side opposite to the right angle in a right-angled triangle is

called the hypotenuse.

23. An obtuse-angled triangle is a

triangle which has an obtuse anijle.

24. An acute-angled triangle is a triangh

which lias tJwee acute an<j:les.

[It will be seen hereafter (Book I. ^Proposition 17) that every

triangle must have at least ttco acute angles.]

25. Parallel straight lines are such as, being in the

same plane, do not meet, however far they are produced in

either direction.

26. A Parallelogram is a four-sided

figure wiiich lias its opposite sides pa-

rallel.

28. A square is a four-sided figure which

has all its sides equal and all its angles right

angles.

[It may easily be shewn that if a quadrilateral

has all its sides equal and one angle a right angle,

then all its angles will be right angles.]

29. A rhombus is a four-sided figure

which has all its sides equal, but its

angles are not right angles.

30. A trapezium is a four-sided figure

which has two of its sides parallel.

27. A rectangle is a parallelogram which r

has one of its angles a right angle. |

EUCLID'S ELEMENTS.

ON THE POSTULATES.

In order to effect the constructions necessary to the study of

geometry, it must be supposed that certain instruments are

available; but it has always been held that such instruments

should be as few in number, and as simple in character as

possible.

For the purposes of the first Six Books a straight ruler and

a pair of comi^asses are all that are needed ; and in the follow-

ing Postulates, or requests, Euclid demands the use of such

instruments, and assumes that they sufl&ce, theoretically as well

lis practically, to carry out the processes mentioned below.

Postulates.

Let it be granted,

1. That a straight line may be drawn from any one

point to any other point.

When we draw a straight line from the point A to the point B, we

are said io join AB.

2. That a JinitCj that is to say, a terminated straight

line may be produced to any length in that straight line.

3. That a circle may be described from any centre, at

any distance from that centre, that is, with a radius equal

to any finite straight line drawn from the centre.

It is important to notice that the Postulates include no means of

direct measurement : hence the straight ruler is not supposed to be

graduated ; and the compasses, in accordance with EucHd's use, are

not to be employed for transferring distances from one part of a figure

to another.

ON THE AXIOMS.

The science of Geometry is based upon certain simple state-

ments, the truth of which is assumed at the outset to be self-

evident.

These self-evident truths, called by Euclid Common Notions^

are now known as the Axioms.

GENERAL AXIOMS. <

The necessary characteristics of an Axiom are

(i) That it should be self-evident; that is, that its truth

should be immediately accepted without proof.

(ii) That it should be fundamental; that is, that its truth

should not be derivable from any other truth more simple than

itself.

(iii) That it should supply a basis for the establishment of

further truths.

These characteristics may be summed up in the following

definition.

Definition. An Axiom is a self-evident truth, which neither

requires nor is capable of proof, but which serves as a founda-

tion for future reasoning.

Axioms are of two kinds, general and geometrical.

General Axioms apply to magnitudes of all kinds. Geometri-

cal Axioms refer exclusively to geometrical magnitudes^ such as

have been already indicated in the definitions.

General Axioms.

1. Things which are equal to the same thing are equal

to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be taken from equals, the remainders are

equal.

4. If equals be added to unequals, the wholes are un-

equal, the greater sum being that which includes the greater

of the unequals.

5. If equals be taken from unequals, the remainder.4

are unequal, the greater remainder being that which is left

from the greater of the unequals.

6. Things which are double of the same thing, or

of equal things, are equal to one another.

7. Things which are halves of the same thing, or ot

equal things, are equal to one another.

9.* The whole is greater than its part.

* To preserve the classification of general and geometrical axioms,

we have placed Euclid's ninth axiom before the eighth.

EUCLID'S ELEMENTS.

Geometrical Axioms.

8. Magnitudes wliich can be made to coincide with one

another, are equal.

This axiom affords the ultimate test of the equahty of two geome-

trical magnitudes. It implies that any liue, angle, or figure, may be

supposed to be taken up from its position, and without change in

size or form, laid down upon a second line, angle, or figure, for the

purpose of comparison.

This process is called superposition, and the first magnitude is

said to be applied to the other.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

[The statement that all right angles are equal, admits of proof,

and is therefore perhaj^s out of place as an Axiom.]

12. If a straight line meet two straight lines so as tv

make the interior angles on one side of it together less

than two right angles, these straight lines will meet if con-

tinually produced on the side on which are the angles wliicli

are together less than two right angles.

That is to say, if the two straight

lines AB and CD are met by the straight

line EH at F and G, in such a way that

the angles BFG, DGF are together less

than two right angles, it is asserted that

AB and CD will meet if continually pro-

* OF THE

UNIVERSITY OF CALIFORNIA.

CIms

,1^ - ^

Â«^

.Â»â€¢

>.'^i*f^

1v-'

vÂ«Â«'.

Wr'^^m

^^mr-:/

^'

.\^>^

:>s

A TEXT-BOOK

OF

EUCLID'S ELEMENTS.

A TEXT -BOOK

OF

EUCLID'S ELEMENTS

FOR THE USE OF SCHOOLS

BOOKS Lâ€” VI. AND XL

BY

IH. S. HALL, M.A.

FORMKRLY SCHOLAR OF CHRISt's COLLEGE, CAMBRIDGE ;

AND

F. H. STEVENS, M.A.

FORMERLY SCHOLAR OP QUEEN's COLLEGE, OXFORD ;

MASTERS OF THE MILITARY AND ENGINEERING SIDE, CLIFTON COLLEGE,

SECOND EDITION REVISED AND ENLARGED.

Hontron: frrWT VT?t? ^T^

MACMILLAN AND COT fj * ^ ^^^ \

AND NEW YOKE. ^^^tlf^"^!^

iÂ«9i ^^

\A It" Rights reserved.]

Richard Clay and Sons, Limited,

london and bungay.

FirÂ»t Edition 18S8.

Second Edition (Book XI. added) 18S9.

Reprinted 1890. 1891.

PEEFACE TO THE FIEST EDITION.

This volume contains the first Six Books of Euclid's

Elements, together with Appendices giving the most im-

portant elementary developments of Euclidean Geometry.

The text has been carefully revised, and special atten-

tion given to those points which experience has shewn to

present difficulties to beginners.

In the course of this revision the Enunciations have

been altered as little as possible; and, except in Book V.,

very few departures have been made from Euclid's proofs:

in each case changes have been adopted only where the old

text has been generally found a cause of difficulty; and

such changes are for the most part in favour of well-recog-

nised alternatives.

For example, the ambiguity has been removed from the

Enunciations of Propositions 18 and 19 of Book I.: the

fact that Propositions 8 and 26 establish the complete

identical equality of the two triangles considered has been

strongly urged; and thus the redundant step has been

removed from Proposition 34. In Book II. Simson's ar-

rangement of Proposition 13 has been abandoned for a

well-known alternative proof. In Book III. Proposition

25 is not given at length, and its place is taken by a

/! /^

VI PREFACE.

simple equivalent. Propositions 35 and 36 have been

treated generally, and it has not been tliought necessary

to do more than call attention in a note to the special

cases. Finally, in Book VI. we have adopted an alterna-

tive proof of Proposition 7, a theorem wliich has been too

much neglected, owing to the cumbrous form in whicli it

has been usually given.

These are the chief deviations from the ordinary text

as regards method and arrangement of proof: they are

points familiar as difficulties to most teachers, and to name

them indicates sufficiently, without further enumeration,

the general principles which have guided our revision.

A few alternative proofs of difficult propositions are

given for the convenience of those teachers who care to

use them.

With regard to Book V. we have established the princi-

pal propositions, both from the algebraical and geometrical

definitions of ratio and proportion, and we have endeavoured

to bring out clearly the distinction between these two modes

of treatment.

In compiling the geometrical section of Book V. we

have followed the system first advocated by the late Pro-

fessor De Morgan ; and here we derived very material

assistance from the exposition of the subject given in the

text-book of the Association for the Improvement of Geo-

metrical Teaching. To this source we are indebted for the

improved and more precise wording of definitions (as given

on pages 286, 288 to 291), as well as for the order and

substance of most of the propositions which appear between

pages 297 and 306. But as we have not (except in the

points above mentioned) adhered verbally to the text of

the Association, we are anxious, while expressing in the

fullest manner our obligation to their work, to exempt the

PREFACE. Vll

Association from all responsibility for our treatment of the

subject.

One purpose of the book is to gradually familiarise the

student with the use of legitimate symbols and abbrevia-

tions; for a geometrical argument may thus be thrown into

a form which is not only more readily seized by an advanced

reader, but is useful as a guide to the way in which Euclid's

propositions may be handled in written work. On the

other hand, we think it very desirable to defer the intro-

duction of symbols until the beginner has learnt that they

can only be properly used in Pure Geometry as abbrevia-

tions for verbal argument: and we hope thus to prevent

the slovenly and inaccurate habits which are very apt to

arise from their employment before this principle is fully

recognised.

Accordingly in Book I. we have used no contractions

or symbols of any kind, though we have introduced verbal

alterations into the text wherever it appeared that con-

ciseness or clearness would be gained.

In Book II. abbreviated forms of constantly recurring

words are used, and the phrases therefore and is 'equal to

are replaced by the usual symbols.

In the Third and following Books, and in additional

matter throughout the whole, we have employed all such

signs and abbreviations as we believe to add to the clear-

ness of the reasoning, care being taken that the symbols

chosen are compatible with a rigorous geometrical method,

and are recognised by the majority of teachers.

It must be understood that our use of symbols, and the

removal of unnecessary verbiage and repetition, by no

means implies a desire to secure brevity at all hazards.

On the contrary, nothing appears to us more mischievous

than an abridgement which is attained by omitting

Vlli PREFACE.

steps, or condensing two or more steps into one. Such

uses spring from the pressure of examinations; but an

examination is not, or ought not to be, a mere race; and

while we wish to indicate generally in the later books how

a geometrical argument may be abbreviated for the pur-

poses of written work, we have not thought well to reduce

the propositions to the bare skeleton so often presented to

an Examiner. Indeed it does not follow that the form most

suitable for the page of a text-book is also best adapted

to examination purposes; for the object to be attained

in each case is entirely different. The text-book should

present the argument in the clearest possible manner to the

mind of a reader to whom it is new : the written proposition

need only convey to the Examiner the assurance that the

proposition has been thoroughly grasped and remembered

by the pupil.

From first to last we have kept in mind the undoubted

fact that a very small proportion of those who study Ele-

mentary Geometry, and study it with profit, are destined

to become mathematicians in any real sense; and that to

a large majority of students, Euclid is intended to serve

not so much as a first lesson in mathematical reasoning,

as the first, and sometimes the only, model of formal and

rigid argument presented in an elementary education.

This consideration has determined not only the full

treatment of the earlier Books, but the retention of the

formal, if somewhat cumbrous, methods of Euclid in many

places where proofs of greater brevity and mathematical

elegance are available.

We hope that the additional matter introduced into

the book will provide sufficient exercise for pupils whose

study of Euclid is preliminary to a mathematical edu-

cation.

PREFACE. IX

The questions distributed tlirough the text follow very

easily from the propositions to which they are attached,

and we think that teachers are likely to find in them

all that is needed for an average pupil reading the subject

for the first time.

The Theorems and Examples at the end of each Book

contain questions of a slightly more difficult type : they

have been very carefully classified and arranged, and brought

into close connection with typical examples worked out

either partially or in full ; and it is hoped that this section

of the book, on which much thought has been expended,

will do something towards removing that extreme want of

freedom in solving deductions that is so commonly found

even among students who have a good knowledge of the

text of Euclid.

In the course of our work we have made ourselves

acquainted with most modern English books on Euclidean

Geometry : among these we have already expressed our

special indebtedness to the text-book recently published by

the Association for the Improvement of Geometrical Teach-

ing; and we must also mention the Edition of Euclid's Ele-

ments prepared by Mr J. S. Mackay, whose historical notes

and frequent references to original authorities have been of

the utmost service to us.

Our treatment of Maxima and Minima on page 239 is

based upon suggestions deriyed from a discussion of the

subject which took place at the annual meeting of the

Geometrical Association in January 1887.

Of the Riders and Deductions some are original; but

the greater part have been drawn from that large store of

floating material which has furnished Examination Papers

for the last 30 years, and must necessarily form the basis

of any elementary collection. Proofs which have been

"X > PREFACE.

found in two or more books without acknowledgement

liave been regarded as common property.

As regards figures, in accordance with a usage not

uncommon in recent editions of Euclid, we have made a

distinction between given lines and lines of construction.

Throughout tlie book we have italicised those deductions

on which we desired to lay special stress as being in them-

selves important geometrical results : this arrangement we

think will be useful to teachers who have little time to

devote to riders, or who wish to sketch out a suitable course

for revision.

We have in conclusion to tender our thanks to many of

our friends for the valuable criticism and advice which we

received from them as the book was passing through the

press, and especially to the Rev. H. C. Watson, of Clifton

College, who added to these services much kind assistance

in the revision of proof-sheets.

H. S. HALL,

F. H. STEVENS.

July, 1888.

PREFACE TO THE SECOND EDITION.

In the Second Edition the text of Books I â€” VI. has

been revised ; and at the request of many teachers we have

added the first twenty-one Propositions of Book XI. together

with a collection of Theorems and Examples illustrating the

elements of Solid Geometry.

September, 1889.

CONTENTS.

BOOK I.

PAGE

Definitions, Postulates, Axioms 1

Section I. Propositions 1 â€” 26 11

Section II. Parallels and Parallelograms.

Propositions 27 â€” 34 50

Section III. The Areas of Parallelograms and Triangles.

Propositions 35â€”48 66

Theorems and Examples on Book I.

Analysis, Synthesis 87

I. On the Identical Equality of Triangles , . . 90

II. On Inequalities 93

III. On Parallels . .95

IV. On Parallelograms 96

V. Miscellaneous Theorems and Examples . . . 100

VI. On the Concurrence of Straight Lines m a Tri-

angle 102

VII. On the Construction of Triangles with given

Parts 107

VIII. On Areas 109

IX. On Loci 114

X. On the Intersection of Loci 117

XI 1 CONTENTS.

BOOK II.

I'AGE

DEriNITIONS, &0 ... 120

Propositions 1 â€” 14 122

Theorems and Examples on Book II 1-14

BOOK m.

Definitions, (fee 149

Propositions 1 â€” 37 153

Note on the Method of Limits as Applied to Tangency . 213

Theorems and Examples on Book III.

I. On the Centre and Chords of a Circle . . 216

n. On the Tangent and the Contact of Circles.

The Common Tangent to Two Circles, Problems on

Tangency, Orthogonal Circles 217

IIL On Angles in Segments, and Angles at the Centres

AND Circumferences op Circles.

The Orthocentre of a Triangle, and properties of the

Pedal Triangle, Loci, Simson's Line . . , 222

IV. On the Circle in Connection with Kectangles.

Further Problems on Tangency 233

V. On Maxima and Minima 239

VI. Harder Miscellaneous Examples 246

BOOK IV.

Definitions, &c. 250

Propositions 1 â€” 16 251

Note on Regular Polygons 274

Theorems and Examples on Book IV

I. On the Triangle and its Circles.

Circumscribed, Inscribed, and Escribed Circles, The

Nine-points Circle 277

II. Miscellaneous Examples 283

CONTENTS. Xlll

BOOK V.

PAGE

Inteoductory 285

Definitions 286

Summary, with Algkbbaical Proofs, op the Principal

Theorems of Book V 292

Proofs of the Propositions derived from the Geometrical

Definition of Proportion 297

BOOK VI.

Definitions 307

Propositions 1 â€” D 308

Theorems and Exaviples on Book VI.

I, On Harmonic Section 359

II. On Centres of Similarity and Similitude . . . 363

III. On Pole and Polar 365

IV. On the Eadical Axis of Two or More Circles . . 371

V. On Transversals 374

VI. Miscellaneous Examples on Book VI 377

BOOK XI.

Definitions 383

Propositions 1â€”21 393

Exercises on Book XI , . . . 418

Theorems and Examples on Book XI. . . . . . 420

- OP THE

IVEB.SITY]

, . OB'

EUCLID'S ELEMENTS.

BOOK I.

Definitions.

1. A point is that which has position, but no mag-

nitude,

2. A line is that which has length without breadth.

The extremities of a line are points, and the intersection of two

lines is a point.

3. A straight line is that which lies evenly between

its extreme points.

Any portion cut off from a straight lineis called a segment of it.

4. A surface is that which has length and breadth,

but no thickness.

The boundaries of a surface are lines.

5. A plane surface is one in which any two points

being taken, the straight line between them lies wholly in

that surface.

A plane surface is frequently referred to simply as a plane.

Note, Euclid regards a point merely as a viark of position, and

he therefore attaches to it no idea of size and shape.

Similarly he considers that the properties of a line arise only from

its length and position, without reference to that minute breadth which

every line must really have if actually draion, even though the most

perfect instruments are used.

The definition of a surface is to be understood in a similar way.

II. ]â€¢:. 1.

2 EUCLID'S KLEMENTS.

6. A plane angle ia the inclination of two straiglit

lines to one another, which meet together, but are not in

the same straight line.

The point at which the straight lines meet is called the vertex of

the angle, and the straight lines themselves the arms of the angle.

When several angles are at one point O, any one

of them is expressed by three letters, of which the

letter that refers to the vertex is put between the

other two. Thus if the straight lines OA, OB, OC

meet at the point O, the angle contained by the

straight lines OA, OB is named the angle AOB or

BOA ; and the angle contained by OA, OC is named

the angle AOC or COA. Similarly the angle con-

tained by OB, 00 is referred to as the angle BOG

or COB. But if there be only one angle at a point,

it may be expressed by a single letter, as the angle

atO.

Of the two straight lines OB, OC shewn in the

adjoining figure, we recognize that OC is rnore in-

clined than OB to the straight line OA : this we

express by saying that the angle AOC is greater

than the angle AOB. Thus an angle must be

regarded as having magnitude. q

It should be observed that the angle AOC is the sum of the

angles AOB and BOC ; and that AOB is the difference of the angles

AOC and BOC.

The beginner is cautioned against supposing that the size of an

angle is altered either by increasing or diminishing the length of its

arms.

[Another view of an angle is recognized in many branches of

mathematics ; and though not employed by Euclid, it is here given

because it furnishes more clearly than any other a conception of what

is meant by the magnitxidc of an angle.

Suppose that the straight line OP in the figure

is capable of revolution about the point O, like the

hand of a watch, but in the opposite direction ; and

suppose that in this way it has passed successively

from the position OA to the positions occupied by

OB and OC.

Such a line must have undergone more turning

in passing from OA to OC, than in passing from OA to OB; and

consequently the angle AOC is said to be greater than the angle AOB.]

DEFINITIONS.

7. When a straight line standing on

another straight line makes the adjacent

angles equal to one another, each of the an-

gles is called a right angle ; and the straight

line which stands on the other is called a

perpendicular to it.

8. An obtuse angle is an angle whicli

is greater than one right angle, but less

than two right angles.

9. An acute angle ,is an angle which is

less than a right angle.

B

[In the adjoining figure the straight line

OB may be supposed to have arrived at

its present position, from the position occu-

pied by OA, by revolution about the point O

m either of the two directions indicated by

the arrows : thus two straight lines drawn

from a point may be considered as forming

tico angles, (marked (i) and (ii) in the figure)

of which the greater (ii) is said to be reflex.

If the arms OA, OB are in the same

straight line, the angle formed by them

on either side is called a straight angle.]

10. Any portion of a plane surface bounded by one

or more lines, straight or curved, is called a plane figure.

The sum of the bounding lines is called the perimeter of the figure.

Two figures are said to be equal in area, when they enclose equal

portions of a plane surface. <,..â€¢

11. A circle is a plane figure contained

by one line, which is called the circum-

ference, and is such that all straight lines

drawn from a certain point within the

figure to the circumference are equal to one

another : this point is called the centre of

the circle.

A radius of a circle is a straight line drawn from the

centre to the circumference.

1â€”2

4 Euclid's elements.

12. A diameter of a circle is a straiglit line drawn

ilirougli the centre, and terminated both ways by the

circumference.

13. A semicircle is tlie figure bounded by a diameter

of a circle and the jmrt of the circumference cut off by the

diameter.

14. A segment of a circle is the figure bounded by

a straight line and the part of the circumference which it

cuts oft".

15. Rectilineal figures are those which are bounded

by straight lines.

16. A triangle is a plane figure ])ounded by three

straight lines.

Any one of the angular points of a triangle may be regarded as its

vertex ; and the opposite side is then called the base.

17. A quadrilateral is a plane figure bounded by

four straight lines.

The straight line which joins opposite angular points in a qiiadri-

lateral is called a diagonal.

18. A polygon is a plane figure bounded by more

than four straight lines.

19. An equilateral triangle is a triangle

whose three sides are equal,

L

20. An isosceles triangle is a triangle two

of whose sides are equal.

21. A scalene triangle is a triangle which

has three unequal sides.

DEFINITIONS.

22. A right-angled triangle is a triangle

which has a right angle.

The side opposite to the right angle in a right-angled triangle is

called the hypotenuse.

23. An obtuse-angled triangle is a

triangle which has an obtuse anijle.

24. An acute-angled triangle is a triangh

which lias tJwee acute an<j:les.

[It will be seen hereafter (Book I. ^Proposition 17) that every

triangle must have at least ttco acute angles.]

25. Parallel straight lines are such as, being in the

same plane, do not meet, however far they are produced in

either direction.

26. A Parallelogram is a four-sided

figure wiiich lias its opposite sides pa-

rallel.

28. A square is a four-sided figure which

has all its sides equal and all its angles right

angles.

[It may easily be shewn that if a quadrilateral

has all its sides equal and one angle a right angle,

then all its angles will be right angles.]

29. A rhombus is a four-sided figure

which has all its sides equal, but its

angles are not right angles.

30. A trapezium is a four-sided figure

which has two of its sides parallel.

27. A rectangle is a parallelogram which r

has one of its angles a right angle. |

EUCLID'S ELEMENTS.

ON THE POSTULATES.

In order to effect the constructions necessary to the study of

geometry, it must be supposed that certain instruments are

available; but it has always been held that such instruments

should be as few in number, and as simple in character as

possible.

For the purposes of the first Six Books a straight ruler and

a pair of comi^asses are all that are needed ; and in the follow-

ing Postulates, or requests, Euclid demands the use of such

instruments, and assumes that they sufl&ce, theoretically as well

lis practically, to carry out the processes mentioned below.

Postulates.

Let it be granted,

1. That a straight line may be drawn from any one

point to any other point.

When we draw a straight line from the point A to the point B, we

are said io join AB.

2. That a JinitCj that is to say, a terminated straight

line may be produced to any length in that straight line.

3. That a circle may be described from any centre, at

any distance from that centre, that is, with a radius equal

to any finite straight line drawn from the centre.

It is important to notice that the Postulates include no means of

direct measurement : hence the straight ruler is not supposed to be

graduated ; and the compasses, in accordance with EucHd's use, are

not to be employed for transferring distances from one part of a figure

to another.

ON THE AXIOMS.

The science of Geometry is based upon certain simple state-

ments, the truth of which is assumed at the outset to be self-

evident.

These self-evident truths, called by Euclid Common Notions^

are now known as the Axioms.

GENERAL AXIOMS. <

The necessary characteristics of an Axiom are

(i) That it should be self-evident; that is, that its truth

should be immediately accepted without proof.

(ii) That it should be fundamental; that is, that its truth

should not be derivable from any other truth more simple than

itself.

(iii) That it should supply a basis for the establishment of

further truths.

These characteristics may be summed up in the following

definition.

Definition. An Axiom is a self-evident truth, which neither

requires nor is capable of proof, but which serves as a founda-

tion for future reasoning.

Axioms are of two kinds, general and geometrical.

General Axioms apply to magnitudes of all kinds. Geometri-

cal Axioms refer exclusively to geometrical magnitudes^ such as

have been already indicated in the definitions.

General Axioms.

1. Things which are equal to the same thing are equal

to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be taken from equals, the remainders are

equal.

4. If equals be added to unequals, the wholes are un-

equal, the greater sum being that which includes the greater

of the unequals.

5. If equals be taken from unequals, the remainder.4

are unequal, the greater remainder being that which is left

from the greater of the unequals.

6. Things which are double of the same thing, or

of equal things, are equal to one another.

7. Things which are halves of the same thing, or ot

equal things, are equal to one another.

9.* The whole is greater than its part.

* To preserve the classification of general and geometrical axioms,

we have placed Euclid's ninth axiom before the eighth.

EUCLID'S ELEMENTS.

Geometrical Axioms.

8. Magnitudes wliich can be made to coincide with one

another, are equal.

This axiom affords the ultimate test of the equahty of two geome-

trical magnitudes. It implies that any liue, angle, or figure, may be

supposed to be taken up from its position, and without change in

size or form, laid down upon a second line, angle, or figure, for the

purpose of comparison.

This process is called superposition, and the first magnitude is

said to be applied to the other.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

[The statement that all right angles are equal, admits of proof,

and is therefore perhaj^s out of place as an Axiom.]

12. If a straight line meet two straight lines so as tv

make the interior angles on one side of it together less

than two right angles, these straight lines will meet if con-

tinually produced on the side on which are the angles wliicli

are together less than two right angles.

That is to say, if the two straight

lines AB and CD are met by the straight

line EH at F and G, in such a way that

the angles BFG, DGF are together less

than two right angles, it is asserted that

AB and CD will meet if continually pro-

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