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IN MEMORIAM

FLORIAN CAJORI

EUCLID'S

ELEMENTS OF GEOMETRY

BOOKS Iâ€” VI.

SonDon: C j. clay and SONS,

CAMBKIDGE UNIVEKSITY PEESS WAREHOUSE,

AVE MARIA LANE.

CAMBRIDGE: DEIGHTON, BELL, AND CO.

LEIPZIG: F. A. BROCKHAUS.

NEW YORK : MACMILLAN AND CO.

SPECIMEN COZY,

^itt iress iMatftematiral ^mt&

EUCLID'S

ELEMENTS OF GEOMETKY

EDITED FOR THE SYNDICS OF THE PRESS

H. M. TAYLOR, M.A.

FELLOW AND FORMERLY TUTOR OF TRINITY COLLEGE, CAMBRIDGE.

BOOKS Iâ€” VI.

CAMBRIDGE

AT THE UNIVEESITY PRESS

1893

[All Rifihts I'eserved.]

Cambritifie :

PBINTED BY C. J. CLAY, M.A. AND SONS,

AT THE UNIVERSITY PRESS.

\

r3

NOTE.

The Special Board for Mathematics in the University

of Cambridge in a Eeport on Geometrical Teaching dated

May 10, 1887, state as follows:

' The majority of the Board are of opinion that the rigid adherence

to Euclid's texts is prejudicial to the interests of education, and

that greater freedom in the method of teaching Geometry is

desirable. As it appears that this greater freedom cannot be

attained while a knowledge of Euclid's text is insisted upon in

the examinations of the University, they consider that such

alterations should be made in the regulations of the examina-

tions as to admit other proofs besides those of Euclid, while

following however his general sequence of propositions, so that

no proof of any proposition occurring in Euclid should be

accepted in which a subsequent proposition in Euclid's order

is assumed.'

On March 8, 1888, Amended Regulations for the

Previous Examination, which contained the following

provision, were approved by the Senate :

' Euclid's definitions will be required, and no axioms or postulates

except Euclid's may be assumed. The actual proofs of propo-

sitions as given in Euclid will not be required, but no proof of

any proposition occurring in Euclid will be admitted in which

use is made of any proposition which in Euclid's order occurs

subsequently.'

And in the Regulations for the Local Examinations

conducted by the University of Cambridge it is provided

that :

â€¢Proofs other than Euclid's will be admitted, but Euclid's Axioms

will be required, and no proof of any proposition will be

accepted which assumes anything not proved in preceding

propositions in Euclid.'

PREFACE TO BOOKS I. AND II.

IT was with extreme diffidence that I accepted an invi-

tation from the Syndics of the Cambridge University

Press to undertake for them a new edition of the Elements

of Euclid. Though I was deeply sensible of the honour,

which the invitation conferred, I could not but recognise

the great responsibility, which the acceptance of it would

entail.

The invitation of the Syndics was in itself, to my mind,

a sign of a widely felt conviction that the editions in

common use were capable of improvement. Now improve-

ment necessitates change, and every change made in a

work, which has been a text book for centuries, must run

the gauntlet of severe criticism, for while some will view

every alteration with aversion, others will consider that

every change demands an apology for the absence of more

and greater changes.

I will here give a short account of the chief points, in

which this edition differs from the best known editions of

the Elements of Euclid at present in use in England.

While the texts of the editions of Potts and Todhunter

are confessedly little more than reprints of Simson's English

version of the Elements published in 175G, the text of the

present edition does not profess to be a translation from

the Greek. I began by retranslating the First Book : but

there proved to be so many points, in which I thought it

M3061G5 *

vi PREFACE.

desirable to depart from the original, tiiat it seemed best

to give up all idea of simple translation and to retain

merely the substance of the work, following closely Euclid's

sequence of Propositions in Books I. and II. at all events.

Some of the definitions of Euclid, for instance trapezium^

rhomboid, gnomon are omitted altogether as unnecessary.

The word trapezium is defined in the Greek to mean " ant/

four sided figure other than those already defined,^' but in

many modern works it is defined to be "Â« quadrilateral,

which has one pair of parallel sides." The first of these

definitions is obsolete, the second is not universally ac-

cepted. On the other hand definitions are added of several

words in general use, such as ^^eWme^er, parallelogram,

diagonal, which do not occur in Euclid's list.

The chief alteration in the definitions is in that of the

word figure, which is in the Greek text defined to be " that

ivhich is enclosed hy one or more boundaries." I have

preferred to define a figure as "a combination of points,

lines and surfaces." That Euclid's definition leads to diffi-

culty is seen from the fact that, though Euclid defines a

circle as "a figure contained by one line...", he demands in

his postulate that "a circle may be described...". Now it is

the circumference of a circle which is described and not

the surface. Again, when two circles intersect, it is the

circumferences which intersect and not the surfaces.

I have rejected the ordinarily received definition of a

square as " a quadrilateral, whose sides are equal, and whose

angles are right angles." There is no doubt that, when we

define any geometrical figure, we postulate the possibility

of the figure ; but it is useless to embrace in the definition

more properties than are requisite to determine the figure.

The word axiom is used in many modern works as

applicable both to simple geometrical propositions, such as

" two straight lines cannot enclose a space," and to proposi-

PREFACE. vii

tions, other than geometrical, accepted without demonstra-

tion and true universally, such as ^Hhe whole of a thing is

greater than a part " These two classes of propositions are

often distinguished by the terms "geometrical axioms" and

"general axioms." I prefer to use the word axiom as appli-

cable to the latter class only, that is, to simple propositions,

true of magnitudes of all kinds (for instance 'Hhings which

are equal to the same thing are equal to one anothefi'"'^), and

to use the term postulate for a simple geometrical proposi-

tion, whose truth we assume.

When a child is told that A weighs exactly as much

as B, and B weighs exactly as much as C, he without

hesitation arrives at the conclusion that A weighs exactly

as much as C. His conviction of the validity of his

conclusion would not be strengthened, and possibly his

confidence in his conclusion might be impaired, by his

being directed to appeal to the authority of the general

proposition " things which are equal to the same thing are

equal to one another." I have therefore, as a rule, omitted

in the text all reference to the general statements of

axioms, and have only introduced such a statement occa-

sionally, where its introduction seemed to me the shortest

way of explaining the nature of the next step in the

demonstration.

If it be objected that all axioms used should be clearly

stated, and that their number should not be unnecessarily

extended, my reply is that neither the Greek text nor any

edition of it, with Avhich I am acquainted, has attempted

to make its list of axioms perfect in either of these respects.

The lists err in excess, inasmuch as some of the axioms

therein can be deduced from others : they err in defect,

inasmuch as in the demonstrations of Propositions conclu-

sions are often drawn, to support the validity of which no

appeal can be made to any axiom in the lists.

viii PREFACE.

Under the term postulate I have included not only what

may be called the postulates of geometrical operation, such

as ^Ht is assumed that a straight line may he drawn from

any point to any other point^^ but also geometrical theorems,

the truth of which we assume, such as ''Hwo straight lines

cannot have a common part.^^

The postulates of this edition are nine in number.

Postulates 3, 4, 6 are the postulates of geometrical

operation, which are common to all editions of the Elements

of Euclid. Postulates 1, 5, 9 are the Axioms 10, 11, 12 of

modern editions. Postulates 2, 7, 8 do not appear under

the head either of axioms or of postulates in Euclid's text,

but the substance of them is assumed in the demonstrations

of his propositions.

Postulate 9 has been postponed until page 51, as it

seemed undesirable to trouble the student with an attempt

to unravel its meaning, until he was prepared to accept it

as the converse of a theorem, with the proof of which he

had already been made acquainted.

It may be mentioned that a proof of Postulate 5, ^^all

right angles are equal" is given in the text (Proposition lOB),

and that therefore the number of the Postulates might

have been diminished by one : it was however thought

necessary to retain this Postulate in the list, so that it

might be used as a postulate by any person who might

prefer to adhere closely to the original text of Euclid.

One important feature in the present edition is the

greater freedom in the direct use of "the method of super-

position " in the proofs of the Propositions. The method is

used directly by Euclid in his proof of Proposition 4 of

Book I., and indirectly in his proofs of Proposition 5 and

of every other Proposition, in which the theorem of

Proposition 4 is quoted. It seems therefore but a slight

alteration to adopt the direct use of this method in the

PREFACE. ix

proofs of any theorems, in the proofs of which, in Euclid's

text, the theorem of Proposition 4 is quoted.

It may of course be fairly objected that it would be

more logical for a writer, who uses with freedom the

method of superposition, to omit the first three Propo-

sitions of Book I. To this objection my reply must be

that it is considered undesirable to alter the numbering

of the Propositions in Books I. and II. at all events. No

doubt a work written merely for the teaching of geometry,

without immediate reference to the requirements of candi-

dates preparing for examination, might well omit the first

three Propositions and assume as a postulate that "a circle

may he described with any point as centre^ and with a

length equal to any given straight line as radius^" instead

of the postulate of Euclid's text (Postulate 6 of the pre-

sent edition), " a circle may he described with any point as

centre and with any straight line drawn from that j)oi7it as

radius."

The use of the words "each to each" has been aban-

doned. The statement that two things are equal to two

other things each to each, seems to imply, according to the

natural meaning of the words, that all four things are

equal to each other. Where we wish to state briefly that

A has a certain relation to a, B has the same relation to

6, and C has the same relation to c, we prefer to say that

A, B, C have this relation to Â«, h, c respectively.

The enunciations of the Propositions in Books I. and

II. have been, with some few slight exceptions, retained

throughout, and the order of the Propositions remains

unaltered, but different methods of proof have been adopted

in many cases. The chief instances of alteration are to be

found in Propositions 5 and 6 of Book I., and in Book II.

The use of what may be called impossible figures, such

as occurred in Euclid's text in the proofs of Propositions

X PREFACE.

6 and 7 of Book I. has been avoided. It seems better to

prove that a line cannot be drawn satisfying a certain condi-

tion without making a pretence of doing what is impossible.

Two Propositions (10 A and 10 B), have been introduced

to shew that, if the method of superposition be used, we

need not take as a postulate ^^ all rigid angles are equal to

one another,'^ but that we may deduce this theorem from

other postulates which have been already assumed.

Another new Proposition introduced into the text is

Proposition 26 A, '[if two triangles have two sides equal to

two sides, and the angles opposite to one ^;azV of equal sides

equal, the angles opposite to the other pair are either equal or

supplementary/," which may be described, with reference to

Euclid's text, as the missing case of the equality of two

triangles. It is intimately connected with what is called

in Trigonometry " the ambiguous case " in the solution of

triangles.

Another new Proposition (41 A) is the solution of the

problem " to construct a triangle equal to a given rectilineal

figure." It appears to be a more practical method of solving

the general problem of Proposition 45 "^o construct aj)aral-

lelogram equal to a given rectilineal figure, having a side

equal to a given straight line, and having an angle equal to

a given angle," to begin with the construction of a triangle

equal to the given figure rather than to follow the exact

sequence of Euclid's propositions.

In the notes a few ''Additional Propositions" have

been introduced containing important theorems, which did

not occur in Euclid's text, but with which it is desirable

that the student should become familiar as early as

possible. Also outlines have been given of some of tho

many different proofs which have been discovered of

â– Pythagoras's Theorem. They may be found interesting and

useful as exercises for the student.

PREFACE. xi

Euclid's proofs of many of the Propositions of Book II.

are unnecessarily long. His use of the diagonal of the

square in his constructions in Propositions 4 to 8 can

scarcely be considered elegant.

It is curious to notice that Euclid after giving a

demonstration of Proposition 1 makes no use whatever

of the theorem. It seems more logical to deduce from

Proposition 1 those of the subsequent Propositions whicli

can be readily so deduced.

In Book II. outlines of alternative proofs of several of

the Propositions have been given, which may be developed

more fully and used in examinations, in place of the

proofs given in the text. Some of these proofs are not,

so far as I know, to be found in English text books.

The most interesting ones are those of Propositions 12

and 13. Some, which I thought at first were new, I have

since found in foreign text books.

The Propositions in the text have not been distin-

guished by the words "Theorem" and "Problem." The

student may be informed once for all that the word

theorem is used of a geometrical truth which is to be

demonstrated, and that the word problem is used of a

geometrical construction which is to be performed.

Although Euclid always sums up the result of a Propo-

sition by the words oinp ISct Sct^ai or ottc/j t3et Trotryo-ai,

there seems to be no utility in putting the letters q.e.d.

or Q.E.F. at the end of a Proposition in an English text-

book. The words " Quod erat demonstrandum " or " Quod

erat faciendum " in a Latin text were not out of place.

"When the book is opened, the reader will see as a rule

on the left hand page a Pi-oposition, and on the opposite

page notes or exercises. The notes are either appropriate

to the Proposition they face or introductory to the one

next succeeding. The exercises on the right hand page are.

xii PREFACE,

it is hoped, in all cases capable of being solved by means of

the Proposition on the adjoining page and of preceding

Propositions. They have been chosen with care and with

the special view of inducing the student from the com-

mencement of his reading to attempt for himself the

solution of exercises.

For many Propositions it has been difficult to find

suitable exercises : consequently many of the exercises have

been specially manufactured for the Propositions to which

they are attached. Great pains have been taken to verify

the exercises, but notwithstanding it can scarcely be hoped

that all trace of error has been eliminated.

It is with pleasure that I record here my deep sense

of obligation to many friends, who have aided me by

valuable hints and suggestions, and more especially to

A. R. Forsyth, M.A., Fellow and Assistant Tutor of

Trinity College, Charles Smith, M.A., Fellow and Tutor of

Sidney Sussex College, R. T. Wright, M.A., formerly

Fellow and Tutor of Christ's College, my brother-in-law

the Reverend T. J. Sanderson, M.A., formerly Fellow of

Clare College, and my brother- W. W. Taylor, M.A.,

formerly Scholar of Queen's College, Oxford, and after-

wards Scholar of Trinity College, Cambridge. The time

and trouble ungrudgingly spent by these gentlemen on

this edition have saved it from many blemishes, which

would otherwise have disfigured its pages.

I shall be grateful for any corrections or criticisms,

which may be forwarded to me in connection either with

the exercises or with any other part of the work.

H. M. TAYLOR.

Trinity College, Cambridge,

October 1, 1889.

PREFACE TO BOOKS III AND IV.

IN Book III. the chief deviation from Euclid's text will

be found in the first twelve Propositions, where a good

deal of rearrangement has been thought desirable. This

rearrangement has led to some changes in the sequence

of Propositions as well as in the Propositions themselves ;

but, even with these changes, the first twelve Propositions

will be found to include the substance of the whole of the

first twelve of Euclid's text.

The Propositions from 13 to 37 are, except in unim-

portant details, unchanged in substance and in order.

The enunciation of the theorem of Proposition 36 has

been altered to make it more closely resemble that of the

complementary theorem of Proposition 35.

An additional Proposition has been introduced on page

186 involving the principle of the rotation of a plane

figure about a point in its plane. It is a principle of

which extensive use might with advantage be made in the

proof of some of the simpler' properties of the circle.

It has not however been thought desirable to do more

in this edition than to introduce the student to this

method and by a selection of exercises, which can readily

be solved by its means, to indicate the importance of the

method.

PREFACE TO BOOKS V. AND VI.

of the Greek text is printed in brackets at the lieacl of each

Proposition.

In Book VI a slight departure from Euclid's text is

made in the treatment of similar figures. The definition of

similar polygons which is adopted in this work brings into

prominence the important property of the fixed ratio of

their corresponding sides. Its use has the great merit of

tending at once to simplicity and brevity in the proofs of

many theorems.

The numbering of the Propositions in Book VI remains

unchanged : Propositions 27, 28, 29 are omitted as in many

of the recent English editions of Euclid, and iti several cases

a Proposition which consists of a theorem and its converse

is divided into two Parts. Proposition 32 of Euclid's text,

which is a very special case of no great interest, has been

replaced by a simple but important theorem in the theory

of similar and similarly situate figures.

The chief difficulty with respect to the additions which

have been made to Book VI was the immense number of

known theorems from which a selection had to be made.

I have attempted by means of two or three series of

Propositions arranged in something like logical sequence to

introduce the student to important general methods or

well-known interesting results.

One series gives a sketch of the theory of transversals,

and the properties of harmonic and anharmonic ranges and

pencils, and leads up to Pascal's Theorem. Another series

deals with similar and similarly situate figures and leads up

PREFACE TO BOOKS V. AND VI.

to Gergoime's elegant solution of the problem to describe a

circle to touch three given circles. These are followed by

an introduction to the method of Inversion, an account of

Casey's extension of Ptolemy's Theorem, some of the im-

portant properties of coaxial circles, and Poncelet's Theorems

relating to the porisms connected with a series of coaxial

circles.

No attempt has been made to represent the very large

and still increasing collection of theorems connected with

the "Modern Geometry of the Triangle."

I hereby acknowledge the great help I have received in

this portion of my work from friends, and especially from

Dr Forsyth and from my brother Mr J. H. Taylor. To the

latter I am indebted for the Index to Books I â€” VI, which

I hope may prove of some assistance to persons using this

edition.

H. M. T.

TuiNiTY College, Cambridge,

March 16, 1893.

THE ELEMENTS OF GEOMETRY.

T. E.

BOOK I.

Definition 1. That 'which has ^wsition but not magni-

tude is called a point.

The word point is used in many different senses. We speak in

ordinary language of the point of a pin, of a pen or of a pencil.

Any mark made with such a point on paper is of some definite size

and is in some definite position. A small mark is often called a spot

or a dot. Suppose such a spot to become smaller and smaller ; the

smaller it becomes the more nearly it resembles a geometrical point :

but it is only when the spot has become so small that it is on the

point of vanishing altogether, i.e. when in fact the spot still has

position but has no magnitude, that it answers to the geometrical

definition of a point.

A point is generally denoted by a single letter of the alphabet:

for instance we speak of the point A.

Definition 2. That which has position and length but

neither breadth nor thickness is called a line.

The extremities of a line are points.

The intersections of lines are points

DEFINITIONS. 3

The word line also is used in many different senses in ordinary

language, and in most of these senses the main idea suggested is that

of length. For instance we speak of a line of railway as connecting

two distant towns, or of a sounding line as reaching from the bottom

of the sea to the surface, and in so speaking we seldom think of the

breadth of the railway or of the thickness of the sounding line.

When we speak of a geometrical line, we regard merely the length:

we exclude the idea of breadth and thickness altogether : in fact we

consider that the cross-section of the line is of no size, or in other

words that the cross-section is a geometrical point.

If a point move with a continuous motion from one position to

another, the path which it describes during the motion is a line.

Definition 3. That which has position, length and

breadth hut not thickness is called a surface.

The boundaries of a surface are lines.

The intersections of surfaces are lines.

The word surface in ordinary language conveys the idea of ex-

tension in two directions : for instance we speak of the surface of the

Earth, the surface of the sea, the surface of a sheet of paper.

Although in some cases the idea of the thickness or the depth of the

thing spoken of may be present in the speaker's mind, yet as a rule

no stress is laid on depth or thickness. When we speak of a geome-

trical surface we put aside the idea of depth and thickness altogether.

We are told that it takes more than 300,000 sheets of gold leaf to

make an inch of thickness ; but although the gold leaf is so thin, it

must not be regarded as a geometrical surface. In fact each leaf

however thin has always two bounding surfaces. The geometrical

surface is to be regarded as absolutely devoid of thickness, and no

number of surfaces put together would make any thickness whatever.

Definition 4. That tohich has jjosition, length, breadth

and thickness is called a solid.

The boundaries of solids are surfaces.

1â€”2

4 BOOK I.

Definition 5. Any combination of points, lines, and

surfaces is called a figure.

Definition 6. A line which lies evenly between points

on it is called a straight line.

This is Euclid's definition of a straight line. It cannot be turned

to practical use by itself. We supplement the definition, as Euclid

did, by making some assumptions the nature of which wUl be seen

hereafter.

Postulates. There are a few geometrical propositions

so obvious that we take the truth of them for granted, and

a few geometrical operations so simple that we assume we

may perform them when we please without giving any

explanation of the process. The claim we make to use any

one of these propositions, or to perform any one of these

operations, is called a postulate.

Postulate 1. Two straight lines cannot enclose a space.

This postulate is equivalent to

Two straight lines cannot intersect in more than one point.

Postulate 2. Two straight lines cannot have a common

pa/rt.

If two straight lines have two points ^ , jB in common, they must

coincide between A and 5, since, if they did not, the two straight

lines would enclose a space. Again, they must coincide beyond A

IN MEMORIAM

FLORIAN CAJORI

EUCLID'S

ELEMENTS OF GEOMETRY

BOOKS Iâ€” VI.

SonDon: C j. clay and SONS,

CAMBKIDGE UNIVEKSITY PEESS WAREHOUSE,

AVE MARIA LANE.

CAMBRIDGE: DEIGHTON, BELL, AND CO.

LEIPZIG: F. A. BROCKHAUS.

NEW YORK : MACMILLAN AND CO.

SPECIMEN COZY,

^itt iress iMatftematiral ^mt&

EUCLID'S

ELEMENTS OF GEOMETKY

EDITED FOR THE SYNDICS OF THE PRESS

H. M. TAYLOR, M.A.

FELLOW AND FORMERLY TUTOR OF TRINITY COLLEGE, CAMBRIDGE.

BOOKS Iâ€” VI.

CAMBRIDGE

AT THE UNIVEESITY PRESS

1893

[All Rifihts I'eserved.]

Cambritifie :

PBINTED BY C. J. CLAY, M.A. AND SONS,

AT THE UNIVERSITY PRESS.

\

r3

NOTE.

The Special Board for Mathematics in the University

of Cambridge in a Eeport on Geometrical Teaching dated

May 10, 1887, state as follows:

' The majority of the Board are of opinion that the rigid adherence

to Euclid's texts is prejudicial to the interests of education, and

that greater freedom in the method of teaching Geometry is

desirable. As it appears that this greater freedom cannot be

attained while a knowledge of Euclid's text is insisted upon in

the examinations of the University, they consider that such

alterations should be made in the regulations of the examina-

tions as to admit other proofs besides those of Euclid, while

following however his general sequence of propositions, so that

no proof of any proposition occurring in Euclid should be

accepted in which a subsequent proposition in Euclid's order

is assumed.'

On March 8, 1888, Amended Regulations for the

Previous Examination, which contained the following

provision, were approved by the Senate :

' Euclid's definitions will be required, and no axioms or postulates

except Euclid's may be assumed. The actual proofs of propo-

sitions as given in Euclid will not be required, but no proof of

any proposition occurring in Euclid will be admitted in which

use is made of any proposition which in Euclid's order occurs

subsequently.'

And in the Regulations for the Local Examinations

conducted by the University of Cambridge it is provided

that :

â€¢Proofs other than Euclid's will be admitted, but Euclid's Axioms

will be required, and no proof of any proposition will be

accepted which assumes anything not proved in preceding

propositions in Euclid.'

PREFACE TO BOOKS I. AND II.

IT was with extreme diffidence that I accepted an invi-

tation from the Syndics of the Cambridge University

Press to undertake for them a new edition of the Elements

of Euclid. Though I was deeply sensible of the honour,

which the invitation conferred, I could not but recognise

the great responsibility, which the acceptance of it would

entail.

The invitation of the Syndics was in itself, to my mind,

a sign of a widely felt conviction that the editions in

common use were capable of improvement. Now improve-

ment necessitates change, and every change made in a

work, which has been a text book for centuries, must run

the gauntlet of severe criticism, for while some will view

every alteration with aversion, others will consider that

every change demands an apology for the absence of more

and greater changes.

I will here give a short account of the chief points, in

which this edition differs from the best known editions of

the Elements of Euclid at present in use in England.

While the texts of the editions of Potts and Todhunter

are confessedly little more than reprints of Simson's English

version of the Elements published in 175G, the text of the

present edition does not profess to be a translation from

the Greek. I began by retranslating the First Book : but

there proved to be so many points, in which I thought it

M3061G5 *

vi PREFACE.

desirable to depart from the original, tiiat it seemed best

to give up all idea of simple translation and to retain

merely the substance of the work, following closely Euclid's

sequence of Propositions in Books I. and II. at all events.

Some of the definitions of Euclid, for instance trapezium^

rhomboid, gnomon are omitted altogether as unnecessary.

The word trapezium is defined in the Greek to mean " ant/

four sided figure other than those already defined,^' but in

many modern works it is defined to be "Â« quadrilateral,

which has one pair of parallel sides." The first of these

definitions is obsolete, the second is not universally ac-

cepted. On the other hand definitions are added of several

words in general use, such as ^^eWme^er, parallelogram,

diagonal, which do not occur in Euclid's list.

The chief alteration in the definitions is in that of the

word figure, which is in the Greek text defined to be " that

ivhich is enclosed hy one or more boundaries." I have

preferred to define a figure as "a combination of points,

lines and surfaces." That Euclid's definition leads to diffi-

culty is seen from the fact that, though Euclid defines a

circle as "a figure contained by one line...", he demands in

his postulate that "a circle may be described...". Now it is

the circumference of a circle which is described and not

the surface. Again, when two circles intersect, it is the

circumferences which intersect and not the surfaces.

I have rejected the ordinarily received definition of a

square as " a quadrilateral, whose sides are equal, and whose

angles are right angles." There is no doubt that, when we

define any geometrical figure, we postulate the possibility

of the figure ; but it is useless to embrace in the definition

more properties than are requisite to determine the figure.

The word axiom is used in many modern works as

applicable both to simple geometrical propositions, such as

" two straight lines cannot enclose a space," and to proposi-

PREFACE. vii

tions, other than geometrical, accepted without demonstra-

tion and true universally, such as ^Hhe whole of a thing is

greater than a part " These two classes of propositions are

often distinguished by the terms "geometrical axioms" and

"general axioms." I prefer to use the word axiom as appli-

cable to the latter class only, that is, to simple propositions,

true of magnitudes of all kinds (for instance 'Hhings which

are equal to the same thing are equal to one anothefi'"'^), and

to use the term postulate for a simple geometrical proposi-

tion, whose truth we assume.

When a child is told that A weighs exactly as much

as B, and B weighs exactly as much as C, he without

hesitation arrives at the conclusion that A weighs exactly

as much as C. His conviction of the validity of his

conclusion would not be strengthened, and possibly his

confidence in his conclusion might be impaired, by his

being directed to appeal to the authority of the general

proposition " things which are equal to the same thing are

equal to one another." I have therefore, as a rule, omitted

in the text all reference to the general statements of

axioms, and have only introduced such a statement occa-

sionally, where its introduction seemed to me the shortest

way of explaining the nature of the next step in the

demonstration.

If it be objected that all axioms used should be clearly

stated, and that their number should not be unnecessarily

extended, my reply is that neither the Greek text nor any

edition of it, with Avhich I am acquainted, has attempted

to make its list of axioms perfect in either of these respects.

The lists err in excess, inasmuch as some of the axioms

therein can be deduced from others : they err in defect,

inasmuch as in the demonstrations of Propositions conclu-

sions are often drawn, to support the validity of which no

appeal can be made to any axiom in the lists.

viii PREFACE.

Under the term postulate I have included not only what

may be called the postulates of geometrical operation, such

as ^Ht is assumed that a straight line may he drawn from

any point to any other point^^ but also geometrical theorems,

the truth of which we assume, such as ''Hwo straight lines

cannot have a common part.^^

The postulates of this edition are nine in number.

Postulates 3, 4, 6 are the postulates of geometrical

operation, which are common to all editions of the Elements

of Euclid. Postulates 1, 5, 9 are the Axioms 10, 11, 12 of

modern editions. Postulates 2, 7, 8 do not appear under

the head either of axioms or of postulates in Euclid's text,

but the substance of them is assumed in the demonstrations

of his propositions.

Postulate 9 has been postponed until page 51, as it

seemed undesirable to trouble the student with an attempt

to unravel its meaning, until he was prepared to accept it

as the converse of a theorem, with the proof of which he

had already been made acquainted.

It may be mentioned that a proof of Postulate 5, ^^all

right angles are equal" is given in the text (Proposition lOB),

and that therefore the number of the Postulates might

have been diminished by one : it was however thought

necessary to retain this Postulate in the list, so that it

might be used as a postulate by any person who might

prefer to adhere closely to the original text of Euclid.

One important feature in the present edition is the

greater freedom in the direct use of "the method of super-

position " in the proofs of the Propositions. The method is

used directly by Euclid in his proof of Proposition 4 of

Book I., and indirectly in his proofs of Proposition 5 and

of every other Proposition, in which the theorem of

Proposition 4 is quoted. It seems therefore but a slight

alteration to adopt the direct use of this method in the

PREFACE. ix

proofs of any theorems, in the proofs of which, in Euclid's

text, the theorem of Proposition 4 is quoted.

It may of course be fairly objected that it would be

more logical for a writer, who uses with freedom the

method of superposition, to omit the first three Propo-

sitions of Book I. To this objection my reply must be

that it is considered undesirable to alter the numbering

of the Propositions in Books I. and II. at all events. No

doubt a work written merely for the teaching of geometry,

without immediate reference to the requirements of candi-

dates preparing for examination, might well omit the first

three Propositions and assume as a postulate that "a circle

may he described with any point as centre^ and with a

length equal to any given straight line as radius^" instead

of the postulate of Euclid's text (Postulate 6 of the pre-

sent edition), " a circle may he described with any point as

centre and with any straight line drawn from that j)oi7it as

radius."

The use of the words "each to each" has been aban-

doned. The statement that two things are equal to two

other things each to each, seems to imply, according to the

natural meaning of the words, that all four things are

equal to each other. Where we wish to state briefly that

A has a certain relation to a, B has the same relation to

6, and C has the same relation to c, we prefer to say that

A, B, C have this relation to Â«, h, c respectively.

The enunciations of the Propositions in Books I. and

II. have been, with some few slight exceptions, retained

throughout, and the order of the Propositions remains

unaltered, but different methods of proof have been adopted

in many cases. The chief instances of alteration are to be

found in Propositions 5 and 6 of Book I., and in Book II.

The use of what may be called impossible figures, such

as occurred in Euclid's text in the proofs of Propositions

X PREFACE.

6 and 7 of Book I. has been avoided. It seems better to

prove that a line cannot be drawn satisfying a certain condi-

tion without making a pretence of doing what is impossible.

Two Propositions (10 A and 10 B), have been introduced

to shew that, if the method of superposition be used, we

need not take as a postulate ^^ all rigid angles are equal to

one another,'^ but that we may deduce this theorem from

other postulates which have been already assumed.

Another new Proposition introduced into the text is

Proposition 26 A, '[if two triangles have two sides equal to

two sides, and the angles opposite to one ^;azV of equal sides

equal, the angles opposite to the other pair are either equal or

supplementary/," which may be described, with reference to

Euclid's text, as the missing case of the equality of two

triangles. It is intimately connected with what is called

in Trigonometry " the ambiguous case " in the solution of

triangles.

Another new Proposition (41 A) is the solution of the

problem " to construct a triangle equal to a given rectilineal

figure." It appears to be a more practical method of solving

the general problem of Proposition 45 "^o construct aj)aral-

lelogram equal to a given rectilineal figure, having a side

equal to a given straight line, and having an angle equal to

a given angle," to begin with the construction of a triangle

equal to the given figure rather than to follow the exact

sequence of Euclid's propositions.

In the notes a few ''Additional Propositions" have

been introduced containing important theorems, which did

not occur in Euclid's text, but with which it is desirable

that the student should become familiar as early as

possible. Also outlines have been given of some of tho

many different proofs which have been discovered of

â– Pythagoras's Theorem. They may be found interesting and

useful as exercises for the student.

PREFACE. xi

Euclid's proofs of many of the Propositions of Book II.

are unnecessarily long. His use of the diagonal of the

square in his constructions in Propositions 4 to 8 can

scarcely be considered elegant.

It is curious to notice that Euclid after giving a

demonstration of Proposition 1 makes no use whatever

of the theorem. It seems more logical to deduce from

Proposition 1 those of the subsequent Propositions whicli

can be readily so deduced.

In Book II. outlines of alternative proofs of several of

the Propositions have been given, which may be developed

more fully and used in examinations, in place of the

proofs given in the text. Some of these proofs are not,

so far as I know, to be found in English text books.

The most interesting ones are those of Propositions 12

and 13. Some, which I thought at first were new, I have

since found in foreign text books.

The Propositions in the text have not been distin-

guished by the words "Theorem" and "Problem." The

student may be informed once for all that the word

theorem is used of a geometrical truth which is to be

demonstrated, and that the word problem is used of a

geometrical construction which is to be performed.

Although Euclid always sums up the result of a Propo-

sition by the words oinp ISct Sct^ai or ottc/j t3et Trotryo-ai,

there seems to be no utility in putting the letters q.e.d.

or Q.E.F. at the end of a Proposition in an English text-

book. The words " Quod erat demonstrandum " or " Quod

erat faciendum " in a Latin text were not out of place.

"When the book is opened, the reader will see as a rule

on the left hand page a Pi-oposition, and on the opposite

page notes or exercises. The notes are either appropriate

to the Proposition they face or introductory to the one

next succeeding. The exercises on the right hand page are.

xii PREFACE,

it is hoped, in all cases capable of being solved by means of

the Proposition on the adjoining page and of preceding

Propositions. They have been chosen with care and with

the special view of inducing the student from the com-

mencement of his reading to attempt for himself the

solution of exercises.

For many Propositions it has been difficult to find

suitable exercises : consequently many of the exercises have

been specially manufactured for the Propositions to which

they are attached. Great pains have been taken to verify

the exercises, but notwithstanding it can scarcely be hoped

that all trace of error has been eliminated.

It is with pleasure that I record here my deep sense

of obligation to many friends, who have aided me by

valuable hints and suggestions, and more especially to

A. R. Forsyth, M.A., Fellow and Assistant Tutor of

Trinity College, Charles Smith, M.A., Fellow and Tutor of

Sidney Sussex College, R. T. Wright, M.A., formerly

Fellow and Tutor of Christ's College, my brother-in-law

the Reverend T. J. Sanderson, M.A., formerly Fellow of

Clare College, and my brother- W. W. Taylor, M.A.,

formerly Scholar of Queen's College, Oxford, and after-

wards Scholar of Trinity College, Cambridge. The time

and trouble ungrudgingly spent by these gentlemen on

this edition have saved it from many blemishes, which

would otherwise have disfigured its pages.

I shall be grateful for any corrections or criticisms,

which may be forwarded to me in connection either with

the exercises or with any other part of the work.

H. M. TAYLOR.

Trinity College, Cambridge,

October 1, 1889.

PREFACE TO BOOKS III AND IV.

IN Book III. the chief deviation from Euclid's text will

be found in the first twelve Propositions, where a good

deal of rearrangement has been thought desirable. This

rearrangement has led to some changes in the sequence

of Propositions as well as in the Propositions themselves ;

but, even with these changes, the first twelve Propositions

will be found to include the substance of the whole of the

first twelve of Euclid's text.

The Propositions from 13 to 37 are, except in unim-

portant details, unchanged in substance and in order.

The enunciation of the theorem of Proposition 36 has

been altered to make it more closely resemble that of the

complementary theorem of Proposition 35.

An additional Proposition has been introduced on page

186 involving the principle of the rotation of a plane

figure about a point in its plane. It is a principle of

which extensive use might with advantage be made in the

proof of some of the simpler' properties of the circle.

It has not however been thought desirable to do more

in this edition than to introduce the student to this

method and by a selection of exercises, which can readily

be solved by its means, to indicate the importance of the

method.

PREFACE TO BOOKS V. AND VI.

of the Greek text is printed in brackets at the lieacl of each

Proposition.

In Book VI a slight departure from Euclid's text is

made in the treatment of similar figures. The definition of

similar polygons which is adopted in this work brings into

prominence the important property of the fixed ratio of

their corresponding sides. Its use has the great merit of

tending at once to simplicity and brevity in the proofs of

many theorems.

The numbering of the Propositions in Book VI remains

unchanged : Propositions 27, 28, 29 are omitted as in many

of the recent English editions of Euclid, and iti several cases

a Proposition which consists of a theorem and its converse

is divided into two Parts. Proposition 32 of Euclid's text,

which is a very special case of no great interest, has been

replaced by a simple but important theorem in the theory

of similar and similarly situate figures.

The chief difficulty with respect to the additions which

have been made to Book VI was the immense number of

known theorems from which a selection had to be made.

I have attempted by means of two or three series of

Propositions arranged in something like logical sequence to

introduce the student to important general methods or

well-known interesting results.

One series gives a sketch of the theory of transversals,

and the properties of harmonic and anharmonic ranges and

pencils, and leads up to Pascal's Theorem. Another series

deals with similar and similarly situate figures and leads up

PREFACE TO BOOKS V. AND VI.

to Gergoime's elegant solution of the problem to describe a

circle to touch three given circles. These are followed by

an introduction to the method of Inversion, an account of

Casey's extension of Ptolemy's Theorem, some of the im-

portant properties of coaxial circles, and Poncelet's Theorems

relating to the porisms connected with a series of coaxial

circles.

No attempt has been made to represent the very large

and still increasing collection of theorems connected with

the "Modern Geometry of the Triangle."

I hereby acknowledge the great help I have received in

this portion of my work from friends, and especially from

Dr Forsyth and from my brother Mr J. H. Taylor. To the

latter I am indebted for the Index to Books I â€” VI, which

I hope may prove of some assistance to persons using this

edition.

H. M. T.

TuiNiTY College, Cambridge,

March 16, 1893.

THE ELEMENTS OF GEOMETRY.

T. E.

BOOK I.

Definition 1. That 'which has ^wsition but not magni-

tude is called a point.

The word point is used in many different senses. We speak in

ordinary language of the point of a pin, of a pen or of a pencil.

Any mark made with such a point on paper is of some definite size

and is in some definite position. A small mark is often called a spot

or a dot. Suppose such a spot to become smaller and smaller ; the

smaller it becomes the more nearly it resembles a geometrical point :

but it is only when the spot has become so small that it is on the

point of vanishing altogether, i.e. when in fact the spot still has

position but has no magnitude, that it answers to the geometrical

definition of a point.

A point is generally denoted by a single letter of the alphabet:

for instance we speak of the point A.

Definition 2. That which has position and length but

neither breadth nor thickness is called a line.

The extremities of a line are points.

The intersections of lines are points

DEFINITIONS. 3

The word line also is used in many different senses in ordinary

language, and in most of these senses the main idea suggested is that

of length. For instance we speak of a line of railway as connecting

two distant towns, or of a sounding line as reaching from the bottom

of the sea to the surface, and in so speaking we seldom think of the

breadth of the railway or of the thickness of the sounding line.

When we speak of a geometrical line, we regard merely the length:

we exclude the idea of breadth and thickness altogether : in fact we

consider that the cross-section of the line is of no size, or in other

words that the cross-section is a geometrical point.

If a point move with a continuous motion from one position to

another, the path which it describes during the motion is a line.

Definition 3. That which has position, length and

breadth hut not thickness is called a surface.

The boundaries of a surface are lines.

The intersections of surfaces are lines.

The word surface in ordinary language conveys the idea of ex-

tension in two directions : for instance we speak of the surface of the

Earth, the surface of the sea, the surface of a sheet of paper.

Although in some cases the idea of the thickness or the depth of the

thing spoken of may be present in the speaker's mind, yet as a rule

no stress is laid on depth or thickness. When we speak of a geome-

trical surface we put aside the idea of depth and thickness altogether.

We are told that it takes more than 300,000 sheets of gold leaf to

make an inch of thickness ; but although the gold leaf is so thin, it

must not be regarded as a geometrical surface. In fact each leaf

however thin has always two bounding surfaces. The geometrical

surface is to be regarded as absolutely devoid of thickness, and no

number of surfaces put together would make any thickness whatever.

Definition 4. That tohich has jjosition, length, breadth

and thickness is called a solid.

The boundaries of solids are surfaces.

1â€”2

4 BOOK I.

Definition 5. Any combination of points, lines, and

surfaces is called a figure.

Definition 6. A line which lies evenly between points

on it is called a straight line.

This is Euclid's definition of a straight line. It cannot be turned

to practical use by itself. We supplement the definition, as Euclid

did, by making some assumptions the nature of which wUl be seen

hereafter.

Postulates. There are a few geometrical propositions

so obvious that we take the truth of them for granted, and

a few geometrical operations so simple that we assume we

may perform them when we please without giving any

explanation of the process. The claim we make to use any

one of these propositions, or to perform any one of these

operations, is called a postulate.

Postulate 1. Two straight lines cannot enclose a space.

This postulate is equivalent to

Two straight lines cannot intersect in more than one point.

Postulate 2. Two straight lines cannot have a common

pa/rt.

If two straight lines have two points ^ , jB in common, they must

coincide between A and 5, since, if they did not, the two straight

lines would enclose a space. Again, they must coincide beyond A