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Full text of "The first six books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters .."



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Library 

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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, land-furveying, 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 pre-eminence 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 four-fided 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 Jlraight-edge 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 felf-evident 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 bet-ween 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- 43-J 



(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, 



l-l 



>■••» • •■ 



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. 3-pr. 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 



o-o 



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-) 
(B-3.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.3-pr.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. 3-pr. 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 one-third of two right 

angles; (B. i. pr. 32) but ^L m = f I 1 

(B. 1. pr. 13); 

/. ^ = W = ^W = one-third 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 M|Uw 
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 '5F-A-)^ 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 i-v^ 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 ifM|C 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. s-pr. 13); 



1 92 BOOK V. PROP. XXI. THEOR. 

•• • =] ♦, 

that is ^ CI | . 

Secondly, let = | | ; then {hall ^ = ) . 

For becaufe — |, 

* : * = ■ : A (B. 5-F- 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 ^ ; 

for|C ', 
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 Jp-f-Q-l- beCB+ + U J 
becaufe ^ -f- Q : £ -j- : : Q : 

•*• V ' M-- W+O- ■+ (B. 5 .pr. 19), 
bu < * + D C ■ + (hyp.), 

.'• * 1= B(B- 5-pr-A); 
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 — - s-jZ= 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. 



For-i 
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 XDXEXi-XG 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 -5-P>'-9); 



but they are on the fame bafe -■■■■••■■• and at the 
fame fide of it, and 
- II (B- i-pr. 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.) 

.■ uj-jj 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 B-L—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. 



cmiswick: PRiNTrn by c. whitti.noiiam. 



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j^zser. 



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^^ 



- 



&&^> 



4>Ts jjpF' ^gS 



^5g . 









^ ■ j^^ 



J3H 









^ 



■ y'yj t 

if mjfA 



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