A text-book of Euclid's Elements : for the use of schools : Books I-VI and XI online

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\A It" Rights reserved.]

Richard Clay and Sons, Limited,
london and bungay.

Fir»t Edition 18S8.

Second Edition (Book XI. added) 18S9.

Reprinted 1890. 1891.


This volume contains the first Six Books of Euclid's
Elements, together with Appendices giving the most im-
portant elementary developments of Euclidean Geometry.

The text has been carefully revised, and special atten-
tion given to those points which experience has shewn to
present difficulties to beginners.

In the course of this revision the Enunciations have
been altered as little as possible; and, except in Book V.,
very few departures have been made from Euclid's proofs:
in each case changes have been adopted only where the old
text has been generally found a cause of difficulty; and
such changes are for the most part in favour of well-recog-
nised alternatives.

For example, the ambiguity has been removed from the
Enunciations of Propositions 18 and 19 of Book I.: the
fact that Propositions 8 and 26 establish the complete
identical equality of the two triangles considered has been
strongly urged; and thus the redundant step has been
removed from Proposition 34. In Book II. Simson's ar-
rangement of Proposition 13 has been abandoned for a
well-known alternative proof. In Book III. Proposition
25 is not given at length, and its place is taken by a

/! /^


simple equivalent. Propositions 35 and 36 have been
treated generally, and it has not been tliought necessary
to do more than call attention in a note to the special
cases. Finally, in Book VI. we have adopted an alterna-
tive proof of Proposition 7, a theorem wliich has been too
much neglected, owing to the cumbrous form in whicli it
has been usually given.

These are the chief deviations from the ordinary text
as regards method and arrangement of proof: they are
points familiar as difficulties to most teachers, and to name
them indicates sufficiently, without further enumeration,
the general principles which have guided our revision.

A few alternative proofs of difficult propositions are
given for the convenience of those teachers who care to
use them.

With regard to Book V. we have established the princi-
pal propositions, both from the algebraical and geometrical
definitions of ratio and proportion, and we have endeavoured
to bring out clearly the distinction between these two modes
of treatment.

In compiling the geometrical section of Book V. we
have followed the system first advocated by the late Pro-
fessor De Morgan ; and here we derived very material
assistance from the exposition of the subject given in the
text-book of the Association for the Improvement of Geo-
metrical Teaching. To this source we are indebted for the
improved and more precise wording of definitions (as given
on pages 286, 288 to 291), as well as for the order and
substance of most of the propositions which appear between
pages 297 and 306. But as we have not (except in the
points above mentioned) adhered verbally to the text of
the Association, we are anxious, while expressing in the
fullest manner our obligation to their work, to exempt the


Association from all responsibility for our treatment of the

One purpose of the book is to gradually familiarise the
student with the use of legitimate symbols and abbrevia-
tions; for a geometrical argument may thus be thrown into
a form which is not only more readily seized by an advanced
reader, but is useful as a guide to the way in which Euclid's
propositions may be handled in written work. On the
other hand, we think it very desirable to defer the intro-
duction of symbols until the beginner has learnt that they
can only be properly used in Pure Geometry as abbrevia-
tions for verbal argument: and we hope thus to prevent
the slovenly and inaccurate habits which are very apt to
arise from their employment before this principle is fully

Accordingly in Book I. we have used no contractions
or symbols of any kind, though we have introduced verbal
alterations into the text wherever it appeared that con-
ciseness or clearness would be gained.

In Book II. abbreviated forms of constantly recurring
words are used, and the phrases therefore and is 'equal to
are replaced by the usual symbols.

In the Third and following Books, and in additional
matter throughout the whole, we have employed all such
signs and abbreviations as we believe to add to the clear-
ness of the reasoning, care being taken that the symbols
chosen are compatible with a rigorous geometrical method,
and are recognised by the majority of teachers.

It must be understood that our use of symbols, and the
removal of unnecessary verbiage and repetition, by no
means implies a desire to secure brevity at all hazards.
On the contrary, nothing appears to us more mischievous
than an abridgement which is attained by omitting


steps, or condensing two or more steps into one. Such
uses spring from the pressure of examinations; but an
examination is not, or ought not to be, a mere race; and
while we wish to indicate generally in the later books how
a geometrical argument may be abbreviated for the pur-
poses of written work, we have not thought well to reduce
the propositions to the bare skeleton so often presented to
an Examiner. Indeed it does not follow that the form most
suitable for the page of a text-book is also best adapted
to examination purposes; for the object to be attained
in each case is entirely different. The text-book should
present the argument in the clearest possible manner to the
mind of a reader to whom it is new : the written proposition
need only convey to the Examiner the assurance that the
proposition has been thoroughly grasped and remembered
by the pupil.

From first to last we have kept in mind the undoubted
fact that a very small proportion of those who study Ele-
mentary Geometry, and study it with profit, are destined
to become mathematicians in any real sense; and that to
a large majority of students, Euclid is intended to serve
not so much as a first lesson in mathematical reasoning,
as the first, and sometimes the only, model of formal and
rigid argument presented in an elementary education.

This consideration has determined not only the full
treatment of the earlier Books, but the retention of the
formal, if somewhat cumbrous, methods of Euclid in many
places where proofs of greater brevity and mathematical
elegance are available.

We hope that the additional matter introduced into
the book will provide sufficient exercise for pupils whose
study of Euclid is preliminary to a mathematical edu-


The questions distributed tlirough the text follow very
easily from the propositions to which they are attached,
and we think that teachers are likely to find in them
all that is needed for an average pupil reading the subject
for the first time.

The Theorems and Examples at the end of each Book
contain questions of a slightly more difficult type : they
have been very carefully classified and arranged, and brought
into close connection with typical examples worked out
either partially or in full ; and it is hoped that this section
of the book, on which much thought has been expended,
will do something towards removing that extreme want of
freedom in solving deductions that is so commonly found
even among students who have a good knowledge of the
text of Euclid.

In the course of our work we have made ourselves
acquainted with most modern English books on Euclidean
Geometry : among these we have already expressed our
special indebtedness to the text-book recently published by
the Association for the Improvement of Geometrical Teach-
ing; and we must also mention the Edition of Euclid's Ele-
ments prepared by Mr J. S. Mackay, whose historical notes
and frequent references to original authorities have been of
the utmost service to us.

Our treatment of Maxima and Minima on page 239 is
based upon suggestions deriyed from a discussion of the
subject which took place at the annual meeting of the
Geometrical Association in January 1887.

Of the Riders and Deductions some are original; but
the greater part have been drawn from that large store of
floating material which has furnished Examination Papers
for the last 30 years, and must necessarily form the basis
of any elementary collection. Proofs which have been


found in two or more books without acknowledgement
liave been regarded as common property.

As regards figures, in accordance with a usage not
uncommon in recent editions of Euclid, we have made a
distinction between given lines and lines of construction.

Throughout tlie book we have italicised those deductions
on which we desired to lay special stress as being in them-
selves important geometrical results : this arrangement we
think will be useful to teachers who have little time to
devote to riders, or who wish to sketch out a suitable course
for revision.

We have in conclusion to tender our thanks to many of
our friends for the valuable criticism and advice which we
received from them as the book was passing through the
press, and especially to the Rev. H. C. Watson, of Clifton
College, who added to these services much kind assistance
in the revision of proof-sheets.



July, 1888.


In the Second Edition the text of Books I — VI. has
been revised ; and at the request of many teachers we have
added the first twenty-one Propositions of Book XI. together
with a collection of Theorems and Examples illustrating the
elements of Solid Geometry.

September, 1889.




Definitions, Postulates, Axioms 1

Section I. Propositions 1 — 26 11

Section II. Parallels and Parallelograms.

Propositions 27 — 34 50

Section III. The Areas of Parallelograms and Triangles.

Propositions 35—48 66

Theorems and Examples on Book I.

Analysis, Synthesis 87

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

II. On Inequalities 93

III. On Parallels . .95

IV. On Parallelograms 96

V. Miscellaneous Theorems and Examples . . . 100

VI. On the Concurrence of Straight Lines m a Tri-
angle 102

VII. On the Construction of Triangles with given

Parts 107

VIII. On Areas 109

IX. On Loci 114

X. On the Intersection of Loci 117




DEriNITIONS, &0 ... 120

Propositions 1 — 14 122

Theorems and Examples on Book II 1-14


Definitions, (fee 149

Propositions 1 — 37 153

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

Theorems and Examples on Book III.

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

n. On the Tangent and the Contact of Circles.

The Common Tangent to Two Circles, Problems on

Tangency, Orthogonal Circles 217

IIL On Angles in Segments, and Angles at the Centres
AND Circumferences op Circles.
The Orthocentre of a Triangle, and properties of the
Pedal Triangle, Loci, Simson's Line . . , 222
IV. On the Circle in Connection with Kectangles.

Further Problems on Tangency 233

V. On Maxima and Minima 239

VI. Harder Miscellaneous Examples 246


Definitions, &c. 250

Propositions 1 — 16 251

Note on Regular Polygons 274

Theorems and Examples on Book IV

I. On the Triangle and its Circles.

Circumscribed, Inscribed, and Escribed Circles, The

Nine-points Circle 277

II. Miscellaneous Examples 283



Inteoductory 285

Definitions 286

Summary, with Algkbbaical Proofs, op the Principal

Theorems of Book V 292

Proofs of the Propositions derived from the Geometrical

Definition of Proportion 297


Definitions 307

Propositions 1 — D 308

Theorems and Exaviples on Book VI.

I, On Harmonic Section 359

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

III. On Pole and Polar 365

IV. On the Eadical Axis of Two or More Circles . . 371
V. On Transversals 374

VI. Miscellaneous Examples on Book VI 377


Definitions 383

Propositions 1—21 393

Exercises on Book XI , . . . 418

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



, . OB'



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

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

The extremities of a line are points, and the intersection of two
lines is a point.

3. A straight line is that which lies evenly between
its extreme points.

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

4. A surface is that which has length and breadth,
but no thickness.

The boundaries of a surface are lines.

5. A plane surface is one in which any two points
being taken, the straight line between them lies wholly in
that surface.

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

Note, Euclid regards a point merely as a viark of position, and
he therefore attaches to it no idea of size and shape.

Similarly he considers that the properties of a line arise only from
its length and position, without reference to that minute breadth which
every line must really have if actually draion, even though the most
perfect instruments are used.

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

II. ]•:. 1.


6. A plane angle ia the inclination of two straiglit
lines to one another, which meet together, but are not in
the same straight line.

The point at which the straight lines meet is called the vertex of
the angle, and the straight lines themselves the arms of the angle.

When several angles are at one point O, any one
of them is expressed by three letters, of which the
letter that refers to the vertex is put between the
other two. Thus if the straight lines OA, OB, OC
meet at the point O, the angle contained by the
straight lines OA, OB is named the angle AOB or
BOA ; and the angle contained by OA, OC is named
the angle AOC or COA. Similarly the angle con-
tained by OB, 00 is referred to as the angle BOG
or COB. But if there be only one angle at a point,
it may be expressed by a single letter, as the angle

Of the two straight lines OB, OC shewn in the
adjoining figure, we recognize that OC is rnore in-
clined than OB to the straight line OA : this we
express by saying that the angle AOC is greater
than the angle AOB. Thus an angle must be
regarded as having magnitude. q

It should be observed that the angle AOC is the sum of the
angles AOB and BOC ; and that AOB is the difference of the angles
AOC and BOC.

The beginner is cautioned against supposing that the size of an
angle is altered either by increasing or diminishing the length of its

[Another view of an angle is recognized in many branches of
mathematics ; and though not employed by Euclid, it is here given
because it furnishes more clearly than any other a conception of what
is meant by the magnitxidc of an angle.

Suppose that the straight line OP in the figure
is capable of revolution about the point O, like the
hand of a watch, but in the opposite direction ; and
suppose that in this way it has passed successively
from the position OA to the positions occupied by
OB and OC.

Such a line must have undergone more turning
in passing from OA to OC, than in passing from OA to OB; and
consequently the angle AOC is said to be greater than the angle AOB.]


7. When a straight line standing on
another straight line makes the adjacent
angles equal to one another, each of the an-
gles is called a right angle ; and the straight
line which stands on the other is called a
perpendicular to it.

8. An obtuse angle is an angle whicli
is greater than one right angle, but less
than two right angles.

9. An acute angle ,is an angle which is
less than a right angle.


[In the adjoining figure the straight line
OB may be supposed to have arrived at
its present position, from the position occu-
pied by OA, by revolution about the point O
m either of the two directions indicated by
the arrows : thus two straight lines drawn
from a point may be considered as forming
tico angles, (marked (i) and (ii) in the figure)
of which the greater (ii) is said to be reflex.

If the arms OA, OB are in the same
straight line, the angle formed by them
on either side is called a straight angle.]

10. Any portion of a plane surface bounded by one
or more lines, straight or curved, is called a plane figure.

The sum of the bounding lines is called the perimeter of the figure.
Two figures are said to be equal in area, when they enclose equal
portions of a plane surface. <,..•

11. A circle is a plane figure contained
by one line, which is called the circum-
ference, and is such that all straight lines
drawn from a certain point within the
figure to the circumference are equal to one
another : this point is called the centre of
the circle.

A radius of a circle is a straight line drawn from the
centre to the circumference.


4 Euclid's elements.

12. A diameter of a circle is a straiglit line drawn
ilirougli the centre, and terminated both ways by the

13. A semicircle is tlie figure bounded by a diameter
of a circle and the jmrt of the circumference cut off by the

14. A segment of a circle is the figure bounded by
a straight line and the part of the circumference which it
cuts oft".

15. Rectilineal figures are those which are bounded
by straight lines.

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

Any one of the angular points of a triangle may be regarded as its
vertex ; and the opposite side is then called the base.

17. A quadrilateral is a plane figure bounded by
four straight lines.

The straight line which joins opposite angular points in a qiiadri-
lateral is called a diagonal.

18. A polygon is a plane figure bounded by more
than four straight lines.

19. An equilateral triangle is a triangle
whose three sides are equal,


20. An isosceles triangle is a triangle two
of whose sides are equal.

21. A scalene triangle is a triangle which
has three unequal sides.


22. A right-angled triangle is a triangle
which has a right angle.

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

23. An obtuse-angled triangle is a

triangle which has an obtuse anijle.

24. An acute-angled triangle is a triangh
which lias tJwee acute an<j:les.

[It will be seen hereafter (Book I. ^Proposition 17) that every
triangle must have at least ttco acute angles.]

25. Parallel straight lines are such as, being in the
same plane, do not meet, however far they are produced in
either direction.

26. A Parallelogram is a four-sided
figure wiiich lias its opposite sides pa-

28. A square is a four-sided figure which
has all its sides equal and all its angles right

[It may easily be shewn that if a quadrilateral
has all its sides equal and one angle a right angle,
then all its angles will be right angles.]

29. A rhombus is a four-sided figure
which has all its sides equal, but its
angles are not right angles.

30. A trapezium is a four-sided figure
which has two of its sides parallel.

27. A rectangle is a parallelogram which r
has one of its angles a right angle. |



In order to effect the constructions necessary to the study of
geometry, it must be supposed that certain instruments are
available; but it has always been held that such instruments
should be as few in number, and as simple in character as

For the purposes of the first Six Books a straight ruler and
a pair of comi^asses are all that are needed ; and in the follow-
ing Postulates, or requests, Euclid demands the use of such
instruments, and assumes that they sufl&ce, theoretically as well
lis practically, to carry out the processes mentioned below.


Let it be granted,

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

When we draw a straight line from the point A to the point B, we
are said io join AB.

2. That a JinitCj that is to say, a terminated straight
line may be produced to any length in that straight line.

3. That a circle may be described from any centre, at
any distance from that centre, that is, with a radius equal
to any finite straight line drawn from the centre.

It is important to notice that the Postulates include no means of
direct measurement : hence the straight ruler is not supposed to be
graduated ; and the compasses, in accordance with EucHd's use, are
not to be employed for transferring distances from one part of a figure
to another.


The science of Geometry is based upon certain simple state-
ments, the truth of which is assumed at the outset to be self-

These self-evident truths, called by Euclid Common Notions^
are now known as the Axioms.


The necessary characteristics of an Axiom are

(i) That it should be self-evident; that is, that its truth
should be immediately accepted without proof.

(ii) That it should be fundamental; that is, that its truth
should not be derivable from any other truth more simple than

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

These characteristics may be summed up in the following

Definition. An Axiom is a self-evident truth, which neither
requires nor is capable of proof, but which serves as a founda-
tion for future reasoning.

Axioms are of two kinds, general and geometrical.

General Axioms apply to magnitudes of all kinds. Geometri-
cal Axioms refer exclusively to geometrical magnitudes^ such as
have been already indicated in the definitions.

General Axioms.

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

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

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

4. If equals be added to unequals, the wholes are un-
equal, the greater sum being that which includes the greater
of the unequals.

5. If equals be taken from unequals, the remainder.4
are unequal, the greater remainder being that which is left
from the greater of the unequals.

6. Things which are double of the same thing, or
of equal things, are equal to one another.

7. Things which are halves of the same thing, or ot
equal things, are equal to one another.

9.* The whole is greater than its part.

* To preserve the classification of general and geometrical axioms,
we have placed Euclid's ninth axiom before the eighth.


Geometrical Axioms.

8. Magnitudes wliich can be made to coincide with one
another, are equal.

This axiom affords the ultimate test of the equahty of two geome-
trical magnitudes. It implies that any liue, angle, or figure, may be
supposed to be taken up from its position, and without change in
size or form, laid down upon a second line, angle, or figure, for the
purpose of comparison.

This process is called superposition, and the first magnitude is
said to be applied to the other.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

[The statement that all right angles are equal, admits of proof,
and is therefore perhaj^s out of place as an Axiom.]

12. If a straight line meet two straight lines so as tv
make the interior angles on one side of it together less
than two right angles, these straight lines will meet if con-
tinually produced on the side on which are the angles wliicli
are together less than two right angles.

That is to say, if the two straight
lines AB and CD are met by the straight
line EH at F and G, in such a way that
the angles BFG, DGF are together less
than two right angles, it is asserted that
AB and CD will meet if continually pro-

Online LibraryEuclidA text-book of Euclid's Elements : for the use of schools : Books I-VI and XI → online text (page 1 of 27)