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Aaa
BYRNE'S EUCLID
THE FIRST SIX BOOKS OF
THE ELEMENTS OF EUCLID
WITH COLOURED DIAGRAMS
AND SYMBOLS
?
THE FIRST SIX BOOKS OF
THE ELEMENTS OF EUCLID
IN WHICH COLOURED DIAGRAMS AND SYMBOLS
ARE USED INSTEAD OF LETTERS FOR THE
GREATER EASE OF LEARNERS
BY OLIVER BYRNE
SURVEYOR OF HER MAJESTY'S SETTLEMENTS IN THE FALKLAND ISLANDS
AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS
LONDON
WILLIAM PICKERING
1847
TO THE
RIGHT HONOURABLE THE EARL FITZWILLIAM,
ETC. ETC. ETC.
THIS WORK IS DEDICATED
BY HIS LORDSHIP S OBEDIENT
AND MUCH OBLIGED SERVANT,
OLIVER BYRNE.
Digitized by the Internet Archive
in 2011 with funding from
University of Toronto
http://www.archive.org/details/firstsixbooksofeOOeucl
SS3
INTRODUCTION.
HE arts and fciences have become fo extenfive,
that to facilitate their acquirement is of as
much importance as to extend their boundaries.
Illuftration, if it does not fhorten the time of
ftudy, will at leaft make it more agreeable. This Work
has a greater aim than mere illuftration ; we do not intro
duce colours for the purpofe of entertainment, or to amuie
by certain cottibinations of tint and form, but to affift the
mind in its refearches after truth, to increafe the facilities
of instruction, and to diffufe permanent knowledge. If we
wanted authorities to prove the importance and ufefulnefs
of geometry, we might quote every philofopher fince the
days of Plato. Among the Greeks, in ancient, as in the
fchool of Peftalozzi and others in recent times, geometry
was adopted as the beft gymnaftic of the mind. In fact,
Euclid's Elements have become, by common confent, the
bafis of mathematical fcience all over the civilized globe.
But this will not appear extraordinary, if we coniider that
this fublime fcience is not only better calculated than any
other to call forth the fpirit of inquiry, to elevate the mind,
and to ftrengthen the reafoning faculties, but alfo it forms
the beft introduction to moft of the ufeful and important
vocations of human life. Arithmetic, landfurveying, men
furation, engineering, navigation, mechanics, hydroftatics,
pneumatics, optics, phyfical aftronomy, &c. are all depen
dent on the propofitions of geometry.
viii INTRODUCTION.
Much however depends on the firfr. communication of
any fcience to a learner, though the beft and mod eafy
methods are feldom adopted. Propositions are placed be
fore a fludent, who though having a fufficient understand
ing, is told juft as much about them on entering at the
very threshold of the fcience, as gives him a prepofleffion
molt unfavourable to his future fludy of this delightful
fubjedl ; or " the formalities and paraphernalia of rigour are
fo oftentatioufly put forward, as almoft to hide the reality.
Endlefs and perplexing repetitions, which do not confer
greater exactitude on the reafoning, render the demonftra
tions involved and obfcure, and conceal from the view of
the fludent the confecution of evidence." Thus an aver
fion is created in the mind of the pupil, and a fubjecl fo
calculated to improve the reafoning powers, and give the
habit of clofe thinking, is degraded by a dry and rigid
courfe of inftruction into an uninteresting exercife of the
memory. To raife the curiofity, and to awaken the liftlefs
and dormant powers of younger minds mould be the aim
of every teacher ; but where examples of excellence are
wanting, the attempts to attain it are but few, while emi
nence excites attention and produces imitation. The objecT:
of this Work is to introduce a method of teaching geome
try, which has been much approved of by many fcientific
men in this country, as well as in France and America.
The plan here adopted forcibly appeals to the eye, the mofl
fenfitive and the moft comprehenfive of our external organs,
and its preeminence to imprint it fubjecT: on the mind is
fupported by the incontrovertible maxim expreffed in the
well known words of Horace : —
Segnius irritant animus dcmijfa per aurem
§>uam quts funt oculis fubjetta fidelibus.
A feebler imprefs through the ear is made,
Than what is by the faithful eye conveyed.
INTRODUCTION. ix
All language confifts of reprefentative figns, and thofe
figns are the beft which effect their purpofes with the
greateft precifion and difpatch. Such for all common pur
pofes are the audible figns called words, which are ftill
confidered as audible, whether addreffed immediately to the
ear, or through the medium of letters to the eye. Geo
metrical diagrams are not figns, but the materials of geo
metrical fcience, the objecl: of which is to fhow the relative
quantities of their parts by a procefs of reafoning called
Demonftration. This reafoning has been generally carried
on by words, letters, and black or uncoloured diagrams ;
but as the ufe of coloured fymbols, figns, and diagrams in
the linear arts and fciences, renders the procefs of reafon
ing more precife, and the attainment more expeditious, they
have been in this inftance accordingly adopted.
Such is the expedition of this enticing mode of commu
nicating knowledge, that the Elements of Euclid can be
acquired in lefs than one third the time ufually employed,
and the retention by the memory is much more permanent;
thefe facts have been afcertained by numerous experiments
made by the inventor, and feveral others who have adopted
his plans. The particulars of which are few and obvious ;
the letters annexed to points, lines, or other parts of a dia
gram are in fadl but arbitrary names, and reprefent them in
the demonftration ; inftead of thefe, the parts being differ
ently coloured, are made g
to name themfelves, for
their forms in correfpond
ing colours represent them
in the demonftration.
In order to give a bet
ter idea of this fyftem, and A
of the advantages gained by its adoption, let us take a right
I.
x INTRODUCTION.
angled triangle, and exprefs fome of its properties both by
colours and the method generally employed.
Some of the properties of the right angled triangle ABC,
exprejfed by the method generally employed.
1. The angle BAC, together with the angles BCA and
ABC are equal to two right angles, or twice the angle ABC.
2. The angle CAB added to the angle ACB will be equal
to the angle ABC.
3. The angle ABC is greater than either of the angles
BAC or BCA.
4. The angle BCA or the angle CAB is lefs than the
angle ABC.
5. If from the angle ABC, there be taken the angle
BAC, the remainder will be equal to the angle ACB.
6. The fquare of AC is equal to the fum of the fquares
of AB and BC.
The fame properties expreffed by colouring the different parts.
That is, the red angle added to the yellow angle added to
the blue angle, equal twice the yellow angle, equal two
right angles.
Or in words, the red angle added to the blue angle, equal
the yellow angle.
▲
^ C JK* or
The yellow angle is greater than either the red or blue
angle.
INTRODUCTION. xi
▲
4 ^B or Ml Zl
Either the red or blue angle is lefs than the yellow angle.
▲
5 pp minus
In other terms, the yellow angle made lefs by the blue angle
equal the red angle.
+
That is, the fquare of the yellow line is equal to the fum
of the fquares of the blue and red lines.
In oral demonstrations we gain with colours this impor
tant advantage, the eye and the ear can be addreffed at the
fame moment, fo that for teaching geometry, and other
linear arts and fciences, in claries, the fyftem is the beft ever
propofed, this is apparent from the examples juft given.
Whence it is evident that a reference from the text to
the diagram is more rapid and fure, by giving the forms
and colours of the parts, or by naming the parts and their
colours, than naming the parts and letters on the diagram.
Befides the fuperior fimplicity, this fyftem is likewife con
fpicuous for concentration, and wholly excludes the injuri
ous though prevalent practice of allowing the ftudent to
commit the demonftration to memory; until reafon, and fact,
and proof only make impreflions on the underftanding.
Again, when lecturing on the principles or properties of
figures, if we mention the colour of the part or parts re
ferred to, as in faying, the red angle, the blue line, or lines,
&c. the part or parts thus named will be immediately feen
by all in the clafs at the fame inftant ; not fo if we fay the
angle ABC, the triangle PFQ^the figure EGKt, and fo on ;
xii INTRODUCTION.
for the letters mud be traced one by one before the ftudents
arrange in their minds the particular magnitude referred to,
which often occafions confufion and error, as well as lofs of
time. Alfo if the parts which are given as equal, have the
fame colours in any diagram, the mind will not wander
from the object before it ; that is, fuch an arrangement pre
fents an ocular demonstration of the parts to be proved
equal, and the learner retains the data throughout the whole
of the reafoning. But whatever may be the advantages of
the prefent plan, if it be not fubftituted for, it can always
be made a powerful auxiliary to the other methods, for the
purpofe of introduction, or of a more fpeedy reminifcence,
or of more permanent retention by the memory.
The experience of all who have formed fyftems to im
prefs fadts on the understanding, agree in proving that
coloured reprefentations, as pictures, cuts, diagrams, &c. are
more eafily fixed in the mind than mere fentences un
marked by any peculiarity. Curious as it may appear,
poets feem to be aware of this fadl more than mathema
ticians ; many modern poets allude to this vifible fyftem of
communicating knowledge, one of them has thus expreffed
himfelf :
Sounds which addrefs the ear are loft and die
In one fhort hour, but thefe which ftrike the eye,
Live long upon the mind, the faithful fight
Engraves the knowledge with a beam of light.
This perhaps may be reckoned the only improvement
which plain geometry has received fince the days of Euclid,
and if there were any geometers of note before that time,
Euclid's fuccefs has quite eclipfed their memory, and even
occafioned all good things of that kind to be afilgned to
him ; like JEfop among the writers of Fables. It may
alfo be worthy of remark, as tangible diagrams afford the
only medium through which geometry and other linear
INTRODUCTION. xiii
arts and fciences can be taught to the blind, this vifible fys
tem is no lefs adapted to the exigencies of the deaf and
dumb.
Care mult be taken to {how that colour has nothing to
do with the lines, angles, or magnitudes, except merely to
name them. A mathematical line, which is length with
out breadth, cannot pofiefs colour, yet the jun&ion of two
colours on the fame plane gives a good idea of what is
meant by a mathematical line ; recoiled: we are fpeaking
familiarly, fuch a junction is to be underftood and not the
colour, when we fay the black line, the red line or lines, &c.
Colours and coloured diagrams may at firft. appear a
clumfy method to convey proper notions of the properties
and parts of mathematical figures and magnitudes, how
ever they will be found to afford a means more refined and
extenfive than any that has been hitherto propofed.
We fhall here define a point, a line, and a furface, and
demonftrate apropofition in order to fhow the truth of this
alfertion.
A point is that which has pofition, but not magnitude ;
or a point is pofition only, abfiradled from the confideration
of length, breadth, and thicknefs. Perhaps the follow
ing defcription is better calculated to explain the nature of
a mathematical point to thofe who have not acquired the
idea, than the above fpecious definition.
Let three colours meet and cover a
portion of the paper, where they meet
is not blue, nor is it yellow, nor is it
red, as it occupies no portion of the
plane, for if it did, it would belong
to the blue, the red, or the yellow
part ; yet it exifts, and has pofition
without magnitude, fo that with a little reflection, this June
XIV
INTRODUCTION.
tioii of three colours on a plane, gives a good idea of a
mathematical point.
A line is length without breadth. With the afliftance
of colours, nearly in the fame manner as before, an idea of
a line may be thus given : —
Let two colours meet and cover a portion of the paper ;
where they meet is not red, nor is it
blue ; therefore the junction occu
pies no portion of the plane, and
therefore it cannot have breadth, but
only length : from which we can
readily form an idea of what is meant by a mathematical
line. For the purpofe of illu fixation, one colour differing
from the colour of the paper, or plane upon which it is
drawn, would have been fufficient; hence in future, if we
fay the red line, the blue line, or lines, &c. it is the junc
tions with the plane upon which they are drawn are to be
underftood.
Surface is that which has length and breadth without
thicknefs.
When we confider a folid body
(PQ), we perceive at once that it
has three dimenfions, namely : —
length, breadth, and thicknefs ;
S fuppofe one part of this folid (PS)
to be red, and the other part (QR)
yellow, and that the colours be
diftincr. without commingling, the
blue furface (RS) which feparates
thefe parts, or which is the fame
2 thing, that which divides the folid
without lofs of material, mufr. be
without thicknefs, and only poffeffes length and breadth ;
R
1
INTRODUCTION.
xv
this plainly appears from reafoning, limilar to that juft em
ployed in defining, or rather defcribing a point and a line.
The propofition which we have felefted to elucidate the
manner in which the principles are applied, is the fifth of
the firft Book.
In an ifofceles triangle ABC, the
° A
internal angles at the bafe ABC,
ACB are equal, and when the fides
AB, AC are produced, the exter
nal angles at the bafe BCE, CBD
are alio equal.
Produce — — — and
make — — — — —
Draw «— — — and
in
we have
and
^^ common :
and
Again
in
= ^ (B. ,. pr. + .)
Z 7 ^ \ ,
xvi INTRODUCTION.
and =: (B. i. pr. 4).
But
^
Q. E. D.
5y annexing Letters to the Diagram.
Let the equal fides AB and AC be produced through the
extremities BC, of the third fide, and in the produced part
BD of either, let any point D be aflumed, and from the
other let AE be cut off equal to AD (B. 1. pr. 3). Let
the points E and D, fo taken in the produced fides, be con
nected by ftraight lines DC and BE with the alternate ex
tremities of the third fide of the triangle.
In the triangles DAC and EAB the fides DA and AC
are reflectively equal to EA and AB, and the included
angle A is common to both triangles. Hence (B. 1 . pr. 4.)
the line DC is equal to BE, the angle ADC to the angle
AEB, and the angle ACD to the angle ABE ; if from
the equal lines AD and AE the equal fides AB and AC
be taken, the remainders BD and CE will be equal. Hence
in the triangles BDC and CEB, the fides BD and DC are
refpectively equal to CE and EB, and the angles D and E
included by thofe fides are alfo equal. Hence (B. 1 . pr. 4.)
INTRODUCTION. xvii
the angles DBC and ECB, which are thofe included by
the third fide BC and the productions of the equal fides
AB and AC are equal. Alfo the angles DCB and EBC
are equal if thofe equals be taken from the angles DCA
and EBA before proved equal, the remainders, which are
the angles ABC and ACB oppofite to the equal fides, will
be equal.
Therefore in an ifofceles triangle, &c.
Q^E. D.
Our object in this place being to introduce the fyftem
rather than to teach any particular fet of propofitions, we
have therefore feledted the foregoing out of the regular
courfe. For fchools and other public places of inftruclion,
dyed chalks will anfwer to defcribe diagrams, 6cc. for private
ufe coloured pencils will be found very convenient.
We are happy to find that the Elements of Mathematics
now forms a confiderable part of every found female edu
cation, therefore we call the attention of thofe interefted
or engaged in the education of ladies to this very attractive
mode of communicating knowledge, and to the fucceeding
work for its future developement.
We fhall for the prefent conclude by obferving, as the
fenfes of fight and hearing can be fo forcibly and inftanta
neously addreffed alike with one thoufand as with one, the
million might be taught geometry and other branches of
mathematics with great eafe, this would advance the pur
pofe of education more than any thing that might be named,
for it would teach the people how to think, and not what
to think ; it is in this particular the great error of education
originates.
XV1U
THE ELEMENTS OF EUCLID.
BOOK I.
DEFINITIONS.
I.
A point is that which has no parts.
II.
A line is length without breadth.
III.
The extremities of a line are points.
IV.
A ftraight or right line is that which lies evenly between
its extremities.
V.
A furface is that which has length and breadth only.
VI.
The extremities of a furface are lines.
VII.
A plane furface is that which lies evenly between its ex
tremities.
VIII.
A plane angle is the inclination of two lines to one ano
ther, in a plane, which meet together, but are not in the
fame direction.
IX.
A plane rectilinear angle is the inclina
tion of two ftraight lines to one another,
which meet together, but are not in the
fame ftraight line.
BOOK I. DEFINITIONS. xix
A
X.
When one ftraight line (landing on ano
ther ftraight line makes the adjacent angles
equal, each of thefe angles is called a right
angle, and each of thefe lines is faid to be
perpendicular to the other.
XI.
An obtufe angle is an angle greater
than a right angle.
XII.
An acute angle is an angle lefs than a
right angle.
XIII.
A term or boundary is the extremity of any thing.
XIV.
A figure is a furface enclofed on all fides by a line or lines.
XV.
A circle is a plane figure, bounded
by one continued line, called its cir
cumference or periphery ; and hav
ing a certain point within it, from
which all ftraight lines drawn to its
circumference are equal.
XVI.
This point (from which the equal lines are drawn) is
called the centre of the circle.
*•*•• •
xx BOOK I. DEFINITIONS.
XVII.
A diameter of a circle is a flraight line drawn
through the centre, terminated both ways
in the circumference.
XVIII.
A femicircle is the figure contained by the
diameter, and the part of the circle cut off
by the diameter.
XIX.
* A fegment of a circle is a figure contained
by a flraight line, and the part of the cir
\ J cumference which it cuts off.
XX.
A figure contained by flraight lines only, is called a recti
linear figure.
XXI.
A triangle is a rectilinear figure included by three fides.
XXII.
A quadrilateral figure is one which is bounded
by four fides. The flraight lines — «— —
and !■.■■■ connecting the vertices of the
oppofite angles of a quadrilateral figure, are
called its diagonals.
XXIII.
A polygon is a rectilinear figure bounded by more than
four fides.
BOOK I. DEFINITIONS.
xxi
XXIV.
A triangle whofe three fides are equal, is
faid to be equilateral.
XXV.
A triangle which has only two fides equal
is called an ifofceles triangle.
XXVI.
A fcalene triangle is one which has no two fides equal.
XXVII.
A right angled triangle is that which
has a right angle.
XXVIII.
An obtufe angled triangle is that which
has an obtufe angle.
XXIX.
An acute angled triangle is that which
has three acute angles.
XXX.
Of fourfided figures, a fquare is that which
has all its fides equal, and all its angles right
angles.
XXXI.
A rhombus is that which has all its fides
equal, but its angles are not right angles.
XXXII.
An oblong is that which has all its
angles right angles, but has not all its
fides equal.
u
xxii BOOK 1. POSTULATES.
XXXIII.
A rhomboid is that which has its op
pofite fides equal to one another,
but all its fides are not equal, nor its
angles right angles.
XXXIV.
All other quadrilateral figures are called trapeziums.
XXXV.
Parallel flraight lines are fuch as are in
■'^^^ m ^ mmm ^ m ^ mmi ^ the fame plane, and which being pro
duced continually in both directions,
would never meet.
POSTULATES.
I.
Let it be granted that a flraight line may be drawn from
any one point to any other point.
II.
Let it be granted that a finite flraight line may be pro
duced to any length in a flraight line.
III.
Let it be granted that a circle may be defcribed with any
centre at any diflance from that centre.
AXIOMS.
I.
Magnitudes which are equal to the fame are equal to
each other.
II.
If equals be added to equals the fums will be equal.
BOOK I. AXIOMS.
xxin
III.
If equals be taken away from equals the remainders will
be equal.
IV.
If equals be added to unequals the fums will be un
equal.
V.
If equals be taken away from unequals the remainders
will be unequal.
VI.
The doubles of the fame or equal magnitudes are equal.
VII.
The halves of the fame or equal magnitudes are equal.
VIII.
Magnitudes which coincide with one another, or exactly
fill the fame fpace, are equal.
IX.
The whole is greater than its part.
X.
Two flraight lines cannot include a fpace.
XI.
All right angles are equal.
XII.
If two ftraight lines ( Z^ZI
flraight line («
) meet a third
■ ) fo as to make the two interior
angles ( and i ^ ) on the fame fide lefs than
two right angles, thefe two ftraight lines will meet if
they be produced on that fide on which the angles
are lefs than two right angles.
XXIV
BOOK I. ELUCIDATIONS.
The twelfth axiom may be expreffed in any of the fol
lowing ways :
i . Two diverging ftraight lines cannot be both parallel
to the fame ftraight line.
2. If a flraight line interfecT: one of the two parallel
ftraight lines it muft alfo interfecl the other.
3. Only one flraight line can be drawn through a given
point, parallel to a given ftraight line.
Geometry has for its principal objects the expofition and
explanation of the properties of figure, and figure is defined
to be the relation which fubfifts between the boundaries of
fpace. Space or magnitude is of three kinds, linear, Juper
ficial, &n&folid.
Angles might properly be confideret" as a fourth fpecies
of magnitude. Angular magnitude evidently confifts of
parts, and muft therefore be admitted to be a fpecies ol
quantity The ftudent muft not fuppofe that the magni
tude of an angle is affected by the length
of the ftraight lines which include it, and
of whofe mutual divergence it is the mea
fure. The vertex of an angle is the point
where the fides or the legs of the angle
meet, as A.
An angle is often defignated by a fingle letter when its
legs are the only lines which meet to
gether at its vertex. Thus the red and
blue lines form the yellow angle, which
in other fyftems would be called the
angle A. But when more than two
B lines meet in the fame point, it was ne
ceffary by former methods, in order to
avoid confufion, to employ three letters
to defignate an angle about that point,
A
BOOK I. ELUCIDATIONS. xxv
the letter which marked the vertex of the angle being
always placed in the middle. Thus the black and red lines
meeting together at C, form the blue angle, and has been
ufually denominated the angle FCD or DCF The lines
FC and CD are the legs of the angle; the point C is its
vertex. In like manner the black angle would be defignated
the angle DCB or BCD. The red and blue angles added
together, or the angle HCF added to FCD, make the angle
HCD ; and fo of other angles.
When the legs of an angle are produced or prolonged
beyond its vertex, the angles made by them on both fides
of the vertex are faid to be vertically oppofite to each other :
Thus the red and yellow angles are faid to be vertically
oppofite angles.
Superpojition is the procefs by which one magnitude may
be conceived to be placed upon another, fo as exactly to
cover it, or fo that every part of each fhall exactly coin
cide.
A line is faid to be produced, when it is extended, pro
longed, or has its length increafed, and the increafe of
length which it receives is called its produced part, or its
production.
The entire length of the line or lines which enclofe a
figure, is called its perimeter. The firft fix books of Euclid
treat of plain figures only. A line drawn from the centre
of a circle to its circumference, is called a radius. The
lines which include a figure are called its Jides. That fide
of a right angled triangle, which is oppofite to the right
angle, is called the hypotenufe. An oblong is defined in the
fecond book, and called a rectangle. All the lines which
are conlideied in the firft. fix books of the Elements are
fuppofed to be in the fame plane.
The Jlraightedge and compajfes are the only inftruments,
xxvi BOOK I. ELUCIDATIONS.
the ufe of which is permitted in Euclid, or plain Geometry.
To declare this reftriction is the object of the populates.
The Axioms of geometry are certain general propofitions,
the truth of which is taken to be felfevident and incapable
of being eftabliflied by demonftration.
Propofitions are thofe refults which are obtained in geo
metry by a procefs of reafoning. There are two fpecies of
propofitions in geometry, problems and theorems.
A Problem is a propofition in which fomething is pro
pofed to be done ; as a line to be drawn under fome given
conditions, a circle to be defcribed, fome figure to be con
firucted, 5cc.
The folution of the problem confifts in fhowing how the
thing required may be done by the aid of the rule or ftraight
edge and compafies.
The demonjlration confifts in proving that the procefs in
dicated in the folution really attains the required end.
A Theorem is a propofition in which the truth of fome
principle is aflerted. This principle muft be deduced from
the axioms and definitions, or other truths previously and
independently eftablifhed. To fhow this is the object of
demonstration.
A Problem is analogous to a poftulate.
A Theorem refembles an axiom.
A Pojlulate is a problem, the folution of which is afiumed.
An Axiom is a theorem, the truth of which is granted
without demonfbration.
A Corollary is an inference deduced immediately from a
propofition.
A Scholium is a note or obfervation on a propofition not
containing an inference of fufficient importance to entitle it
to the name of a corollary.
A Lemma is a propofition merely introduced for the pur
pole of efiablifhing fome more important propofition.
XXV11
SYMBOLS AND ABBREVIATIONS.
,*. exprefies the word therefore.
V becaufe.
— equal. This fign of equality may
be read equal to, or is equal to, or are equal to ; but
any difcrepancy in regard to the introduction of the
auxiliary verbs Is, are, &c. cannot affect the geometri
cal rigour.
d\p means the fame as if the words ' not equal' were written.
r~ fignifies greater than.
33 ... . lefs than.
if ... . not greater than.
~h .... not lefs than.
j is vezdplus (more), the fign of addition ; when interpofed
between two or more magnitudes, fignifies their fum.
— is read minus (lefs), fignifies fubtraction ; and when
placed between two quantities denotes that the latter
is to be taken from the former.
X this fign exprefies the product of two or more numbers
when placed between them in arithmetic and algebra ;
but in geometry it is generally ufed to exprefs a rect
angle, when placed between " two fixaight lines which
contain one of its right angles." A reclangle may alfo
be reprefented by placing a point between two of its
conterminous fides.
2 exprefies an analogy or proportion ; thus, if A, B, C
and D, reprefent four magnitudes, and A has to
B the fame ratio that C has to D, the propofition
is thus briefly written,
A : B : : C : D,
A : B = C : D,
A C
° r B=D.
This equality or famenefs of ratio is read,
xxviii STMBOLS AND ABBREVIATIONS.
as A is to B, fo is C to D ;
or A is to B, as C is to D.
 fignifies parallel to.
_L . . . . perpendicular to.
. angle.
. right angle.
m
two right angles.
Xi x or I > briefly defignates a point.
\ . =, or ^ flgnities greater, equal, or lefs than.
The lquare defcribed on a line is concifely written thus,
In the fame manner twice the fquare of, is expreffed by
2 2 .
def. fignifies definition.
pos pojlulate.
ax axiom.
hyp hypothefis. It may be neceffary here to re
mark, that the hypothefis is the condition affumed or
taken for granted. Thus, the hypothefis of the pro
pofition given in the Introduction, is that the triangle
is ifofceles, or that its legs are equal.
conft confiriiolion. The confiruSlion is the change
made in the original figure, by drawing lines, making
angles, defcribing circles, &c. in order to adapt it to
the argument of the demonftration or the folution of
the problem. The conditions under which thefe
changes are made, are as indisputable as thofe con
tained in the hypothefis. For inftance, if we make
an angle equal to a given angle, thefe two angles are
equal by confbruction.
Q^ E. D Quod erat demonfirandum.
Which was to be demonftrated.
CORRIGENDA. xxix
Faults to be correSled before reading this Volurne.
Page 13, line 9, for def. 7 read def. 10.
45, laft line, for pr. 19 raz^pr. 29.
54, line 4 from the bottom, /or black and red line read blue
and red line.
59, line 4, /or add black line fquared read add blue line
fquared.
60, line 17, /or red line multiplied by red and yellow line
read red line multiplied by red, blue, and yellow line.
76, line 11, for def. 7 read def. 10.
81, line 10, for take black line r*W take blue line.
105, line 11, for yellow black angle add blue angle equal red
angle read yellow black angle add blue angle add red
angle.
129, laft line, for circle read triangle.
141, line 1, for Draw black line read Draw blue line.
196, line 3, before the yellow magnitude infert M.
©ttclto
BOOK I. •
PROPOSITION I. PROBLEM.
N a given finite
Jlraight line ( )
to defcribe an equila
teral triangle.
Defcribe I —J and
©
(poftulate 3.); draw and — — (port. 1.).
then will \ be equilateral.
For ^— = (def. 15.);
and therefore * \ is the equilateral triangle required.
Q^E. D.
BOOK I. PROP. II. PROB.
ROM a given point ( ■■ ),
to draw ajiraight line equal
to a given finite Jlraight
line ( ).
■ (port, i.), defcribe
A(pr. i.), produce — — (port.
©
2.), defcribe
(poft. 3.), and
(poft. 3.) ; produce — — — (poft. 2.), then
is the line required.
For
and
(def. 15.),
(conft.), ,\
(ax. 3.), but (def. 15.)
drawn from the given point (
)>
is equal the given line
Q. E. D.
BOOK I. PROP. III. PROB.
ROM the greater
( — ■) of
two given Jiraight
lines, to cut off a part equal to
the lefs ( ) .
Draw
(pr. 2.) ; defcribe
(port. 3 .), then
For
and
(def. 15.),
(conft.) ;
(ax. 1.).
Q. E. D.
BOOK I. PROP. IF. THEOR.
F two triangles
have two Jides
of the one
reJpecJively
equal to two Jides of the
other, ( ■ to — ■—
and — — to ■ ) tfW
//$*• rf«£/<?j ( and )
contained by thofe equal
fides alfo equal ; then their bafes or their fdes (■ and
— ■ ) are alfo equal : and the remaining and their remain
ing angles oppofte to equal fdes are refpeSlively equal
i J^ =z ^^ and ^^ =  f^ ) : and the triangles are
equal in every reJpecJ.
Let the two triangles be conceived, to be fo placed, that
the vertex of the one of the equal angles, or $
— — to coincide
— coincide with ■ if ap
will coincide with — ■— — ,
or two ftraight lines will enclofe a fpace, which is impoflible
fliall fall upon that of the other, and r
with 9 then will 
plied : confequently — — —
(ax. 10), therefore
> = >
and
^L = ^L , and as the triangles / \ and /V
coincide, when applied, they are equal in every refpedl:.
Q. E. D.
BOOK I. PROP. V. THEOR.
N any ifofceles triangle
A
if the equal fides
be produced, the external
angles at the bafe are equal, and the
internal angles at the bafe are alfo
equal.
Produce
; and
(poft. 2.), take
j (P r  3);
draw
Then in
common to
(conft), ^
(hyp.) /. Jk = k
A = ±,A=A
(pr. 4.) but
(ax. 3.)
Q.E. D.
BOOK I. PROP. VI. THEOR.
and
N any triangle (
A
)if
two angles ( and j^L )
are equal \t lie fides ( ....
■"■ ' ) oppojite to them are alfo
equal.
For if the fides be not equal, let one
of them ■■■■ be greater than the
other — , and from it cut off
■ = — — ■ — (pr. 3.), draw
Then
(conft.)
in
L^A,
(hyp.) and
common,
.*. the triangles are equal (pr. 4.) a part equal to the whole,
which is abfurd ; .'. neither of the fides ■— » or
■ ■ m i is greater than the other, /. hence they are
equal
Q. E. D.
BOOK I. PROP. VII. THEOR.
7
N the fame bafe (>
■), and
on
the fame fide of it there cannot be two
triangles having their conterminous
fdes ( and ■ — ■— ,
— ■— ■« — ■» #«</ ■■»■■■■■») at both extremities of
the bafe, equal to each other.
When two triangles ftand on the fame bafe,
and on the fame fide of it, the vertex of the one
(hall either fall outfide of the other triangle, or
within it ; or, laflly, on one of its fides.
If it be poflible let the two triangles be con
f = 1
firucted fo that
draw
0=*
J
and,
(P r  5)
, then
and
▼ =>
but (pr. 5.)
s
which is abfurd,
therefore the two triangles cannot have their conterminous
fides equal at both extremities of the bafe.
Q. E. D.
BOOK I. PROP. VIII. THEOR.
F two triangles
have two Jides
of the one reflec
tively equal to
two fides of the other
and — — = ),
and alfo their bafes (
rr — ■"■)> equal ; then the
angles ("^^B and "^^H')
contained by their equal fides
are alfo equal.
If the equal bafes
and
be conceived
to be placed one upon the other, fo that the triangles fhall
lie at the fame fide of them, and that the equal fides
«. __» and — _ , _ _____ and _____,__,_. be con
terminous, the vertex of the one muft fall on the vertex
of the other ; for to fuppofe them not coincident would
contradict the laft propofition.
Therefore the fides
cident with
and „
and
A = A
,« being coin
Q. E. D.
BOOK I. PROP. IX. PROB.
bifeSl a given reSlilinear
angle 4 ).
Take
(P r  3)
draw
, upon which
defcribe ^f
draw — ^—
(pr. i.),
Becaufe — — — . = ___ (confl:.)
and ^— i — common to the two triangles
and
(confl:.),
A
( P r. 8.)
Q. E. D.
10
BOOK I. PROP. X. PROB.
O bifefi a given finite Jiraight
line ( ««■■■■).
and
common to the two triangles.
Therefore the given line is bifecled.
Qj. E. D.
BOOK I. PROP. XL PROB.
ii
( ;
a perpendicular.
ROM a given
point (^— ™ ')>
in a given
Jlraight line
— ), to draw
Take any point (■
cut off ■
) in the given line,
— (P r  3)»
A
conftrucl: £_ \ (pr. i.),
draw and it fhall be perpendicular to
the given line.
For
(conft.)
(conft.)
and
common to the two triangles.
Therefore Jj ~
(pr. 8.)
(def. io.).
Q^E. D.
12
500A: /. PROP. XII. PROD.
O draw a
Jiraight line
perpendicular
to a given
/ indefinite Jiraight line
(«a^_ ) from a given
{point /Y\ ) 'without.
With the given point x\ as centre, at one fide of the
line, and any diftance — — — capable of extending to
the other fide, defcribe
Make
draw —
(pr. 10.)
and
then
For (pr. 8.) fince
(conft.)
and
common to both,
= (def. 15.)
and
(def. io.).
Q. E. D.
BOOK I. PROP. XIII. THEOR.
*3
HEN a Jiraight line
( ) Jlanding
upon another Jiraight
line ( )
makes angles with it; they are
either two right angles or together
equal to two right angles.
If
be J_ to
then,
and
*=0\
(def. 7.),
But if
draw
+
+ jm =
be not JL to ,
L ;(pr. 11.)
= ( I J (conft.),
: mm + V+mk(zx.2.)
Q. E. D.
H
BOOK I. PROP. XIV. THEOR.
IF two jlr aight lines
( and "~*"^),
meeting a third Jlr aight
line ( ), «/ //tf
yZras* ^w«/, tfW ^/ oppojite fides of
it, make with it adjacent angles
(
and
A
) equal to
two right angles ; thefe fraight
lines lie in one continuous fraight
line.
For, if pomble let
j and not
be the continuation of
then
but by the hypothecs
,. 4 = A
+
(ax. 3.); which is abfurd (ax. 9.).
is not the continuation of
and
the like may be demonftrated of any other flraight line
except , .*. ^^— ^— is the continuation
of
Q. E. D.
BOOK I. PROP. XV. THEOR.
15
F two right lines (
and ) interfeSl one
another, the vertical an
gles
and
and
<4
are equal.
<4 +
► 4
In the fame manner it may be lhown that
Q^_E. D.
i6
BOO A' /. PROP. XVI. THEOR.
F a fide of a
trian \
is produced, the external
angle ( V..„\ ) «
greater than either of the
internal remote angles
(
A " A
Make
Draw
=  (pr. io.).
and produce it until
— : draw  .
In \ and ^*^f .
► 4
(conft. pr. 15.), .'. ^m = ^L (pr. 4.),
In like manner it can be mown, that if ^^— ■ 
be produced, ™ ^ IZ ^^ . and therefore
which is = ft is C ^ ft .
Q. E. D.
BOOK I. PROP. XVII. THEOR. 17
NY tiao angles of a tri
A
angle ^___Jk are to
gether lefs than two right angles.
Produce
A
+
, then will
= £D
But
CZ Jk (pr. 16.)
and in the fame manner it may be mown that any other
two angles of the triangle taken together are lefs than two
right angles.
Q. E. D.
D
i8
BOOK I. PROP. XVIII. THEOR.
A
N any triangle
if one fide «■■*» be
greater than another
, the angle op
pofite to the greater fide is greater
than the angle oppofite to the lefs.
1. e.
*
Make
(pr. 3.), draw
Then will J/i R ~ J ^ (pr. 5.);
but MM d (pr. 16.)
,*. £ ^ C and much more
, s ^c >.
Q. E. D.
BOOK I. PROP. XIX. THEOR.
*9
A
F in any triangle
one angle mm be greater
than another J ^ the fide
which is oppofite to the greater
angle, is greater than the Jide
oppofite the lefs.
If
be not greater than
or
then mull
If
then
A
(p r  5) ;
which is contrary to the hypothefis.
— is not lefs than — — ■ — ; for if it were,
(pr. 1 8.)
which is contrary to the hypothefis :
Q. E. D.
20
BOOK I. PROP. XX. THEOR.
4
I NY two fides
and •^^— of a
triangle
Z\
taken together are greater than the
third fide (■
')•
Produce
and
draw
(P r  3);
Then becaufe —'
(conft.).
^ = 4 (pr 
*c4
(ax. 9.)
+
and .'.
+
(pr. 19.)
Q.E.D
BOOK I. PROP. XXL THEOR.
21
•om
any
point ( / )
A
within a triangle
Jiraight lines be
drawn to the extremities of one fide
(_.... ), thefe lines tnujl be toge
ther lefs than the other twofdes, but
tnujl contain a greater angle.
Produce
mm— mm f mmmmmm C «^— ■» (pr. 20.),
add ..... to each,
\ __... C ■■— ■  ...... (ax. 4.)
In the fame manner it may be mown that
.— + C h
which was to be proved.
Again
and alfo
4c4
c4
(pr. 16.),
(pr. 16.),
QJE.D.
22
BOOK I. PROP. XXII. THEOR.
[IVEN three right
lines < ■•••■■
the fum of any
two greater than
the third, to conJlruEl a tri
angle whoje fides fliall be re
fpeSlively equal to the given
lines.
BOOK I. PROP. XXIII. PROB. 23
T a given point ( ) in a
given firaight line (^— ■■— ),
to make an angle equal to a
given reel i lineal angle (jgKm )
Draw
between any two points
in the legs of the given angle.
fo that
Conftruct
and
A
(pr. 22.).
Then
(pr. 8.).
Q. E. D.
24
BOOK I. PROP. XXIV. THEOR.
F two triangles
have two fides of
the one reflec
tively equal to
two fides of the other (
to ————— and 
to ), and if one of
the angles ( < jl ^ ) contain
ed by the equal fides be
[L m \)> the fide ( — — ^ ) which is
oppofite to the greater angle is greater than the fide ( )
which is oppofte to the lefs angle.
greater than the other (L. m \), the fide (
Make C3 = / N (pr. 23.),
and — ^— = (pr. 3.),
draw ■■■■■■■•■ and •—.
Becaufe — — — =: — — — (ax. 1. hyp. conft.)
.'. ^ = ^f (pr 
but '^^ Z2 * »
.*. ^J Z] £^'
/. — — CI (pr. 19.)
but ■ = (p r 4)
.. c
Q. E. D.
BOOK I. PROP. XXV. THEOR.
25
F two triangles
have two Jides
( and
— ) of the
one refpeSlively equal to two
Jides ( and )
of the other, but their bafes
unequal, the angle fubtended
by the greater bafe (^— — )
of the one, mujl be greater
than the angle fubtended by
the lefs bafe ( ) of the other.
▲ A A A
= , CZ or Z2 mk > s not equal to ^^
= ^ then ^— — ss (pr. 4.)
for if
zz ^^ then ■— — « =
which is contrary to the hypothefis ;
is not lefs than
for if
A
A=A
then
(pr. 24.),
which is alfo contrary to the hypothefis
1= m*
Q^E. D.
26 BOOK I. PROP. XXVI. THEOR.
Case I.
F two triangles
have two angles
of the one re
fpeflively equal
to two angles of tlie other,
(
and
Case II.
AA
\), and a fide
of the one equal to a fide of
the other fimilarly placed
with reJpecJ to the equal
angles, the remaining fdes
and angles are refpeclively
equal to one another.
CASE I.
Let ■ ..!■■ — and ....■■ ■■ which lie between
the equal angles be equal,
then i^BHI ~ MMMMMItM .
For if it be poffible, let one of them
greater than the other ;
be
make
In
and
draw
we
have
A = A
(pr. 4.)
BOOK I. PROP. XXVI. THEOR. 27
but JA = Mm (hyp.
and therefore ^Bl = ■ &. which is abfurd ;
hence neither of the fides ■""■""■ and ——•■■• is
greater than the other ; and .*. they are equal ;
., and </] = ^j ? (pr<
4.).
CASE II.
Again, let «^— — • — «— — ^— ? which lie oppofite
the equal angles MmL and 4Hk>. If it be poflible, let
, then take — — — ■ =: « ■ ■" — ■,
draw
Then in * ^ and Lm~. we have
— = and = ,
.'. mk. = Mi (pr 4)
but Mk = mm (hyp)
.*. Amk. = AWL which is abfurd (pr. 16.).
Confequently, neither of the fides •— — ■• or ■—•••• is
greater than the other, hence they muft be equal. It
follows (by pr. 4.) that the triangles are equal in all
refpedls.
Q. E. D.
28
BOOK I. PROP. XXVII. THEOR.
are parallel.
F ajlraight line
( ) meet
ing two other
Jlraight lines,
and ) makes
•with them the alternate
angles (
and
) equal, thefe two Jlraight lines
If
be not parallel to
they (hall meet
when produced.
If it be poffible, let thofe lines be not parallel, but meet
when produced ; then the external angle ^w is greater
than flftk. (pr. 16), but they are alfo equal (hyp.), which
is abfurd : in the fame manner it may be fhown that they
cannot meet on the other fide ; .*. they are parallel.
Q. E. D.
BOOK I. PROP. XXVIII. THEOR. 29
F aflraight line
ting two other
Jlraight lines
and ),
makes the external equal to
the internal and oppojite
angle, at the fame fide of
the cutting line {namely,
(
A A
or
), or if it makes the two internal angles
at the fame fide ( V ■ and ^^ , or  ^ tfW ^^^)
together equal to two right angles, thofe two fraight lines
are parallel.
Firft, if
mL = jik then A = W
A = W • 1
= (pr'i 5)»
(pr. 27.).
A II
Secondly, if J £ }  =
then ^ + ^F = L— JL. J(pr i3)»
(ax. 3.)
* = ▼
(pr. 27.)
BOOK I. PROP. XXIX. THEOR.
STRAIGHT line
( ) falling on
two parallel Jlraight
lines ( and
), makes the alternate
angles equal to one another ; and
alfo the external equal to the in
ternal and oppofite angle on the
fame fide ; and the two internal
angles on the fame fide together
equal to tivo right angles.
For if the alternate angles
draw
■, making
Therefore
and J^ ^ be not equal,
Am (p r  2 3)
(pr. 27.) and there
fore two flraight lines which interfect are parallel to the
fame ftraight line, which is impoffible (ax. 12).
Hence the alternate angles
and
are not
unequal, that is, they are equal: = J m. (pr. 15);
.*. J f^ = J^ ^ , the external angle equal to the inter
nal and oppofite on the fame fide : if M ^r be added to
both, then
+
*
=£D
(P 1 "^)
That is to fay, the two internal angles at the fame fide of
the cutting line are equal to two right angles.
Q. E. D.
BOOK I. PROP. XXX. THEOR.
3 1
TRAlGHT/mes( mmm " m )
which are parallel to the
fame Jlraight line ( ),
are parallel to one another.
interfed:
Then,
(=)•
= ^^ = Mm (pr. 29.),
II
(pr. 27.)
Q. E. D.
32
BOOK I. PROP. XXXI. PROD.
ROM a given
point f to
draw a Jlr aight
line parallel to a given
Jlraight line ( ).
Draw — ^— • from the point / to any point /
in
make
then —
(pr. 23.),
 (pr. 27.).
Q, E. D.
BOOK I. PROP. XXXII. THEOR. 33
F any fide ( )
of a triangle be pro
duced, the external
am
T
'gle ( ) is equal
to the fum of the two internal and
oppofite angles ( and ^ Rt, ) ,
and the three internal angles of
every triangle taken together are
equal to two right angles.
Through the point / draw
II (pr 3 0
Then < ^^^ ( (pr. 29.),
(pr. 13.). 4
+ Km*. = ^^ (ax. 2.),
and therefore
(pr. 13.).
O. E. D.
34 BOOK I. PROP. XXXIII. THEOR.
fRAIGHT lines (
and ) which join
the adjacent extremities of
two equal and parallel Jlraight
* ), are
them/elves equal and parallel.
Draw
the diagonal.
(hyp.)
and
— — common to the two triangles ;
= — — , and^J = ^L (pr. 4.);
and .".
(pr. 27.).
Q. E. D.
BOOK I. PROP. XXXIV. THEOR. 35
HE oppofite Jides and angles of
any parallelogram are equal,
and the diagonal ( )
divides it into two equal parts.
Since
(pr. 29.)
and
■— — common to the two triangles.
/. \
> (pr. 26.)
and m J = m (ax.) :
Therefore the oppofite fides and angles of the parallelo
gram are equal : and as the triangles \^ and \^ /
are equal in every refpect (pr. 4,), the diagonal divides
the parallelogram into two equal parts.
Q. E. D.
36 BOOK I. PROP. XXXV. THEOR.
ARALLELOGRAMS
on the fame bafe, and
between the fame paral
lels, are [in area) equal.
On account of the parallels,
and
But,
Kpr. 29.)
' (pr 34)
(pr. 8.)
and
U
minus
minus
\ ■
Q. E. D.
BOOK I. PROP. XXXVI. THEOR. 37
ARALLELO
GRAMS
1 a
equal bafes, and between the
fame parallels, are equal.
Draw
and ..— ,
» b y (P r  34> and hyp.);
= and II (pr 33)
And therefore
X
is a parallelogram :
but
!>■
(P r  35)
II
(ax. 1.).
Q. E. D.
38 BOOK I. PROP. XXXVII. THEOR.
RIANGLES
k
and
i
on the fame bafe (■— «■— )
and between the fame paral
lels are equal.
Draw
Produce
\ (pr 3 1 )
L and A
and are parallelograms
on the fame bafe, and between the fame parallels,
and therefore equal, (pr. 35.)
~ twice
=r twice
i
> (P r  34)
i
Q.E D.
BOOK I. PROP. XXXVIII. THEOR. 39
RIANGLES
II and H
) on :
f^wrt/ ^rf/^j and between "•**
the fame parallels are equal.
Draw ......
and II > (pf  3 ''
AM
(pr. 36.);
i . ,„, 1
but i cs twice ^H (pr. 34.),
# i
and ^jv = twice ^ (pr. 34.),
A A
(ax. 7.).
Q^E. D.
4o BOOK I. PROP. XXXIX. THEOR.
QUAL triangles
W
\
and ^ on the fame bafe
( ) and on the fame fide of it, are
between the fame parallels.
If — ■— — » , which joins the vertices
of the triangles, be not  — ^— ,
draw —  (pr.3i.)>
meeting  .
Draw
Becaufe
II
(conft.)
(pr 37):
(hyp.) ;
, a part equal to the whole,
which is abfurd.
U — ^— ; and in the fame
manner it can be demonftrated, that no other line except
is  ; ..  .
O. E. D.
BOOK I. PROP. XL. THEOR.
QUAL trian
gles
41
(
and
L
)
on equal bafes, and on the
fame Jide, are between the
fame parallels.
If ..... which joins the vertices of triangles
be not 1 1 ■' ,
draw — —  — — (pr. 31.),
meeting
Draw
Becaufe
(conft.)
.  = > , a part equal to the whole,
which is abfurd.
1 ' f ^— — : and in the fame manner it
can be demonftrated, that no other line except
is  : /. 
Q. E. D.
42 BOOK L PROP. XLI. THEOR.
F a paral
lelogram
A
V
Draw
and a triangle are upon
the fame bafe ^^^^^— and between
the fame parallels  and
— ^— ^— , the parallelogram is double
the triangle.
the diagonal ;
Then
V=J
z= twice
(P r  37)
(P r  34)
^^ = twice £J .
.Q.E.D.
BOOK I. PROP. XLII. THEOR. 43
O conflruSl a
parallelogram
equal to a given
4
triangle ■■■^ L and hav
ing an angle equal to a given
rectilinear angle ^ .
Make — i^^^— zz  (pr. 10.)
Draw .
Make J^ = (P n 2 3*)
Draw  " jj ~  (pr. 31.)
4
= twi ce y (pr. 41.)
but T = A (pr. 38.)
,V.4.
Q. E. D.
44 BOOK I. PROP. XLIII. THEOR.
HE complements
and ^ ^f of
the parallelograms which are about
the diagonal of a parallelogram are
equal.
1
(pr 34)
and
V = >
(pr. 34)
(ax. 3.)
Q. E. D.
BOOK I. PROP. XLIV. PROB.
45
O a given
Jlraight line
( ) to ap
ply a parallelo
gram equal to a given tri
angle ( \ ), and
having an angle equal to
a given reSlilinear angle
Make
w.
with
(pr. 42.)
and having one of its fides — — —  conterminous
with and in continuation of — ^— — ».
Produce — — till it meets  •■•»•
draw prnHnpp it till it mpptc — »—■•» continued ;
draw I ■» meeting
produced, and produce ••»•••»•
but
A=T
(pr 43J
(conft.)
(pr.19. and conft.)
Q. E. D.
BOOK I. PROP. XLV. PROB.
O conjlruSl a parallelogram equal
to a given reftilinear figure
( ) and having an
angle equal to a given reftilinear angle
Draw
and
tl.
dividing
the rectilinear figure into triangles.
Conftruft
having = £ (pr. 42.)
*~\
#=►
and
to — — — a ppiy
having mW = AW (pr. 44)
man, apply £ =z
having HF = AW (P^ 44)
is a parallelogram, (prs. 29, 14, 30.)
having ,fl7 =
Q. E. D.
BOOK I. PROP. XLVI. PROB. 47
PON a given Jlraight line
(— ■■ — ) to confiruB a
fquare.
Draw
Draw ■
ing
» _L and =
(pr. 1 1. and 3.)
II
drawn
• , and meet
W ~W
In 1_
M
(conft.)
S3 a right angle (conft.)
M — = a ri g h t angle (pr. 29.),
and the remaining fides and angles muft
be equal, (pr. 34.)
and .*.
mk is a fquare. (def. 27.)
Q. E. D.
48 BOOK I. PROP. XLVIL THEOR.
N a right angled triangle
thefquare on the
hypotenufe — — — is equal to
the fum of the fquares of the fides, («
and ).
On
and
defcribe fquares, (pr. 46.)
Draw ■■■ »i
alfo draw
— — (pr. 31.)
 and — ^— .
To each add
= —  and
Again, becauje
BOOK I. PROP. XLVII. THEOR. 49
and
:= twice
twice
In the fame manner it may be fhown
that
#
hence
++
Q E. D.
H
5°
BOOK I. PROP. XLVIIL THEOR.
F the fquare
of one fide
( — ; — ) of
a triangle is
equal to the fquares of the
other two fides ( n
and ), the angle
( )fubtended by that
fide is a right angle.
Draw —
■ and =
(prs.11.3.)
ind draw •»•»•■■• alfo.
Since
(conft.)
2 +
+
but ■
and
— 8 +
+
= — " — ' (P r  47).
' = 2 (hyp.)
and .*.
confequently
(pr. 8.),
is a right angle.
Q. E. D.
BOOK II.
DEFINITION I.
RECTANGLE or a
right angled parallelo
gram is faid to be con
tained by any two of its adjacent
or conterminous fides.
Thus : the right angled parallelogram ■
be contained by the fides ■— ^— and «
or it may be briefly defignated by
is faid to
If the adjacent fides are equal; i. e. — — — — s ■—■"■""■"j
then — ^— — • m ■ i which is the expreflion
for the redlangle under
is a fquare, and
is equal to J
and
■ or
■ or
52
BOOK II. DEFINITIONS.
DEFINITION II.
N a parallelogram,
the figure compoied
of one 01 the paral
lelograms about the diagonal,
together with the two comple
ments, is called a Gnomon.
Thus
and
are
called Gnomons.
BOOK II. PROP. I. PROP,.
53
HE reclangle contained
by two Jlraight lines,
one of which is divided
into any number of parts,
i +
is equal to the fum of the rectangles
contained by the undivided line, and the fever al parts of the
divided line.
complete the parallelograms, that is to fay,
Draw < ......
> (pr. 31.B. i.)
■ =i + l + l
I
I
I
+
 +
Q.E. D.
54
BOOK II. PROP. II. THEOR.
Draw
I
I
F a Jlraight line be divided
into any two parts ■■ * > 9
the fquare of the whole line
is equal to the fum of the
rectangles contained by the whole line and
each of its parts.
+
Defcribe
parallel to 
(B. i.pr. 46.)
(B. i.pr. 31 )
II
Q. E. D.
BOOK II. PROP. III. THEOR.
55
F a jlraight line be di
vided into any two parts
■ ■ ■■' , the rectangle
contained by the whole
line and either of its parts, is equal to
the fquare of that part, together with
the reBangle under the parts.
= — 2 +
or,
Defcribe
(pr. 46, B. 1.)
Complete (pr. 31, B. 1.)
Then
+
I
but
and
I
+
In a limilar manner it may be readily mown
Q.E.D
56
BOOK II. PROP. IV. THEOR.
F a Jlraight line be divided
into any two parts > ,
the fquare of the whole line
is equal to the fquare s of the
parts, together with twice the rectangle
contained by the parts.
twice
+
+
Defcribe
draw 
(pr. 46, B. 1.)
■ (port. 1.),
and
(pr. 31, B. 1.)
4 +
4,4
(pr. 5, B. 1.),
(pr. 29, B. 1.)
*,4
BOOK II. PROP. IV. THEOR. 57
E
.*. by (prs.6,29, 34. B. 1.) ^^J is a fquare m
For the fame reafons r I is a fquare ss ■ " B ,
B.
but e_j = EJ+M+ +
twice ■■ » ■— ■ .
Q. E. D.
58
BOOK II. PROP. V. PROB.
F a Jlraight
line be divided
into two equal
parts and alfo — — — — —
into two unequal parts,
the rectangle contained by
the unequal parts, together with the fquare of the line between
the points offeclion, is equal to the fquare of half that line
Defcribe (pr. 46, B. 1.), draw ■ and
^ — 11
)
(pr. 3 i,B.i.)
I
(p. 36, B. 1.)
(p. 43, B. 1.)
(ax. 2.
..
BOOK II. PROP. V. THEOR. 59
but
and
(cor. pr. 4. B. 2.)
2 (conft.)
.*. (ax. 2.)
■  H
+
Q. E. D.
6o
BOOK II. PROP. VI. THEOR.
1
1
1
1
1
/
l^HHMHHUUHHHmHr
F a Jlraight line be
bifecled ■
and produced to any
point — «^»—— ,
the reSlangle contained by the
•whole line fo increafed, and the
part produced, together with the
fquare of half the line, is equal
to the fquare of the line made up
of the half and the produced part .
— +
^
Defcribe (pr. 46, B. i.)» draw
anc
(pr. 31, B.i.)
(prs. 36, 43, B. 1 )
but z=
(cor. 4, B. 2.)
A
+
(conft.ax.2.)
t
Q. E. D.
BOOK II. PROP. VII. THEOR.
F a Jiraight line be divided
into any two parts mi ,
the fquares of the whole line
and one of the parts are
equal to twice the reSlangle contained by
the whole line and that part, together
with the fquare of the other parts.
wmw— — 2 I — — 2 ""■■
61
Defcribe
Draw — i
, (pr. 46, B. i.)
(pott. 1.),
(pr. 31, B. i.)
= I (P r  43> B. 1.),
* to both, (cor. 4, B. 2.)
I
(cor. 4, B. 2.)
I
+ ■ +
— +
+
+ — * = 2
+
Q. E. D.
62
BOOK II. PROP. Fill. THEOR.
: iy
w^ ■
■ •••■■■I ■iiitiiiiniii ■■■^■■■■■■■m
F ajlraight line be divided
into any two parts
, the fquare of
thefum of the whole line
and any one of its parts, is equal to
four times the reclangle contained by
the whole line, and that part together
with the fquare of the other part.
— +
Produce
and make
Conftrudt.
draw
(pr. 46, B. 1.);
(pr. 7, B. 11.)
= 4
— +
Q. E. D.
BOOK II. PROP. IX. THEOR.
63
F a ftraight
line be divided
into two equal
parts mm — .
and alfo into two unequal
parts , the
fquares of the unequal
parts are together double
the fquares of half the line, *■
and of the part between the points offeSlion.
 + 2 = 2 » + 2
Make 1
Draw
— II
_L and =
or
and
4
and draw
9 II —9
(pr. 5, B. 1.) rr half a right angle,
(cor. pr. 32, B. 1.)
(pr. 5, B. 1.) rs half a right angle,
(cor. pr. 32, B. 1.)
= a right angle.
t
hence
(prs. 5, 29, B. 1.).
aaa^, ■»■■■ '.
(prs. 6, 34, B. 1.)
+
or J
(pr. 47, B. 1.)
+
+ *
Q. E. D.
6 4
BOOK II. PROP. X. THEOR.
! +
F a Jlraight line
fec7ed and pro
duced to any point
— — » 9 thefquaresofthe
whole produced line, and of
the produced part, are toge
ther double of the fquares of
the half line, and of the line
made up of the half and pro
duced part.
Make
and
■5— J_ and = to
draw " ■■in., and
• •*«■«• i
II
draw ■
— or — — «
...... 9
(pr. 31, B. 1.);
alfo.
jk (pr. 5, B. 1.) = half a right angle,
(cor. pr. 32, B. 1 .)
(pr. 5, B. 1.) = half a right angle
(cor. pr. 32, B. 1.)
zz a right angle.
BOOK II. PROP. X. THEOR. 65
t =^=i
half a right angle (prs. 5, 32, 29, 34, B. 1.),
and .___ — «...•■■■» ... — ■ — —
_....., ("prs. 6, 34, B. 1.). Hence by (pr. 47, B. 1.)
Q. E. D.
K
66
BOOK II. PROP. XL PROB.
O divide a given Jlraight line
in Juch a manner, that the reft angle
contained by the whole line and one
of its parts may be equal to the
fquare of the other.
!■••■• mm** a
Defcribe
make —
I
(pr. 46, B. I.),
 (pr. 10, B. 1.),
draw
take
(pr. 3, B. 1.),
on
defcribe
(pr. 46, B. 1.),
Produce —
Then, (pr. 6, B. 2.)
>•«■■■■■
— (port. 2.).
+
— ' »■■■*, or,
ll
>■••» • •■
Q.E. D.
BOOK II. PROP. XII. THEOR.
67
N any obtufe angled
triangle, thefquare
of the fide fubtend
ing the obtufe angle
exceeds the fum of the fquares
of the fides containing the ob
tufe angle, by twice the rec
tangle contained by either of
thefe fides and the produced 'parts
of the fame from the obtufe
angle to the perpendicular let
fall on it from the oppofi'ce acute
angle.
+
* by
By pr. 4, B. 2.
„..* = 2 _j 3 __ 2 „ .
add ^— — 2 to both
2 + 2 = 8 (pr. 47f B.i.)
2 •
+
+
■ or
+
(pr. 47, B. 1.). Therefore,
»" = 2 • ■
~ : hence ■
by 2
+
 +
+
Q. E. D.
68
BOOK II. PROP. XIII. THEOR.
FIRST
SECOND.
N any tri
angle, the
fqnareofthe
Jide fubtend
ing an acute angle, is
lefs than the fum of the
fquares of the Jides con
taining that angle, by twice the rectangle contained by either
of thefe fides, and the part of it intercepted between the foot of
the perpendicular let fall on it from the oppofite angle, and the
angular point of the acute angle.
FIRST.
'■  2 by 2 .
SECOND.
2 \ 2 by 2
Firrt, fuppofe the perpendicular to fall within the
triangle, then (pr. j, B. 2.)
■»■■■ 2 J — ^^ 2 ZZZ 2 • ^"—"m • <^^^  ■■■■«■ ',
add to each _ ' then,
I 2 1 2 ^^ ~
.*. (pr. 47 » B  O
BOOK II. PROP. XIII. THEOR. 69
Next fuppofe the perpendicular to fall without the
triangle, then (pr. 7, B. 2.)
add to each — — 2 then
«■». '' j  { — —  — 2 • .... • — — .
+ „„.= + . /# ( pr . 47 , B. i.),
_«a» '  «^— 2 ~ 2 • •■■■■■■■■ • — — j i ■■ I i e
Q. E. D.
7°
BOOK II. PROP. XIV. PROB.
O draw a right line of
which the fquare fliall be
equal to a given recJi
linear figure.
fuch that,
*
Make
(pr. 45, B. i.),
produce
take ■■••
until
(pr. 10, B. i.),
Defcribe
and produce —
2
(P°ft. 3).
to meet it : draw
Or "■""■■"™ ~~ m »■»• • ■ ••••— ^ —I— taamia*
(pr. 5, B. 2.),
but " zz ii ~ \ ■•••«■•« (pr. 47, B. i.);
mm*m*mm— — f ■■•«■• mm ^ «■■■■■■ • ■ mm * ■■■■■■■1 —ft— ««•••#«■
— , and
■■■ • ••■>■
Q. E. D.
BOOK III.
DEFINITIONS.
I.
QUAL circles are thofe whofe diameters are
equal.
II.
A right line is said to touch a circle
when it meets the circle, and being
produced does not cut it.
III.
Circles are faid to touch one an
other which meet but do not cut
one another.
IV.
Right lines are faid to be equally
diflant from the centre of a circle
when the perpendiculars drawn to
them from the centre are equal.
7 2
DEFINITIONS.
V.
And the ftraight line on which the greater perpendi
cular falls is faid to be farther from the centre.
VI.
A fegment of a circle is the figure contained
by a ftraight line and the part of the circum
ference it cuts off.
VII.
An angle in a fegment is the angle con
tained by two ftraight lines drawn from any
point in the circumference of the fegment
to the extremities of the ftraight line which
is the bafe of the fegment.
VIII.
An angle is faid to ftand on the part of
the circumference, or the arch, intercepted
between the right lines that contain the angle.
IX.
A fedtor of a circle is the figure contained
by two radii and the arch between them.
DEFINITIONS.
73
X.
Similar fegments of circles
are thofe which contain
equal angles.
Circles which have the fame centre are
called concentric circles.
74
BOOK III. PROP. I. PROB.
O find the centre of a given
circle
o
Draw within the circle any ftraight
draw — — _L ■■ •
biledt — wmmmmm ■ , and the point of
bifecfion is the centre.
For, if it be pofhble, let any other
point as the point of concourfe of — — — , »■— ■ '
and —«■«■■» be the centre.
Becaufe in
and
\/
—— ss  (hyp. and B. i, def. 15.)
zr »■■•— (conft.) and —■««■■ common,
^B. 1, pr. 8.), and are therefore right
angles ; but
ym = £2 ( c °»ft) yy =
(ax. 1 1 .)
which is abfurd ; tberefore the afTumed point is not the
centre of the circle ; and in the fame manner it can be
proved that no other point which is not on — — ^— is
the centre, therefore the centre is in ' , and
therefore the point where < is bifecled is the
centre.
Q. E. D.
BOOK III. PROP. II. THEOR.
75
STRAIGHT line ( ■■ )
joining two points in the
circumference of a circle
, lies wholly within the circle.
Find the centre of
o
(B. 3 .pr.i.);
from the centre draw
to any point in
meeting the circumference from the centre ;
draw and ■ .
Then
= ^ (B. i.pr. 5.)
but
or
\ (B. i.pr. 16.)
 (B. 1. pr. 19.)
but
.*. every point in
lies within the circle.
Q. E. D.
76 BOOK III. PROP. III. THEOR.
F a jlraight line ( — — )
drawn through the centre of a
circle
o
SifecJs a chord
( •"•) which does not pafs through
the centre, it is perpendicular to it; or,
if perpendicular to it, it bifeSls it.
Draw
and
to the centre of the circle.
In ^ I and  .„^S.
common, and
« ■ ■ ■ • . • ■ ■
and .'.
= KB. i.pr.8.)
JL (B. i.def. 7.)
Again let ______ _L ..
Then
,. ^d  b*>
(B. i.pr. 5.)
(hyp)
and
ind .*.
(B. 1. pr. 26.)
bifefts
Q. E. D.
BOOK III. PROP. IV. THEOR.
77
F in a circle tivojlraight lines
cut one another, which do
not bath pafs through the
centre, they do not hifecJ one
another.
If one of the lines pafs through the
centre, it is evident that it cannot be
bifedled by the other, which does not
pafs through the centre.
But if neither of the lines
or
pafs through the centre, draw — ■—
from the centre to their interfedlion.
If
. be bifedled, ........ J_ to it (B. 3. pr. 3.)
ft = i _^ and if be
bifefted, ...... J_
( B  3 P r  3)
and .*. j P^ = ^ ; a part
equal to the whole, which is abfurd :
.*. — — ■ — and ii
do not bifect one another.
Q. E. D.
78
BOOK III. PROP. V. THEOR.
F two circles
interfetl, they have not the
©
fame centre.
Suppofe it poffible that two interfering circles have a
common centre ; from fuch fuppofed centre draw
to the interfering point, and ^^^^....... • meeting
the circumferences of the circles.
(B. i.def. 15.)
...... (B. 1. def. 15.)
_... • a part
equal to the whole, which is abfurd :
.*. circles fuppofed to interfedt in any point cannot
have the fame centre.
Q. E. D.
BOOK III. PROP. VI. THEOR.
79
F two circles
©
touch
one another internally, they
have not the fame centre.
For, if it be poffible, let both circles have the fame
centre ; from fuch a fuppofed centre draw i
cutting both circles, and to the point of contact.
Then
and
«»•»•■■
 (B. i.def. 15.)
 (B. i.def. 15.)
equal to the whole, which is abfurd ;
therefore the afTumed point is not the centre of both cir
cles ; and in the fame manner it can be demonftrated that
no other point is.
Q. E. D.
8o
BOOK III. PROP. VII. THEOR.
FIGURE I.
FIGURE II.
F from any point within a circle
which is not the centre, lines
are drawn to the circumference ; the greatejl of thofe
lines is that (—■•■■■■) which pajfes through the centre,
and the leaf is the remaining part ( — ) of the
diameter.
Of the others, that ( — — — ) which is nearer to
the line pafjing through the centre, is greater than that
( «^ » ) which is more remote.
Fig. 2. The two lines ('
and
)
which make equal angles with that paffing through the
centre, on oppoftefdes of it, are equal to each other; and
there cannot be drawn a third line equal to them, from
the fame point to the circumference.
FIGURE I.
To the centre of the circle draw  and «■■■—•
then  — . (B. i. def. 15.)
vmmwmmmam = — — j ■■ C — — — (B.I. pr. 20.)
in like manner ■■« .1 ±1 may be fhewn to be greater than
M 1 ■ ; or any other line drawn from the fame point
to the circumference. Again, by (B. 1. pr. 20.)
take — — from both ; .*. — — — C (ax.),
and in like manner it may be fhewn that is lefs
BOOK III. PROP. VII. THEOR. 81
than any other line drawn from the fame point to the cir
cumference. Again, in **/ and
common, m £2 ? anc ^
(B. i. pr. 24.) and
may in like manner be proved greater than any other line
drawn from the fame point to the circumference more
remote from — ^■m—— «.
FIGURE II.
If ^^ rz then .... — ■ , if not
take — — = — — — draw , then
s^ I A , y
in ^^ I and , ■ common,
= and
(B. i.pr. 4.)
a part equal to the whole, which is abfurd :
— — =1 *■■■■»..*.; and no other line is equal to
— drawn from the fame point to the circumfer
ence ; for if it were nearer to the one paffing through the
centre it would be greater, and if it were more remote it
would be lefs.
Q. E. D.
M
82
BOOK III. PROP. Fill. THEOR.
The original text of this propofition is here divided into
three parts.
F from a point without a circle, Jlraight
f:
lines
are drawn to the cir
cumference ; of thofe falling upon the concave circum
ference the greatejl is that (— ^.«.) which pajfes
through the centre, and the line ( ' " ) ^hich is
nearer the greatejl is greater than that ( )
which is more remote.
Draw ■■•••••• and •■■■••■■■■ to the centre.
Then, ■— which palTes through the centre, is
greateit; for fince — — ™ =  . if — ^— ^—
be added to both, ■■» :=z •■ ^"™" p **" ?
but [Z (B. i. pr. 20.) .*. ^— «  is greater
than any other line drawn from the fame point to the
concave circumference.
Again in
and
BOOK III. PROP. VIII. THEOR
and i common, but ^ CZ
(B. i. pr. 24.);
and in like manner
may be fhewn C than any
other line more remote from
II.
Of thofe lines falling on the convex circumference the
leaf is that (———■) which being produced would
pafs through the centre, and the line which is nearer to
the leaf is lefs than that which is more remote.
For, lince — — \~
and
ciiitiifl
'. And fo of others
III.
Alfo the lines making equal angles with that which
paff'es through the centre are equal, whether falling on
the concave or convex circumference ; and no third line
can be drawn equal to them from the fame point to the
circumference.
For if ■■■
make
r~ »■•■■ 9 but making rr L ;
= ■■■»■■ ? and draw ■■■■■■  ,
84
BOOK III. PROP. Fill. THEOR.
Then
in
> and /
we have
and
L A
common, and alio ^ = ,
 = (B. i. pr. 4.);
but
which is abfurd.
.....<>... is not :z:
_ * •>■>
■■■■•■■ nor to any part
of ...___ 9 /. ■■■ ■ is not CZ —.
Neither is ■•• ■• C ■•"•■— ~, they are
.*. = to each other.
And any other line drawn from the fame point to the
circumference mull lie at the fame fide with one of thefe
lines, and be more or lefs remote than it from the line pair
ing through the centre, and cannot therefore be equal to it.
Q. E. D.
BOOK III. PROP. IX. THEOR.
85
F a point b" taken . within a
from which
ctr„ie
o
wore than two equal ftraight lines
can be drawn to the circumference, that
point mujl be the centre of the circle.
For, if it be fuppofed that the point ^
in which more than two equal ftraight
lines meet is not the centre, lbme other
point — ' mult be; join thefe two points by
and produce it both ways to the circumference.
Then fince more than two equal ftraight lines are drawn
from a point which is not the centre, to the circumference,
two of them at leaft muft lie at the fame fide of the diameter
'j and fince from a point
A,
which is
not the centre, ftraight lines are drawn to the circumference ;
the greateft is ^— ■•■ », which paffes through the centre :
and — «~— which is nearer to »«~« ? r~ — — —
which is more remote (B. 3. pr. 8.) ;
but = (hyp) which is abfurd.
The fame may be demonftrated of any other point, dif
ferent from / \ 9 which muft be the centre of the circle,
Q. E. D.
86
BOOK III. PROP. X. THEOR.
NE circle I ) cannot inter fe£i another
rv
J in more points than two.
For, if it be poflible, let it interfedt in three points ;
from the centre of I J draw
O
to the points of interferon ;
(B. i. def. 15.),
but as the circles interfec~t, they have not the fame
centre (B. 3. pr. 5.) :
.*. the affumed point is not the centre of ^ J , and
O
and
are drawn
from a point not the centre, they are not equal (B. 3.
prs. 7, 8) ; but it was mewn before that they were equal,
which is abfurd ; the circles therefore do not interfedt. in
three points.
Q. E. D.
BOOK III. PROP. XL THEOR.
87
O
F two circles and
I 1 touch one another
internally, the right line joining their
centres, being produced, jliall pafs through
a point of contact.
For, if it be poffible, let
join their centres, and produce it both
ways ; from a point of contact draw
11 to the centre of f J , and from the fame point
of contadl draw •■■•■■•«• to the centre of I I.
k
Becaufe in
+
(B. 1. pr. 20.),
I "••!•••••,
and
O
as they are radii of
88 BOOK III. PROP. XL THEOR.
but — — " — — — C — ; t ak e
away — ^— ^ which is common,
and ^— ^ d ;
but — ^— =  —
•
becaufe they are radii of
O
and .*. CZ a part greater than the
whole, which is abfurd.
The centres are not therefore fo placed, that a line
joining them can pafs through any point but a point of
contact.
Q. E. D.
BOOK III. PROP. XII. THEOR.
89
F two circles
o
titer externally, the Jlraight line
——■■i»—   joining their centres,
pajfes through the point of contact.
touch one ano
If it be poffible, let
join the centres, and
not pafs through a point of contact; then from a point of
contact draw and to the centres.
Becaufe
and «
and 
+
(B. 1. pr. 20.),
= (B. 1. def. 15.),
= (B. i.def.15.),
+
, a part greater
than the whole, which is abfurd.
The centres are not therefore fo placed, thai «"he line
joining them can pafs through any point but the point of
contact.
Q. E. D.
N
9 o
BOOK III. PROP. XIII. THEOR.
FIGURE I.
FIGURE II.
NE circle can
not touch ano
ther, either
externally or
internally, in more points
than one.
FIGURE III.
Fig. i . For, if it be poffible, let
and f j touch one
another internally in two points ;
draw ... . i. joining their cen
tres, and produce it until it pafs
through one of the points of contadl (B. 3. pr. 11.);
draw — — ^— and ~ ^— ^— ,
But = (B. 1. def. 15.),
.*. if
be added to both,
+
but
and .*.
+
+
which is abfurd.
(B. 1. def. 15.),
= — — ; but
— (B. 1. pr. 20.),
BOOK III. PROP. XIII. THEOR. ot
Fig. 2. But if the points of contact be the extremities
of the right line joining the centres, this ftraight line mull
be bifedled in two different points for the two centres ; be
caufe it is the diameter of both circles, which is abfurd.
Fig. 3. Next, if it be pomble, let
O and O
touch externally in two points; draw ——..... joining
the centres of the circles, and pamng through one of the
points of contact, and draw — — — ■ and ^^—^— .
— = (B. 1. def. 15.);
and  — — — — (B. 1. def. 15.):
+ — — — = — — — ; but
+ — — • [Z — — (B. 1. pr. 20.),
which is abfurd.
There is therefore no cafe in which two circles can
touch one another in two points.
Q E. D.
9 2
BOOK III. PROP. XIV. THEOR.
QUALfraight lines (^ ")
infcribed in a circle are e
qually diji ant from the centre ;
and alfo t Jiraight lines equally
dijlant from the centre are equal.
From the centre of
o
draw
to ■■■» and •>
, join ■■^— and — —
Then
and
hnce
= half (B. 3. pr. 3.)
= 1 — ( B  3 P r 3)
= ..... (hyp.)
and
(B. i.def. 15.)
and
but iince
is a right angle
+ ' ' (B.i.pr.47.)
,... 2  M , 2 for the
 2 +
fame reafon,
+
BOOK III. PROP. XIV. THEOR. 93
t
....«<.« • »
Alfo, if the lines ....... and ........ be
equally diftant from the centre ; that is to fay, if the per
pendiculars ■■ •«•■ and .......... be given equal, then
For, as in the preceding cafe,
1 + 2 = 2 +
but ■■amuin " ^Z ■■•■•■■■« "
= g , and the doubles of thefe
i. and •«_,.... are alfo equal.
Q. E. D.
94
BOOK III. PROP. XV. THEOR.
FIGURE I.
but
HE diameter is the greatejl jlraight
line in a circle : and, of all others,
that which is nearejl to the centre is
greater than the more remote.
FIGURE I.
The diameter — — — is C any line
For draw > — — — and —— <
and ■ ■ = •
— I— i
(B. i . pr. 20.)
Again, the line which is nearer the centre is greater
than the one more remote.
Firft, let the given lines be — and ,
winch are at the fame fide of the centre and do
not interfedl ;
draw
s
\
BOOK III. PROP. XV. THEOR.
95
In
and \
►
and •■
but
\/
and
(B. I. pr. 24.)
FIGURE II.
Let the given lines be — — and — — >
which either are at different fides of the centre,
or interfec~t ; from the centre draw ......——
and  _L and 9
make ........ zz ••, and
draw — — — J_ >— •— .
FIGURE II.
Since
and
the centre,
but ■
are equally diftant from
(B. 3. pr. 14.);
[Pt. i.B. 3. pr. 15.),
Q. E. D.
9 6
BOOK III. PROP. XVI. THEOR.
HEJlraight
line ■
drawn
from the
extremity of the diame
ter i of a circle
perpendicular to it falls
*'•... ., without the circle.
Jl.*''*" * And if any Jlraight
line ■■■■■■■ be
drawn from a point
i within that perpendi
cular to the point of contact, it cuts the circle.
PART I
If it be poffible, let
which meets the circle
again, be J_
', and draw
Then, becauie
^ = ^ (B.i.pr. 5 ),
and .*. each of these angles is acute. (B. i. pr. 17.)
but = _j (hyp.), which is abfurd, therefore
_____ drawn _L — — —  does not meet
the circle again.
BOOK III. PROP. XVI. THEOR. 07
PART II.
Let be J_ — — ■^ and let  be
drawn from a point *•" between and the
circle, which, if it be poflible, does not cut the circle.
Becaufe  i =  _j >
^ is an acute angle ; fuppofe
............... J_ ........ 9 drawn from the centre of the
circle, it mull: fall at the fide of ^ the acute angle.
.*. m^> which is fuppofed to be a right angle, is C Ik ,
but •«■•»•••«••. ~ — — ■— ■ .
and .'. ■•■•>. C •••••■••■■■■, a part greater than
the whole, which is abfurd. Therefore the point does
not fall outfide the circle, and therefore the ftraight line
........... cuts the circle.
Q.E.D.
98
BOOK III. PROP. XVII. THEOR.
O draw a tangent to a given
circle f rom a
o
given point, either in or outjide of its
circumference.
If the given point be in the cir
cumference, as at „.„ , it is plain that
the ftraight line ' mmm "™ J_ — — —
the radius, will be the required tan
gent (B. 3. pr. 16.) But if the given point
outfide of the circumference, draw —
be
from it to the centre, cutting
draw
concentric with
then
o
( J; and
 , defcribe
radius zz •■— ,
will be the tangent required.
BOOK III. PROP. XVII. THEOR.
zx  A
99
For in
__ zz •■•■ ■— , jttk common,
and (•■•■■■■■•■ ~ «■.
(B. i. pr. 4.) = = a right angle,
.*. — — — • is a tangent to
o
ioo BOOK III. PROP. XVIII. THEOR.
F a right line •..... fa
a tangent to a circle, the
fir aight line — ■ — drawn
from the centre to the
point of contatt, is perpendicular to it.
For, if it be pomble,
let ™ ^™" •••■ be _]_ ■•••
then becaufe
4 = ^
is acute (B. i . pr. 17.)
C
(B. 1. pr. 19.);
but
and .*. — — ■ —  £2 — i
the whole, which is abfurd.
►•►••• , a part greater than
.". — — is not _L  ; and in the fame man
ner it can be demonitrated, that no other line except
— ■ — — is perpendicular to ■■■■■
Q. E. D.
BOOK III PROP. XIX. THEOR.
101
F a Jlraight line mmKmmmm ^ m
be a tangent to a circle,
the Jlraight line » ,
drawn perpendicular to it
from point of the contact, pajfes through
the centre of the circle.
For, if it be poifible, let the centre
be without
and draw
■ ••■ from the fuppofed centre
to the point of contact.
Becaufe
(B. 3. pr. 18.)
= 1 1 , a right angle ;
but ^^ = I 1 (hyp.), and ,\ =
a part equal to the whole, which is abfurd.
Therefore the arTumed point is not the centre ; and in
the fame manner it can be demonftrated, that no other
point without m ^ mm ^ m is the centre.
Q. E. D.
102
BOOK III. PROP. XX. THEOR.
FIGURE I
HE angle at the centre of a circle, is double
the angle at the circumference, when they
have the fame part of the circumference for
their bafe.
FIGURE I.
Let the centre of the circle be on ■ .....
a fide of
Becaufe
k = \
But
(B. i. pr. 5.).
or
+
:= twice (B. 1. pr. 32).
FIGURE 11.
FIGURE II.
Let the centre be within
circumference ; draw ^—
4
j the angle at the
from the angular
point through the centre of the circle ;
^ = A
then ^ = W 9 a °d = ,
becaufe of the equality of the fides (B. 1. pr. 5).
BOOK III. PROP. XX. THEOR. 103
Hence
_i_ 4 + + = twke 4
But ^f = 4 + V 9 and
twice
FIGURE III.
Let the centre be without ▼ and
__— . the diameter.
FIGURE III.
draw
Becaufe
= twice
:= twice
▲
ZZ twice
(cafe 1.) ;
and
Q. E. D.
io4 BOOK III. PROP. XXI. THEOR.
FIGURE I.
HE angles ( 4& 9 4^ ) in the fame
fegment of a circle are equal.
FIGURE I.
Let the fegment be greater than a femicircle, and
draw — ^— ^^— and — — — — to the centre.
twice 4Pt or twice ;n
(B. 3. pr. 20.) ;
4=4
4
FIGURE II.
FIGURE II.
Let the fegment be a femicircle, 01 lefs than a
femicircle, draw — ■— — ■ the diameter, alfo draw
< = 4 > = *
(cafe 1.)
Q. E. D.
BOOK III. PROP. XXII. THEOR. 105
f
FIE oppofite angles Afc
and ^ j «l «"/,/
o/~ tf«y quadrilateral figure in
ferred in a circle, are together equal to
two right angles.
Draw
and
the diagonals ; and becaufe angles in
the fame fegment are equal ^r — JP^
and ^r = ^f ;
add ^^ to both.
two right angles (B. 1. pr. 32.). In like manner it may
be fhown that,
Q. E. D.
io6 BOOK III. PROP. XXIII. THEOR.
PON the fame
Jlraight line,
and upon the
fame fide of it,
two fimilar fegments of cir
cles cannot he conflrutled
which do not coincide.
For if it be poffible, let two fimilar fegments
Q
and
be constructed ;
draw any right line
draw .
cutting both the fegments,
and — .
Becaufe the fegments are fimilar,
(B. 3. def. 10.),
but (Z ^^ (B. 1. pr. 16.)
which is abfurd : therefore no point in either of
the fegments falls without the other, and
therefore the fegments coincide.
O. E. D.
BOOK III PROP. XXIV. THEOR.
107
IMILAR
fegments
and
9 of cir
cles upon equal Jlraight
lines ( •— ^— ■ and » )
are each equal to the other.
For, if 'j^^ 1^^ be fo applied to
that — — — — may fall on , the extremities of
— — — may be on the extremities — ^^— and
at the fame fide as
becaufe
muft wholly coincide with
and the fimilar fegments being then upon the fame
ftraight line and at the fame fide of it, muft
alfo coincide (B. 3. pr. 23.), and
are therefore equal.
Q. E. D.
io8
BOOK III. PROP. XXV. PROB.
SEGMENT of a circle
being given, to defcribe the
circle of which it is the
fegment.
From any point in the fegment
draw mmmmmmmm and — — — bifedl
them, and from the points of bifecfion
draw L — ■ — ■ — —
and — ■— — — i J ™^™^^
where they meet is the centre of the circle.
Becaufe __ — _ terminated in the circle is bifecled
perpendicularly by  , it paffes through the
centre (B. 3. pr. I.), likewile — _ paffes through
the centre, therefore the centre is in the interferon of
thefe perpendiculars.
CLE. D.
BOOK III. PROP. XXVI. THEOR. 109
N equal circles
the arcs
O w o
on
'which
Jland equal angles, •whether at the centre or circum
ference, are equal.
Firft, let
draw
at the centre,
and —
Then fince
OO
.«•
an d ^VC...........*,';^ have
and
But
k=k
(B. 1. pr. 4.).
(B. 3pr. 20.);
• O and o
are fimilar (B. 3. def. 10.) ;
they are alio equal (B. 3. pr. 24.)
no BOOK III. PROP. XXVI. THEOR.
If therefore the equal fegments be taken from the
equal circles, the remaining fegments will be equal ;
lence
(ax. 3.);
and .*.
But if the given equal angles be at the circumference,
it is evident that the angles at the centre, being double
of thofe at the circumference, are alfo equal, and there
fore the arcs on which they ftand are equal.
Q. E. D.
BOOK III. PROP. XXVII. THEOR. 1 1 1
N equal circles,
oo
the angles
^v
and
k
which Jland upon equal
arches are equal, whether they be at the centres or at
the circumferences.
For if it be poflible, let one of them
▲
be greater than the other
and make
k=k
▲
.*. N*_^ = Sw* ( B  3 P r  26.)
but V^^ = ♦♦.....,.♦ (hyp.)
.". ^ , * = V Lj d/ a part equal
to the whole, which is abfurd ; .*. neither angle
is greater than the other, and
.*. they are equal.
Q.E.D
*••■■■•••
ii2 BOOK III. PROP. XXVIII. TIIEOR.
N equal circles
equa
oo
iitil chords
arches.
cut off equal
From the centres of the equal circles,
draw ^^— , — — — and ■ ■■■■■■■■■■ ■ , «■■■■
and becaufe
=
alib
(hyp.)
(B. 3. pr. 26.)
and
.0=0
(ax. 3.)
Q. E. D.
BOOK III. PROP. XXIX. THEOR. 113
N equal circles
O w O
the chords — ^— and
tend equal arcs are equal.
which fub
If the equal arcs be femicircles the propofition is
evident. But if not,
let
and
■5 . anu ,
be drawn to the centres ;
becaufe
and
but
and
(hyp)
(B3.pr.27.);
— .......... and «
•• (B. 1. pr. 4.);
but thefe are the chords fubtending
the equal arcs.
Q. E. D.
ii4
BOOK III. PROP. XXX. PROB.
O bifecl a given
arc
C)
Draw
make
draw
Draw
■■■« , and it bifedls the arc.
and — — — — .
and
(conft.),
is common,
(conft.)
(B. i. pr. 4.)
= ,*■%■ (B. 3. pr. 28.),
and therefore the given arc is bifedred.
Q. E. D.
BOOK III. PROP. XXXI. THEOR. 115
N a circle the angle in afemicircle is a right
angle, the angle in a fegment greater than a
femicircle is acute, and the angle in a feg
ment lefs than afemicircle is obtufe.
FIGURE I.
FIGURE I.
The angle ^ in a femicircle is a right angle.
V
Draw
and
JB = and Mk = ^ (B. 1. pr. 5.)
+
A= V
the half of two
right angles = a right angle. (B. 1. pr. 32.)
FIGURE II.
The angle ^^ in a fegment greater than a femi
circle is acute.
▲
Draw
the diameter, and
= a right angle
▲
is acute.
FIGURE II.
n6 BOOK III. PROP. XXXI. THEOR.
FIGURE III.
FIGURE III.
The angle v ^k in a fegment lefs than femi
circle is obtufe.
Take in the oppofite circumference any point, to
which draw — «— — — and ■■ .
*
Becaufe f
(B. 3. pr. 22.)
= m
but
a
(part 2.),
is obtufe.
Q. E. D.
BOOK III. PROP. XXXII. THEOR. i
F a right line ■—■— —
be a tangent to a circle,
and from the point of con
tact a right line — — — 
be drawn cutting the circle, the angle
I made by this line with the tangent
is equal to the angle in the alter
ate fegment of the circle.
If the chord fhould pafs through the centre, it is evi
dent the angles are equal, for each of them is a right angle.
(B. 3. prs. 16, 31.)
But if not, draw
from the
point of contact, it muft pafs through the centre of the
circle, (B. 3. pr. 19.)
w + f = zLJ = f (b. i.pr.32.)
= (ax.).
Again O =£Dk= +4
(B. 3. pr. 22.)
a*
= ^m , (ax.), which is the angle in
the alternate fegment.
Q. E. D.
1 1 8 BOOK III. PROP. XXXIII. PROB.
N agivenjlraight line — —
to dejcribe a fegment of a
circle that Jhall contain an
angle equal to a given angle
^a,
If the given angle be a right angle,
bifedl the given line, and defcribe a
femicircle on it, this will evidently
contain a right angle. (B. 3. pr. 31.)
If the given angle be acute or ob
tufe, make with the given line, at its extremity,
, draw
and
make
with
= ^ , defcribe I I
— or as radius,
for they are equal.
is a tangent to
o
(B. 3. pr. 16.)
divides the circle into two fegments
capable of containing angles equal to
l W and which were made refpedlively equal
■o£7
and
(B. 3 .pr. 32.)
Q. E. D.
BOOK III. PROP. XXXIV. PROB. 119
O cut off from a given cir
cle I 1 a fegment
o
which Jljall contain an angle equal to a
given angle
Draw —
(B. 3. pr. 17.),
a tangent to the circle at any point ;
at the point of contact make
the given angle ;
contains an angle := the given angle.
V
Becaufe ■ is a tangent,
and — ^—m m cuts it, the
ingle
angle in
(B. 3. pr. 32.),
but
(conft.)
Q. E. D.
120
BOOK III. PROP. XXXV. THEOR.
FIGURE I.
FIGURE II.
F two chords
circle
I ... .^_ I tn a cir
interject each other, the recJangle contained
by the fegments of the one is equal to the
re El angle contained by the fegments of the other.
FIGURE I.
If the given right lines pafs through the centre, they are
bifedled in the point of interfedtion, hence the rectangles
under their fegments are the fquares of their halves, and
are therefore equal.
FIGURE II.
Let —■»——■— pafs through the 'centre, and
__..... not; draw and .
Then
X
(B. 2. pr. 6.),
or
X
x =
(B. 2. pr. 5.).
X
FIGURE III.
FIGURE III.
Let neither of the given lines pafs through the
centre, draw through their interfection a diameter
and X = X
...... (Part. 2.),
alfo   X = X
(Part. 2.) ;
X
X
Q. E. D.
BOOK III. PROP. XXXVI. THEOR. 121
F from a point without a FIGURE I.
circle twojiraight lines be
drawn to it, one of which
— mm is a tangent to
the circle, and the other ^— —— .
cuts it ; the rectangle under the whole
cutting line — «■•" and the
external fegment — is equal to
the fquare of the tangent — — — .
FIGURE I.
Let —."•• pafs through the centre;
draw from the centre to the point of contact ;
minus 2 (B. 1. pr. 47),
2
or
minus
•~~ ^ HH (Liitf BMMW ^Q
(B. 2. pr. 6).
FIGURE II.
If •"••■ do not
pafs through the centre, draw
FIGURE II.
and — — ■ ,
Then
minus "
(B. 2. pr. 6), that is,
 X
minus %
,* (B. 3 .pr. 18).
Q. E. D.
122 BOOK III. PROP. XXXVII. THEOR.
F from a point out fide of a
circle twojlraight lines be
drawn, the one ^^—
cutting the circle, the
other — — — meeting it, and if
the recJangle contained by the whole
cutting line ■ ■' • and its ex
ternal fegment »• — •• be equal to
thejquare of the line meeting the circle,
the latter < is a tangent to
the circle.
Draw from the given point
___ j a tangent to the circle, and draw from the
centre , .....••••, and — ■■—  ?
* = X (fi.3pr.36)
but ___ 2 = — X — — — (hyp.),
and .*.
Then in
and — —
and
J
and
.*■«»— and
is common,
but
and .'.
^ = (B. i.pr. 8.);
ZS L_j a right angle (B. 3. pr. 18.),
a right angle,
is a tangent to the circle (B. 3. pr. 16.).
Q. E. D.
BOOK IV.
DEFINITIONS.
RECTILINEAR figure is
faid to be infcribedin another,
when all the angular points
of the infcribed figure are on
the fides of the figure in which it is faid
to be infcribed.
II.
A figure is faid to be defcribed about another figure, when
all the fides of the circumfcribed figure pafs through the
angular points of the other figure.
III.
A rectilinear figure is faid to be
infcribed in a circle, when the vertex
of each angle of the figure is in the
circumference of the circle.
IV.
A rectilinear figure is faid to be cir
cumfcribed about a circle, when each of
its fides is a tangent to the circle.
124 BOOK IF. DEFINITIONS.
A circle is faid to be infcribed in
a rectilinear figure, when each fide
of the figure is a tangent to the
circle.
VI.
A circle is faid to be circum
fcribed about a rectilinear figure,
when the circumference panes
through the vertex of each
angle of the figure.
¥
is circumfcribed.
VII.
A straight line is faid to be infcribed in
a circle, when its extremities are in the
circumference.
The Fourth Book of the Elements is devoted to the folution of
problems, chiefly relating to the infcription and circumfcrip
tion of regular polygons and circles.
A regular polygon is one whofe angles and fides are equal.
BOOK IF. PROP. I. PROP,.
125
N a given circle
O
to place ajlraight line,
equal to agivenfiraight line ( ),
not greater than the diameter of the
circle.
Draw ..i..*— 5 the diameter of ;
and if  — z= , then
the problem is folved.
But if — — ■— « — be not equal to 9
— iz ( h yp);
make «»«..... — — — (B. 1. pr. 3.) with
 as radius,
defcribe f 1, cutting , and
draw 7 which is the line required.
For — ZZ ■••■•»■■•■ — —~ mmmm ^
(B. 1. def. 15. conft.)
Q. E. D.
126
BOOK IF. PROP. II. PROB.
N a given circle
O
to tn
fcribe a triangle equiangular
to a given triangle.
To any point of the given circle draw
 , a tangent
(B. 3. pr. 17.); and at the point of contact
make A m = ^^ (B. 1. pr. 23.)
and in like manner
draw
— , and
Becaufe
and
J^ = ^ (conft.)
j£ = ^J (B. 3. pr. 32.)
.\ ^^ = ^P ; alfo
\/ 5S for the fame reafon.
/. ▼ = ^ (B. i.pr. 32.),
and therefore the triangle infcribed in the circle is equi
angular to the given one.
Q. E. D.
BOOK IV. PROP. III. PROB.
12,7
BOUT a given
circle
O
to
circumfcribe a triangle equi
angular to a given triangle.
Produce any fide
, of the given triangle both
ways ; from the centre of the given circle draw
any radius.
Make = A (B. 1. pr. 23.)
and
At the extremities of the three radii, draw
and — — .— ? tangents to the
given circle. (B. 3. pr. 17.)
The four angles of
Z.
9 taken together, are
equal to four right angles. (B. 1. pr. 32.)
128 BOOK IV. PROP. III. PROB.
but  and ^^^ are right angles (conft.)
, two right angles
but 4 = L_l_Ji (^ '■ P r  I 3)
and = (conft.)
%
and .*.
In the fame manner it can be demonstrated that
&=a.
4 = 4
(B. i. pr. 32.)
and therefore the triangle circumfcribed about the given
circle is equiangular to the given triangle.
Q, E. D.
BOOK IV. PROP. IV. PROB.
1 2Q
N a given triangle
A
to in
fer i be a circle.
Bifedl
J and ^V.
(B. i.pr. 9.) by
and •— ■ ^—
from the point where thefe lines
meet draw ■■■■ ?
and •••■• refpectively per
pendicular to — — — — ,
and
y 1
In
M
A'"'
>
common, .*. ~ ■■
and  *•—
(B. 1. pr. 4 and 26.)
In like manner, it may be mown alfo
that ..—.—.. = —  ,
■*#•••»•■••
hence with any one of thefe lines as radius, defcribe
and it will pafs through the extremities of the
o
other two ; and the fides of the given triangle, being per
pendicular to the three radii at their extremities, touch the
circle (B. 3. pr. 16.), which is therefore inferibed in the
given circle.
Q. E. I).
13°
BOOK IV. PROP. V. PROB.
O defcribe a circle about a given triangle.
and
........ (B. i . pr. 10.)
From the points of bifedtion draw
_L «— ^— and — —
— — — and
— refpec
tively (B. i. pr. 11.), and from their point of
concourfe draw — — — , •■«■•■■— and
and defcribe a circle with any one of them, and
it will be the circle required.
In
(confl.),
common,
4 (conft.),
(B. i.pr. 4 .)
■■■■■a ••■■>»
In like manner it may be fhown that
a # ..■....■.. ^iz ^^^^^■^■■^ — — "^^^~ \ and
therefore a circle defcribed from the concourfe of
thefe three lines with any one of them as a radius
will circumfcribe the given triangle.
Q. E. D.
BOOK IV. PROP. VI. PROP,. 131
O
N a given circle ( J to
infer ibe afquare.
Draw the two diameters of the
circle _L to each other, and draw
. — — , — — and —
s>
is a fquare.
For, fince and fl^ are, each of them, in
a femicircle, they are right angles (B. 3. pr. 31),
(B. i.pr. 28)
and in like manner — — — II
And becaufe fl — ^ (conft.), and
«•••»»••»•« zzz >■■■■■■■■■» g — »■■•■•■■•■■• (B. 1. def icV
.*. = (B. i.pr. 4);
and fince the adjacent fides and angles of the parallelo
gram S X are equal, they are all equal (B. 1 . pr. 34) ;
o
and .*. S ^ ? inferibed in the given circle, is a
fquare. Q. £. D.
132
BOOK IV. PROP. VII. PROB.
BOUT a given circle
I 1 to circumfcribe
a fquart
Draw two diameters of the given
circle perpendicular to each other,
and through their extremities draw
1 "> ^^^ 9
tangents to the circle ;
and
.Q
C
alio
II ■
be demonftrated that
that i and
an d LbmmJ i s a fquare.
a right angle, (B. 3. pr. 18.)
= LA (conft.),
••»• 5 in the fame manner it can
»•»■ . and alfo
C
is a parallelogram, and
becaufe
they are all right angles (B. 1. pr. 34) :
it is alfo evident that
and
" 9 "9
are equal.
,c
is a fquare.
Q. E. D.
BOOK IV. PROP. Fill. PROB.
J 33
O infcribe a circle in a
given fquare.
Make
and
draw  —
and —  
(B. i. pr. 31.)
and fince
is a parallelogram ;
= ( h yp)
is equilateral (B. 1. pr. 34.)
In like manner, it can be ihown that
are equilateral parallelograms ;
■■■■■«■■■■
and therefore if a circle be defcribed from the concourle
of thefe lines with any one of them as radius, it will be
infcribed in the given fquare. (B. 3. pr. 16.)
Q^E. D.
*3+
BOOK IF. PROP. IX. PROS.
]Q defer ibe a circle about a
given fquare
Draw the diagonals ^— — ...
and — — ■ interfering each
other ; then,
becaufe
1 and k
)ave
their fides equal, and the bafe
■— — common to both,
or
t
It
(B. i.pr. 8),
is bifedled : in like manner it can be mown
that
is bifecled ;
hence
\ =
v = r
their halves,
'. ■ = — — — ; (B. i. pr. 6.)
and in like manner it can be proved that
If from the confluence of thefe lines with any one of
them as radius, a circle be defcribed, it will circumfcribe
the given fquare.
Q. E. D.
BOOK IF. PROP. X. PROB.
O conJiruSi an ifofceles
triangle, in which each of
the angles at the bafe fliail
n
[ be double of the vertical
an
Take any ftraight line —
and divide it fo that
4. x =
(B. 2. pr. 1 1.)
With
■■■■■ as radius, defcribe
o
and place
in it from the extremity of the radius,
(B. 4. pr. 1) ; draw
Then
\ is the required triangle.
For, draw
and defcribe
I ) about /
(B. 4. pr. 5.)
.*. —  — is a tangent to I ) (B. 3. pr. 37.)
= y\ (B. 3. pr. 32),
136 BOOK IF. PROP. X. PROP.
add ^r to each,
l! ' ▼ + W = A i B  '• P r  5) :
fince = ..... (B. 1. pr. 5.)
confequently J^ = /^  ^ = M^
(B. 1. pr. 32.)
.*. «■■"■» = (B. 1. pr. 6.)
.*. ■ — ^^— ^— iz: — — —  (conft.)
.'. y\ = ▼ (B. 1. pr. 5.)
=: twice x\ * 9 and confequently each angle at
the bafe is double of the vertical angle.
Q. E. D.
BOOK IV. PROP. XL PROB.
*37
N a given circle
o
to infcribe an equilateral and equi
angular pentagon.
Conftrud: an ifofceles triangle, in
which each of the angles at the bafe
ihall be double of the angle at the
vertex, and infcribe in the given
▲
circle a triangle equiangular to it ; (B. 4. pr. 2.)
^ and m^ ( B<I 'P r 9)
Bifedt
draw
and
Becaufe each of the angles
> +k
and
A
are equal,
the arcs upon which they ftand are equal, (B. 3. pr. 26.)
and .*. i^—^— , — — ■—■ , ■ , and
■■■■»«■ which fubtend thefe arcs are equal (B.3.pr. 29.)
and .*. the pentagon is equilateral, it is alfo equiangular,
as each of its angles ftand upon equal arcs. (B. 3. pr. 27).
Q^E. D.
■38
BOOK IF. PROP. XII. PROB.
O defcribe an equilateral
and equiangular penta
gon about a given circle
O
Draw five tangents through the
vertices of the angles of any regular
pentagon infcribed in the given
o
(B. 3. pr. 17).
Thefe five tangents will form the required pentagon.
Draw
f—
i
In
and
(B. i.pr. 47),
and — — — common ;
,7 =
\A
= twice
and ▼ = (B. i.pr. 8.)
and ^ ^ twice
In the fame manner it can be demonftrated that
:= twice ^^ , and W = twice fe.:
but = (B. 3pr. 27),
£1
BOOK IF. PROP. XII. PROB. 139
,*, their halves = &. alfo (__ sr _J,
and
..»>•> common ;
and — ■— rr — — ■— »
twice — — ;
In the fame manner it can be demonftrated
that ^— ■— — twice — — ,
In the fame manner it can be demonftrated that the
other fides are equal, and therefore the pentagon is equi*
lateral, it is alfo equiangular, for
£^l r= twice flfct. and \^^ r= twice
and therefore
•'• AHw = \^B 1 m the fame manner it can be
demonftrated that the other angles of the defcribed
pentagon are equal.
Q.E.D
'1°
BOOK IF. PROP. XIII. PROB.
O infcribe a circle in a
given equiangular and
equilateral pentagon.
Let tx J be a given equiangular
and equilateral pentagon ; it is re
quired to infcribe a circle in it.
Make y=z J^. and ^ =="
(B. i.pr. 9.)
Draw
Becaufe
and
9 9
=  ,r=A,
common to the two triangles
, &c.
and
/.
A;
Z= ••••« and =: J^ (B. I. pr. 4.)
And becaufe =
.*. = twice
hence
#
rz twice
is bifedted by
In like manner it may be demonftrated that \^j is
bifedled by ■« « , and that the remaining angle of
the polygon is bifedted in a fimilar manner.
BOOK IV. PROP. XIII. PROP,. 141
Draw «^^^^— , .., &c. perpendicular to the
lides of the pentagon.
Then in the two triangles ^f and
A
we have ^T = mm 1 (conft.), ^^— — common,
and ^^ =41 =r a right angle ;
.*. — — — = .......... (B. 1. pr. 26.)
In the fame way it may be mown that the five perpen
diculars on the fides of the pentagon are equal to one
another.
O
Defcribe with any one of the perpendicu
lars as radius, and it will be the infcribed circle required.
For if it does not touch the fides of the pentagon, but cut
them, then a line drawn from the extremity at right angles
to the diameter of a circle will fall within the circle, which
has been fhown to be abfurd. (B. 3. pr. 16.)
Q^E. D.
H 2
BOOK IV. PROP. XIV. PROB.
Bifetf:
O defcribe a circle about a
given equilateral and equi
angular pentagon.
T and
by — and • , and
from the point of fedtion, draw
 := ....... (B. i. pr. 6) ;
and fince in
common,
(B. i.pr. 4).
In like manner it may be proved that
=: = <— — • , and
therefore nr — — — :
a a 1 ••»• ti«t *
Therefore if a circle be defcribed from the point where
thefe five lines meet, with any one of them
as a radius, it will circumfcribe
the given pentagon.
Q. E. I).
BOOK IV PROP. XV PROP.
O infcribe an equilateral and equian
gular hexagon in a given circle
H3
O
From any point in the circumference of
the given circle defcribe ( pamng
O
through its centre, and draw the diameters
and
draw
9 9
......... , ... ? ......... 9 &c. and the
required hexagon is infcribed in the given
circle.
Since
paries through the centres
of the circles, <£ and ^v are equilateral
[
triangles, hence ^^ ' = j ^r sr onethird of two right
angles; (B. i. pr. 32) but ^L m = f I 1
(B. 1. pr. 13);
/. ^ = W = ^W = onethird of I I 1
(B. 1. pr. 32), and the angles vertically oppolite to theie
are all equal to one another (B. 1. pr. 15), and ftand on
equal arches (B. 3. pr. 26), which are fubtended by equal
chords (B. 3. pr. 29) ; and fince each of the angles of the
hexagon is double of the angle of an equilateral triangle,
it is alfo equiangular. O E D
i44
BOOK IV PROP. XVI. PROP.
O infcribe an equilateral and
equiangular quindecagon in
a given circle.
and
be
the fides of an equilateral pentagon
infcribed in the given circle, and
the fide of an inscribed equi
lateral triangle.
The arc fubtended by
. and _____
_6_
1 4
of the whole
circumference.
The arc fubtended by
5
1 4
Their difference __: T V
.*. the arc fubtended by
the whole circumference.
of the whole
circumference.
__: T V difference of
Hence if firaight lines equal to ■■—.■■■■■ be placed in the
circle (B. 4. pr. 1), an equilateral and equiangular quin
decagon will be thus infcribed in the circle.
Q. E. D.
BOOK V.
DEFINITIONS.
LESS magnitude is faid to be an aliquot part or
fubmultiple of a greater magnitude, when the
lefs meafures the greater; that is, when the
lefs is contained a certain number of times ex
actly in the greater.
II.
A greater magnitude is faid to be a multiple of a lefs,
when the greater is meafured by the lefs ; that is, when
the greater contains the lefs a certain number of times
exactly.
III.
Ratio is the relation which one quantity bears to another
of the fame kind, with refpedl to magnitude.
IV.
Magnitudes are faid to have a ratio to one another, when
they are of the fame kind ; and the one which is not the
greater can be multiplied fo as to exceed the other.
The other definitions will be given throughout the book
where their aid is fir ft required,
v
146
AXIOMS.
QUIMULTIPLES or equifubmultiples of the
fame, or of equal magnitudes, are equal.
If A = B, then
twice A := twice B, that is,
2 A = 2 B;
3A = 3 B;
4 A = 4B;
&c. &c.
and 1 of A = i of B ;
iofA = iofB;
&c. &c.
II.
A multiple of a greater magnitude is greater than the fame
multiple of a lefs.
Let A C B, then
2AC2B;
3 ACZ3B;
4 A C 4 B;
&c. &c.
III.
That magnitude, of which a multiple is greater than the
fame multiple of another, is greater than the other.
Let 2 A m 2 B, then
ACZB;
or, let 3 A C 3 B, then
ACZB;
or, let m A C m B, then
ACB.
BOOK V. PROP. I. THEOR.
i*7
F any number of magnitudes be equimultiples of as
many others, each of each : what multiple soever
any one of the fir Jl is of its part, the fame multiple
Jhall of the fir Jl magnitudes taken together be of all
the others taken together.
LetQQQQQ be the fame multiple of Q,
that WJFW is of f .
that OOOOO « of O.
Then is evident that
QQQQQ1 [Q
is the fame multiple of 4
OQOOQ
[Q
which that QQQQQ isofQ ;
becaufe there are as many magnitudes
in 4
QQQQQ
fffff >
L OOOOO
V
o
as there are in QQQQQ = Q .
The fame demonftration holds in any number of mag
nitudes, which has here been applied to three.
.*. If any number of magnitudes, &c.
1 48 BOOK V. PROP. II. THEOR.
F the jirjl magnitude be the fame multiple of the
fecond that the third is of the fourth, and the fifth
the fame multiple of the fecond that the fix th is oj
the fourth, then foall the firjl, together with the
fifth, be the fame multiple of the fecond that the third, together
with the fixth, is of the fourth.
Let \ , the firft, be the fame multiple of ) ,
the fecond, that O0>O> tne tnu 'd> is of <j>, the fourth;
and let 00^^, the fifth, be the fame multiple of ) ,
the fecond, that OOOOj l ^ e ^ xtn > 1S °f 0>> l ^ e
fourth.
Then it is evident, that J > , the firft and
fifth together, is the fame multiple of , the fecond,
that l \ \, the third and fixth together, is of
looooj
the fame multiple of (J> , the fourth ; becaufe there are as
many magnitudes in j _ z= as there are
m looooj  ° ■
/. If the firft magnitude, &c.
BOOK V. PROP. III. THEOR.
149
F the jirjl of four magnitudes be the fame multiple
of the fecond that the third is of the fourth, and
if any equimultiples whatever of the fir ft and third
be taken, thofe Jliall be equimultiples ; one of the
fecond, and the other of the fourth.
The First.
The Second.
Let i
take \
• be the lame multiple of
The Third. The Fourth.
which J I is of A ;
y the fame multiple of <
♦ ♦♦♦
which <;
is of
♦ ♦'
that <!
Then it is evident,
The Second.
► is the fame multiple of 
i jo BOOK V. PROP. III. THEOR.
♦ ♦♦♦
which <
♦♦♦♦
♦♦♦♦
The Fourth.
• is of A ;
becaufe <
> contains <
> contains
as many times as
y contains
♦ ♦
♦ ♦
> contains ^
♦♦♦♦
♦♦♦♦
♦♦♦♦
The fame reafoning is applicable in all cafes.
.'. If the firft four, &c.
BOOK V. DEFINITION V.
'5 1
DEFINITION V.
Four magnitudes, £», , ^ , ^, are laid to he propor
tionals when every equimultiple of the firft and third be
taken, and every equimultiple of the fecond and fourth, as,
of the firft
&c.
of the fecond
of the third + ^
♦♦♦
♦ ♦♦♦
♦ ♦♦♦♦
♦♦♦♦♦♦
&c.
of the fourth
If <
&c. &c.
Then taking every pair of equimultiples of the firft and
third, and every pair of equimultiples of the fecond and
fourth,
= °rZ, ■■
= o rZ
SOT" 3
: or ^
: or ^
;, = or 3
:. = or 3
;, = or 3
!» — or ~l
♦ ♦
♦ ♦
then will ^ ^
♦ ♦
I 5 2
BOOK V. DEFINITION V.
That is, if twice the firft be greater, equal, or lefs than
twice the fecond, twice the third will be greater, equal, or
lefs than twice the fourth ; or, if twice the firft be greater,
equal, or lefs than three times the fecond, twice the third
will be greater, equal, or lefs than three times the fourth,
and so on, as above exprelfed.
in
then
will
• •• c,
= or Zl
• •• c,
= or Zl
• #• c,
^ or Z3
• •• d,
= or n
••• 1=,
= or Z]
&c.
[♦♦♦ c=,
= or Zl
♦♦♦ c,
= or Zl
 ♦♦♦ c,
= or ^
♦ ♦♦ &
= or z:
,♦♦♦ c
= or Zl
&c.
&c.
&c.
In other terms, if three times the firft be greater, equal,
or lefs than twice the fecond, three times the third will be
greater, equal, or lefs than twice the fourth ; or, if three
times the firft be greater, equal, or lefs than three times the
fecond, then will three times the third be greater, equal, or
lefs than three times the fourth ; or if three times the firft
be greater, equal, or lefs than four times the fecond, then
will three times the third be greater, equal, or lefs than four
times the fourth, and so on. Again,
BOOK V. DEFINITION V.
J 53
If <
then
will
tiff
cz>
™ ^™
or
or
^1
•••#
c
__
or
ZJ
••••
cz,
__
or
Z]
••••
c
:m
or
Z]
&c.
♦♦♦♦
IZ,
^^
or
Zl
♦ ♦♦♦
I— 9
=
or
Z]
•♦♦♦♦
L— >
—
or
Z]
♦♦♦♦
L~ 9
=
or
Zl
[♦♦♦♦
c,
—
or
Z3
&c.
&c.
And so on, with any other equimultiples of the four
magnitudes, taken in the fame manner.
Euclid exprefles this definition as follows : —
The firft of four magnitudes is faid to have the fame
ratio to the fecond, which the third has to the fourth,
when any equimultiples whatfoever of the firft and third
being taken, and any equimultiples whatfoever of the
fecond and fourth ; if the multiple of the firft be lefs than
that of the fecond, the multiple of the third is alfo lefs than
that of the fourth ; or, it the multiple of the firft be equal
to that of the fecond, the multiple of the third is alfo equal
to that of the fourth ; or, ir the multiple of the firft be
greater than that of the fecond, the multiple of the third
is alfo greater than that of the fourth.
In future we fhall exprefs this definition generally, thus :
If M # C, = or Zl m ,
when M ▲ CZ, = or "1 w ^
154 BOOK V. DEFINITION V.
Then we infer that % , the firft, has the fame ratio
to  , the fecond, which ^, the third, has to ^P the
fourth : expreffed in the fucceeding demonstrations thus :
• :■ :: ♦: V;
or thus, # : = ♦ : 9 9
or thus, — = —  : and is read,
V
" as £ is to , so is ^ to ^.
And if # : :: ^ : f we mall infer if
M § C, =: or ^] //; , then will
M ^ C = or Z3 /« ^.
That is, if the firft be to the fecond, as the third is to the
fourth ; then if M times the firft be greater than, equal to,
or lefs than tn times the fecond, then fhall M times the
third be greater than, equal to, or lefs than m times the
fourth, in which M and m are not to be confidered parti
cular multiples, but every pair of multiples whatever;
nor are fuch marks as Q, ^, , &c. to be confidered
any more than reprefentatives of geometrical magnitudes.
The ftudent fhould thoroughly underftand this definition
before proceeding further.
BOOK V. PROP. IV. THEOR. 155
F the fir jl of four magnitudes have the fame ratio to
the fecond, which the third has to the fourth, then
any equimultiples whatever of the firfi and third
shall have the fame ratio to any equimultiples of
the fecond and fourth ; viz., the equimultiple of the firfl fliall
have the fame ratio to that of the fecond, which the equi
multiple of the third has to that of the fourth.
Let :>:.*♦ :^, then 3 :2::34:2f,
every equimultiple of 3 and 3 ^ are equimultiples of
and ^ , and every equimultiple of 2   and 2 JP , are
equimultiples of 1 1 and ^ (B. 5, pr. 3.)
That is, M times 3 and M times 3 ^ are equimulti
ples of and ^ , and m times 2 1 1 and m 2 S are equi
multiples of 2 I I and 2 ^ • but • I I • • ^ • W
(hyp); .*. if M 3 EZ, =, or —j «/ 2 , then
M 3 ^ CZ . =, or ^ ;« 2 f (def. 5.)
and therefore 3 : z   :: 3 ♦ ; 2 ^ (def. 5.)
The fame reafoning holds good if any other equimul
tiple of the firft and third be taken, any other equimultiple
of the fecond and fourth.
.*. If the firft four magnitudes, &c.
i 5 6
BOOK V. PROP. V. THEOR.
F one magnitude be the fame multiple of another,
which a magnitude taken from thefirjl is of a mag
nitude taken from the other, the remainder Jhall be
the fame multiple of the remainder, that the whole
is of the whole.
Q
Let OQ = M '
D
and
= M'.,
o
<^>Q> minus = M' minus M' ■,
O
/. & = M' (* minus ■),
and .*. Jp^ =M' A.
,*. If one magnitude, &c.
BOOK V. PROP. VI. THEOR. 157
IBg<BI
Km
/*llo
* ■P* 1
g\y/^a
Mm
Hr 's ^V
t Vara
F /wo magnitudes be equimultiples of two others,
and if equimultiples of t lief e be taken from the fir ft
two, the remainders are either equal to thefe others,
or equimultiples of them.
Q
Let = M' ■ ; and QQ = M' a ;
o
then minus m m =
M' * minus m m = (M' minus /»') ■ ,
ar >d OO mmus w ' A = M' a minus m a =
(M' minus /»') a •
Hence, (M' minus tri) ■ and (M' minus rri) a are equi
multiples of ■ and a , and equal to ■ and a ,
when M' minus m sr 1 .
.'. If two magnitudes be equimultiples, &c.
i 5 8
BOOK V. PROP. A. THEOR.
F the fir Jl of the four magnitudes has the fame ratio
to the fecond which the third has to the fourth,
then if the firfi be greater than the fecond, the
BfeSSi] third is a/fo greater than the fourth ; and if equal,
equal; ij fiefs, lefs.
Let £ :   : : qp : ; therefore, by the fifth defini
tion, if f C H, then will f f C
but if # EI ■, then ## [= ■■
and ^ CO,
and .*. ^ C ► •
Similarly, if £ z=, or ^] , then will f =,
or ^ ► .
.*. If the firft of four, &c.
DEFINITION XIV.
Geometricians make ufe of the technical term " Inver
tendo," by inverfion, when there are four proportionals,
and it is inferred, that the fecond is to the firft: as the fourth
to the third.
Let \ : B : : C : D , then, by " invertendo" it is inferred
B : A :: U : C.
BOOK V. PROP. B. THEOR.
'59
F four magnitudes are proportionals, they are pro
portionals alfo when taken inverfely.
Let ^ : Q : : ■ : { ► ,
then, inverfely, Q:f :: : ■ .
If M qp ID ot Q, then MUw
by the fifth definition.
Let M ■ Zl ^ O, that is,ffl[jCMf ,
,'. M 1 H ;» , or, /» EM;
.*. iffflQCMf , then will m EM
In the fame manner it may be mown,
that if m Q = or Z3 M ^ ,
then will /» ;=, or 13 M   ;
and therefore, by the fifth definition, we infer
that Q : ^ : # : ■.
.*. If four magnitudes, &c.
160 ROOKV. PROP. C. THEOR.
F the jirjl he the fame multiple of the fecond, or the
fame part of it, that the third is of the fourth ;
the firjl is to the fecond, as the third is to the
fourth.
Let _ _ , the firft,be the fame multiple of Q, the fecond,
that , the third, is of A, the fourth.
Then ■■ : * :: il : *
♦ ♦'
becaufe J is the fame multiple of
that is of Wk (according to the hypothcfis) ;
■ ■ • ■■
and M  ; is taken the fame multiple of"
that M is of J ,
.*. (according to the third propofition),
M _ is the fame multiple of £
that M is of £ .
BOOK V. PROP. C. THEOR. 161
Therefore, if M . be of £ a greater multiple than
m £ is, then M is a greater multiple of £ tnan
m £ is ; that is, if M 5 \ be greater than w 0, then
M will be greater than m ^ ; in the fame manner
it can be fhewn, if M ! be equal m Q. then
M will be equal ;« £.
And, generally, if M f CZ, = or ZD m £
then M will be CZ, = or ^ m 6 ;
.*. by the fifth definition,
■ ■•'♦♦•••
■ ■
Next, let be the fame part of !
that 4k is of r .
In this cafe alfo : j :: A : T.
For, becaufe
A is the fame part of ! ! that A is of
■ ■ ♦ ♦
1 62 BOOK V. PROP. C. THEOR.
therefore J . is the fame multiple of
that is of £ .
Therefore, by the preceding cafe,
■ ■ . a •• • ▲ •
■■'•"♦♦ "■•
and .*. £ : . . :: £ : . ,
by proportion B.
/. If the firft be the fame multiple, &c.
BOOK V. PROP. D. THEOR. 163
the fit ft be to the fecond as the third to the fourth,
and if the Jirji be a multiple, or a part of the
fecond ; the third is the fame multiple, or the fame
part of the fourth.
L >•• ■
and firft, let
•V
je a multiple  .
(hall b
e the fame multiple of ■■ .
First.
•
Second.
■
Third. Fourth.
♦ ♦ w
O
QQ
QQ
OO
Take
a
QQ
_ •
Whatever
multiple
: ^L isofH
take OO
OO
the fam<
; multiple of ■ ,
then, becaufe
and of the fecond and fourth, we have taken equimultiples,
and yT/C> therefore (B. 5. pr. 4),
1 64 BOOK V. PROP. D. THEOR.
: QQ :: JJ : OO' but(C0nft)>
QQ ••( B '5FA)^ 4  oc
and /Ty\ is the fame multiple of ^
that is of .
Next, Id  : : : JP : £,
and alfo   a part of ;
then <9 mail be the fame part of ^ .
nverfely (B
•5).
••"
..♦♦
■"♦♦
but
 is a part
.*.
that is,
•i
is a multiple of   ;
♦♦
ic fr\** lorviP iv^ ii
/. by the preceding cafe, . is the fame multiple of
that is, ^ is the fame part of ,
that   is of .
.*. If the firft be to the fecond, &c.
BOOK V. PROP. VII. THEOR
165
QUAL magnitudes have the fame ratio to the fame
magnitude t and the fame has the fame ratio to equal
magnitudes.
Let $ = 4 and any other magnitude ;
then # : = + : and : # = : 4
Becaufe £ = ^ ,
.. M • = M 4 ;
.\ if M # CZ, = or ^ w , then
M + C, = or 31 m I,
and .. • : I = ^ :  (B. 5. def. 5).
From the foregoing reafoning it is evident that,
if m C> = or ^ M 0, then
m C = or Zl M ^
/.■•=■ 4 (B. 5. def. 5).
/. Equal magnitudes, &c.
1 66 ROOK V. DEFINITION VII.
DEFINITION VII.
When of the equimultiples of four magnitudes (taken as in
the fifth definition), the multiple of the firfl: is greater than
that of the fecond, but the multiple of the third is not
greater than the multiple of the fourth ; then the firfl is
laid to have to the fecond a greater ratio than the third
magnitude has to the fourth : and, on the contrary, the
third is laid to have to the fourth a lefs ratio than the firfl:
has to the fecond.
If, among the equimultiples of four magnitudes, com
pared as in the fifth definition, we fhould find
• ####[Z ,but
+ ♦ ♦ ♦ ♦ s or Zl ffff,orifwe fhould
find any particular multiple M' of the firfl: and third, and
a particular multiple tri of the fecond and fourth, fuch,
that M' times the firfl: is C tri times the fecond, but M'
times the third is not [Z tri times the fourth, /. e. = or
~1 tri times the fourth ; then the firfl is faid to have to
the fecond a greater ratio than the third has to the fourth ;
or the third has to the fourth, under fuch circumftances, a
lefs ratio than the firfl has to the fecond : although feveral
other equimultiples may tend to fhow that the four mag
nitudes are proportionals.
This definition will in future be exprefled thus : —
If M' ^ CI tri O, but M' 1 = or Z tri ► ,
then ^P : Q rZ H : ► •
In the above general exprefllon, M' and tri are to be
confidered particular multiples, not like the multiples M
BOOK V. DEFINITION VII.
167
and m introduced in the fifth definition, which are in that
definition confidered to be every pair of multiples that can
be taken. It muff, alio be here obferved, that ^P , £~J, 1 1 ,
and the like fymbols are to be confidered merely the repre
fentatives of geometrical magnitudes.
In a partial arithmetical way, this may be fet forth as
follows :
Let us take the four numbers, : , 7, i;, and
Firft.
Second.
Third.
Fourth.
8
7
10
9
16
H
20
I O
24
21
3°
27
32
28
40
36
40
35
5°
45
48
42
60
54
56
49
7°
6 3
64
5°
80
72
72
63
90
8:
80
70
100
90
88
V
no
99
96
84
120
108
104
9 1
'3°
117
112
98
j 40
126
&C.
&c.
&c
Sec.
Among the above multiples we find r C 14 and z
tZ that is, twice the firft is greater than twice the
lecond, and twice the third is greater than twice the fourth;
and i 6 ^ 2 1 and 2 ^3 that is, twice the firft is lefs
than three times the fecond, and twice the third is lefs than
three times the fourth ; and among the fame multiples we
can find Hi 56 and v IZ that is, 9 times the firft
is greater than 8 times the fecond, and 9 times the third is
greater than 8 times the fourth. Many other equimul
1 68 BOOK V. DEFINITION VII.
tiples might be selected, which would tend to fliow that
the numbers ?, 7, 10, were proportionals, but they are
not, for we can find a multiple of the firft: £Z a multiple of
the fecond, but the fame multiple of the third that has been
taken of the firft: not [Z the fame multiple of the fourth
which has been taken of the fecond; for inftance, 9 times
the firft: is Q 10 times the fecond, but 9 times the third is
not CI I0 times the fourth, that is, 72 EZ 70, but 90
not C or 8 times the firft: we find C 9 times the
fecond, but 8 times the third is not greater than 9 times
the fourth, that is, 64 C 63, but So is not C When
any fuch multiples as thefe can be found, the firft: ( !)is
faid to have to the fecond (7) a greater ratio than the third
(10) has to the fourth and on the contrary the third
(10) is faid to have to the fourth a lefs ratio than the
firft: 3) has to the fecond (7).
BOOK V. PROP. VIII. THEOR.
169
F unequal magnitudes the greater has a greater
ratio to the fame than the lefs has : and the fame
magnitude has a greater ? atio to the lefs than it
has to the greater.
Let I I and be two unequal magnitudes,
and £ any other.
We mail firft prove that   which is the greater of the
two unequal magnitudes, has a greater ratio to £ than ,
the lefs, has to A j
that is, ■ : £ CZ r : # ;
A
take M' 1 1 , /»' , M' , and m ;
fuch, that M' a and M' g mail be each C ;
alfo take m £ the lean: multiple of £ ,
which will make m'
M' =M'
.*. M' is not
;;/
but M' I I is ~ m £ , for,
as m' is the firft multiple which fir ft becomes CZ M' ,
than (m minus 1) orw' ^ minus Q is not I M' 1 1 .
and % is not C M' A,
/. tri
minus
that
+ muft be Z2 M'  + M' a ;
A
is, m
muft be — 1 M'
.'. M' I I is C *»' j but it has been ftiown above that
z
170 BOOK V. PROP. VIII. THEOR.
M' is not C»'§, therefore, by the feventh definition,
A
 has to £ a greater ratio than : .
Next we mall prove that £ has a greater ratio to , the
lefs, than it has to , the greater;
or, % : I c # : ■•
Take m £ , M' , ni %, and M' ,
the fame as in the firff. cafe, fuch, that
M' a and M'   will be each CZ > ar >d f» % the leaft
multiple of £ , which firfr. becomes greater
than M' p = M' ■ .
.". ml % minus £ is not d M' j  ,
and f is not C M' ▲ ; confequently
ot' minus # + # is Zl M'  + M' ▲ ;
.'. »z' is ^ M'   , and .*. by the feventh definition,
A has to a greater ratio than Q has to  .
.*. Of unequal magnitudes, &c.
The contrivance employed in this proportion for finding
among the multiples taken, as in the fifth definition, a mul
tiple of the firft greater than the multiple of the fecond, but
the fame multiple of the third which has been taken of the
firft, not greater than the fame multiple of the fourth which
has been taken of the fecond, may be illuftrated numerically
as follows : —
The number 9 has a greater ratio to 7 than has to 7 :
that is, 9 : 7 CI : 7 5 or, b } 1 : 7 fZ  '7
BOOK V. PROP. Fill. THEOR, 171
The multiple of 1, which firft becomes greater than 7,
is 8 times, therefore we may multiply the firft and third
by 8, 9, 10, or any other greater number ; in this cafe, let
us multiply the firft and third by 8, and we have 64^8
and : again, the firft multiple of 7 which becomes
greater than 64 is 10 times; then, by multiplying the
fecond and fourth by 10, we fhall have 70 and 70 ; then,
arranging thefe multiples, we have —
8 times 10 times 8 times 10 times
the first. the second. the third. the fourth.
64+ 8 70 70
Confequently , « 8, or 72, is greater than  : , but
is not greater than 70, .\ by the feventh definition, 9 has a
greater ratio to 7 than has to  .
The above is merely illuftrative of the foregoing demon
ftration, for this property could be fhown of thefe or other
numbers very readily in the following manner ; becaufe, if
an antecedent contains its confequent a greater number of
times than another antecedent contains its confequent, or
when a fraction is formed of an antecedent for the nu
merator, and its confequent for the denominator be greater
than another fraction which is formed of another antece
dent for the numerator and its confequent for the denomi
nator, the ratio of the firft antecedent to its confequent is
greater than the ratio of the laft antecedent to its confe
quent.
Thus, the number 9 has a greater ratio to 7, than 8 has
to 7, for  is greater than .
Again, 17 : 19 is a greater ratio than 13:15, becaufe
17 17 X 15 25,5 , 13 13 X 19 247 ,
5  ^T>TTi  isi' and I5 = T^T 9 = «? hence « IS
evident that ?f is greater than ~ t ..  is greater than
172 BOOK V. PROP. VIII. THEOR.
— , and, according to what has been above fhown, \j has
to 19 a greater ratio than 13 has to 15.
So that the general terms upon which a greater, equal,
or lefs ratio exifts are as follows : —
A C
If g be greater than ■=, A is faid to have to B a greater
A C
ratio than C has to D ; if ^ be equal to jt, then A has to
B the fame ratio which C has to D ; and if ^ be lefs than
c
^, A is faid to have to B a lefs ratio than C has to D.
The ftudent mould underftand all up to this propofition
perfectly before proceeding further, in order fully to com
prehend the following propofitions of this book. We there
fore ftrongly recommend the learner to commence again,
and read up to this {lowly, and carefully reafon at each flep,
as he proceeds, particularly guarding againft the mifchiev
ous fyflem of depending wholly on the memory. By fol
lowing thefe inftruclions, he will find that the parts which
ufually prefent confiderable difficulties will prefent no diffi
culties whatever, in profecuting the ftudy of this important
book.
BOOK V. PROP. IX. THEOR. 173
AGNITUDES which have the fame ratio to the
fame magnitude are equal to one another ; and
thofe to which the fame magnitude has the fame
ratio are equal to one another.
Let ▲ : I I : : £ : 1 1, then ^ =f .
For, if not, let ▲ C • > then will
♦ : € C # : ■ (B. 5 pr 8),
which is abfurd according to the hypothecs.
.*. ^ is not C % '
In the fame manner it may be mown, that
£ is not CZ t '
Again, let  : ▲ : : # ? then will ^ = .
For (invert.) + :  # •  ?
therefore, by the firfl cafe, ▲ =0.
.*. Magnitudes which have the fame ratio, 6cc.
This may be fhown otherwife, as follows : —
Let \ : B ZZZ ' : C> then Br:C, for, as the fraction
— = the fraction — , and the numerator of one equal to the
B c *
numerator of the other, therefore the denominator of thefe
fractions are equal, that is BrC.
Again, if B : = C : A, B = C. For, as  = ^,
B muft = C
*74
BOOK V. PROP. X. THEOR.
HAT magnitude which has a greater ratio than
another has unto the fame magnitude, is the greater
of the two : and that magnitude to which the fame
has a greater ratio than it has unto another mag
nitude, is the lefs of the two.
Let jp : C # : 1 1, then ^ C # •
For if not, let W — or ~l ^ ;
then, qp : = # : ( B  5 P r  7) or
^ : 1 13 9 : (B. 5. pr. 8) and (invert.),
which is abfurd according to the hypothecs.
.*. ^p is not = or ^ £ , and
.*. ^ muftbe CZ ••
Again, let ? : # C ! : JP,
then, ^ H V
For if not, £ muft be C or = ^ ,
then :^ : JP (B. 5. pr. 8) and (invert.) ;
== I : ■ (B. 5. pr. 7), which is abfurd (hyp.);
/. £ is not CZ or = ^P,
and .*. A muft be 13 ••
or
.*. That magnitude which has, 6cc.
BOOK V. PROP. XL THEOR.
l 75
ATI OS that are the fame to the fame ratio, are the
fame to each other.
Let ♦ : ■ r= % : and : = A : •,
then will ^ :   = A : •.
For if M # Cf => or 13 » ■ ,
then M £ C» =» or 3 z» p ,
and if M C =:, or ^ /« p ,
then M A CZ, :=, or ^ m •, (B. 5. def. 5) ;
\ if M ♦ C, =, or 33 m ■ , M A CZ, =, or 3 w • .
and .*. (B. 5. def. 5) + : B = A : ••
.*. Ratios that are the fame, &c.
i 7 6
BOOK V. PROP. XII. THEOR.
F any number of magnitudes be proportionals, as
one of the antecedents is to its confequent, Jo Jhall
all the antecedents taken together be to all the
confequents.
Let H : • = U : O = ► : ' = •:▼ = *:•;
then will   : £ ss
■ +D + +• + *:# + <>+ +▼ + ••
For ifMC m % , then M Q [Z m £>,
and M . C m M • C m ▼ ,
alfo MaC« •• (B. 5. def. 5.)
Therefore, if M   CZ m , then will
M+MQ + M +M. + Mi,
or M J + O + + • + A ) be grater
than m £ 4" w C 4" m "f" m T "I" w •'
or^«(#+0+ +▼+•)■
In the fame way it may be mown, if M times one of the
antecedents be equal to or lefs than m times one of the con
fequents, M times all the antecedents taken together, will
be equal to or lefs than m times all the confequents taken
together. Therefore, by the fifth definition, as one of the
antecedents is to its confequent, fo are all the antecedents
taken together to all the confequents taken together.
.*. If any number of magnitudes, &c.
BOOK V. PROP. XIII. THEOR.
[ 77
F the jirji has to the fecond the fame ratio which
the third has to the fourth, but the third to the
fourth a greater ratio than the fifth has to the
fixth ; the firjijhall alfo have to the fecond a greater
ratio than the fifth to the fixth.
Let 9 : Q = ■ : >, but ■ : C O ' •>
then f:OCO:l
For, becaufe   : CO : i) t ^ iere are *° me mu ^"
tiples (M' and ni) of j  and <^, and of and £ .
fuch that M'  CZ ni ,
but M' <^ not C ni £, by the feventh definition.
Let thefe multiples be taken, and take the fame multiples
of fM and f^.
/. (B. 5. def. 5.) if M' 9 C, =, or Z\ ni Q ;
then will M' ■ IZ, =, or ^2 m ' ,
but M' I C m ' (connruclion) ;
.. m ' qp tz ni Q,
but M' <^> is not CZ ni £ (conftrudtion) ;
and therefore by the feventh definition,
.*. If the firft has to the fecond, &c.
A A
i/8
BOOK V. PROP. XIV. THEOR.
F the firji has the fame ratio to the fecond which the
third has to the fourth ; then, if the fir j} be greater
than the third, the fecond foall be greater than the
fourth; and if equal, equal ; and if lefs, lefs.
Let ^ : Q : : : + , and firft fuppofe
V CZ , then will O CZ #.
For f : O C I U (B. 5 pr 8). and by the
hypothefis, ^ I Q = : + ;
.*.■:♦ CB:D(B. 5 .pr. i3).
.*. ♦ Zl D (B. S pr. io.), orOCf
Secondly, let ■ = , then will ^J zz 4 .
For * : (J = : Q (B. 5 . pr. 7 ),
and fl : Q = 9 : ^ (hyp.) ;
.\ ■ :Q= M :♦ (B. 5 . P r. n),
and ,\ Q = + (B. 5, pr. 9).
Thirdly, if JP Z] , then will O Z] ♦ ;
becaufe  CI ^ and : + = ^ : O ;
.*. ♦ c O? by tne ^ft ca ^" e »
that is, Q Zl ♦ •
'. If the firft has the fame ratio, &c.
BOOK V. PROP. XV. THEOR.
179
A.GNITUDES have the fame ratio to one another
which their equimultiples have.
Let £ and be two magnitudes ;
then, # : ft :: M' % : M' I.
For
• :■
.*. • "• I "4 • : 4 • (B. 5 P r  I2 )
And as the fame reafoning is generally applicable, we have
• : ■ : : M' A : M' ■ .
.*. Magnitudes have the fame ratio, &,c.
180 BOOK V. DEFINITION XIII.
DEFINITION XIII.
The technical term permutando, or alternando, by permu
tation or alternately, is ufed when there are four propor
tionals, and it is inferred that the firfl has the fame ratio to
the third which the fecond has to the fourth ; or that the
tirft is to the third as the fecond is to the fourth : as is
ihown in the following propofition : —
Let : + ::?:■'
by '* permutando" or "alternando" it is
inferred . : ^ :: ^ : .
It may be neceffary here to remark that the magnitudes
, A, M, , muft be homogeneous, that is, of the
fame nature or fimilitude of kind ; we muft therefore, in
fuch cafes, compare lines with lines, furfaces with furfaces,
folids with folids, &c. Hence the ftudent will readily
perceive that a line and a furface, a furface and a folid, or
other heterogenous magnitudes, can never ftand in the re
lation of antecedent and confequent.
BOOK V. PROP. XVI. THEOR.
81
F four magnitudes of the fame kind be proportionals,
they are alfo proportionals when taken alternately.
Let <p : Q :: : 4 , then
::0'#.
ForM 9 : M Q :: * : O ( B  5 P r  I 5)>
and M:MQ:: : + (hyp.) and (B. 5. pr. 1 1 ) ;
alfo m : ;;/ ^ :: : ▲ (B. 5. pr. 15);
,\ M qp : M Q :: « : ;» 4 (B. 5. pr. 14),
and .*. if M ^ C» =» or I] w  ,
then will M Q d, => or ^ /// ^ (B. 5. pr. 14) ;
therefore, by the fifth definition,
v m o: ♦•
.*. If four magnitudes of the fame kind, &c.
1 82 BOOK V. DEFINITION XVI.
DEFINITION XVI.
Dividendo, by divifion, when there are four proportionals,
and it is inferred, that the excefs of the firfr. above the fecond
is to the fecond, as the excefs of the third above the fourth,
is to the fourth.
Let A : B : : C : D ;
by " dividendo " it is inferred
A minus B : B : : i minus ) : D.
According to the above, A is fuppofed to be greater than
B, and C greater than ; if this be not the cafe, but to
have B greater than A, and greater than C > B and
can be made to ftand as antecedents, and A and C as
confequents, by " invertion "
B : A : •. D : C ;
then, by "dividendo," we infer
B minus A : A : : minus C : C •
BOOK V. PROP. XVII. THEOR.
183
jF magnitudes, taken jointly, be proportionals, they
fliall alfo be proportionals when taken feparately :
that is, if two magnitudes together have to one of
them the fame ratio which two others have to one
of thefe, the remaining one of the fir ft two foall have to the other
the fame ratio which the remaining one of the lafi two has to the
other of thefe.
Let f + O: Q:: : + ♦ : ♦,
then will
:Q:: % : ♦■
Take M V IZ m Q to each add M Q,
then we have M ^ + M U[Z>«U + M Q ?
orM(^ + 0) c (« + M)Q:
but becaufe ^P + Q:Q::B + 4:^ (hyp.),
andM(^P + 0)EZ(* + M)D;
.*. M( +^)CI( W + M)4 (B. 5. def. 5 );
/. M ■ + M + C m + + M # ;
.*. M i tZ w ♦ , by taking M + from both fides :
that is, when MfC* O, then M Cw^,
In the fame manner it may be proved, that if
M ^P = or ^ /« Q, then will M = or ^ « 4
and /. ^ : Q : : > : 4 (B. 5. def. 5).
.*. If magnitudes taken jointly, &c.
l8 4 BOOK V. DEFINITION XV.
DEFINITION XV.
The term componendo, by compofition, is ufed when there
are four proportionals ; and it is inferred that the firft toge
ther with the fecond is to the fecond as the third together
with the fourth is to the fourth.
Let A : B : : : D ;
then, by the term " componendo," it is inferred that
A + B : B :: + D : D.
By " invertion" B and p may become the firlt and third,
A and the fecond and fourth, as
B : A : : D : ,
then, by " componendo," we infer that
B + A : A : : D + : .
BOOK V. PROP. XVIII. THEOR.
185
F magnitudes, taken feparately, be proportionals,
they Jhall alfo be proportionals when taken jointly :
that is, if the firjl be to the fecond as the third is
to the fourth, the firjl and fecond together Jliall be
to the fecond as the third and fourth together is to the fourth.
Let * : O
then * + Q : Q
for if not, let ^ { Q
fuppofing Q
.'. W : O ' =
but
;■ + ♦:♦;
not = ^ ;
• (B. 5. pr. 17);
.Q:: : ♦ (hyp.);
: :: I : 4 ( B  5 P r  JI );
••••=♦ (B 5 P^ 9).
which is contrary to the fuppofition ;
.*. £ is not unequal to ^ ;
that is =1 4 5
'. If magnitudes, taken feparately, &c.
B B
i86
BOOK V. PROP. XIX. THEOR.
F a whole magnitude be to a whole, as a magnitude
taken from the firft, is to a magnitude taken from
the other ; the remainder Jhall be to the remainder,
as the whole to the whole.
then will Q : :: ^ + D : + »,
.\ G :
again Q :
but * + O
therefore ^J ".
If a whole magnitude be to a whole, &c.
*■■ :■(<!
ivid.),
:: W :■ (alter.),
:■+ # .'V
: ■ hyp.)
♦ "* + U
■ +♦
(B. 5. pr. 11).
DEFINITION XVII.
The term " convertendo," by converfion, is made ufe of
by geometricians, when there are four proportionals, and
it is inferred, that the firft. is to its excefs above the fecond,
as the third is to its excefs above the fourth. See the fol
lowing propofition : —
BOOK V. PROP. E. THEOR.
187
F four magnitudes be proportionals, they are alfo
proportionals by converjion : that is, the fir Jl is to
its excefs above the fecond, as the third to its ex
cefs above the fourth.
then lhall # O : • • : ■ : ■ >
Becaufe
therefore 1
.. o
10:0:
:0"B
• ♦
• ::■♦
!♦♦;
(divid.),
I (inver.),
I (compo.).
,*. If four magnitudes, &c.
DEFINITION XVIII.
" Ex aBquali" (fc. diflantia), or ex zequo, from equality of
diftance : when there is any number of magnitudes more
than two, and as many others, fuch that they are propor
tionals when taken two and two of each rank, and it is
inferred that the nrft is to the laft of the firft rank of mag
nitudes, as the firft is to the laft of the others : " of this
there are the two following kinds, which arife from the
different order in which the magnitudes are taken, two
and two."
188 BOOK V. DEFINITION XIX.
DEFINITION XIX.
" Ex aequali," from equality. This term is ufed amply by
itfelf, when the firft magnitude is to the fecond of the firft
rank, as the nrft to the fecond of the other rank ; and as
the fecond is to the third of the hrft rank, fo is the fecond
to the third of the other ; and fo on in order : and the in
ference is as mentioned in the preceding definition ; whence
this is called ordinate proportion. It is demonftrated in
Book 5. pr. 22.
Thus, if there be two ranks of magnitudes,
A, B, 1. , P, E, F, the nrft rank,
and L, M, , , P, Q, the fecond,
fuch that A : B : : L : M, B : ( :: M : ,
: : I : : : , : E : : : P, E : F : : P : Q ;
we infer by the term " ex asquali" that
A : F :: L : Q.
BOOKV. DEFINITION XX. 189
DEFINITION XX.
" Ex squali in proportione perturbata feu inordinata,"
from equality in perturbate, or diforderly proportion. This
term is ufed when the firft magnitude is to the fecond of
the firft rank as the laft but one is to the laft of the fecond
rank ; and as the fecond is to the third of the firft rank, fo
is the laft but two to the laft but one of the fecond rank ;
and as the third is to the fourth of the firft rank, fo is the
third from the laft to the laft but two of the fecond rank ;
and fo on in a crofs order : and the inference is in the 1 8th
definition. It is demonstrated in B. 5. pr. 23.
Thus, if there be two ranks of magnitudes,
A, B, C, D, , , the firft rank,
and , , N , O , P , Q , the fecond,
fuch that A:B::P:Q,B:C::0:P,
C : D :: N : O, D : :: : N, i : F :: : I ;
the term " ex a?quali in proportione perturbata feu inordi
nata" infers that
A : :: : Q.
190 BOOK V. PROP. XX. THEOR.
F there be three magnitudes, and other three, which,
taken two and two, have the fame ratio ; then, if'
the Jirjl be greater than the third, the fourth Jha I I
be greater than the fixth ; and if equal, equal ;
and if lefs, lefs.
Let ^P, {^J, , be the fir ft three magnitudes,
and ^, (3, ( , be the other three,
fuch that V :0 ::+ : C> , an <l O : M ''O '■ O •
Then, if ^ d» => or ^ , then will ^ C ==,
orZ3 t
From the hypothecs, by alternando, we have
andQ :0 ::■ :•;
/. ^ : ♦ ::  : t (B. 5 . pr. n);
•\ if f C» =. or D , tlien will + C =,
or3 # (B. 5. pr. 14).
.*. If there be three magnitudes, 6cc.
BOOK V. PROP. XXI. THEOR.
191
F there be three magnitudes, and other three which
have the fame ratio, taken two and two, but in a
crofs order ; then if the firjl magnitude be greater
than the third, the fourth Jhall be greater than the
jixth ; and if equal, equal ; and if lefs, lefs.
Let p, £ , , be the firft three magnitudes,
and ^, 0>> ( ? the other three,
fuch that \ : £ :: O ••> and £ : ■ :: ♦ : 0>«
Then, if I C =, or Z2 I
will ♦[=,=,=! .
then
Firft, let < be CI ■ :
then, becaufe £ is any other magnitude,
f :iC:i (B. 5. pr. 8);
butO M :: :4 (hyp);
.*. 0> =
(B. 5. pr. 13);
and becaufe jfe : ■ :: ^ : O ( n yp) 5
.*•■ :A O :♦ (in*.).
and it was fhown that £ : C  '• A <
.'. O : < C O =♦ (B. spr. 13);
1 92 BOOK V. PROP. XXI. THEOR.
•• • =] ♦,
that is ^ CI  .
Secondly, let =   ; then {hall ^ = ) .
For becaufe — ,
* : * = ■ : A (B. 5F 7);
but : A = O : 1 (hyp.),
and I I : 4b = O • ^ (hyp and inv.),
.. O : # = : ♦ (B. 5. P r. 11),
•'• ♦ = • (B 5 P^ 9)
Next, let be Z2 ■• then ^ fhall be ^ ;
forC ',
and it has been fhown that  : 4fc = : $,
and ^ : ' s = ; 1 : O;
/. by the firft cafe is CZ ^j
that is, ^ ^ ).
/. If there be three, &c.
BOOK V. PROP. XXII. THEOR.
*93
F there be any number of magnitudes, and as many
others, which, taken two and two in order, have
the fame ratio ; the firft Jhall have to the laji of
the fir Jl magnitudes the fame ratio which the fir /I
of the others has to the laji of the fame.
N.B. — This is ufually cited by the words "ex trqua/i," or
"ex aquo."
Firft, let there be magnitudes f^ , + , 1 1 ,
and as many others ▲ ,(^, ) ,
luch that
V '•♦ "♦ O*
and^ :  :: <^> * I ;
then mail ^ : { : : ^ • O .
Let thefe magnitudes, as well as any equimultiples
whatever of the antecedents and confequents of the ratios,
ftand as follows : —
and
becaufe qp : ^ : : ^ : 0> »
.\ M fp : »i + : : M ^ : /» £> (B. 5. p. 4).
For the fame reafon
m + : N : : m £> : N ;
and becaufe there are three magnitudes,
c c
i 9 4 BOOKV. PROP. XXII. THEOR.
and other three, M ^ , m /\ , N ,
which, taken two and two, have the fame ratio ;
.*. ifMjP CZ, =, orZlN
then will M + C» => or ^ N , by (B. 5. pr. 20) ;
and ,\ V : :: + : 1 (def. 5).
Next, let there be four magnitudes, ■, ^, § , ^ ,
and other four, £>, ^, > A ,
which, taken two and two, have the fame ratio,
that is to fay, ^p : ♦  <2> : Q,
and : ♦ : : 1 1 : ▲ ,
then mall ^ : + : : ^ : ▲ ;
for, becaufe ■ , ^, , are three magnitudes,
and <2>, 0? 5 other three,
which, taken two and two, have the fame ratio ;
therefore, by the foregoing cafe, <p : j :: ^ : .•■,
but I : ♦ :: : ▲ ;
therefore again, by the firfl cafe, ^p : ^ : : ^> : ▲ ;
and {o on, whatever the number of magnitudes be.
.*. If there be any number, Sec.
BOOK V. PROP. XXIII. THEOR.
T 95
F there be any number of magnitudes, and as many
others, which, taken two and two in a crofs order,
have the fame ratio ; the firji fliall have to the lajl
of the firjl magnitudes the fame ratio which the
firji of the others has to the lajl of the fame.
N.B. — This is ufually cited by the words "ex cequali in
proportione perturbatd ;" or " ex aquo perturbato."
Firft, let there be three magnitudes, £, Q ,  ,
and other three, ; O ' £ '
which, taken two and two in a crofs order,
have the fame ratio ;
o,
that is, : O •
=
and rj : ■ :
•♦
then fhall : 1 :
: ▲
Let thefe magnitudes and their refpective equimultiples
be arranged as follows : —
m ,Mrj, w ,M t,«0» w #»
then * : Q :: M : M Q (B. 5. pr. 15) ;
and for the fame reafon
but J :Q ::<> :# (hyp.),
jo6 BOOK V. PROP. XXIII. THEOR.
.. M : MQ ::<> : • (B. 5. P r. n);
and becaufe O : H :: : 0» (hyp.),
,\ M Q : m ■ :: : w £> (B. 5. pr. 4) ;
then, becaufe there are three magnitudes,
M ,MO,«,
and other three, M , m £>, m £,
which, taken two and two in a crofs order, have
the fame ratio ;
therefore, if M CZ, =, or 3 m I?
then will M C =, or ^ w (B. 5. pr. 2 1 ),
and /. : ■ :: : # (B. 5. def. 5).
Next, let there be four magnitudes,
>p,o, ■• #1
and other four, <^>, %, Hi, Jk.,
which, when taken two and two in a crofs order, have
the fame ratio ; namely,
and
then fhall
V
:D :
:m :
D
■ :
:#:
■
^L
•O:
*
•+'
:0:
For, becaufe , Q ,   are three magnitudes,
BOOKV. PROP. XXIII. THEOR. 197
and , ■§, ▲, other three,
which, taken two and two in a crofs order, have
the fame ratio,
therefore, by the firft cafe, >:!!::#:▲,
but ■ : < :: £> : #,
therefore again, by the firft cafe, I : < :: <^) : A J
and fo on, whatever be the number of fuch magnitudes.
.*. If there be any number, &c.
iq8
BOOK V. PROP. XXIV. THEOR.
j]F the firjl has to the fecond the fame ratio which
the third has to the fourth, and the fifth to the
fecond the fame which the fix th has to the fourth,
the firjl and fifth together jhal I have to the fecond
the fame ratio which the third and fix th together have to the
fourth.
First. Second.
Third. Fourth.
V D
■ ♦
Fifth.
Sixth.
o
•
Let jp : Q :
: ■: ,
and <3 : Q :
: • : ►»
then ^ + £> : O
::■+#:♦
For <2> : D :: # : ( h yP)'
and [J : ^ : : : ■ (hyp) and (invert.),
.\ <2>: qp :: #: ■ (B. 5. pr. 22);
and, becaufe thefe magnitudes are proportionals, they are
proportionals when taken jointly,
.'. V+6'O' •+■= #(B. 5 .pr. 18),
but o>: D  # : ( h yp) 5
.'. ¥ + O^O •+ ■• (B. 5. pr. 22).
,\ If the firft, &c.
BOOK V. PROP. XXV. THEOR.
199
F four magnitudes of the fame kind are propor
tionals, the great ejl and leaf of them together are
greater than the other two together.
Let four magnitudes, ^  ^, I   , ^J, and
of the fame kind, be proportionals, that is to fay,
* + 0: ■ + :iQ:f
and let ■  (3) be the greateft of the four, and confe
quently by pr. A and 14 of Book 5, is the leaft ;
then will JpfQl beCB+ + U J
becaufe ^ f Q : £ j : : Q :
•*• V ' M W+O ■+ (B. 5 .pr. 19),
bu < * + D C ■ + (hyp.),
.'• * 1= B(B 5prA);
to each of thefe add ^J } ,
••• * + D + t= «+D+ ♦•
.*. If four magnitudes, &c.
2oo BOOK V. DEFINITION X.
DEFINITION X.
When three magnitudes are proportionals, the firfl is faid
to have to the third the duplicate ratio of that which it has
to the fecond.
For example, if A, , C , be continued proportionals,
that is, A : : : : C , A is faid to have to C the dupli
cate ratio of A : B ;
or — rz the fquare of—.
This property will be more readily feen of the quantities
a ** > > J » for a r"' : . : : : a ',
ar 1 r !
and — rr r* — the fquare of — sr r,
or of , , j  ;
for —  sjZ= the fquare of— =— .
a r ' r
DEFINITION XI.
When four magnitudes are continual proportionals, the
firft is faid to have to the fourth the triplicate ratio of that
which it has to the fecond ; and fo on, quadruplicate, &c.
increafing the denomination ftill by unity, in any number
of proportionals.
For example, let A,B, C, D, be four continued propor
tionals, that is, A : i> :: : C :: C : D; A is faid to have
to D, the triplicate ratio of A to B ;
or  s; the cube of—.
BOOK V. DEFINITION XL
20I
This definition will be better underftood, and applied to
a greater number of magnitudes than four that are con
tinued proportionals, as follows : —
Let a r' , , a r , a, be four magnitudes in continued pro
portion, that is, a r 3 : :: '• a r '•'• a r '• < l >
then  — =r r 3 ~ the cube of — = r.
a
Or, let ar 5 , ar*, ar 3 , ar', ar, a, be fix magnitudes in pro
portion, that is
ar 5 : ar* : : ar* * ar 3 :: ar 3 : ar 1 : : ar : ar :: ar : a,
ar* cir°
then the ratio — ■ =: r° z= the fifth power of — ; =: r.
a r ar*
Or, let a, ar, ar 2 , ar 3 , ar 4 , be five magnitudes in continued
proportion ; then — . =7 — the fourth power of — — .
r r ar* r* r ar r
DEFINITION A.
To know a compound ratio : —
When there are any number of magnitudes of the fame
kind, the firfr. is faid to have to the laft of them the ratio
compounded of the ratio which the firft has to the fecond,
and of the ratio which the fecond has to the third, and of
the ratio which the third has to the fourth ; and fo on, unto
the laft magnitude.
For example, if A, B, C, D,
be four magnitudes of the fame
kind, the firft A is faid to have to
the laft D the ratio compounded
of the ratio of A to B , and of the
ratio of B to C, and of the ratio ofC to D ; or, the ratio of
DD
A
B
C
D
E
F
G
H
K.
L
M N
202 BOOK V. DEFINITION A.
A to 1 ' is faid to be compounded of the ratios of ' to ,
H to ( , and I to I > .
And if \ has to I. the fame ratio which I has to I , and
B to C the fame ratio that G has to H, and ( to h the
fame that K has to L ; then by this definition, \ is said to
have to L) the ratio compounded of ratios which are the
fame with the ratios of E to F, G to H, and K to L« And
the fame thing is to be underftood when it is more briefly
expreffed by faying, \ has to D the ratio compounded of
the ratios of E to F, G to H, and K to  .
In like manner, the fame things being iuppofed ; if
has to the fame ratio which has to I ' , then for fhort
nefs fake, is faid to have to the ratio compounded of
the ratios of E to F, G to H, and K to L.
This definition may be better underftood from an arith
metical or algebraical illuftration ; for, in fact, a ratio com
pounded of feveral other ratios, is nothing more than a
ratio which has for its antecedent the continued product of
all the antecedents of the ratios compounded, and for its
confequent the continued product of all the confequents of
the ratios compounded.
Thus, the ratio compounded of the ratios of
: , : ,6:11,2:5,
is the ratio of  X X 6 X 2 : X X 11 X 5,
or the ratio of 96 : 11 55, or 32 : 385.
And of the magnitudes A, B, C, D, E, F, of the fame
kind, A : F is the ratio compounded of the ratios of
A :B, B : C, C: D, J : E, E : F;
for A X B X X X E : B X v X X E X F,
XX X X E.
or
xx xexf = "' or the ratio of A : F 
BOOK V. PROP. F. THEOR.
203
ATI OS which are compounded of the fame ratios
are the fame to one another.
Let A : B :: F : G,
5 i ( '•'• r '. ri»
::D::H:K,
and ) : E : : I : L.
A B C D E
F G H K L
Then the ratio which is compounded of the ratios of
A : B, B : C, C : D, I : E, or the ratio of A : E, is the
fame as the ratio compounded of the ratios of F : G,
5 : H, H : K, K : L, or the ratio of F : L.
Fori
c
and
E
XXX
F
"'
H'
£ .
1'
XXX
X X XL'
F
or the ratio of \ : I is the fame as the ratio of F : L.
X x e —
and .*. j —
The fame may be demonstrated of any number of ratios
fo circumftanced.
Next, let A : B : : K : L,
1 : C : : i : K,
_ ' I ; ; j • rl»
) : E :: F: G.
2o + BOOK V. PROP. F. THEOR.
Then the ratio which is compounded of the ratios of
A : , : , : , ]) : E, or the ratio of \ : 1 , is the
fame as the ratio compounded of the ratios of :L, i : ,
: H, F : , or the ratio of F :L.
For  = 
'
and — ss — ;
• A X X X X X XF
X X X i I X X X '
and/.  = ,
or the ratio of A : E is the fame as the ratio of F : L.
,*. Ratios which are compounded, 6tc.
BOOK V. PROP. G. THEOR.
205
F feveral ratios be the fame to fever al ratios, each
to each, the ratio which is compounded of ratios
which are the fame to the fir Jl ratios, each to each,
Jhall be the fame to the ratio compounded of ratios
which are the fame to the other ratios, each to each.
ABCDEFGH
P Q R S T
a bed e f g h
V W X ^
If \ : B
: : a : b
and A : B : : P :
Q
a : b:
: :
w
C :D
.: c : d
C:D::Q:
R
c:d:
: W:
X
E :F
:: e :f
E : F : : R
S
e :/:
: : :
Y
and G : II
::g:h
G:H :: S
T
g:h:
: :
Z
then P : T =
'
T A a
For .7 = I = 7
z^z
9
_ C c
. R D d
=:
~~ '
e
s ■— F — /
=
X
3 G y
T H h
=
>
■ x s x x
X
X X
• Q x R x  x r —
J p
and .*.  ^ 
X
 >
X X
z*
rP :T = :
Z.
.*. If feveral ratios, &c.
2o6
BOOK V. PROP. H. THEOR.
F a ratio which is compounded of feveral ratios be
the fame to a ratio which is con pounded of fever al
other ratios ; and if one of the firjl ratios, or the
ratio which is compounded of feveral of them, be
the fame to one of the laft ratios, or to the ratio which is com
pounded of Jeveral of them ; then the remaining ratio ofthefirji,
or, if there be more than one, the ratio compounded of the re
maining ratios, Jliall be the fame to the remaining ratio of the
la/i, or, if there be more than one, to the ratio compounded ofthefe
remaining ratios.
A
B
C
D
E
F
G
H
P
Q
R
S
T
X
Let A :B, B :C, C :D, D : E, E : F, F : G, G :H,
be the nrft ratios, and P : Q^Qj_R, R : S, S : T, T : X,
the other ratios ; alfo, let A : H, which is compounded of
the rirft ratios, be the fame as the ratio of P : X, which is
the ratio compounded of the other ratios ; and, let the
ratio of A : E, which is compounded of the ratios of A : B,
B :C, C :D, D :E, be the fame as the ratio of P : R,
which is compounded of the ratios P : Q^ Qj R.
Then the ratio which is compounded of the remaining
firft ratios, that is, the ratio compounded of the ratios
E : F, F : G, G : H, that is, the ratio of E : H, mail be
the fame as the ratio of R : X, which is compounded of
the ratios of R : v , S : T, T : X, the remaining other
ratios.
BOOK V. PROP. H. THEOR. 207
B ecau f e XEXl XDXEXiXG P X <J X k X S X T
B X C X D X E X F X G X H Q X K X  X I X \'
or xiiXiXd w e x } x b x g w
B XC X D X E * FX(,XH — DXB *
X (, X H ' ^ SXTXX'
and A X B X C X_D LXS,
B X l X D X E — O X R '
. E X F X G R X X
' * F X G XH SXTXX*
. E __
/. E : H = R : X.
.*. If a ratio which, &c.
2o8
BOOK V. PROP. K. THEOR.
F there be any number of ratios, and any number of
other ratios, fuch that the ratio which is com
pounded of ratios, which are the fame to the firji
ratios, each to each, is the fame to the ratio which
is compounded of ratios, which are the fame, each to each, to
the laji ratios — and if one of the fir Jl ratios, or the ratio which
is compounded of ratios, which are the fame to feveral of the
firft ratios, each to each, be the fame to one of the lajl ratios,
or to the ratio which is compounded of ratios, which are the
fame, each to each, to feveral of the lajl ratios — then the re
maining ratio of the firji ; or, if there be tnore than one, the
ratio which is compounded of ratios, which are the fame, each
to each, to the remaining ratios of the firji, Jhall be the fame
to the remaining ratio of the lajl ; or, if there be more than
one, to the ratio which is compounded of ratios, which are the
fame, each to each, to thefe remaining ratios.
h k m n s
AB, CD, EF, GH, KL, MN,
a b c d e f g
p,i , , y,
h k I m n p
abed e f g
Let A:B, C:D, E:F, G:H, K:L, M:N, be the
firft ratios, and : , : , : , ^:W, \ : , the
other ratios ;
and let A
:B
—
a
:b,
C
:D
—
b
:c,
E
:F
—
■
:,!,
G
H
—
d
J
K
L
—
e
:/,
M
:N
.
r
:?•
BOOK V. PROP. K. THEOR. 209
Then, by the definition of a compound ratio, the ratio
of a : ~ is compounded of the ratios of a 'b> b 'c> c : d> d e*
g'.f, f'gt which are the fame as the ratio of A '• B> C : D»
E*: F, G : H, K : L, M : N, each to each.
:P
—
h
:k,
QJR
—
k
:/,
:
—
I:
m,
:W
—
m
: n,
:Y
—
n
•p.
Then will the ratio of h :p be the ratio compounded of
the ratios of // : k, k\l, l\m, m\n, n:p, which are the
fame as the ratios of : , : , } :T»V :W> X *Y »
each to each.
,*. by the hypothefis a : » = h :p.
Alfo, let the ratio which is compounded of the ratios of
A : B , C : D , two of the firft ratios (or the ratios of a : Ci
for A : B = a : j» and C :D = (, : : ), be the fame as the
ratio of a : d, which is compounded of the ratios of a : b,
b : c, c : d, which are the fame as the ratios of 1 : p ,
> • R » J! '• 1 » three of the other ratios.
And let the ratios of h : s, which is compounded of the
ratios of h : k, k : m, m : n, n : s, which are the fame as
the remaining firft ratios, namely, E :F» G :H> K =L»
M : N ; alfo, let the ratio of e : g, be that which is com
pounded of the ratios e : f, f : g, which are the fame, each
to each, to the remaining other ratios, namely, r :W,
[ : Y « Then the ratio of h : s fhall be the fame as the
ratio of e : g ; or h : s — e : g.
p A XC XE X»: XK XM „ X b X c X J X e X /
B X D X F X H XI, X N T b X c X J X •• X / X g '
E E
2io BOOK V. PROP. K. THEOR.
. X X XX h X k X I XmXn
X X X X JX'X«X»X?'
by the compofition of the ratios ;
aX CXcXdX e X f h X k X I X m X n ,, x
bXcXdXtX/Xg kX I XmX * Xp ^y p )>
or g Xi V c X c/ X e X / h X k X I w m X »
iXc ^ dX e X/Xg k X I Xm *> n Xp'
but ° X * r= A X " = X X __ a X b X c __ h Xk X I .
t X i BXD "X X bXcXti " * X / X «» '
. cXdXt X ) m X n
' ' dXt X fXs " Xp'
A „ j C X c X t X f li X k X m X n ,, ,
d Xt
x/xg
k
X m
X n X s
and
TO X »
" X />
e
f
X f
Xg
(hyp.),
•
h X k
X m X
n .
— e f
• I
k X m
X r
X
s
" <V
•
• •
h
s
—
e
.'. h
: s
—
e :
g
'. If there be any number, &c.
Algebraical and Arithmetical expositions of the Fifth Book of Euclid are given in
Byrne's Doctrine of Proportion ; published by Williams and Co. London. 1841.
BOOK VI.
DEFINITIONS.
I.
ECTILINEAR
figures are faid to
be fimilar, when
they have their fe
veral angles equal, each to each,
and the fides about the equal
angles proportional.
II.
Two fides of one figure are faid to be reciprocally propor
tional to two fides of another figure when one of the fides
of the firft is to the fecond, as the remaining fide of the
fecond is to the remaining fide of the firft.
III.
A straight line is faid to be cut in extreme and mean
ratio, when the whole is to the greater fegment, as the
greater fegment is to the lefs.
IV.
The altitude of any figure is the straight line drawn from
its vertex perpendicular to its bafe, or the bafe produced.
2 ;2
BOOK VI. PROP. I. THEOR.
PUR I ANGLES
and parallelo
grams having the
fame altitude are
to one another as their bafes.
I and A
Let the triangles
have a common vertex, and
their bafes — — and ■ »—»
in the fame ftraight line.
Produce — — — — — both ways, take fuccemvely on
1 ' produced lines equal to it ; and on — — • pro
duced lines succeffively equal to it ; and draw lines from
the common vertex to their extremities.
A
The triangles 4lJZ. wi thus formed are all equal
to one another, fince their bafes are equal. (B. i. pr. 38.)
A
and its bafe are refpectively equi
i
multiples of m and the bafe
BOOK VI. PROP. I. THEOR. 2< 3
Lk
In like manner m _ and its bale are refpec
{
tively equimultiples of ^ and the bafe ■ ■* .
.*. Ifwor6times jf ~ = or ^ n or 5 times ■>
then ;« or 6 times ■ CZ == or Z] 8 or 5 times wu« ,
w and « fland for every multiple taken as in the fifth
definition of the Fifth Book. Although we have only
mown that this property exifts when m equal 6, and n
equal 5, yet it is evident that the property holds good for
every multiple value that may be given to m, and to n.
a
(B. 5. def. 5.)
Parallelograms having the fame altitude are the doubles
of the triangles, on their bafes, and are proportional to
them (Part 1), and hence their doubles, the parallelograms,
are as their bafes. (B. 5. pr. 15.)
Q. E. D.
2I 4
BOOK VI PROP. II. THEOR.
*
*
A
F a Jlraight line
be drawn parallel to any
jide .—.—..— of a tri
angle, it Jliall cut the other
fides, or thofe Jides produced, into pro
portional fegments .
And if any Jlraight line
divide the fides of a triangle, or thofe
fides produced, into proportional feg
ments, it is parallel to the remaining
.fide— •
Let
PART I.
., then fhall
• ■••«■»•■»
D
raw
V
and
. (B. i.pr. 37);
\ (B.5.pr. 7 );but
*■*»»■■ ■
(B. 6. pr. i),
■**«•!•■
HiHtllllB' .
(B. 5 .pr. ii).
BOOK VI. PROP. II. THEOR.
2] 5
PART II.
Let
■■■■•■■■a
then
Let the fame conftrudtion remain,
becaufe
and  ?
«*«■■■!■■■ *
> (B. 6. pr. i);
but
!■*■»*■ t I I » I
(hyp),
■7
■ ♦. • •
\ (B. 5 P r  1 1
,7=
3 ( B 5P>'9);
but they are on the fame bafe ■■■■••■■• and at the
fame fide of it, and
 II (B ipr. 39).
Q. E. D.
2l6
BOOK VL PROP. III. THEOR.
RIGHT line ( )
bifecling the angle of a
triangle, divides the op
pojite Jide into fegments
—  ) proportional
to the conterminous Jides (.
)•
And if a Jlraight line (•
)
drawn from any angle of a triangle
divide the oppofte fde (—^— ■■■■■■)
into fegments ( 9 .........)
proportional to the conterminous fdes (—■■■», ___ ),
it bifecls the angle.
PART I.
Draw ■■••■•»  — — — , to meet „„ ;
then, := ^ (B. i. pr. 29),
(B. 1. pr. 6);
and becaufe
(B. 6. pr. 2);
(B. 5. pr. 7 ).
BOOK VI. PROP. III. THEOR. 217
PART II.
Let the fame conftrudlion remain,
and — — — : .— — — . :: 1 : .—
(B. 6. pr. 2);
but — — — : — — — :: 1 : ■ (hyp)
(B. 5. pr. n).
and .*. •■■■«■■■ — (B. 5. pr. 9),
and .*. ^f — ^ (B. 1. pr. 5); but fince
II ; m = t
and zr ^f (B. 1. pr. 29);
.". ^ = T, and = J^.
and .'. .ii ■■ biiedts J^ ,
Q.E. D.
F F
2l8
BOOK VI. PROP. IF. THEOR.
N equiangular tri
angles ( S \
and \ ) the fides
rt^o«/ /7j^ ^«^/ angles are pro
portional, and the Jides which are
^L oppojite to the equal angles are
 .1 ■ S ■   
homologous
Let the equiangular triangles be fo placed that two fides
oppofite to equal angles
and
^^^ may be conterminous, and in the fame ftraight line;
and that the triangles lying at the fame fide of that ftraight
line, may have the equal angles not conterminous,
i. e. jtKk oppofite to
, and
to
Draw and
. Then, becaufe
* *
^— ■—  , . «...— (B.i.pr.28);
and for a like reafon, •■■«■»■■■« 1 1 — — ,
is a parallelogram.
But
(B. 6. pr. 2);
BOOK VI. PROP. IV. THEOR. 219
and fince = ^—^— (B. 1. pr. 34),
— : :: : — ; and by
alternation,
(B. 5. pr. 16).
In like manner it may be mown, that
a> •• •«■■*••»
and by alternation, that
* * a • • • a • a 1 a • a>
but it has been already proved that
am lamiiiaii
and therefore, ex xquali,
(B. 5. pr. 22),
therefore the fides about the equal angles are proportional,
and thofe which are oppofite to the equal angles
are homologous.
Q. E. D.
220 BOOK VI. PROP. V. THEOR.
F two triangles have their Jides propor
tional (••■■«■■ : ........_
:: — m—m l — ) and
(
«■■!■■■■■■■■ •. » ■
: : — — : — — «— ) //z^ art equiangular,
and the equal angles are Jubt ended by the homolo
gous Jides.
From the extremities of 9
and , making
W= M (B. ,. pr. 23);
and confequently ^ = (B. I. pr. 32),
and fince the triangles are equiangular,
draw
(B. 6. pr. 4);
but
(hyp);
and confequently
(B. 5. pr. 9 ).
In the like manner it may be mown that
BOOK VI. PROP. V. THEOR. 221
Therefore, the two triangles having a common bafe
, and their fides equal, have alfo equal angles op
^ = , ¥a„d/l = W
pofite to equal fides, i. e.
s\ m
(B. 1. pr. 8).
But ^F = ^fc (conft.)
and .*. z= mtk ; for the lame
reafon m \ := m A ■ and
confequently £^ z= (B. 1. 32);
and therefore the triangles are equiangular, and it is evi
dent that the homologous fides fubtend the equal angles.
Q. E. D.
2 22
BOOK VI. PROP. VI. THEOR.
s\
F two triangles ( ^S
Z\ "
and
) have one
\
angle ( ^Kk ) of the one, equal to one
A
\ angle ( f \ ) of the other, and the fides
^L about the equal angles proportional, the
,,,,1^ triangles /hall be equiangular, and have
thofe angles equal which the homologous
fides fubt end.
From the extremities of
of
^^— • , one of the fides
S \ , about m \
9 draw
— — . and , making
▼ = A , and ^F — J^ ; then ^ =
(B. i. pr. 32), and two triangles being equiangular,
>•>■>■>■■■■> 
(B. 6. pr. 4);
but
(hyp.) ;
«**••••••»•<•
(B. 5. pr. 11),
and confequently = —
(B. 5. pr. 9);
BOOK VI. PROP. VI. THEOR. 223
z\ = \/
in every refpedt.
(B. 1. pr. 4).
But ^J == j^ (conft.),
and .*. / ■ ZZ J^ : and
fince alio ■ \ — mtk .
' \ ■=. (B. 1. pr. 32);
y\
and .*. jf„.„^k and ^ \ are equiangular, with
their equal angles oppoiite to homologous fides.
Q^E. D.
224
BOOK VI. PROP. VII. THEOR.
/
.♦
'V
F two triangles (
A
and
* ) have one angle in
each equal ( equal to ^ ), the
Jides about two other angles proportional
4
and each of the remaining angles (
and ^J ) either lefs or not lefs than a
right angle y the triangles are equiangular, and thofe angles
are equal about which the fides are proportional.
Firft let it be affumed that the angles ^^  and <^
are each lefs than a right angle : then if it be fuppofed
that itA and ^A contained by the proportional fides,
are not equal, let ^^,\ be the greater, and make
Becaufe 4 = * ( h yP)> and ^A = ^\ (conft.)
= ^^ (B. I. pr. 32);
BOOK VI. PROP. VII THEOR. 225
M ■ ■■«■■■«■■•
(B. 6. pr. 4),
but — ^— : :: : (hyp)
(B. 5. pr. 9),
and .*. '^^ = ^^ (B. 1. pr. 5).
But ^^ I is lefs than a right angle (hyp.)
.*. 4^ is lefs than a right angle ; and .*. mull
be greater than a right angle (B. 1 . pr. 13), but it has been
proved = ^^ and therefore lefs than a right angle,
which is abfurd. .*. ^^\ and ^>\ are not unequal ;
.*. they are equal, and fince "^B — \ (hyp.)
4 = 4
(B. 1. pr. 32), and therefore the tri
angles are equiangular.
^ and ^
But if ^^B and ^^ be affumed to be each not lefs
than a right angle, it may be proved as before, that the
triangles are equiangular, and have the fides about the
equal angles proportional. (B. 6. pr. 4).
Q. E. D.
G G
:26
BOOK VI. PROP. VIII. THEOR.
N a right angled
triangle
)>if
— )
(
a perpendicular (
be drawn from the right angle
to the oppojitejide, the triangles
f ^/j^  j ^ ) on each Jide of it are fimilar to the whole
triangle and to each other.
Becaufe
common to
(B. i. ax. 1 1), and
and
A = 4
(B. i. pr. 32);
and are equiangular ; and
coniequently have their fides about the equal angles pro
portional (B. 6. pr. 4), and are therefore fimilar (B. 6.
def. 1).
In like manner it may be proved that nk is fimilar to
L
; but
has been lliewn to be fimilar
to
k
and L are
9 • •
fimilar to the whole and to each other.
Q. E. D.
BOOK VI. PROP. IX. PROB.
22"
ROM a given jlraight line ( )
to cut off any required part .
From either extremity of the
given line draw ——»"••■... making any
angle with * and produce
■•■•■■I till the whole produced line
■••■••■■•■ contains " as often as
contains the required part.
Draw
>, and draw
II
is the required part of
For fince
(B. 6. pr. 2), and by compofition (B. 5. pr. 18) ;
but
■••"» contains
as often
as
contains the required part (conft.) ;
is the required part.
Q. E. D.
228
BOOK VI. PROP. X. PROB.
and
draw
('
O divide a Jlraight
line ( — )
fimilarly to a
given divided line
)•
From either extremity of
the given line —
draw ■—••..—■.«.
making any angle ; take
............ .......... an( j
■■•■a««a«4
equal to
refpedively (B. i. pr. 2) ;
, and draw —■■■■» and
— — II to it.
or
and
Since j —■■■«« \
are
(B. 6. pr. 2),
(B. 6. pr. 2),
and .*. the given line
fimilarly to
(conft.),
(conft.),
is divided
Q.E. D.
BOOK VI. PROP. XI. PROB.
229
O find a third proportional
to two given Jiraight lines
At either extremity of the given
line ^^— ^ draw ..—
making an angle ; take
......... r= , and
draw ■ ;
make ........ ~
and draw ........
(B. 1. pr. 31.)
.■ ujjj is the third proportional
to — — — and .
For fince
...■a n..i..«
(B. 6pr. 2);
but —
■(conft.) ;
(B. 5. pr. 7).
Q^E. D.
2 3
BOOK VI. PROP. XII. PROB.
O find a fourth pro
portional to three
given lines
and
take
and
alfo
draw
and
I •«««»«•* !«»»«•««««
Draw
making any angle ;
(B. i. pr. 31);
is the fourth proportional.
1
Y
mt J
On account of the parallels,
(B. 6. pr. 2);
■— } = {: Er}( conft ) ;
*. ■■■■■■■■■■■ * »■•■••»■«■ jj •»
(B. 5. pr. 7).
Q^E. D.
BOOK VI. PROP. XIII. PROB.
2 3*
O find a mean propor
tional between two given
Jlraight lines
}•
Draw any ftraight line
make — ■
and
; bifecl
and from the point of bifedtion as a centre, and half the
line as a radius, defcribe a femicircle
draw — ■— «— — — — .
r\
is the mean proportional required.
Draw
and
Since ^ p> is a right angle (B. 3. pr. 31),
ar >d — ^— is J_ from it upon the oppofite fide,
•*• ^~~^ is a mean proportional between
— ^— and '■ (B. 6. pr. 8),
and .*. between  and (conft.).
QE.D
232 BOOK VI. PROP. XIV. THEOR.
QJJ A L parallelograms
\
and
which have one angle in each equal,
have the Jides about the equal angles
reciprocally proportional
II.
And parallelograms which have one angle in each equal,
and the Jides about them reciprocally proportional, are equal.
Let
and
and
J and
and
— ■ — , be (o placed that .
— — may be continued right lines. It is evi
dent that they may affume this pofition. (B. i. prs. 13, 14,
'5)
Complete
%
Since
V
V:\:\
[B. 5. pr. 7.)
BOOK VI PROP. XIV. THEOR. 233
(B. 6. pr. 1.)
The fame conftrudtion remaining
(B. 6. pr. 1.)
— (hyp.)
(B. 6. pr. 1.)
(B. 5. pr. n.)
and .*. = ^^ (B. 5. pr. 9).
Q^E. D.
H H
234
BOOK VI. PROP. XV. THEOR.
I.
QUAL triangles, which have
one angle in each equal
1 ^^ = ^m ), have the
fides about the equal angles reciprocally
proportional
II.
And two triangles which have an angle of the one equal to
an angle of the other, and the Jides about the equal angles reci
procally proportional, are equal.
I.
Let the triangles be fo placed that the equal angles
^^ and ^B may be vertically oppofite, that is to fay,
fo that — — — and — — — may be in the lame
ftraight line. Whence alfo — — — — ■ and muft
be in the fame ftraight line. (B. i. pr. 14.)
Draw ■— — — . then
>
4
(B. 6. pr. 1.)
(B. 5. pr. 7.)
(B. 6. pr. 1.)
BOOK VI. PROP. XV. THEOR. 235
A
(B. 5. pr. 11.)
II.
Let the fame conftruction remain, and
^^^r * '
(B. 6. pr. 1.)
and
A
(B. 6. pr. 1.)
But : ;:  : , (hyp.)
(B.5 pr. 11);
(B. 5. pr. 9.)
Q. E. D.
• • •
A^1
A=A
3 6
BOOK VI. PROP. XVI. THEOR.
PART I.
F four Jh ■ aight lines be proportional
the reclangle (
■ ) contained
by the extremes, is equal to the rectangle
X  —  ) contained by the means.
PART II.
And if the re£l
angle contained by
the extremes be equal
to the reSlangle con
tained by the means,
the four Jlraight lines
are proportional.
PART I.
From the extremities of — and ■— ™
«—o^bb» and i _]_ to them and ~
draw
and
refpe<£tively : complete the parallelograms
and
I
And fince,
(hyp)
(conft.)
(B. 6. pr. 14),
BOOK VI. PROP. XVI. THEOR.
2 37
that is, the redtangle contained by the extremes, equal to
the redlangle contained by the means.
PART II.
Let the fame conltrudtion remain ; becaufe
■ ■a a ■»■***■ vmmwwww
and mmm—mm — ....a......
• •
■■BH
(B. 6. pr. 14).
But = .
and — = — — . (conft.)
■■■■■■■■»■■■■■■ **«■■■■■■•■■• ■■»■■■•■■■■
(B. 5. pr. 7).
Q. E. D.
»■
2 3 8
BOOK VI. PROP. XVII. THEOR.
PART I
jF three jlraight lines be pro
portional (■■■ : __
reSlangle under the extremes
is equal to the fquare of the mean.
PART II.
And if the rettangle under the ex
tremes be equal to the fquare of the mean,
the three Jlraight lines are proportional.
lince
then
PART I.
A flu me
X
., and
(B. 6. pr. 16).
or
But
X
'9
X
___ 2 ; therefore, if the three ftraight lines are
proportional, the redtangle contained by the extremes is
equal to the fquare of the mean.
PART II.
Aflume
 X ■
. , then
X —
(B. 6. pr. 16), and
Q. E. D.
BOOK VI. PROP. XVIII. THEOR. 239
N a given Jlraight line (■
to conjlruB a recJilinear figure
Jimilar to a given one
and Jimilarly placed.
)
Refolve the given figure into triangles by
drawing the lines . . and
At the extremities of
^ = f^V and
again at the extremities of
and ^k —
^ — ^\ T in like manner make
* = \/andV = V
Th
en
is fimilar to
It is evident from the construction and (B. 1. pr. 32) that
the figures are equiangular ; and fince the triangles
W and V
are
equiangular: then by (B. 6. pr.4),
:: —
and
240 BOOK VI. PROP. XVIII. THEOR.
Again, becaule ^ and are equiangular,
^^ md ^B
._ •• *
ex aequali,
(B. 6. pr. 22.)
In like manner it may be fhown that the remaining fides
of the two figures are proportional.
.. by (B. 6. def. i.)
is fimilar to
and fimilarly fituated ; and on the given line
Q^E. D.
BOOK VI. PROP. XIX. THEOR. 241
IMILAR trian
gles {
\
and ^fl B ) are to one
another in the duplicate ratio
of their homologous Jides.
Mk and m
Let 4Bt and ^^ be equal angles, and ....——
and ■ . homologous fides of the fimilar triangles
and ^j m
and , and on — ■■■■ the greater
of thefe lines take ....... a third proportional, fo that
draw
(B. 6. pr. 4) ;
but
(B. 5. pr. 16, alt.),
(conft.),
— confe
1 1
242 BOOK VI. PROP. XIX. THEOR.
A\
quently = ^* for they have the fides about
the equal angles ^^ and 4Bt reciprocally proportional
(B. 6. pr. 15);
■AAA\
(B. 5P r. 7);
but A \ : ^fc :: — — :
(B. 6. P r. 1),
Aa
• • ..>■>.
that is to fay, the triangles are to one another in the dupli
cate ratio of their homologous fides
and (B. 5. def. ii).
Q^ E. D.
BOOK VI. PROP. XX. THEOR.
243
[IMILAR poly
gons may be di
vided into the
fame number of
fimilar triangles, eachfimilar
pair of which are propor
tional to the polygons ; and
the polygons are to each other
in the duplicate ratio of their
homologous /ides .
Draw
and
and
and — — —  j refolving
the polygons into triangles.
Then becaufe the polygons
are fimilar,
and — —
«•■«••■«««•
are fimilar, and ^ — ^
(B. 6. pr. 6);
but
♦ ♦
= w becaufe they are angles of fimilar poly
gons ; therefore the remainders g/^ and ^k are equal ;
hence «■»«•■■■■■ ; ...«...** ;; ._..___._ ; ••■•■•*•••■
on account of the fimilar triangles,
244 BOOK VI. PROP. XX. THEOR.
and .... : _^__. •• • _ «.
on account of the limilar polygons,
■■■■■■a ■■■■
ex sequali (B. 5. pr. 22), and as thefe proportional fides
contain equal angles, the triangles
s M ^ and ^^.
are limilar (B. 6. pr. 6).
In like manner it may be fhown that the
triangles ^^ and ^ W are limilar.
But is to in the duplicate ratio of
•■■■■■■■■. to — — — — (B. 6. pr. 19), and
M^ is to ^^
in like manner, in the duplicate
ratio of ««■■■■■■«■ to «— — — ;
>>
(B. 5. pr. 11);
Again M ^^ is to ^^r in the duplicate ratio of
M^r to ^^~
^r ^W
to — — — , and is to in
BOOK VI. PROP. XX. THEOR. 245
the duplicate ratio of — —  to ■
▼ ▼
and as one of the antecedents is to one of the confequents,
fo is the fum of all the antecedents to the fum of all the
confequents ; that is to fay, the fimilar triangles have to one
another the fame ratio as the polygons (B. 5. pr. 12).
But is to in the duplicate ratio of
to
^
is to ™ WL. in the duplicate
ratio of _________ to _____ .
Q E. D
246
HOOK VI. PROP. XXI. THEOR.
ECTILINEAR figures
(
and
which are fimilar to the fame figure (
are fimilar alfo to each other.
Since Hi B^. and arc fimi
lar, they are equiangular, and have the
fides about the equal angles proportional
(B. 6. def. 1); and fince the figures
and ^^ are alfo fimilar, thev
are equiangular, and have the fides about the equal angles
proportional ; therefore ■■■^ and ■■fet. are alio
equiangular, and have the fides about the equal angles pro
portional (B. 5. pr. 1 1), and are therefore fimilar.
Q.E. D.
BOOK VI. PROP. XXII. THEOR.
247
PART I.
Y four fir aight lines be pro
portional (■■■■■■■ : "™""
:: — : ), the
fimilar rectilinear figures
Jimilarly described on them are alfo pro
portional.
PART 11.
And if four Jimilar rectilinear
figures, Jimilarly defcribed on four
ftraight lines, be proportional, the
firaight lines are alfo proportional.
Take
and —
to
fince
but
and
part 1.
a third proportional to
. and •••■•■■■•« a third proportional
— and (B. 6. pr. 11);
:: : (hyp.),
... ...... ;;  ; >■■■■■■■■■• (conft.)
.*. ex asquali,
(B. 6. pr. 20),
, ••••••••••• *
248 BOOK VI. PROP. XXII. THEOR.
and .*.
(B. 5. pr. 11).
PART II.
Let the fame conftrudlion remain
(hyp).
(conft.)
(B. 5. pr. 11).
Q.E. D.
BOOK VI PROP. XXIII. THEOR. 249
QUIANGULAR parallel
ograms ( and
) are to one another
in a ratio compounded of the ratios of
their jides.
Let two of the fides — — _— m and
about the equal angles be placed
fo that they may form one ftraight
line.
Since ^ + J = f\\
and J^ = (hyp.),
and /.
4 +
■ 11 and <■
— form one ftraight line
(B. 1. P r. 14);
complete £' .
Since
#
and
#
(B. 6. pr. 1),
;B.6. pr. 1),
has to
_. to
a ratio compounded of the ratios of
, and of «^— ■— a to 1 .
K K.
Q^E. D.
2 5 o BOOK VI. PROP. XXIV. THEOR.
{ELJ)
N any parallelogram (^
the parallelograms ( i ^j
B:
and f I ) which are about
the diagonal are Jtmilar to the whole, and
to each other.
As
and
B
have
common angle they are equiangular;
but becaufe ■ I
and
are fimilar (B. 6. pr. 4),
■»•«»■■■
and the remaining oppofite fides are equal to thofe,
.*. B—J and BL—J have the fides about the equal
angles proportional, and are therefore fimilar.
In the fame manner it can be demonftrated that the
rH and B
parallelograms £]_J and f / are fimilar.
Since, therefore, each of the parallelograms
and
B
is fimilar to
to each
BOOK VI. PROP. XXV. PROB.
25 1
O defcribe a rectilinear figure,
which /hall be fimilar to a given
 rectilinear figure (
equal to another \wb )•
),and
Upon defcribe
and upon . defcribe   = tB^ ■
and having zz
(B. i. pr. 45), and then
and
....... will lie in the fame ftraight line
(B. 1. prs. 29, 14),
Between
and
■• find a mean proportional
(B. 6. pr. 13), and upon
defcribe , fimilar to
and fimilarly fituated.
Then
For fince
and
are fimilar, and
j .......... (conft.),
(B. 6. pr. 20) ;
252 BOOK VI. PROP. XXV. PROB.
but
: _ M (B.6.pr. i);
;B. 5 .pr.n);
but
and .\
and
(conft.),
(B. 5. pr. 14);
= ■} (conft.) ; consequently,
which is fimilar to ^^fl Bk is alio = w .
Q. E. D.
BOOK VI. PROP. XXVI. THEOR. 253
F fimilar and Jimilarly
pofited parallelograms
U and (/J)
have a common angle, they are about
the fame diagonal.
(
For, if poffible, let
be the diagonal of
draw
(B. 1. pr. 31
Since
diagonal
L.*13
are about the fame
A
and have common,
they are fimilar (B. 6. pr. 24) ;
but
(hyp)»
and .*.
(B. 5. pr. 9.),
which is abfurd.
is not the diagonal of
a
in the fame manner it can be demonftrated that no other
line is except ===== .
Q. E. D.
2 54
BOOK VI. PROP. XXVII. THEOR.
F all the rectangles
contained by the
fegments of a given
Jlraight line, the
greatejl is the fquare which is
defcribed on half the line.
be the
unequal fegments,
equal fegments ;
For it has been demonftrated already (B. 2. pr. 5), that
the fquare of half the line is equal to the rectangle con
tained by any unequal fegments together with the fquare
of the part intermediate between the middle point and the
point of unequal fection. The fquare defcribed on half the
line exceeds therefore the rectangle contained by any un
equal fegments of the line.
Q.E. D.
BOOK VI. PROP. XXVIII. PROB. 255
O divide a given
Jlraight line
( )
J fo that the rec
tangle contained by its segments
may be equal to a given area, m
not exceeding the fauare of
half the line.
Let the given area be =
Bifedt «.»—
make ••"*
and if — »»
or
the problem is iblved.
But if
muft
% then
(hyp.)
Draw
make 
with ^"
•«■»•■»■»
or
as radius defcribe a circle cutting the
given line ; draw
Then
X
(B. 2. pr. 5.;

But
+
(B. 1. pr. 47);
256 BOOK VI. PROP. XXVIII. PROB.
.. X + '
« I *
j
from both, take — — —
and — X
that X
__ 4
But i— — mm» — (conrt.),
and .*. «■■ i —■«■»» is fo divided
— «
Q^E. D.
BOOK VI. PROP. XXIX. PROB.
257
O produce a givenjlraight
line (— ),fo
that the recJangle con
tained by the fegments
between the extremities of the given
line and the point to which it is pro
duced, may be equal to a given area,
i. e. equal to the fquare on
Make
— , and
draw — ■«—
draw
with the radius
meeting
Then — —
* and
■, defcribe a circle
■ produced.
X
But
1 (B. 2. pr. 6.) = \
......... 9 __ _ 2 (B. 1. pr.47.)
and
* +
from both take
X
but =
+
2 «
= the given area.
C^E. D.
L L
258 BOOK VI. PROP. XXX. PROB.
O cut a given finite jiraight line (— — )
in extreme and mean ratio.
On
(B. i. pr. 46) ; and produce
— x
defcribe the fquare
— — — , (o that
(B. 6. pr. 29);
take
and draw «■
meeting
II
Then
u
X
(B. 1. pr. 31).
»■■■■■■»
and is /. —
1
• and if from both thefe equals
be taken the common part
J , which is the fquare of "
will be zr II, which is ^ ■■■ X
that is
X
and
■•■■••■ is divided in extreme and mean ratio.
(B. 6. def. 3).
Q. E. D.
BOOK VI. PROP. XXXI. THEOR. 259
F any fimilar rectilinear
figures be fimilar ly defer ibed
on the fides of a right an
gled triangle (
/\
), the figure
defer ibed on the fide ( ) fub
tending the right angle is equal to the
fum of the figures on the other fides.
From the right angle draw
to —
then ••■»■ : .
(B. 6. pr. 8).
(B. 6. pr. 20).
perpendicular
but
Hence
but
(B. 6. pr. 20).
»+
+
+
■ as»Mt»»m
ifiiiuai
and ,\
Q. E. D.
260 BOOK VI. PROP. XXXII. THEOR
F two triangles ( A ^ tf «^
/%\ ), have two Jides pro
^k W^mtSn portionai ( ; — — ~
\\ :: .......... ; «.... ), and be Jo placed
\ at an angle that the homologous Jides are pa
rallel, the remaining Jides (
one right line.
and ) Jl
orm
Since
= (B. i. pr. 29) ;
and alfo fince — — —  •• ••
= A (B. 1. pr. 29);
= ^^ ; and fince
(hyp).
the triangles are equiangular (B. 6. pr. 6) ;
A = A
but
—
▲+ +A=±+4+A
m
(B. 1. pr. 32), and .*.
and
lie in the fame flraight line (B. 1. pr. 14).
Q.E. D.
BOOK VI. PROP. XXXIII. THEOR. 261
N equal circles (
00
), angles,
whether at the centre or circumference, are
in the fame ratio to one another as the arcs
on which they Jland (
fo alfo are feci or s.
L:4.
) i
o
Take in the circumference of  I any number
of arcs «■—» , — , &c. each 35 ■— ? and alfo in
the circumference of I take any number of
O
arcs
, &c. each rr
, draw the
radii to the extremities of the equal arcs.
Then fince the arcs
" 9 "■"• . ■—■, &c are all equal,
the angles # , W , \, &c. are alfo equal (B. 3. pr. 27);
.*. mwk is the fame multiple of which the arc
is of «« ; and in the fame manner 4B^
4.
is the fame multiple of ml , which the arc ,»„„„„«»•***
is of the arc
262 BOOK VI. PROP. XXXIII. THEOR.
Then it is evident (B. 3. pr. 27),
if 4V ( or if w times ^ ) EZ> =, ^ 4W
(or « times ^ )
then ^^ ^+ (or m times «*— ) C> =^> ~
*•••• (o r « times ).;
• • ^» ^^ • • ^^^ • 1...  ^
(B. 5. def. 5), or the
angles at the centre are as the arcs on which they ftand ;
but the angles at the circumference being halves of the
angles at the centre (B. 3. pr. 20) are in the fame ratio
(B. 5. pr. 15), and therefore are as the arcs on which they
ftand.
It is evident, that fedtors in equal circles, and on equal
arcs are equal (B. 1. pr. 4; B. 3. prs. 24, 27, and def. 9).
Hence, if the fectors be fubftituted for the angles in the
above demon ftration, the fecond part of the propofition will
be eftablifhed, that is, in equal circles the fedlors have the
fame ratio to one another as the arcs on which they fland.
Q.E. D.
angle
Z
BOOK VI. PROP. A. THEOR.
F the right line (........),
bifeSling an external
angle ll ij/' the trz
meet the oppojite
263
fide ( ) produced, that whole produced fide ( ),
and its external fegment (—«•) will be proportional to the
fides ( —... #»</ ), w^/'c/i contain the angle
adjacent to the external bifecJed angle.
For if ■■ be drawn  >*— •>«« ,
\ /, (B. 1. pr. 29) ;
= ^,(hy P .).
= , (B. 1. pr. 29);
and
, (B. i.pr. 6),
and
(B. 5. pr. 7);
But alfo,
IBIIIIEIIII
*iaani>f • ■ ■■
(B. 6. pr. 2);
and therefore
(B. 5 .pr. 11).
Q. E. D.
264
BOOK VI. PROP. B. THEOR.
F an angle of a triangle be bi
fecJed by a Jlraight line, which
likewife cuts the bafe ; the rec
tangle contained by the fides of
the triangle is equal to the rectangle con
tained by the fegments of the bafe, together
with the fquare of the Jlraight line which
b/fecJs the angle.
Let be drawn, making
^ = £t ; then fhall
... x +
o
(B. 4. pr. 5),
produce to meet the circle, and draw —
Since 4fl = 4Hk (hyp)>
and = ^T (B. 3. pr. 21),
,ZL \
are equiangular (B. 1. pr. 32) ;
y
(B. 6. pr. 4);
ROOK VI. PROP. R. THEOR. 265
• • — — — X ~~^^~ — x
(B.6. pr. 16.)
= X + f
(B. 2. pr. 3);
but X = . w
(B. 3. pr. 35 );
X = X h
Q.E. D.
M M
266
BOOK VI. PROP. C. THEOR.
mall
F from any angle of a triangle a
Jlraight line be drawn perpendi
cular to the bafe ; the reft angle
contained by the fides of the tri
angle is equal to the rectangle contained by
the perpendicular and the diameter of the
circle defcribed about the triangle.
From
draw _L
■ X =
of >**
„*7> _
• s •«•■««*
— ; then
Xthe
diameter of the defcribed circle.
Defcribe
O
(B. 4. pr. 5), draw its diameter
and
and draw — — ; then becaufe
— ■ >> (confl. and B. 3. pr. 31);
= /^ (B. 3. pr. 21);
• j*.
is equiangular to
A
(B. 6. pr. 4);
and .*. • X == X
(B. 6. pr. 16).
Q.E. D.
BOOK VI. PROP. D. THEOR.
267
HE rectangle contained by the
diagonals of a quadrilateral figure
infcribed in a circle, is equal to
both the rectangles contained by
its oppojite Jides.
Let /
figure infcribed in
be any quadrilateral
o
and draw
and
then
X
Make
4^ = W (B.i.pr. 23),
^ = ^ ; and
=
(B. 3. pr. 21);
and .*.
(B. 6. pr. 4);
X
amiiiiB
(B. 6. pr. 16) ; again,
becaufe 4» = W (conft.),
X
268 BOOK VI. PROP. D. THEOR.
and \/ = \y (B. 3. pr. 21)
(B. 6. pr. 4);
and /. <•...•••* ^ «.»— — ;
mi ■■■•■!■■■»
(B. 6. pr. 16);
but, from above,
X = X + X
(B. 2. pr. 1 .
Q. E. D.
THE END.
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