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\^^^

IN MEMORIAM

FLORIAN CAJORl

i

i^^ CX^-'in^

Â«^

\s

The School Edition

EUCLID'S

ELEMENTS OF GEOMETRY,

THE FIEST SIX BOOKS,

CHIEFLY FROM THE TEXT OF De. SIMSON,

WITH EXPLANATORY NOTES ;

A SERIES OF QUESTIONS ON EACH BOOK;

AND A SELECTION OF GEOMETRICAL EXERCISES FROM

THE SENATE-HOUSE AND COLLEGE EXAMINATION

PAPERS : WITH HINTS, &c.

DESIGNED FOE THE USE OP THE JTTITIOE CLASSES Ilf PUBI.IC AND

PEIVATB SCHOOLS.

EOBEET POTTS, M.A.,

IBliriTY COLLEGE.

CORRECTED AND IMPROFED.

LONDON:

JOHN W. PARKER, SON, AND BOURN, 445, WEST STRAND.

MNDCCO^LXnJ?

L02f DON :

WILLIAM STEVENS, PHINTER, 37, BELL TAED,

TEMPLE BAR.

PREFACE TO THE THIRD EDITION. ^^ ^ ^

Some time after the publication of an Octavo Edition of'Ettcua's^^**

Elements with Geometrical Exercises, &c., designed for the use of

Academical Students ; at the request of some schoolmasters of emi-

nence, a duodecimo Edition of the Six Books was put forth on the

same plan for the use of Schools. Soon after its appearance, Pro-

fessor Christie, the Secretary of the Royal Society, in the Preface to

his Treatise on Descriptive Geometry for the use of the Royal Military'

Academy, was pleased to notice these works in the following terms :â€”

" When the greater Portion of this Part of the Course was printed,

and had for some time been in use in the Academy, a new Edition of

Euclid's Elements, by Mr. Robert Potts, M.A., of Trinity College,

Cambridge, which is likely to supersede most others, to the extent, at

least, of the Six Books, was published. From the manner of arrang-

ing the Demonstrations, this edition has the advantages of the

symbolical form, and it is at the same time free from the manifold

objections to which that form is open. The duodecimo edition of this

Work, comprising only the first Six Books of Euclid, with Deductions

from them, having been introduced at this Institution as a text-book,

now renders any other Treatise on Plane Geometry unnecessary in

our course of Mathematics."

For the very favourable reception which both Editions have met

with, the Editor's grateful acknowledgements are due. It has been his

desire in putting forth a revised Edition of the School Euclid, to render

the work in some degree more worthy of the favour which the former

editions have received, tn the present Edition several errors and

oversights have been corrected and some additions made to the notes :

the questions on each book have been considerably augmented and a

better arrangement of the Geometrical Exercises has been attempted :

and lastly, some hints and remarks on them have been given to assist

the learner. The additions made to the present Edition amount to

more than fifty pages, and, it is hoped, that they will render the work

more useful to the learner.

And here an occasion may be taken to quote the opinions of some

able men respecting the use and importance of the Mathematical

Sciences.

On the subject of Education in its most extensive sense, an ancient

writer " directs the aspirant after excellence to commence with the

Science of Moral Culture; to proceed next to Logic ; next to Mathe-

matics ; next to Physics ; and lastly, to Theology." Another writer

on Education would place Mathematics before Logic, which (he

remarks) " seems the preferable course : for by practising itself in the

IV PREFACE.

former, the mind becomes stored with distinctions ; the faculties of

constanc} and firmness are established; and its rule is always to dis-

tinguish between cavilling and investigationâ€” between close reasoning

and cross reasoning ; for the contrary of all which habits, those are for

the most part noted, who apply themselves to Logic without studying

in some department of Mathematics ; taking noise and wrangling for

proficiency, and thinking refutation accomplished by the instancing

of a doubt. This will explain the inscription placed by Plato over the

door of his house : * Whoso knows not Geometry, let him not enter

here.' On the precedence of Moral Culture, however, to all the other

Sciences, the acknowledgement is general, and the agreement entire."

The same writer recommends the study of the Mathematics, for the

cure of "compound ignorance." " Of this," he proceeds to say, " the

essence is opinion not agi'eeable to fact ; and it necessarily involves

another opinion, namely, that we are already possessed of knowledge.

So that besides not knowing, we know not that we know not ; and

hence its designation of compound ignorance. In like manner, as of

many chronic complaints and established maladies, no cure can be

efi*ected by physicians of the body : of this, no cure can be efi'ected by

physicians of the mind : for with a pre-supposal of knowledge in our

own regard, the pursuit and acquirement of further knowledge is not

to be looked for. The approximate cure, and one from which in the

main much benefit may be anticipated, is to engage the patient in the

study of measures (Geometry, computation, &c.); for in such pursuits

the true and the false are separated by the clearest interval, and no

room is left for the intrusions of fancy. From these the mind may

discover the delight of certainty; and when, on returning to his own

opinions, it finds in them no such sort of repose and gratification, it

may discover their erroneous character, its ignorance may become

simple, and a capacity for the acquirement of truth and virtue be

obtained."

Lord Bacon, the founder of Inductive Philosophy, was not insen-

sible of the high importance of the Mathematical Sciences, as appears

in the following passage from his work on " The Advancement of

Learning."

" The Mathematics are either pure or mixed. To the pure Mathe-

matics are those sciences belonging which handle quantity determinate,

merely severed from any axioms of natural philosophy; and these are

two. Geometry, and Arithmetic; the one handling quantity continued,

and the other dissevered. Mixed hath for subject some axioms or

parts of natural philosophy, and considereth quantity determined, as it

is auxiliary and incident unto them. For many parts of nature can

PBEFACE. V

neither be invented with sufficient subtlety, nor demonstrated with

sufficient perspicuity, nor accommodated unto use with sufficient

dexterity, without the aid and intervening of the Mathematics : of

which sort are perspective, music, astronomy, cosmography, archi-

tecture, enginery, and divers others.

" In the Mathematics I can report no deficience, except it be that

men do not sufficiently understand the excellent use of the pure

Mathematics, in that they do remedy and cure many defects in the

wit and faculties intellectual. For, if the wit be dull, they sharpen it ;

if too wandering, they fix it ; if too inherent in the sense, they abstract

it. So that as tennis is a game of no use in itself, but of great use in

respect that it maketh a quick eye, and a body ready to put itself into

all postures ; so in the Mathematics, that use which is collateral and

intervenient, is no less worthy than that which is principal and

intended. And as for the mixed Mathematics, I may only make this I

prediction, that there cannot fail to be more kinds of them, as nature |

grows further disclosed." I

How truly has this prediction been fulfilled in the subsequent

advancement of the Mixed Sciences, and in the applications of the

pure Mathematics to Natural Philosophy!

Dr. Whewell, in his " Thoughts on the Study of Mathematics,"

has maintained, that mathematical studies judiciously pursued, form

one of the most efi'ective means of developing and cultivating the

reason : and that "the object of a liberal education is to develope the

whole mental system of man; â€” to make his speculative inferences

coincide with his practical convictions ; â€” to enable him to render a

reason for the belief that is in him, and not to leave him in the con-

dition of Solomon's sluggard, who is wiser in his own conceit than

seven men that can render a reason." And in his more recent work

entitled, " Of a Liberal Education, &c." he has more fully shewn the

importance of Geometry as one of the most effectual instruments

of intellectual education. In page 55 he thus proceeds: â€” "But

besides the value of Mathematical Studies in Education, as a perfect

example and complete exercise of demonstrative reasoning; Mathe-

matical Truths have this additional recommendation, that they have

always been referred to, by each successive generation of thoughtful

and cultivated men, as examples of truth and of demonstration ; and

have thus become standard points of reference, among cultivated men,

whenever they speak of truth, knowledge, or proof. Thus Mathe-

matics has not only a disciplinal but an historical interest. This is

peculiarly the case with those portions of Mathematics which we have

mentioned. We find geometrical proof adduced in illustration of the

VI PREFACE.

nature of reasoning, in the earliest speculations on this subject, the

Dialogues of Plato ; we find geometrical proof one of the main sub-

jects of discussion in some of the most recent of such speculations, as

those of Dugald Stewart and his contemporaries. The recollection

of the truths of Elementary Geometry has, in all ages, given a meaning

and a reality to the best attempts to explain man's power of arriving

at truth. Other branches of Mathematics have, in like manner,

become recognized examples, among educated men, of man's powers

of attaining truth."

Dr. Pemberton, in the preface to his view of Sir Isaac Newton's

Discoveries, makes mention of the circumstance, " that Newton used

to speak with regret of his mistake, at the beginning of his Mathe-

matical Studies, in having applied himself to the works of Descartes

and other Algebraical writers, before he had considered the Elements

of Euclid with the attention they deserve."

To these we may subjoin the opinion of Mr. John Stuart Mill,

which he has recorded in his invaluable System of Logic, (Vol. li.

p. 180) in the following terms. " The value of Mathematical instruc-

tion as a preparation for those more difficult investigations (physiology,

society, government, &c.) consists in the applicability not of its doc-

trines, but of its method. Mathematics will ever remain the most

perfect type of the Deductive Method in general ; and the applications

of Mathematics to the simpler branches of physics, furnish the only

school in which philosophers can effectually learn the most difficult

and important portion of their art, the employment of the laws of

simpler phenomena for explaining and predicting those of the more

complex. These grounds are quite sufficient for deeming mathemati-

cal training an indispensable basis of real scientific education, and

regarding, with Plato, one who is dytutfiiTpnTo?, as wanting in one of

the most essential qualifications for the successful cultivation of the

higher branches of philosophy."

In addition to these authorities it may be remarked, that the new

Regulations which were confirmed by a Grace of the Senate on the

11th of May, 1846, assign to Geometry and to Geometrical methods,

a more important place in the Examinations both for Honors and

for the Ordinary Degree in this University.

Trinity College, RP.

3farch 1, 1850.

This Edition (the fifth), has been augmented by upwards of forty

pages of additional Notes, Questions and Geometrical Exercises.

Trinity College, R. P.

November 5, 1859.

CONTENTS.

Preface to the Third Edition , c iii

First Book of the Elements 1

Notes to the First Book 42

Questions on the First Book 59

On the Ancient Geometrical Analysis 64

Geometrical Exercises on Book I ,..,â€žÂ« 69

Second Book of the Elements 85

Notes to the Second Book 99

Note on the Algebraical Symbols & Abbreviations used in Geometry 1 09

Questions on the Second Book 110

Geometrical Exercises on Book II 113

Third Book of the Elements 120

Notes to the Third Book 153

Questions on the Third Book 157

Geometrical Exercises on Book III 160

Fourth Book of the Elements 175

Notes to the Fourth Book 190

Questions on the Fourth Book 193

Geometrical Exercises on Book IV 196

Fifth Book of the Elements 204

Notes to the Fifth Book 235

Questions on the Fifth Book 257

Sixth Book of the Elements â– 259

Notes to the Sixth Book 294

Questions on the Sixth Book 299

Geometrical Exercises on Book VI 302

Solutions, Hints, &c. on Book I 313

II 323

Ill 326

IV . 338

VI 345

Index to the Geometrical Exercises 356

â–

LIBER CANTABRIGIENSIS.

PAET. I.

_^^ Account of the Aids afforded to poor Students^

the encouragements offered to diligent Students, and

the rewards conferred on successful Students, in the Uni-

versity of Cambridge ; to which is prefixed a Collection

of Ma/rims, Aphorisms, ^c. Designed for the Use of

Learners. By Egbert Potts, M.A., Trinity College.

Fcap. Svo.,pp. 570^ price 5s. 6d.

" It was not a bad idea to prefix to the many encouragements alForded to students

in the University of Cambridge, a selection of maxims drawn from the writings ot

men who have sliown that learning is to be judged by its fruits in social and

individual life."â€” 2'fte Literary Churchman.

" A work like this was much -wanted.."â€” ClericalJoumal.

*' The book altogether is one of merit and xalne."â€” Guardian.

" The several parts of this book are most interesting and mstT\ictiye."â€”i:ducational

Times.

" No doubt many will thank Mr. Potts for the very valuable information he has

afforded in this laborious compilation.'' â€” Critic.

" A vast amount of information is compressed into a small compass, at the cost

evidently of great labour and pains. The Aphorisms which form a prefix of 174

pages, may suggest useful reflections to earnest students."â€” jTAe Patriot.

John W. Parker, Son, & Bourn, West Strand, London.

PAET II.

Containing an Account (I) of the recent changes in the

Statutes made under the powers of the Act (19 and 20 Vict,

cap. 88).* (2) Of the Minor Scholarships instituted and

open to the competition of Students hefore Residence: (3)

Of the Course of Collegiate and University Studies at

Cambridge. Price 2s. 6d.

EUCLID'S

ELEMENTS OF GEOMETRY.

BOOK L

DEFINITIONS.

A POINT is that which has no parts, or which has no magnitude.

II.

A line is length without breadth.

III.

The extremities of a line are points.

IV.

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

V.

A superficies is that which has only length and breadth.

VI.

The extremities of a superficies are lines.

VII.

A plane superficies is that in which any two points being taken, the

straight line between them lies wholly in that superficies.

VIII.

A plane angle is the inclination of two lines to each other in a

plane, which meet together, but are not in the same direction.

IX.

A plane rectilineal angle is the inclination of two straight lines to

one another, which meet together, but are not in the same straight line.

I2Z

s

Euclid's elements.

N.B. If there be only one angle at a point, it may be expressed by

a letter placed at that point, as the angle at E : but when several angles

are at one point B, either of them is expressed by three letters, of which

the letter that is at the vertex of the angle, that is, at the point in which

the straight lines that contain the angle meet one another, is put between

the other two letters, and one of these two is somewhere upon one of

these straight lines, and the other upon the other line. Thus the angle

which is contained by the straight lines AB, CB, is named the angle

ABCy or CBA ; that which is contained by AB, DB, is named the angle

ABD, or DBA ; and that which is contained by DB^ CB^ is called the

angle DBC, or CBD.

X.

When a straight line standing on another straight line, makes the

adjacent angles equal to one another, each of these angles is called a

right angle ; and the straight line which stands on the other is called

a perpendicular to it.

XL

An obtuse angle is that which is greater than a right angle.

XII.

An acute angle is that which is less than a right angle.

XIII.

A term or boundary is the extremity of any thing.

XIV.

A figure is that which Is enclosed by one or more boundaries.

I

DEFINITIONS.

XV.

A circle is a plane figure contained by one line, which is called the

circumference, and is such that all straight lines drawn from a certain

point within the figure to the circumference, are equal to one another.

XVI.

And this point is called the center of the circle.

XVII.

A diameter of a circle is a straight line drawn through the center,

and terminated both ways by the circumference.

XVIII.

A semicircle is the figure contained by a diameter and the part of

the circumference cut off by the diameter.

XIX.

The center of a semicircle is the same with that of the circle.

XX.

Eectilineal figures are those which are contained by straight lines.

XXI.

Trilateral figures, or triangles, by three straight lines.

XXII.

Jl^uadrilateral, by four straight lines.

XXIII.

Multilateral figures, or polygons, by more than four straight lines.

b2

EUCLID S ELEMENTS.

XXIV.

Of three-sided figures, an equilateral triangle is that which has

three equal sides.

XXV.

An isosceles triangle is that which has two sides equo I.

XXVI.

A scalene triangle is that which has three unequal sides.

XXVII.

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

.^L.

XXVIII.

An obtuse-angled triangle is that which has an obtuse angle.

XXIX.

An acute-angled triangle is that which has three acute angles.

\

z.

XXX.

_\

Of quadrilateral or four-sided figures, a square has all its sides equal

and all its angles right angles.

DEFINITIONS.

XXXI.

An oblong is that which has all its angles right angles, but has not

all its sides equal.

XXXII.

A rhombus has all its sides equal, but its angles are not right angles.

XXXIII.

A rhomboid has its opposite sides equal to each other, but all its

sides are not equal, nor its angles right angles.

XXXIV.

All other four-sided figures besides these, are called Trapeziums.

XXXV.

Parallel straight lines are such as are in the same plane, and which

being produced ever so far both ways, do not meet.

A.

A parallelogram is a four-sided figure, of which the opposite sides

are parallel: and the diameter, or the diagonal is the straight line

joining two of its opposite angles.

POSTULATES.

I.

Let it be granted that a straight line may be drawn from any one

point to any other point.

II.

That a terminated straight line may be produced to any length in

a straight line.

III.

And that a circle may be described from any center, at any distance

from that center.

3 Euclid's elements.

AXIOMS.

I.

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

II.

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

III.

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

IV.

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

V.

If equals be taken from unequals, the remainders are unequal.

VI.

Things which are double of the same, are equal to one another.

VII.

Things which are halves of the same, are equal to one another.

VIII.

Magnitudes which coincide with one another, that is, which exact! \

fill the same space, are equal to one another.

IX.

The whole is greater than its part.

X.

Two straight lines cannot enclose a space.

XI.

All right angles are equal to one another.

XII.

If a straight line meets two straight lines, so as to make the two

interior angles on the same side of it taken together less than two

right angles ; these straight lines being continually produced, shall at

length meet upon that side on which are the angles which are less than

iwo right angles.

BOOK I. PROP. I, U.

PROPOSITION I. PROBLEM.

To describe an equilateral triangle upon a given finite straight line.

Let ABhe the given straight line.

It is required to describe an equilateral triangle upon AJB,

c

From the center A, at the distance AD, describe the circle BCD-,

(post. 3.)

from the center B, at the distance BA, describe the circle A CE ;

and from C, one of the points in which the circles cut one another,

draw the straight lines CA, CB to the points A^ B. (post. 1.)

Then ^^C shall be an equilateral triangle.

Because the point A is the center of the circle BCD,

therefore ^Cis equal to AB ', (def. 15.)

and because the point B is the center of the circle A CE,

therefore BC is equal to AB ;

but it has been proved that ^ C is equal to AB ;

therefore A C, BC are each of them equal to AB ;

but things which are equal to the same thing are equal to one another ;

therefore ACis equal to BC; (ax. 1.)

wherefore AB, BC, CA are equal to one another:

and the triangle ABC is therefore equilateral,

and it is described upon the given straight line AB.

Which was requii*ed to be done.

PROPOSITION II. PROBLEM.

From a given point, to draw a straight line equal to a given straight line.

Let A be the given point, and B C the given straight line.

It is required to di'aw from the point A, a straight line equal to BC.

From the point A to B draw the straight line AB; (post. 1.)

upon AB describe the equilateral triangle ABD, (i. 1.)

and produce the straight lines DA, DB to E and F; (post. 2.)

from the center B, at the distance BC, describe the circle CGH,

(post. 3.) cutting DF in the point G:

and from the center D, at the distance DG, describe the circle GKL,

cutting AE in the point L.

Then the straight line AL shall be equal to BC.

Because the point B is the center of the circle CGII,

therefore BCh equal to BG-, (def. 15.)

and because D is the center of the circle GKL,

therefore DL is equal to DG,

and BA, DB parts of them are equal ; (l. 1.)

therefore the remainder AL is equal to the remainder BG; (ax. 3.)

but it has been shewn that BC is equal to BG,

wherefore AL and ^Care each of them equal to BG;

and things that are equal to the same thing are equal to one another ;

therefore the straight line AL is equal to BC. (ax. 1.)

Wherefore from the given point A, a straight line AL has been drawn

equal to the given straight line BC. Which was to be done.

PROPOSITION III. PROBLEM.

From the greater of two given straight lines to cut off a part equal to the less.

Let AB and Cbe the two given straight lines, of which A.B is the

greater.

It is required to cut off from^ J? the greater, a part equal to C, the less.

D

From the point A draw the straight line AD equal to C; (l. 2.)

and from the center A, at the distance AD, describe the circle DEF

(post. 3.) cutting AB in the point E.

Then AE shall be equal to C.

Because A is the center of the circle DEF,

therefore AE is equal to AD-, (def. 15.)

but the straight line C is equal to AD; (constr.)

whence AE and C are each of them equal to AD;

wherefore the straight line AE is equal to C. (ax. 1.)

And therefore from AB the greater of two straight lines, a part AE

has been cut off equal to C, the less. Which was to be done.

PROPOSITION IV. THEOREM.

If two triangles have tico sides of the one equal to two sides of the other,

each to each, and have likewise the angles contained by those sides equal to

each other ; they shall likewise have their bases or third sides equal, and

the two triangles shall he eqiial, and their other angles shall be equal, each

to each, viz. those to which the equal sides are opposite.

Let ABC, DEF be two triangles, which have the two sides AB,

A C equal to the two sides DE, DF, each to each, viz. AB to DE, and

^ C to DF, and the included angle BA C equal to the included angle

EDF.

BOOK I. PROP, jy, V. 9

Then shall the base ^Cbe equal to the base JEF; and the triangle

ABC to the triangle DEF-, and the other angles to which the equal

sides are opposite shall be equal, each to each, viz. the angle ABC to

the angle JDFF, and the angle A CB to the angle DFF. ^

I ..â€ž,._.â€ž.â€ž..

^^ that the point A may be on J), and the straight line AB on DB\

then the point B shall coincide with the point E^

because AB is equal to EE;

and AB coinciding with EE,

the straight line A C shall fall on EF^

because the angle BA C is equal to the angle EEF\ -^

therefore also the point C shall coincide with the point jP,

because AC\% equal to EF\

but the point B was shewn to coincide with the point E ;

wherefore the base ^C shall coincide with the base EF;

because the point B coinciding with E, and C with F,

IN MEMORIAM

FLORIAN CAJORl

i

i^^ CX^-'in^

Â«^

\s

The School Edition

EUCLID'S

ELEMENTS OF GEOMETRY,

THE FIEST SIX BOOKS,

CHIEFLY FROM THE TEXT OF De. SIMSON,

WITH EXPLANATORY NOTES ;

A SERIES OF QUESTIONS ON EACH BOOK;

AND A SELECTION OF GEOMETRICAL EXERCISES FROM

THE SENATE-HOUSE AND COLLEGE EXAMINATION

PAPERS : WITH HINTS, &c.

DESIGNED FOE THE USE OP THE JTTITIOE CLASSES Ilf PUBI.IC AND

PEIVATB SCHOOLS.

EOBEET POTTS, M.A.,

IBliriTY COLLEGE.

CORRECTED AND IMPROFED.

LONDON:

JOHN W. PARKER, SON, AND BOURN, 445, WEST STRAND.

MNDCCO^LXnJ?

L02f DON :

WILLIAM STEVENS, PHINTER, 37, BELL TAED,

TEMPLE BAR.

PREFACE TO THE THIRD EDITION. ^^ ^ ^

Some time after the publication of an Octavo Edition of'Ettcua's^^**

Elements with Geometrical Exercises, &c., designed for the use of

Academical Students ; at the request of some schoolmasters of emi-

nence, a duodecimo Edition of the Six Books was put forth on the

same plan for the use of Schools. Soon after its appearance, Pro-

fessor Christie, the Secretary of the Royal Society, in the Preface to

his Treatise on Descriptive Geometry for the use of the Royal Military'

Academy, was pleased to notice these works in the following terms :â€”

" When the greater Portion of this Part of the Course was printed,

and had for some time been in use in the Academy, a new Edition of

Euclid's Elements, by Mr. Robert Potts, M.A., of Trinity College,

Cambridge, which is likely to supersede most others, to the extent, at

least, of the Six Books, was published. From the manner of arrang-

ing the Demonstrations, this edition has the advantages of the

symbolical form, and it is at the same time free from the manifold

objections to which that form is open. The duodecimo edition of this

Work, comprising only the first Six Books of Euclid, with Deductions

from them, having been introduced at this Institution as a text-book,

now renders any other Treatise on Plane Geometry unnecessary in

our course of Mathematics."

For the very favourable reception which both Editions have met

with, the Editor's grateful acknowledgements are due. It has been his

desire in putting forth a revised Edition of the School Euclid, to render

the work in some degree more worthy of the favour which the former

editions have received, tn the present Edition several errors and

oversights have been corrected and some additions made to the notes :

the questions on each book have been considerably augmented and a

better arrangement of the Geometrical Exercises has been attempted :

and lastly, some hints and remarks on them have been given to assist

the learner. The additions made to the present Edition amount to

more than fifty pages, and, it is hoped, that they will render the work

more useful to the learner.

And here an occasion may be taken to quote the opinions of some

able men respecting the use and importance of the Mathematical

Sciences.

On the subject of Education in its most extensive sense, an ancient

writer " directs the aspirant after excellence to commence with the

Science of Moral Culture; to proceed next to Logic ; next to Mathe-

matics ; next to Physics ; and lastly, to Theology." Another writer

on Education would place Mathematics before Logic, which (he

remarks) " seems the preferable course : for by practising itself in the

IV PREFACE.

former, the mind becomes stored with distinctions ; the faculties of

constanc} and firmness are established; and its rule is always to dis-

tinguish between cavilling and investigationâ€” between close reasoning

and cross reasoning ; for the contrary of all which habits, those are for

the most part noted, who apply themselves to Logic without studying

in some department of Mathematics ; taking noise and wrangling for

proficiency, and thinking refutation accomplished by the instancing

of a doubt. This will explain the inscription placed by Plato over the

door of his house : * Whoso knows not Geometry, let him not enter

here.' On the precedence of Moral Culture, however, to all the other

Sciences, the acknowledgement is general, and the agreement entire."

The same writer recommends the study of the Mathematics, for the

cure of "compound ignorance." " Of this," he proceeds to say, " the

essence is opinion not agi'eeable to fact ; and it necessarily involves

another opinion, namely, that we are already possessed of knowledge.

So that besides not knowing, we know not that we know not ; and

hence its designation of compound ignorance. In like manner, as of

many chronic complaints and established maladies, no cure can be

efi*ected by physicians of the body : of this, no cure can be efi'ected by

physicians of the mind : for with a pre-supposal of knowledge in our

own regard, the pursuit and acquirement of further knowledge is not

to be looked for. The approximate cure, and one from which in the

main much benefit may be anticipated, is to engage the patient in the

study of measures (Geometry, computation, &c.); for in such pursuits

the true and the false are separated by the clearest interval, and no

room is left for the intrusions of fancy. From these the mind may

discover the delight of certainty; and when, on returning to his own

opinions, it finds in them no such sort of repose and gratification, it

may discover their erroneous character, its ignorance may become

simple, and a capacity for the acquirement of truth and virtue be

obtained."

Lord Bacon, the founder of Inductive Philosophy, was not insen-

sible of the high importance of the Mathematical Sciences, as appears

in the following passage from his work on " The Advancement of

Learning."

" The Mathematics are either pure or mixed. To the pure Mathe-

matics are those sciences belonging which handle quantity determinate,

merely severed from any axioms of natural philosophy; and these are

two. Geometry, and Arithmetic; the one handling quantity continued,

and the other dissevered. Mixed hath for subject some axioms or

parts of natural philosophy, and considereth quantity determined, as it

is auxiliary and incident unto them. For many parts of nature can

PBEFACE. V

neither be invented with sufficient subtlety, nor demonstrated with

sufficient perspicuity, nor accommodated unto use with sufficient

dexterity, without the aid and intervening of the Mathematics : of

which sort are perspective, music, astronomy, cosmography, archi-

tecture, enginery, and divers others.

" In the Mathematics I can report no deficience, except it be that

men do not sufficiently understand the excellent use of the pure

Mathematics, in that they do remedy and cure many defects in the

wit and faculties intellectual. For, if the wit be dull, they sharpen it ;

if too wandering, they fix it ; if too inherent in the sense, they abstract

it. So that as tennis is a game of no use in itself, but of great use in

respect that it maketh a quick eye, and a body ready to put itself into

all postures ; so in the Mathematics, that use which is collateral and

intervenient, is no less worthy than that which is principal and

intended. And as for the mixed Mathematics, I may only make this I

prediction, that there cannot fail to be more kinds of them, as nature |

grows further disclosed." I

How truly has this prediction been fulfilled in the subsequent

advancement of the Mixed Sciences, and in the applications of the

pure Mathematics to Natural Philosophy!

Dr. Whewell, in his " Thoughts on the Study of Mathematics,"

has maintained, that mathematical studies judiciously pursued, form

one of the most efi'ective means of developing and cultivating the

reason : and that "the object of a liberal education is to develope the

whole mental system of man; â€” to make his speculative inferences

coincide with his practical convictions ; â€” to enable him to render a

reason for the belief that is in him, and not to leave him in the con-

dition of Solomon's sluggard, who is wiser in his own conceit than

seven men that can render a reason." And in his more recent work

entitled, " Of a Liberal Education, &c." he has more fully shewn the

importance of Geometry as one of the most effectual instruments

of intellectual education. In page 55 he thus proceeds: â€” "But

besides the value of Mathematical Studies in Education, as a perfect

example and complete exercise of demonstrative reasoning; Mathe-

matical Truths have this additional recommendation, that they have

always been referred to, by each successive generation of thoughtful

and cultivated men, as examples of truth and of demonstration ; and

have thus become standard points of reference, among cultivated men,

whenever they speak of truth, knowledge, or proof. Thus Mathe-

matics has not only a disciplinal but an historical interest. This is

peculiarly the case with those portions of Mathematics which we have

mentioned. We find geometrical proof adduced in illustration of the

VI PREFACE.

nature of reasoning, in the earliest speculations on this subject, the

Dialogues of Plato ; we find geometrical proof one of the main sub-

jects of discussion in some of the most recent of such speculations, as

those of Dugald Stewart and his contemporaries. The recollection

of the truths of Elementary Geometry has, in all ages, given a meaning

and a reality to the best attempts to explain man's power of arriving

at truth. Other branches of Mathematics have, in like manner,

become recognized examples, among educated men, of man's powers

of attaining truth."

Dr. Pemberton, in the preface to his view of Sir Isaac Newton's

Discoveries, makes mention of the circumstance, " that Newton used

to speak with regret of his mistake, at the beginning of his Mathe-

matical Studies, in having applied himself to the works of Descartes

and other Algebraical writers, before he had considered the Elements

of Euclid with the attention they deserve."

To these we may subjoin the opinion of Mr. John Stuart Mill,

which he has recorded in his invaluable System of Logic, (Vol. li.

p. 180) in the following terms. " The value of Mathematical instruc-

tion as a preparation for those more difficult investigations (physiology,

society, government, &c.) consists in the applicability not of its doc-

trines, but of its method. Mathematics will ever remain the most

perfect type of the Deductive Method in general ; and the applications

of Mathematics to the simpler branches of physics, furnish the only

school in which philosophers can effectually learn the most difficult

and important portion of their art, the employment of the laws of

simpler phenomena for explaining and predicting those of the more

complex. These grounds are quite sufficient for deeming mathemati-

cal training an indispensable basis of real scientific education, and

regarding, with Plato, one who is dytutfiiTpnTo?, as wanting in one of

the most essential qualifications for the successful cultivation of the

higher branches of philosophy."

In addition to these authorities it may be remarked, that the new

Regulations which were confirmed by a Grace of the Senate on the

11th of May, 1846, assign to Geometry and to Geometrical methods,

a more important place in the Examinations both for Honors and

for the Ordinary Degree in this University.

Trinity College, RP.

3farch 1, 1850.

This Edition (the fifth), has been augmented by upwards of forty

pages of additional Notes, Questions and Geometrical Exercises.

Trinity College, R. P.

November 5, 1859.

CONTENTS.

Preface to the Third Edition , c iii

First Book of the Elements 1

Notes to the First Book 42

Questions on the First Book 59

On the Ancient Geometrical Analysis 64

Geometrical Exercises on Book I ,..,â€žÂ« 69

Second Book of the Elements 85

Notes to the Second Book 99

Note on the Algebraical Symbols & Abbreviations used in Geometry 1 09

Questions on the Second Book 110

Geometrical Exercises on Book II 113

Third Book of the Elements 120

Notes to the Third Book 153

Questions on the Third Book 157

Geometrical Exercises on Book III 160

Fourth Book of the Elements 175

Notes to the Fourth Book 190

Questions on the Fourth Book 193

Geometrical Exercises on Book IV 196

Fifth Book of the Elements 204

Notes to the Fifth Book 235

Questions on the Fifth Book 257

Sixth Book of the Elements â– 259

Notes to the Sixth Book 294

Questions on the Sixth Book 299

Geometrical Exercises on Book VI 302

Solutions, Hints, &c. on Book I 313

II 323

Ill 326

IV . 338

VI 345

Index to the Geometrical Exercises 356

â–

LIBER CANTABRIGIENSIS.

PAET. I.

_^^ Account of the Aids afforded to poor Students^

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the rewards conferred on successful Students, in the Uni-

versity of Cambridge ; to which is prefixed a Collection

of Ma/rims, Aphorisms, ^c. Designed for the Use of

Learners. By Egbert Potts, M.A., Trinity College.

Fcap. Svo.,pp. 570^ price 5s. 6d.

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in the University of Cambridge, a selection of maxims drawn from the writings ot

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PAET II.

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Of the Course of Collegiate and University Studies at

Cambridge. Price 2s. 6d.

EUCLID'S

ELEMENTS OF GEOMETRY.

BOOK L

DEFINITIONS.

A POINT is that which has no parts, or which has no magnitude.

II.

A line is length without breadth.

III.

The extremities of a line are points.

IV.

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

V.

A superficies is that which has only length and breadth.

VI.

The extremities of a superficies are lines.

VII.

A plane superficies is that in which any two points being taken, the

straight line between them lies wholly in that superficies.

VIII.

A plane angle is the inclination of two lines to each other in a

plane, which meet together, but are not in the same direction.

IX.

A plane rectilineal angle is the inclination of two straight lines to

one another, which meet together, but are not in the same straight line.

I2Z

s

Euclid's elements.

N.B. If there be only one angle at a point, it may be expressed by

a letter placed at that point, as the angle at E : but when several angles

are at one point B, either of them is expressed by three letters, of which

the letter that is at the vertex of the angle, that is, at the point in which

the straight lines that contain the angle meet one another, is put between

the other two letters, and one of these two is somewhere upon one of

these straight lines, and the other upon the other line. Thus the angle

which is contained by the straight lines AB, CB, is named the angle

ABCy or CBA ; that which is contained by AB, DB, is named the angle

ABD, or DBA ; and that which is contained by DB^ CB^ is called the

angle DBC, or CBD.

X.

When a straight line standing on another straight line, makes the

adjacent angles equal to one another, each of these angles is called a

right angle ; and the straight line which stands on the other is called

a perpendicular to it.

XL

An obtuse angle is that which is greater than a right angle.

XII.

An acute angle is that which is less than a right angle.

XIII.

A term or boundary is the extremity of any thing.

XIV.

A figure is that which Is enclosed by one or more boundaries.

I

DEFINITIONS.

XV.

A circle is a plane figure contained by one line, which is called the

circumference, and is such that all straight lines drawn from a certain

point within the figure to the circumference, are equal to one another.

XVI.

And this point is called the center of the circle.

XVII.

A diameter of a circle is a straight line drawn through the center,

and terminated both ways by the circumference.

XVIII.

A semicircle is the figure contained by a diameter and the part of

the circumference cut off by the diameter.

XIX.

The center of a semicircle is the same with that of the circle.

XX.

Eectilineal figures are those which are contained by straight lines.

XXI.

Trilateral figures, or triangles, by three straight lines.

XXII.

Jl^uadrilateral, by four straight lines.

XXIII.

Multilateral figures, or polygons, by more than four straight lines.

b2

EUCLID S ELEMENTS.

XXIV.

Of three-sided figures, an equilateral triangle is that which has

three equal sides.

XXV.

An isosceles triangle is that which has two sides equo I.

XXVI.

A scalene triangle is that which has three unequal sides.

XXVII.

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

.^L.

XXVIII.

An obtuse-angled triangle is that which has an obtuse angle.

XXIX.

An acute-angled triangle is that which has three acute angles.

\

z.

XXX.

_\

Of quadrilateral or four-sided figures, a square has all its sides equal

and all its angles right angles.

DEFINITIONS.

XXXI.

An oblong is that which has all its angles right angles, but has not

all its sides equal.

XXXII.

A rhombus has all its sides equal, but its angles are not right angles.

XXXIII.

A rhomboid has its opposite sides equal to each other, but all its

sides are not equal, nor its angles right angles.

XXXIV.

All other four-sided figures besides these, are called Trapeziums.

XXXV.

Parallel straight lines are such as are in the same plane, and which

being produced ever so far both ways, do not meet.

A.

A parallelogram is a four-sided figure, of which the opposite sides

are parallel: and the diameter, or the diagonal is the straight line

joining two of its opposite angles.

POSTULATES.

I.

Let it be granted that a straight line may be drawn from any one

point to any other point.

II.

That a terminated straight line may be produced to any length in

a straight line.

III.

And that a circle may be described from any center, at any distance

from that center.

3 Euclid's elements.

AXIOMS.

I.

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

II.

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

III.

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

IV.

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

V.

If equals be taken from unequals, the remainders are unequal.

VI.

Things which are double of the same, are equal to one another.

VII.

Things which are halves of the same, are equal to one another.

VIII.

Magnitudes which coincide with one another, that is, which exact! \

fill the same space, are equal to one another.

IX.

The whole is greater than its part.

X.

Two straight lines cannot enclose a space.

XI.

All right angles are equal to one another.

XII.

If a straight line meets two straight lines, so as to make the two

interior angles on the same side of it taken together less than two

right angles ; these straight lines being continually produced, shall at

length meet upon that side on which are the angles which are less than

iwo right angles.

BOOK I. PROP. I, U.

PROPOSITION I. PROBLEM.

To describe an equilateral triangle upon a given finite straight line.

Let ABhe the given straight line.

It is required to describe an equilateral triangle upon AJB,

c

From the center A, at the distance AD, describe the circle BCD-,

(post. 3.)

from the center B, at the distance BA, describe the circle A CE ;

and from C, one of the points in which the circles cut one another,

draw the straight lines CA, CB to the points A^ B. (post. 1.)

Then ^^C shall be an equilateral triangle.

Because the point A is the center of the circle BCD,

therefore ^Cis equal to AB ', (def. 15.)

and because the point B is the center of the circle A CE,

therefore BC is equal to AB ;

but it has been proved that ^ C is equal to AB ;

therefore A C, BC are each of them equal to AB ;

but things which are equal to the same thing are equal to one another ;

therefore ACis equal to BC; (ax. 1.)

wherefore AB, BC, CA are equal to one another:

and the triangle ABC is therefore equilateral,

and it is described upon the given straight line AB.

Which was requii*ed to be done.

PROPOSITION II. PROBLEM.

From a given point, to draw a straight line equal to a given straight line.

Let A be the given point, and B C the given straight line.

It is required to di'aw from the point A, a straight line equal to BC.

From the point A to B draw the straight line AB; (post. 1.)

upon AB describe the equilateral triangle ABD, (i. 1.)

and produce the straight lines DA, DB to E and F; (post. 2.)

from the center B, at the distance BC, describe the circle CGH,

(post. 3.) cutting DF in the point G:

and from the center D, at the distance DG, describe the circle GKL,

cutting AE in the point L.

Then the straight line AL shall be equal to BC.

Because the point B is the center of the circle CGII,

therefore BCh equal to BG-, (def. 15.)

and because D is the center of the circle GKL,

therefore DL is equal to DG,

and BA, DB parts of them are equal ; (l. 1.)

therefore the remainder AL is equal to the remainder BG; (ax. 3.)

but it has been shewn that BC is equal to BG,

wherefore AL and ^Care each of them equal to BG;

and things that are equal to the same thing are equal to one another ;

therefore the straight line AL is equal to BC. (ax. 1.)

Wherefore from the given point A, a straight line AL has been drawn

equal to the given straight line BC. Which was to be done.

PROPOSITION III. PROBLEM.

From the greater of two given straight lines to cut off a part equal to the less.

Let AB and Cbe the two given straight lines, of which A.B is the

greater.

It is required to cut off from^ J? the greater, a part equal to C, the less.

D

From the point A draw the straight line AD equal to C; (l. 2.)

and from the center A, at the distance AD, describe the circle DEF

(post. 3.) cutting AB in the point E.

Then AE shall be equal to C.

Because A is the center of the circle DEF,

therefore AE is equal to AD-, (def. 15.)

but the straight line C is equal to AD; (constr.)

whence AE and C are each of them equal to AD;

wherefore the straight line AE is equal to C. (ax. 1.)

And therefore from AB the greater of two straight lines, a part AE

has been cut off equal to C, the less. Which was to be done.

PROPOSITION IV. THEOREM.

If two triangles have tico sides of the one equal to two sides of the other,

each to each, and have likewise the angles contained by those sides equal to

each other ; they shall likewise have their bases or third sides equal, and

the two triangles shall he eqiial, and their other angles shall be equal, each

to each, viz. those to which the equal sides are opposite.

Let ABC, DEF be two triangles, which have the two sides AB,

A C equal to the two sides DE, DF, each to each, viz. AB to DE, and

^ C to DF, and the included angle BA C equal to the included angle

EDF.

BOOK I. PROP, jy, V. 9

Then shall the base ^Cbe equal to the base JEF; and the triangle

ABC to the triangle DEF-, and the other angles to which the equal

sides are opposite shall be equal, each to each, viz. the angle ABC to

the angle JDFF, and the angle A CB to the angle DFF. ^

I ..â€ž,._.â€ž.â€ž..

^^ that the point A may be on J), and the straight line AB on DB\

then the point B shall coincide with the point E^

because AB is equal to EE;

and AB coinciding with EE,

the straight line A C shall fall on EF^

because the angle BA C is equal to the angle EEF\ -^

therefore also the point C shall coincide with the point jP,

because AC\% equal to EF\

but the point B was shewn to coincide with the point E ;

wherefore the base ^C shall coincide with the base EF;

because the point B coinciding with E, and C with F,

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