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3^



IN MEMORIAM
FLORIAN CAJORI




EUCLID'S
ELEMENTS OF GEOMETRY

BOOKS I— 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 I— 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 :

•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 "« 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 «, 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
■ 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 — 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.

1—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



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