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IN MEMORIAM

FLORIAN CAJORl

â– â– â„¢ â„¢*â„¢ ^

E U C L I 13

1 BOOKS I, 11.

1 C, L. DODGSON,

EUCLID

BOOKS I, II.

EUCLID, BOOK I.

Arranged in Iiogical Sequence.

5 6

7

8

9 11

12 10 15

16

18 27

19

20

23 21

17

24 31

25 I

28

29

32 33

13

26

14

. â€” ' 11 31 I

30 34 I

, ' , â€” ' 41

35 43 46 -

36 37

38 41 39

40

â€” V â€”

42

47

48

44

30

45

:0

^ J

EUCLID

BOOKS I, II

EDITED BY

CHARLES L. DODGSON, M.A.

STUOENT AND LATE MATHEMATICAL LECTURER

OF CH. CH., OXFORD

SECOND EDITION

WITH WORDS SUBSTITUTED FOR THE ALGEBRAICAL SYMBOLS

USED IN THE FIRST EDITION

MACMILLAN AND CO.

1883

[All rights reserved '\

CAJORI

OXFORD:

BV E. PICKARD HALL, M.A,, AND J. H. STACY,

Printers to the University,

(?A45

INTRODUCTION.

PART I.

ADDRESSED TO THE TEACHER.

In preparing this edition of the first Two Books of Euclid,

my aim has been to show what Euclid's method really is in

itself, when stripped of all accidental verbiage and repetition.

With this object, I have held myself free to alter and abridge

the language wherever it seemed desirable, so long as I made no

real change in his methods of proof, or in his logical sequence.

This logical sequence, which has been for so many centuries

familiar to students of Geometry â€” so that ' The Forty-Seventh

Proposition ' is as clear a reference as if one were to quote the

enuntiation in full â€” it has lately been proposed to supersede :

partly from the instmctive passion for novelty which, even if

Euclid's system were the best possible, would still desire a

change; partly from the tacitly assumed theory that modern

lights are necessarily better than ancient ones. I am not now

speaking of writers who retain unaltered Euclid's sequence and

numbering of propositions, and merely substitute new proofs,

or interpolate new deductions, but of those who reject his

system altogether, and, taking up the subject de novo, attempt

to teach Geometry by methods of their own.

Some of these rival systems 1 have examined with much care

(I may specify Chauvenet, Cooley, Cuthbertson, Henrici, Le-

gendre, Loomis, Morell, Pierce, Reynolds, Willock, Wilson,

Wright, and the Syllabus put forth by the Association for the

e

VI INTR OD UCTION,

Improvement of Geometrical Teaching), and I feel deeply con-

vinced that, for purposes of teaching, no treatise has yet appeared

worthy to supersede that of Euclid.

It can never be too constantly, or too distinctly, stated that,

for the purpose of teaching beginners the subject-matter of

Euclid I, II, we do not need a complete collection of all known

propositions (probably some thousands) which come within that

limit, but simply a selection of some of the best of them, in a

logically arranged sequence. In both these respects, I hold that

Euclid's treatise is, at present, not only unequalled, but un-

approached.

For the diagrams used in this book I am indebted to the

great kindness of Mr. Todhunter, who has most generously

allowed me to make use of the series prepared for his own

edition of Euclid.

I will here enumerate, under the three headings of ' Additions '

'Omissions' and 'Alterations,' the chief points of difference

between this and the ordinary editions of Euclid, and will state

my reasons for adopting them.

1. Additions.

Def. &c. Â§ 11. The Axiom 'Two different right Lines cannot

have a common segment ' (in 3 equivalent forms). This is tacitly

assumed by Euclid, all through the two Books (see Note to

Prop. 4), and it is so distinctly analogous to his 'two right

Lines cannot enclose a Superficies' that it seems desirable to

have it formally stated.

Def. &c. Â§ 20. Here, to Euclid's Postulate ' A Circle can be

described about any Centre, and at any distance from it,' I

have added the words ' i.e. so that its Circumference shall pass

through any given Point.' This I believe to be Euclid's real

meaning. Modern critics have attempted to identify this given

' distance ' wath ' length of a given right Line,' and have then

very plausibly pointed to Props. 2, 3, as an instance of un-

necessary length of argument. 'Why does he not,' they say,

INTRODUCTION. vii

'solve Prop. 3 by simply drawing a Circle with radius equal

to the given Line ? ' All this involves the tacit assumption that

the * distance ' (duia-Trjfiaf i.e. * interval ' or ' difference of position ')

between two Points is equal to the length of the right Line

joining them. Now it may be granted that this ' distance ' is

merely an abbreviation for the phrase ' length of the shortest

path by which a Point can pass from one position to the other : '

and also that this path is (as any path would be) a Line : but

that it is a rtgh^ Line is just what Euclid did no( mean to assume :

for this would make Prop. 20 an Axiom. Euclid contemplates

the ' distance ' between two Points as a magnitude that exists

quite independently of any Line being drawn to join them (in

Prop. 12 he talks of the 'distance CD' without joining the points

C, D), and, as he has no means of measuring this distance, so

neither has he any means oi transferring \t, as the critics would

suggest. Hence Props. 2, 3, are logically necessary to prove the

possibility, with the given Postulates, of cutting off a Line equal

to a given Line. When once this has been proved, it can be

done practically in any way that is most convenient.

Axioms, Â§ 9. This is quite as axiomatic as the one tacitly

assumed by Euclid (in Props. 7, 18, 21, 24), viz. ' If one magnitude

be greater than a second, and the second greater than a third :

the first is greater than the third.' Mine is shorter, and has also

the advantage of saving a step in the argument : e.g. in Prop.

7, Euclid proves that the angle ADC is greater than the angle

BCD, a fact that is of no use in itself, and is only needed as a

step to another fact : this step I dispense with.

Prop. 8. Here I assert of all three angles what Euclid

asserts of one only. But his Proposition virtually contains

mine, as it may be proved three times over, with different sets

of bases.

Prop. 24- Euclid contents himself with proving the first

case, no doubt assuming that the reader can prove the rest

for himself. The ordinary way of making the argument com-

plete, viz. to interpolate 'of the two sides DE, DF, let DE

be not greater than DF,' is very unsatisfactory : for, though it

is true that, on this hypothesis, F will fall outside the Triangle

viii INTRODUCTION.

DEG, yet no proof of this is given. The Theorem, as here

completed, is distinctly analogous to Prop. 7.

Book II, Def. Â§ 4. The introduction of this one word

* projection' enables us to give, in Props. 12, 13, alternative

enuntiations which will, I think, be found much more easy to

grasp than the existing ones.

Book II, Prop. 8. Considering that this Proposition, with the

ordinary proof, is now constantly omitted by Students, under

the belief that Examiners never set it, I venture to suggest this

shorter method of proving it, in hopes of recalling attention to a

Theorem which, though not quoted in the Six Books of Euclid,

is useful in Conic Sections,

{Another proof of Euc. II. 8.)

[Instead of ' On AD describe ' &c, read as follows : â€”

A C B D

Square of ^Z> is equal to

squares oi AB, BD, with twice rectangle of AB, BD; [II. 4.

i. e. to squares of AB, BC, with twice rectangle of AB, BC;

i. e. to twice rectangle of AB, BC, with square of AC, [II. 7.

with twice rectangle of AB, BC ;

i. e. to four times rectangle of AB, BC, with square of AC.

Q^E.D.]

2. Omissions.

Euclid gives separate Definitions for ' plane angle ' and * plane

rectilineal angle.' I have ignored the existence of any angles

other than rectilineal, as I see no reason for mentioning them

in a book meant for beginners.

Prop. II. Here I omit the Corollary (introduced by Simson)

' Two Lines cannot have a common segment,' for several reasons.

First, it is not EucHd's: secondly, it is assumed as an Axiom,

at least as early as Prop. 4 : thirdly, the proof, offered for it,

is illogical, since, in order to draw a Line from B at right angles

to AB, we must produce AB ; and as this can, ex hypothesi, be

done in two different ways, we shall have two constructions, and

therefore two perpendiculars to deal with.

INTRODUCTION. ix

Prop. 46. Here instead of drawing a Line, at right angles

to AB^ longer than it, and then cutting off a piece equal to it,

I have combined the two processes into one, following the ex-

ample which Euclid himself has set in Prop. i6.

3. Alterations.

Definitions, &c. Â§ 7. Instead of the usual * A straight Line

is that which lies evenly between its extreme Points,' I have

expressed it 'A right Line,' (* right' is more in harmony, than

'straight,' with the term * rectilineal ') *is one that lies evenly

as to Points in it.' This is Euclid's expression: and it is ap-

plicable (which the other is not) to infinite Lines.

Prop. 12. Here I bisect the angle FCG instead of the Line

FG'. i.e. I use Prop. 9 instead of Prop. 10. The usual con-

struction really uses both, for Prop. 10 requires Prop. 9.

Prop. 16. Here, instead of saying that it *may be proved,

by bisecting BC &c. that the angle BCG is greater than the

angle ABC^ I simply point out that it has been proved â€” on

the principle that, when a Theorem has once been proved for

one case, it may be taken as proved for all similar cases.

Prop. 30. Here Euclid has contented himself, as he often

does, with proving one case only. But unfortunately the one he

has chosen is the one that least needs proof: for, if it be given

that neither of the outside Lines cuts the (infinitely producible)

middle Line, it is obvious that they cannot meet each other.

Book II, Prop. 14. Here, instead of producing DE to //,

I have drawn EH at right angles to BF, This at once supplies

us with the fact that GEH is a right angle, without the necessity

of tacitly assuming, as Euclid does, that * if one of the two ad-

jacent angles, which one Line makes with another, be right, so

also is the other.'

PART II.

ADDRESSED TO THE STUDENT.

The student is recommended to read the Two Books in the

following order, making sure that he has thoroughly mastered

each Section before beginning the next.

BOOK I.

Â§ 1. Magnitude. page

Axioms, Â§Â§ I to 9 7

Â§ 2. Triangles, &c.

Definitions, Â§Â§ i to 20 i

Props. I to X II

Â§ 3. Right angles, &c.

Definitions, &c. Â§Â§ 21 to 23 4

Props. XI to XXVI 25

Â§ 4. Parallel Lines, &c.

Definitions, Â§ 24 4

Props. XXVII, XXVIII 44

Axioms, Â§16 9

Props. XXIX to XXXII 46

Â§ 5. Parallelograms, &c.

Definitions, &c. Â§Â§ 25 to 27 4

Props. XXXIII to XLV 53

Â§ 6. Squares, &c.

Definitions, &c. Â§Â§ 28 to 31 5

Props. XLVI to XLVIII 70

INTRODUCTION. xi

BOOK II.

Â§ 1. Rectangles, &c. page

Definitions, Â§Â§ i to 3 75

Axioms, Â§ I 76

Props. I to XI 77

Â§ 2. Triangles, &c.

Definitions, Â§ 4 75

Axioms, Â§ 2 76

Props. XII to XIV . . . . . . .96

BOOK I.

DEFINITIONS, &c.

[N. B. Certain Postulates and Axioms are here inserted^ to

exhibit their logical connection with the Definitions : all these

will be repeated^ for convenience of reference^ under the separate

headings of ''Postulates'' and ^Axio7ns.^

They are here printed in italics^ in order to keep them distmct

fro7n the Definitions.

All interpolated matter will be enclosed in square brackets?^

Â§ I.

A Point has [position but] no magnitude.

Â§ 2-

A Line has [position and] length, but no breadth or

thickness.

Â§ 3. Axiom.

The extremities of a finite Line are Points.

Â§ 4.

A Superficies has [position,] length, and breadth, but no

thickness.

Â§ 5.

[A finite Superficies is called a Figure.]

2 EUCLID'S ELEMENTS,

Â§ 6. Axiom.

TJie boundaries of a Figure are Lines.

Â§7-

A right Line is one that lies evenly as to Points in it.

Â§ 8. Postulate.

A right Line can be drawn from any Point to any

other Point.

Â§ 9. Axiom.

(In 3 equivalent forms.)

(i). \07ily one such line can be drawn.'\

(2). (Euclid's form) Two right Lines cannot enclose

a Superficies.

(3). [Two right Lines, which coincide at two differ-

ent Points, coincide between them^

Â§ 10. Postulate.

A right Line, terminated at a Point, can be produced

beyond it.

Â§ II. Axiom.

(In 3 equivalent forms.)

(i). \Only 07ie such produced portion can be drazvn.]

(2). [Two different right Lines cannot have a com-

mon segment?^

(3). [Two right Lines, which coincide at two different

Points^ coincide if produced beyond them.']

â– BOOK I. 3

Â§ 12.

A plane Superficies, or a Plane, is such that, whatever

two Points in it be taken, the right Line passing through

them lies wholly in it.

Â§ 13-

An angle is the declination of two right Lines from each

other, which are terminated at the same Point but are not

in the same right Line. [The Lines are called its Arms,

and the Point its Vertex.]

Â§ 14-

A Figure, bounded by right Lines, is called rectilinear.

[The bounding Lines are called its Sides, and the vertices

of its angles are called its Vertices. One of the sides is

sometimes called the Base.]

Â§ 15.

A plane rectilinear Figure with three Sides is called a

trilateral Figure, or a Triangle ; with four, a quadri-

lateral Figure.

Â§ 16.

If two opposite Vertices of a quadrilateral Figure be

joined ; the joining Line is called a Diagonal of the Figure.

Â§ 17.

If a plane rectilinear Figure have all its Sides equal, it is

called equilateral : if all its angles, equiangular.

Â§ 18.

A Triangle with two Sides equal, is called isosceles ;

with all unequal, scalene.

B 2

4 EUCLID'S ELEMENTS.

Â§ 19-

A Circle is a plane Figure bounded by one Line, and

such that all right Lines, drawn from a certain Point within

it to the bounding Line, are equal. The bounding Line

is called its Circumference, and the Point its Centre.

Â§ 20. Postulate.

A Circle can be described about any Centre^ and at

any distance from it [i.e. so that its Circumference

shall pass through any given Poini\.

Â§ 21.

When a right Line, meeting another, makes the adjacent

angles equal, each of them is called a right angle, [and the

first Line is said to be at right angles to, or perpendicular

to, the other].

Â§ 22. Axiom.

All right angles are equal. \See Appendix A. Â§ 2.

Â§ 23.

If an angle be greater than a right angle, it is said to be

obtuse ; if less, acute.

Â§ 24.

Lines which, being in the same Plane, do not meet, how-

ever far produced, are said to be parallel to each other.

Â§ 25.

A Parallelogram is a quadrilateral Figure whose opposite

sides are parallel.

BOOK I. 5

Â§ 26.

[A Parallelogram is said to be about any Line which

passes through two opposite vertices.]

Â§ 27.

[If, through a Point in a Diagonal of a Parallelogram,

Lines be drawn parallel to the Sides : of the four Parallelo-

grams so formed, the two which are not about the Diagonal

are called the Complements.]

Â§ 28.

A Parallelogram, having all its angles right, is called a

rectangular Parallelogram, or a Rectangle.

Â§ 29.

A Square is an equilateral Rectangle.

Â§ 30.

[If a certain Line be given, the phrase " the square of the

Line " denotes the magnitude of any Square which has each

of its sides equal to the Line.]

Â§ 31-

If a Triangle have one angle right, it is called right-

angled ; if one obtuse, obtuse-angled; if all acute,

acute-angled.

Â§ 32.

In a right-angled Triangle, the side opposite to the right

angle is called the Hypotenuse.

EUCLID S ELEMENTS.

POSTULATES.

Â§ I.

A right Line can be drawn from any Point to any other

Point.

Â§ 2.

A right Line, terminated at a Point, can be produced

beyond it.

Â§ 3-

A Circle can be described about any Centre, and at any

distance from it [i.e. so that its Circumference shall pass

through any given Point].

BOOK I,

AXIOMS.

I. Axioms of Magnitude.

Â§ I-

Things which are equal to the same are equal to one

another.

Â§ 2-

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

Â§ 3-

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

Â§4-

If equals be added to unequals, the wholes are unequal.

Â§5-

If equals be taken from unequals, the # remainders are

unequal.

Â§6.

Things which are doubles of the same are equal.

Â§7-

Things which are halves of the same are equal.

8 EUCLID'S ELEMENTS.

Â§8.

A whole is greater than a part.

Â§9-

[A thing, which is greater than one of two equals, is

greater than a thing which is less than the other.]

II. Geometrical Axioms.

Â§ lo.

The extremities of a finite Line are Points.

Â§ II-

The boundaries of a Figure are Lines.

Â§ 12.

Lines, and Figures, which can be so placed as to coincide,

[and angles which can be so placed that the arms of the one

lie along those of the other,] are equal.

Â§ 13-

{In 3 equivalent forms)

(i). [From one Point to another only one right Line can

be drawn.]

(2). (Euclid's form) Two right Lines cannot enclose a

Superficies.

(3). [Two right Lines, which coincide at two different

Points, coincide between them.]

BOOK I, 9

Â§ 14.

{In 3 equivalent forms ^

(i). [If a right Line, terminated at a Point, be produced

beyond it, only one such produced portion can be drawn.]

(2). [Two different right Lines cannot have a common

segment.]

(3). [Two right Lines, which coincide at two different

Points, coincide if produced beyond them.] [Appendix A. Â§ i.

Â§ 15.

All right angles are equal. [Appendix A. Â§ 2.

Â§ 16.

If a right Line, meeting two others, make two interior

angles on one side of it together less than two right angles :

these two Lines, produced if necessary, will meet on that

side.

EUCLID'S ELEMENTS.

CONVENTIONS.

Â§ I

All Points, Lines, and Figures, hereafter discussed, will be

considered to be in one and the same Plane.

Â§ 2-

The word ' Line' will mean * right Line/

Â§ 3.

The word ' Circle * will mean ' Circumference of Circle,*

whenever it is obvious that the Circumference is intended,

e. g. when Circles are said to ' intersect.'

Â§ 4.

The words ' because,' ' therefore,' will be represented by

the symbols â€¢.', .'.

BOOK /. I r

PROP. I. Problem.

On a given Line to describe an equilateral Triangle.

Let AB be given Line. It is required to describe on it

an equilateral Triangle.

About centre A, at distance AB, describe Circle BCD;

about centre B, at distance BA, describe Circle ACE] and

join C, where they intersect, to A and B, It is to be proved

that the Triangle ABC is equilateral.

â€¢.â€¢ A is centre of Circle BCD,

.-. ylC is equal to ^^; [Def. Â§ 19.

also, â€¢.* B is centre of Circle A CE,

.*. BC is equal to AB.

.*. Triangle ABC is equilateral;

and it is described on given Line AB.

Q.E.F.

EUCLID'S ELEMENTS.

PROP. II. Problem.

From a given Point to draw a L ine equal to a given

Line.

Let A be given Point, and BC given Line. It is required

to draw from A a Line equal to BC

Join AB\ on AB describe equilateral Triangle ABC;

produce DA, DB, to U and F; about centre B, at distance

BC, describe Circle CGFT, cutting I>F at G ; and about

centre Z>, at distance DG, describe Circle GLF, cutting

DF at Z. It is to be proved that ^Z is equal to BC.

'.' B is centre of Circle CGH,

.-. BC is equal to BG ;

also, â€¢.* D is centre of Circle GZF,

.'. DG is equal to DL ;

but DB, DA, parts of these, are equal ;

.*. the remainders, BG, AL, are equal ;

but BG is equal to BC;

.'. AL is equal to BC, the given Line ;

and it is drawn from A, the given Point.

Q.E.F.

[Def. Â§ 19.

[Ax. Â§ 3.

[Ax. Â§

BOOK I.

13

PROP. III. Problem.

From the greater of two given unequal Lines to cut

off a part equal to the less.

Let AB, C, be given Lines, of which AB is the greater.

It is required to cut off from AB 2. part equal to C.

From A draw AD equal to C; and about centre A, at

distance AD, describe Circle DEF, cutting AB at F. It is

to be proved that ^^ is equal to C. [Prop. 2.

â€¢â€¢â€¢ A is centre of Circle DEF,

.'. AE is equal to AD) [Def. Â§ 19.

.-. it is equal to C, the given lesser Line ; [Ax. Â§ i.

and it is cut off from AB, the given greater Line.

Q.E.F.

14 EUCLID'S ELEMENTS.

PROP. IV. Theorem.

//", in two Triangles^ two sides and the included angle

of the one be respectively equal to two sides and the

included angle of the other: then the base and the

remaining angles of the one are respectively equal to the

base and the remainhig angles of the other ^ those angles

being equal which are opposite to equal sides ; and the

Triangles are equal.

Let ABC, DEF be two Triangles, in which AB, AC Sire

respectively equal to DE, DF, and angle A is equal to angle

D. It is to be proved that BC is equal to EF \ that angles

B, C, are respectively equal to angles F^ F ; and that the

Triangles are equal.

If Triangle ABC be applied to Triangle DFF, so that

A may fall on D, and yl^ along DE,

then, '.' AB is equal to DE^

.'. B falls on E;

and *.' angle A is equal to angle Z>,

.-. ^C falls along Z>i^;

and *.â€¢ ^ C is equal to JDF,

.'. C falls on F;

BOOK I. 15

and *.* B falls on E, and C on F,

.'. BC coincides with EF\

.-. BC is equal to EF\ [Ax. Â§ 12.

also remaining angles coincide, and therefore are equal ;

and Triangles coincide, and therefore are equal.

Therefore if, in two Triangles, two sides, &c. Q.E.D.

{Appendix A. Â§ 1 .

1 6 EUCLID'S ELEMENTS.

PROP. V. Theorem.

The angles at the base of an isosceles Triangle are

equal ; and, if the equal sides be produced, the angles

at the other side of the base are equal.

Let ABC be an isosceles Triangle, having AB,AC, equal ;

and let them be produced to D and J^. It is to be proved

that angles ABC, ACB are equal, and also angles DBC,

ECB.

In BD take any Point F-, from AE cut off ^G^ equal to

AF', and join ^(?, CF.

In Triangles ABG, ACF,

( AB, AG, are respectively equal to AC, AF,

\ and angle A is common,

r BG is equal to CF

.-. -| and angles ABG, AGB, are respectively equal to

( angles ^Ci^,^i^C.

[Prop. 4.

BOOK I. 17

f AF is equal to A G,

' * I and AB, AC, parts of them, are equal,

.-. remainders BF, CG, are equal. [Ax. Â§ 3.

Next, in Triangles BFC, CGB,

^ ^ I BF, FC, are respectively equal to CG, GB,

\ and angle BFC is equal to angle CGB,

.'. angles FBC, FCB, are respectively equal to

angles GCB, GBC. [Prop. 4.

( angle ABG is equal to angle ACF,

' * I and angles CBG, BCF, parts of them, are equal,

.*. remaining angles ^^C,^C^, are equal; [Ax. Â§ 3.

and these are angles at base.

Also it has been proved that the angles at the other side of

base, namely BBC, FCB, are equal.

Therefore the angles &c.

Q.E.D.

Corollary.

An equilateral Triangle is equiangular.

1 8 EUCLID'S ELEMENTS,

PROP. VI. Theorem.

If a Triangle have two angles equal: the opposite

sides are equal-

Let ABC be a Triangle, having angles B, ACB^ equal. It

is to be proved that AC^ AB, are equal.

For if they be unequal, one must be the greater ; let ^^ be

the greater; from it cut o^DB equal to ^C; and join DC,

Then in Triangles DBC, ABC,

( DB is equal to ^C,

â€¢.* < BC is common,

( and angle B is equal to angle ACB^

.'. the Triangles are equal, [Prop. 4.

BOOK /.

19

the part equal to the whole, which is absurd ;

[Ax. Â§ 8.

.-. AC^ AB, are not unequal;

i.e. they are equal.

Therefore, if a Triangle, &c.

Q.E.D.

Corollary.

An equiangular Triangle is equilateral.

c 2

20 EUCLID'S ELEMENTS.

PROP. VII. Theorem.

On the same base, and on the same side of it, there

cannot be two different Triangles, in which the sides

termi7iated at one end of the base are equal, and like-

wise those terminated at the other end.

If possible, let there be two such triangles, ABC, ABD,

in which ^C is equal to AD, and BC to BD.

First, let the vertex of each be without the other.

Join CD.

'.' AC is equal to AD,

.*. angle ADC is equal to angle A CD ; [Prop. 5.

( angle BDC is greater than one of these equals,

\ and angle BCD is less than the other,

/. angle BDC is greater than angle BCD; [Ax. Â§ 9.

again, â€¢.* BC is equal to BD,

.'. angle BDC is equal to angle BCD ;

but it is also greater ; which is absurd.

BOOK I. 21

Secondly, let Z), the vertex of one, be within the other.

Join CD', and produce ACy AD, to E, F.

'.- AC \s equal to AD,

.'. angle ECD is equal to angle FDC; [Prop 5.

AA

^

451

o

D6

I8S3

^0^

\d

m

'

en

UC-NRLF

$B 5E7 77a

00

IN MEMORIAM

FLORIAN CAJORl

â– â– â„¢ â„¢*â„¢ ^

E U C L I 13

1 BOOKS I, 11.

1 C, L. DODGSON,

EUCLID

BOOKS I, II.

EUCLID, BOOK I.

Arranged in Iiogical Sequence.

5 6

7

8

9 11

12 10 15

16

18 27

19

20

23 21

17

24 31

25 I

28

29

32 33

13

26

14

. â€” ' 11 31 I

30 34 I

, ' , â€” ' 41

35 43 46 -

36 37

38 41 39

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EUCLID

BOOKS I, II

EDITED BY

CHARLES L. DODGSON, M.A.

STUOENT AND LATE MATHEMATICAL LECTURER

OF CH. CH., OXFORD

SECOND EDITION

WITH WORDS SUBSTITUTED FOR THE ALGEBRAICAL SYMBOLS

USED IN THE FIRST EDITION

MACMILLAN AND CO.

1883

[All rights reserved '\

CAJORI

OXFORD:

BV E. PICKARD HALL, M.A,, AND J. H. STACY,

Printers to the University,

(?A45

INTRODUCTION.

PART I.

ADDRESSED TO THE TEACHER.

In preparing this edition of the first Two Books of Euclid,

my aim has been to show what Euclid's method really is in

itself, when stripped of all accidental verbiage and repetition.

With this object, I have held myself free to alter and abridge

the language wherever it seemed desirable, so long as I made no

real change in his methods of proof, or in his logical sequence.

This logical sequence, which has been for so many centuries

familiar to students of Geometry â€” so that ' The Forty-Seventh

Proposition ' is as clear a reference as if one were to quote the

enuntiation in full â€” it has lately been proposed to supersede :

partly from the instmctive passion for novelty which, even if

Euclid's system were the best possible, would still desire a

change; partly from the tacitly assumed theory that modern

lights are necessarily better than ancient ones. I am not now

speaking of writers who retain unaltered Euclid's sequence and

numbering of propositions, and merely substitute new proofs,

or interpolate new deductions, but of those who reject his

system altogether, and, taking up the subject de novo, attempt

to teach Geometry by methods of their own.

Some of these rival systems 1 have examined with much care

(I may specify Chauvenet, Cooley, Cuthbertson, Henrici, Le-

gendre, Loomis, Morell, Pierce, Reynolds, Willock, Wilson,

Wright, and the Syllabus put forth by the Association for the

e

VI INTR OD UCTION,

Improvement of Geometrical Teaching), and I feel deeply con-

vinced that, for purposes of teaching, no treatise has yet appeared

worthy to supersede that of Euclid.

It can never be too constantly, or too distinctly, stated that,

for the purpose of teaching beginners the subject-matter of

Euclid I, II, we do not need a complete collection of all known

propositions (probably some thousands) which come within that

limit, but simply a selection of some of the best of them, in a

logically arranged sequence. In both these respects, I hold that

Euclid's treatise is, at present, not only unequalled, but un-

approached.

For the diagrams used in this book I am indebted to the

great kindness of Mr. Todhunter, who has most generously

allowed me to make use of the series prepared for his own

edition of Euclid.

I will here enumerate, under the three headings of ' Additions '

'Omissions' and 'Alterations,' the chief points of difference

between this and the ordinary editions of Euclid, and will state

my reasons for adopting them.

1. Additions.

Def. &c. Â§ 11. The Axiom 'Two different right Lines cannot

have a common segment ' (in 3 equivalent forms). This is tacitly

assumed by Euclid, all through the two Books (see Note to

Prop. 4), and it is so distinctly analogous to his 'two right

Lines cannot enclose a Superficies' that it seems desirable to

have it formally stated.

Def. &c. Â§ 20. Here, to Euclid's Postulate ' A Circle can be

described about any Centre, and at any distance from it,' I

have added the words ' i.e. so that its Circumference shall pass

through any given Point.' This I believe to be Euclid's real

meaning. Modern critics have attempted to identify this given

' distance ' wath ' length of a given right Line,' and have then

very plausibly pointed to Props. 2, 3, as an instance of un-

necessary length of argument. 'Why does he not,' they say,

INTRODUCTION. vii

'solve Prop. 3 by simply drawing a Circle with radius equal

to the given Line ? ' All this involves the tacit assumption that

the * distance ' (duia-Trjfiaf i.e. * interval ' or ' difference of position ')

between two Points is equal to the length of the right Line

joining them. Now it may be granted that this ' distance ' is

merely an abbreviation for the phrase ' length of the shortest

path by which a Point can pass from one position to the other : '

and also that this path is (as any path would be) a Line : but

that it is a rtgh^ Line is just what Euclid did no( mean to assume :

for this would make Prop. 20 an Axiom. Euclid contemplates

the ' distance ' between two Points as a magnitude that exists

quite independently of any Line being drawn to join them (in

Prop. 12 he talks of the 'distance CD' without joining the points

C, D), and, as he has no means of measuring this distance, so

neither has he any means oi transferring \t, as the critics would

suggest. Hence Props. 2, 3, are logically necessary to prove the

possibility, with the given Postulates, of cutting off a Line equal

to a given Line. When once this has been proved, it can be

done practically in any way that is most convenient.

Axioms, Â§ 9. This is quite as axiomatic as the one tacitly

assumed by Euclid (in Props. 7, 18, 21, 24), viz. ' If one magnitude

be greater than a second, and the second greater than a third :

the first is greater than the third.' Mine is shorter, and has also

the advantage of saving a step in the argument : e.g. in Prop.

7, Euclid proves that the angle ADC is greater than the angle

BCD, a fact that is of no use in itself, and is only needed as a

step to another fact : this step I dispense with.

Prop. 8. Here I assert of all three angles what Euclid

asserts of one only. But his Proposition virtually contains

mine, as it may be proved three times over, with different sets

of bases.

Prop. 24- Euclid contents himself with proving the first

case, no doubt assuming that the reader can prove the rest

for himself. The ordinary way of making the argument com-

plete, viz. to interpolate 'of the two sides DE, DF, let DE

be not greater than DF,' is very unsatisfactory : for, though it

is true that, on this hypothesis, F will fall outside the Triangle

viii INTRODUCTION.

DEG, yet no proof of this is given. The Theorem, as here

completed, is distinctly analogous to Prop. 7.

Book II, Def. Â§ 4. The introduction of this one word

* projection' enables us to give, in Props. 12, 13, alternative

enuntiations which will, I think, be found much more easy to

grasp than the existing ones.

Book II, Prop. 8. Considering that this Proposition, with the

ordinary proof, is now constantly omitted by Students, under

the belief that Examiners never set it, I venture to suggest this

shorter method of proving it, in hopes of recalling attention to a

Theorem which, though not quoted in the Six Books of Euclid,

is useful in Conic Sections,

{Another proof of Euc. II. 8.)

[Instead of ' On AD describe ' &c, read as follows : â€”

A C B D

Square of ^Z> is equal to

squares oi AB, BD, with twice rectangle of AB, BD; [II. 4.

i. e. to squares of AB, BC, with twice rectangle of AB, BC;

i. e. to twice rectangle of AB, BC, with square of AC, [II. 7.

with twice rectangle of AB, BC ;

i. e. to four times rectangle of AB, BC, with square of AC.

Q^E.D.]

2. Omissions.

Euclid gives separate Definitions for ' plane angle ' and * plane

rectilineal angle.' I have ignored the existence of any angles

other than rectilineal, as I see no reason for mentioning them

in a book meant for beginners.

Prop. II. Here I omit the Corollary (introduced by Simson)

' Two Lines cannot have a common segment,' for several reasons.

First, it is not EucHd's: secondly, it is assumed as an Axiom,

at least as early as Prop. 4 : thirdly, the proof, offered for it,

is illogical, since, in order to draw a Line from B at right angles

to AB, we must produce AB ; and as this can, ex hypothesi, be

done in two different ways, we shall have two constructions, and

therefore two perpendiculars to deal with.

INTRODUCTION. ix

Prop. 46. Here instead of drawing a Line, at right angles

to AB^ longer than it, and then cutting off a piece equal to it,

I have combined the two processes into one, following the ex-

ample which Euclid himself has set in Prop. i6.

3. Alterations.

Definitions, &c. Â§ 7. Instead of the usual * A straight Line

is that which lies evenly between its extreme Points,' I have

expressed it 'A right Line,' (* right' is more in harmony, than

'straight,' with the term * rectilineal ') *is one that lies evenly

as to Points in it.' This is Euclid's expression: and it is ap-

plicable (which the other is not) to infinite Lines.

Prop. 12. Here I bisect the angle FCG instead of the Line

FG'. i.e. I use Prop. 9 instead of Prop. 10. The usual con-

struction really uses both, for Prop. 10 requires Prop. 9.

Prop. 16. Here, instead of saying that it *may be proved,

by bisecting BC &c. that the angle BCG is greater than the

angle ABC^ I simply point out that it has been proved â€” on

the principle that, when a Theorem has once been proved for

one case, it may be taken as proved for all similar cases.

Prop. 30. Here Euclid has contented himself, as he often

does, with proving one case only. But unfortunately the one he

has chosen is the one that least needs proof: for, if it be given

that neither of the outside Lines cuts the (infinitely producible)

middle Line, it is obvious that they cannot meet each other.

Book II, Prop. 14. Here, instead of producing DE to //,

I have drawn EH at right angles to BF, This at once supplies

us with the fact that GEH is a right angle, without the necessity

of tacitly assuming, as Euclid does, that * if one of the two ad-

jacent angles, which one Line makes with another, be right, so

also is the other.'

PART II.

ADDRESSED TO THE STUDENT.

The student is recommended to read the Two Books in the

following order, making sure that he has thoroughly mastered

each Section before beginning the next.

BOOK I.

Â§ 1. Magnitude. page

Axioms, Â§Â§ I to 9 7

Â§ 2. Triangles, &c.

Definitions, Â§Â§ i to 20 i

Props. I to X II

Â§ 3. Right angles, &c.

Definitions, &c. Â§Â§ 21 to 23 4

Props. XI to XXVI 25

Â§ 4. Parallel Lines, &c.

Definitions, Â§ 24 4

Props. XXVII, XXVIII 44

Axioms, Â§16 9

Props. XXIX to XXXII 46

Â§ 5. Parallelograms, &c.

Definitions, &c. Â§Â§ 25 to 27 4

Props. XXXIII to XLV 53

Â§ 6. Squares, &c.

Definitions, &c. Â§Â§ 28 to 31 5

Props. XLVI to XLVIII 70

INTRODUCTION. xi

BOOK II.

Â§ 1. Rectangles, &c. page

Definitions, Â§Â§ i to 3 75

Axioms, Â§ I 76

Props. I to XI 77

Â§ 2. Triangles, &c.

Definitions, Â§ 4 75

Axioms, Â§ 2 76

Props. XII to XIV . . . . . . .96

BOOK I.

DEFINITIONS, &c.

[N. B. Certain Postulates and Axioms are here inserted^ to

exhibit their logical connection with the Definitions : all these

will be repeated^ for convenience of reference^ under the separate

headings of ''Postulates'' and ^Axio7ns.^

They are here printed in italics^ in order to keep them distmct

fro7n the Definitions.

All interpolated matter will be enclosed in square brackets?^

Â§ I.

A Point has [position but] no magnitude.

Â§ 2-

A Line has [position and] length, but no breadth or

thickness.

Â§ 3. Axiom.

The extremities of a finite Line are Points.

Â§ 4.

A Superficies has [position,] length, and breadth, but no

thickness.

Â§ 5.

[A finite Superficies is called a Figure.]

2 EUCLID'S ELEMENTS,

Â§ 6. Axiom.

TJie boundaries of a Figure are Lines.

Â§7-

A right Line is one that lies evenly as to Points in it.

Â§ 8. Postulate.

A right Line can be drawn from any Point to any

other Point.

Â§ 9. Axiom.

(In 3 equivalent forms.)

(i). \07ily one such line can be drawn.'\

(2). (Euclid's form) Two right Lines cannot enclose

a Superficies.

(3). [Two right Lines, which coincide at two differ-

ent Points, coincide between them^

Â§ 10. Postulate.

A right Line, terminated at a Point, can be produced

beyond it.

Â§ II. Axiom.

(In 3 equivalent forms.)

(i). \Only 07ie such produced portion can be drazvn.]

(2). [Two different right Lines cannot have a com-

mon segment?^

(3). [Two right Lines, which coincide at two different

Points^ coincide if produced beyond them.']

â– BOOK I. 3

Â§ 12.

A plane Superficies, or a Plane, is such that, whatever

two Points in it be taken, the right Line passing through

them lies wholly in it.

Â§ 13-

An angle is the declination of two right Lines from each

other, which are terminated at the same Point but are not

in the same right Line. [The Lines are called its Arms,

and the Point its Vertex.]

Â§ 14-

A Figure, bounded by right Lines, is called rectilinear.

[The bounding Lines are called its Sides, and the vertices

of its angles are called its Vertices. One of the sides is

sometimes called the Base.]

Â§ 15.

A plane rectilinear Figure with three Sides is called a

trilateral Figure, or a Triangle ; with four, a quadri-

lateral Figure.

Â§ 16.

If two opposite Vertices of a quadrilateral Figure be

joined ; the joining Line is called a Diagonal of the Figure.

Â§ 17.

If a plane rectilinear Figure have all its Sides equal, it is

called equilateral : if all its angles, equiangular.

Â§ 18.

A Triangle with two Sides equal, is called isosceles ;

with all unequal, scalene.

B 2

4 EUCLID'S ELEMENTS.

Â§ 19-

A Circle is a plane Figure bounded by one Line, and

such that all right Lines, drawn from a certain Point within

it to the bounding Line, are equal. The bounding Line

is called its Circumference, and the Point its Centre.

Â§ 20. Postulate.

A Circle can be described about any Centre^ and at

any distance from it [i.e. so that its Circumference

shall pass through any given Poini\.

Â§ 21.

When a right Line, meeting another, makes the adjacent

angles equal, each of them is called a right angle, [and the

first Line is said to be at right angles to, or perpendicular

to, the other].

Â§ 22. Axiom.

All right angles are equal. \See Appendix A. Â§ 2.

Â§ 23.

If an angle be greater than a right angle, it is said to be

obtuse ; if less, acute.

Â§ 24.

Lines which, being in the same Plane, do not meet, how-

ever far produced, are said to be parallel to each other.

Â§ 25.

A Parallelogram is a quadrilateral Figure whose opposite

sides are parallel.

BOOK I. 5

Â§ 26.

[A Parallelogram is said to be about any Line which

passes through two opposite vertices.]

Â§ 27.

[If, through a Point in a Diagonal of a Parallelogram,

Lines be drawn parallel to the Sides : of the four Parallelo-

grams so formed, the two which are not about the Diagonal

are called the Complements.]

Â§ 28.

A Parallelogram, having all its angles right, is called a

rectangular Parallelogram, or a Rectangle.

Â§ 29.

A Square is an equilateral Rectangle.

Â§ 30.

[If a certain Line be given, the phrase " the square of the

Line " denotes the magnitude of any Square which has each

of its sides equal to the Line.]

Â§ 31-

If a Triangle have one angle right, it is called right-

angled ; if one obtuse, obtuse-angled; if all acute,

acute-angled.

Â§ 32.

In a right-angled Triangle, the side opposite to the right

angle is called the Hypotenuse.

EUCLID S ELEMENTS.

POSTULATES.

Â§ I.

A right Line can be drawn from any Point to any other

Point.

Â§ 2.

A right Line, terminated at a Point, can be produced

beyond it.

Â§ 3-

A Circle can be described about any Centre, and at any

distance from it [i.e. so that its Circumference shall pass

through any given Point].

BOOK I,

AXIOMS.

I. Axioms of Magnitude.

Â§ I-

Things which are equal to the same are equal to one

another.

Â§ 2-

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

Â§ 3-

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

Â§4-

If equals be added to unequals, the wholes are unequal.

Â§5-

If equals be taken from unequals, the # remainders are

unequal.

Â§6.

Things which are doubles of the same are equal.

Â§7-

Things which are halves of the same are equal.

8 EUCLID'S ELEMENTS.

Â§8.

A whole is greater than a part.

Â§9-

[A thing, which is greater than one of two equals, is

greater than a thing which is less than the other.]

II. Geometrical Axioms.

Â§ lo.

The extremities of a finite Line are Points.

Â§ II-

The boundaries of a Figure are Lines.

Â§ 12.

Lines, and Figures, which can be so placed as to coincide,

[and angles which can be so placed that the arms of the one

lie along those of the other,] are equal.

Â§ 13-

{In 3 equivalent forms)

(i). [From one Point to another only one right Line can

be drawn.]

(2). (Euclid's form) Two right Lines cannot enclose a

Superficies.

(3). [Two right Lines, which coincide at two different

Points, coincide between them.]

BOOK I, 9

Â§ 14.

{In 3 equivalent forms ^

(i). [If a right Line, terminated at a Point, be produced

beyond it, only one such produced portion can be drawn.]

(2). [Two different right Lines cannot have a common

segment.]

(3). [Two right Lines, which coincide at two different

Points, coincide if produced beyond them.] [Appendix A. Â§ i.

Â§ 15.

All right angles are equal. [Appendix A. Â§ 2.

Â§ 16.

If a right Line, meeting two others, make two interior

angles on one side of it together less than two right angles :

these two Lines, produced if necessary, will meet on that

side.

EUCLID'S ELEMENTS.

CONVENTIONS.

Â§ I

All Points, Lines, and Figures, hereafter discussed, will be

considered to be in one and the same Plane.

Â§ 2-

The word ' Line' will mean * right Line/

Â§ 3.

The word ' Circle * will mean ' Circumference of Circle,*

whenever it is obvious that the Circumference is intended,

e. g. when Circles are said to ' intersect.'

Â§ 4.

The words ' because,' ' therefore,' will be represented by

the symbols â€¢.', .'.

BOOK /. I r

PROP. I. Problem.

On a given Line to describe an equilateral Triangle.

Let AB be given Line. It is required to describe on it

an equilateral Triangle.

About centre A, at distance AB, describe Circle BCD;

about centre B, at distance BA, describe Circle ACE] and

join C, where they intersect, to A and B, It is to be proved

that the Triangle ABC is equilateral.

â€¢.â€¢ A is centre of Circle BCD,

.-. ylC is equal to ^^; [Def. Â§ 19.

also, â€¢.* B is centre of Circle A CE,

.*. BC is equal to AB.

.*. Triangle ABC is equilateral;

and it is described on given Line AB.

Q.E.F.

EUCLID'S ELEMENTS.

PROP. II. Problem.

From a given Point to draw a L ine equal to a given

Line.

Let A be given Point, and BC given Line. It is required

to draw from A a Line equal to BC

Join AB\ on AB describe equilateral Triangle ABC;

produce DA, DB, to U and F; about centre B, at distance

BC, describe Circle CGFT, cutting I>F at G ; and about

centre Z>, at distance DG, describe Circle GLF, cutting

DF at Z. It is to be proved that ^Z is equal to BC.

'.' B is centre of Circle CGH,

.-. BC is equal to BG ;

also, â€¢.* D is centre of Circle GZF,

.'. DG is equal to DL ;

but DB, DA, parts of these, are equal ;

.*. the remainders, BG, AL, are equal ;

but BG is equal to BC;

.'. AL is equal to BC, the given Line ;

and it is drawn from A, the given Point.

Q.E.F.

[Def. Â§ 19.

[Ax. Â§ 3.

[Ax. Â§

BOOK I.

13

PROP. III. Problem.

From the greater of two given unequal Lines to cut

off a part equal to the less.

Let AB, C, be given Lines, of which AB is the greater.

It is required to cut off from AB 2. part equal to C.

From A draw AD equal to C; and about centre A, at

distance AD, describe Circle DEF, cutting AB at F. It is

to be proved that ^^ is equal to C. [Prop. 2.

â€¢â€¢â€¢ A is centre of Circle DEF,

.'. AE is equal to AD) [Def. Â§ 19.

.-. it is equal to C, the given lesser Line ; [Ax. Â§ i.

and it is cut off from AB, the given greater Line.

Q.E.F.

14 EUCLID'S ELEMENTS.

PROP. IV. Theorem.

//", in two Triangles^ two sides and the included angle

of the one be respectively equal to two sides and the

included angle of the other: then the base and the

remaining angles of the one are respectively equal to the

base and the remainhig angles of the other ^ those angles

being equal which are opposite to equal sides ; and the

Triangles are equal.

Let ABC, DEF be two Triangles, in which AB, AC Sire

respectively equal to DE, DF, and angle A is equal to angle

D. It is to be proved that BC is equal to EF \ that angles

B, C, are respectively equal to angles F^ F ; and that the

Triangles are equal.

If Triangle ABC be applied to Triangle DFF, so that

A may fall on D, and yl^ along DE,

then, '.' AB is equal to DE^

.'. B falls on E;

and *.' angle A is equal to angle Z>,

.-. ^C falls along Z>i^;

and *.â€¢ ^ C is equal to JDF,

.'. C falls on F;

BOOK I. 15

and *.* B falls on E, and C on F,

.'. BC coincides with EF\

.-. BC is equal to EF\ [Ax. Â§ 12.

also remaining angles coincide, and therefore are equal ;

and Triangles coincide, and therefore are equal.

Therefore if, in two Triangles, two sides, &c. Q.E.D.

{Appendix A. Â§ 1 .

1 6 EUCLID'S ELEMENTS.

PROP. V. Theorem.

The angles at the base of an isosceles Triangle are

equal ; and, if the equal sides be produced, the angles

at the other side of the base are equal.

Let ABC be an isosceles Triangle, having AB,AC, equal ;

and let them be produced to D and J^. It is to be proved

that angles ABC, ACB are equal, and also angles DBC,

ECB.

In BD take any Point F-, from AE cut off ^G^ equal to

AF', and join ^(?, CF.

In Triangles ABG, ACF,

( AB, AG, are respectively equal to AC, AF,

\ and angle A is common,

r BG is equal to CF

.-. -| and angles ABG, AGB, are respectively equal to

( angles ^Ci^,^i^C.

[Prop. 4.

BOOK I. 17

f AF is equal to A G,

' * I and AB, AC, parts of them, are equal,

.-. remainders BF, CG, are equal. [Ax. Â§ 3.

Next, in Triangles BFC, CGB,

^ ^ I BF, FC, are respectively equal to CG, GB,

\ and angle BFC is equal to angle CGB,

.'. angles FBC, FCB, are respectively equal to

angles GCB, GBC. [Prop. 4.

( angle ABG is equal to angle ACF,

' * I and angles CBG, BCF, parts of them, are equal,

.*. remaining angles ^^C,^C^, are equal; [Ax. Â§ 3.

and these are angles at base.

Also it has been proved that the angles at the other side of

base, namely BBC, FCB, are equal.

Therefore the angles &c.

Q.E.D.

Corollary.

An equilateral Triangle is equiangular.

1 8 EUCLID'S ELEMENTS,

PROP. VI. Theorem.

If a Triangle have two angles equal: the opposite

sides are equal-

Let ABC be a Triangle, having angles B, ACB^ equal. It

is to be proved that AC^ AB, are equal.

For if they be unequal, one must be the greater ; let ^^ be

the greater; from it cut o^DB equal to ^C; and join DC,

Then in Triangles DBC, ABC,

( DB is equal to ^C,

â€¢.* < BC is common,

( and angle B is equal to angle ACB^

.'. the Triangles are equal, [Prop. 4.

BOOK /.

19

the part equal to the whole, which is absurd ;

[Ax. Â§ 8.

.-. AC^ AB, are not unequal;

i.e. they are equal.

Therefore, if a Triangle, &c.

Q.E.D.

Corollary.

An equiangular Triangle is equilateral.

c 2

20 EUCLID'S ELEMENTS.

PROP. VII. Theorem.

On the same base, and on the same side of it, there

cannot be two different Triangles, in which the sides

termi7iated at one end of the base are equal, and like-

wise those terminated at the other end.

If possible, let there be two such triangles, ABC, ABD,

in which ^C is equal to AD, and BC to BD.

First, let the vertex of each be without the other.

Join CD.

'.' AC is equal to AD,

.*. angle ADC is equal to angle A CD ; [Prop. 5.

( angle BDC is greater than one of these equals,

\ and angle BCD is less than the other,

/. angle BDC is greater than angle BCD; [Ax. Â§ 9.

again, â€¢.* BC is equal to BD,

.'. angle BDC is equal to angle BCD ;

but it is also greater ; which is absurd.

BOOK I. 21

Secondly, let Z), the vertex of one, be within the other.

Join CD', and produce ACy AD, to E, F.

'.- AC \s equal to AD,

.'. angle ECD is equal to angle FDC; [Prop 5.