Application to the Menſuratzon

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'A

ELEMEN ^{T i .}

OF

'GEOMETRYz

WITH T'HEIR

Application to the Menſuratzon

of SUPERFICIES and SOLIDS,

, 1- T O T H E

'Determination of the MAX'IMA and MINI'MA of Geomctrical Wantitie's; ' \ *

AN'D 'r_0 THE

Cainſtruction of a great Variety 'of GEOME TRICAL PROBLEMS.

. x/Q -

By THO'MA'S SIMPSON, F.R.s£'

r * v V

And Number oſ the Royz1l,Academy of' Sciences at

STOCKHOLM. >

A

. The FIFTH EDITION,'

Carefully Reviſed.- '

L 0 N 1) 0 N: ,

Printed by Luke Hanflxrd, Great TuHſti/e, Lincoln's-ImzFields,

For F.. YV IN G R'A VſſE, Succcſſor to Mr. NOURSE,

x in the drrand. -

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ſ w. de THE HONOURABLE _ "

Charles Frederz'ck, Eſq;

Surveyor-Ge'neral oſ His M'AJESTY'S' ORDNANCE,&©*c. &Ye. &Ft. 'A

\

HONOURABLE SIR,

H E ſubject of the ſheets which I' here beg leave to layi befoi'e you, is of ſo much conſequence to mankind, as juſtly to

claim the regard and ſanction of the Great.

Geometry is, not only a moſt accurate, but a very extenſive ſcience, Whoſe application and ' great utility, as well in the artsof peace as of

war, are well known to You:

\ But though this work, iſ the manner in which it is executed be correſpondent to the importance of the ſubject, may not want ſufficient' merit to render it worthy of' the

approbation of a Gentleman, who, amidſt a

multiplicity of public employments, pre

ſſſerves an undiminiſh'd ardor for the ſciences,

A 2 - and

ivl DEDlCATION.

'and a knowkdge of the works of art and nature; yet I have; Sir, ſtill farther motives for this addreſs: Yourſigreat influence and ' zeal to promote the good of an inſtitution under Which I am Placed; and the ſavours that I have received at your hands, make me - earneſt to, embrnce this opportunity of teſti

fying publickly, that I am,

HONOURABLE SIR,1 i WVith great Reſpect,

Royal Academy, Hard) 3, 1760

Ydur much obliged, _ and mofl dbedient

humble Servant,

x Thomas Simpſon,

r

A; Mr, , r

x ._.. C

4.. a. H

'*\

YP RE F A CȜUE.

I' defign in writing upon the ſuly'ect of Geo metry, was to open an eeffy way for young heg'inners to arrive at a proficiency in that aſſ'fulſtience; without either heing ohliged to go thro' a-numher of unneceſſary propoſitions, or having re eoutſt to the ungeometrical methods of demonſtra tion, that ahound in moſt modern compoſitions o'f this ſſ

nature. ' -

. The difficulty of the underta/cing, I was not unap- ſ pri/'ed of; and ohjections eeeurred that were not etſſy to he removed ; Mvertheleſt, I have grounds to hope, from the reception my ſiiſt attempt has met with, that my endeavours have not heen entirely unſucceſſful. No Pains have,indeed, heenſpared to render the work uſeful : And Iflatter myſel , thatthejþirit and rigour of de monſtration, ſo eſſential to the ſuhject, are alſo tolera hly well preſerved z though I have not heen ſo intent to guard againſt the attach of Criticks, as to loſe ſight of my main deſign of furniſhing a plain, effl inſti tution for learners : Tet I have ſtrong hopes, that there will not he found in theſe ſheets, any inaeeu racies, or overſights, that are ahſolutely unpardonahle.

To expect a faultliffs piece is impoffihle : And I well [enow that the moſt elahorate and heſt-approved ſyſ-' tems of Geometryextant, are not without nzanyimpeifee tions. But, were theſmalle/t impeiflction to he a real fault, my amhition would rather he, to ſhew ſome de; ' gree of judgment, hy avoiding a multitude' ofſhch faults, than hy expoſing and magnifying the flaws of

other writers. ]t is more eeſſy to ſte a fault, than to avoid one: And thoſemen who are the maſt/anguine to di/linguiſh therto/elves at the expence iſ others, are v- A_ 3 , e genera/iy

vi

PREFACE. ſi

v theſuhject.

generally obſerved to ſtand in need iſ greater indulgen cies, than even the' per/ons whom they unmerciſulzy attach. But Iſhall put an end to this digreffon hy painting out one ohjection, that may he hronght again/t1 this war/e; which is, that in demonſtrations admitting ofſe<veral caſes, the moſt eaſy ones areſometimes omit

ted ; and that the converſe of ſome propoſitions is

not at all demonſtrated. But this, 1 conceive, will _ he found a real advantage to the learner; wiihout xwhich, it would have heen impqfflhle to have compriſed the Elements in the compaſs they now take up. Be fides, the greateſt part of the demonſtrations omitted

heingſnch as may he inferredfrom thoſe given, hy means _cffi_dxioms only; they may, therefore, he eaſily ſap

plied hy any reader, ſhould they happen to heceme ne ceſſary, which I haveſcarce ever found to he the ceffe.

But, even allowing this to he a defect, it is ahun-' dantly compenſated hy the extenſi-ve application given

> in the three laſt jections; which'is infinitely more uſeſnl, in itſef, and more neceſſary to the forming an ahle Geometrician, than any thing of the kind we have heenſheahing iſ. *

In this, ſecond, edition (which is, in a manner, 'a t new worſe) many conſiderahle alterations and additions have heen made. ſi The order of ſome of the ſir/i pro poſitions is changed : Andſome difficult projxffitions in

the ſecond hook are rendered more plain. ]n the fourth hooleſe-veral new Theorems on proper-'tions are ſ added. The ſolid Geometry is now connected with the plane, and is demonſtrated with the ſame accuracy.

The meiffnration zff Superfieies and Solids -is alſo,inore explicitly handled 3 and the demonſtration of theſe veral rules is here eſſahliſhedr on a hetterfonndation, than e'ven in authors who have wrote- proſeſſedly on The Maxima and Minima, and the con ſtruction of Geometrical Prohlems, are likewiſe conſi derahly extended and improved. And, at the end,

* 4 Mtes

_._..___. A.

"P R\ E*F A*C E. vii Notes geometrical and critical, 'very zyZ'ful to 'improve ' the judgment ofyoung ſtudents, are now added.

But, 'whiiſt I am talking of improve-ments and inatters of eritieiſm, I am called upon to anſwer to a Jharge, which, ſhould it appear to dfferve credit, -

,wonld indeed leave me hut 'little room to paſs myſelf 'upon the world'for a judge in theſe matters. As the ' gentleman hy whom I ſtand acciy'ed, is known to the i

world hy his holding one of the moſt conſiderahle ma thematical poſts' in the kingdom 3 Iſhall, in order to - do all, due honour to the manner and importance of his ſ

writing, gie-eyou his own words.

U - There hat lately heen puhliſhed a hook under the

" title of Elements of Plane Geometry, deſigned for

"- the uſe ofſehools, which is an incorrect copy of the

" firſt' eight-ſections of this work, lent the pretended

"- author on a particular occaſion, and printed in a

" ſpurious manner, without my knowledge or'conſtnt;

_ " an action too ſcandalous for any man oſ-honour to

" he uilty-ef. The Editor imagined, Iſuppoſe, that

" the changing ſome propoſitions, and mangling the -" deinonſtraticns of others, was aſufficient dfflguſſe

'* to make it paſs for his own performance; hat how

" far this willj'nſtify ſuch a piece ofpiracy, muſt he

" leſt to thejudgment ofthepuhlitk."

- Id/ere I to attempt to deſerihe the ideas excited in my mind hy the ſingular modeſty of this important and ſolemn appeal to the puhliek, Iſhould he at a loſs for ſit words to expreſſ; them, without tran/grcſſng the hounds of deceney. But I hope that I have not, de ſervedſo ill of the' puhliek, to he thought'capahle of

acting ſo 'very humhle a part, as that of copying from this author, and of mangling his demonſtrations, in order to make them paſs-for my own-That amanu ſcript of his (containing hetween 20 and 30 of the

_ principal

t ſirſt maſter.

' \

'P R E F A ClctEn

principal Theorems in Geonzetry, extfemely ill digeſted)

came intomy hande, is indeed true; hat it was,not lent me, hnt ſorted upon me, hy hinffle/f (the fvery firſt night after my removal to Woolwich) in virtue of an article in the original rules and inſtructions for the Academy ; wherehy it is ordered, that the ſecond - maſter ſhall teach Geometry under the direction of the

But this Well intended article, which has heen madeſnlſernient to the pnrpoſes of ignorant tyranny, and daring eelnnzſiny, has ſince, in conſequence of a pahlic/z examination, heen annulled hy an expreſſ ſiorder of theft/laſten General ofthe Or'dnance.---I could mention ſome particulars, ſupported hy good authority, that oeenr'nezi in the eoznſſz of that examination', which would hnt ill agree with the importance he aſſume;

in hi: rely/"dent accuſation ; hat I do not thin/e it worth while: This Gentleman haj, him elſ, hy his different puhlieations, ſh well convinced the world of his ahili

tie-s, as to 'render any farther comment on that head intirely nnneeeffizry and ineffi'ſihctual.

an m

I? (5

A'DVERTISEMEN-Tſſ.

a

As in every work- of_this nature,\deſigned to contain whatever mity be moſt requifite 'to the forming of *a regular and complete ſyſtem of Geometry, a number oſ propoſitions muſt neceſ ſarily havc a place, vyhoſe chief uſe and application lie in the higher. branches of the Mathematics;

and there being many perſons, particularly young gentlemen in publick ſchool-s, who want to 'learn * ſo much Geometry only, aſis is neceſſaryſſ to give them a proper introduction into the practical and moſt common 'applicatione thereof ;'- ſuſſch as Men ſuration, Trigonometry, Navigation, Fortificaltion, Perſpectilve, &do. ſſ For theſe reaſons, I thought

that it might be of ſervice, to point out to ſuch Readers, what propoſitions in theſe elements may be omitted, als-leaſt uſeſul to them; without either hurting the connection, or taking away from the >

evidence of the other demonſtrations; The num tbers of theſe propofitions, in the ſeveral books,

are. as follow. _ \

> . " -

. ' . In Book I. the 6, 17, 19, 21', 22, 23, and

29th. i '

In Book Il. the a, 5; Io, I I, 12, 13th, and the

ed Corol. to the 9tli. 7 ,

In Book III. the 4, 5, 6," 7, 8, 9, 15, 18, 19, i 20, 25, 26, 27, and 28th.

In Book IV._the 4, 5, 6, 9, 11, 13, 16, 17, 20, 21, 22, 23, 25, 26, 27, 28, and 29th.

*7 'In

_ * -(.\x0- )

In Book V. the I, 2, 16, 17, 18, 19, 20, 25, 26, 28, and 3v1ſt. '

ln Book VI. the two or three firſt propoſitions only, need be read; except by thoſe. who are concerned in ſurveying and dividing oſlands; to ' whom-the whole Book will be highly uſeſul,

Alſo, with regard to the ſeventh book, iſ Per;

ſhettive be \the only application in view (which I have known frequently to be the-etaſe) the Iſt, ed, 4th_, and rath propoſitions may ſuffice. But , if a more general idea - oſ the properties oſ in terſecting planes ſhould-be required, ſuch as is ne ceſſary in the doctrine of' ſolids and ſpheric geo metry; then all the propoſitions, tothe lflth, ought to be taken, ,

The 17th, 19th, Qoth, ziſt, zed, and 23d pro ' poſitions of this ſeventh book ſhould alſo be read

by thoſe who would be able to find the content and

proportion oſ ſolid bodies 5 as ſhould, likewiſe, th'e

whole eighth book; except, perhaps, the firſt' and ninth propoſitions; together with-the three firſt lemmasſſ; Which may be thought too plain', by , thoſe who are not very. ſolicitous about geometri;

cal rigour, to need a demonſtration. ſſ

\ (xi)

An iN D E X or T A B L E referring lfl the places in the,"

Blements, rwhere all the my? material propoſitions in the ſi'ſt ſix, and' in the cle-venth and travel/'th hookt of Euclid, are demwſtraled.

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bPr.l.4 . 22) 124,ſi waſ, p.B.Xl. 1' B, 'xgt\ A

5 12' 15 v I.4l r i 2 5 38

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\_,E:_L*E MxE zNſſ T S

Lu-L-_.__ ,

T /_

OF

GEOMETRY.

.BO oſſK I;

*

DEFINITIONS.

I. GEOMETRY is that ſcience, by which we compare ſuch quantities together as have extenſion. '

Extenſion diſtinguiſhed into length, hreadthz and i

ſi thickneſs.

2. A Line is that, which has, length Without

breadth. * \

The terms, hounds, or extremes of a Line are

points. - , ( '

3. A Surface is that, which \

has length and breadth, only, as

C. ſi

\

C

The hounds of a Surface are lines.i B _ ,

"4..A if'

breadth, and thickneſs, as D.

1

\

Elements of Gſeoenetry.

4.; A Solid is that, which has length, The hounds of a Solid are ſue-faces.

5. AlR'rght (or ſtrait)_lineis that, which'lies evenly between its extremes, or which every-Where tends i the ſameſi way, aZ AB. A._-_______Bl

6. A Plane-ſurſace 'is that,which is evhry-whtZre

perfectly flat and even, or which touches, in every

part, any right-line extended betWeen points any where taken in that ſurſace. ' '

7. An Angle is the inclination, or opening oſ two right-lines meet ing in apoint, as D.

D i8. When one 'right-line - C

DC, ſtanding upon another ſ >

AB, makeh the angles on both fides equal, thoſe an

gles me called 'right-an- _ _

gl'es; and that Line >CD is

ſaid to be perpendicular to A I) . . b' 3 theſiother A B on which it

inſiſts. ' -

. 'I A \

9. An Acute-angle is that, which is leſs than a right-angle, as Ei

IO. An Obſſtuſe-angle is that, 1

, which 15 greater than aright-* ' . _ p

angle, as F , ' - \ I F

r I'I. The

Book r the ſi

11.' The diſtance of two points, is the Right- -

line reaching from the one to the other. .

l

12. The diſtance of a point fromv a line, is a Right-line drawn from that point, perpendicular 10, and terminating in, the line given. ' 13.- Parallel (or equi- . C D ' diſtant) right-lineslAB, , -

CD are ſuch, which be-k

ing in the ſame plane ſurſace, if infinitely pro

duced,wouldnevermeet.r A_ ' v B

P

14. A Figure is a bounded ſpace, and is either

a ſurſace, or a ſolid.

I 5. Aright-lined 'plane Figure is that, formed in a plane-ſurſace, whoſe terms, or bounds, are

right-lines. ct

16. All plane Figures bounded by three right -lines, are called Tctriangles. _

17. An equilateral Triangle

is that, whoſe bounds'or fides are all equal, as A.

f

_ liZ. An iſoſceles Triangle, is, (when two ſides are equal, as B. v ' ,

'called a Parallelogram, as D.

\ A

E/e'zzems of' Geometrjy:

\

\

. * 19. 'A ſcalene Triangle is:when A. a all the three ſides are unequal, as A -

C. .

-_ 20. A right angled Triangle is ct B- - that, which has one, right-angle, ' as ACB; whereof the ſide AB

oppoſite to the right-angle, is call

dh- h. ſ. _

t teHypocwue A' XO

21'. An obtuſe-angled Triangle is than-which',

has one,o_btu_ſe_ angle. - -' -

22. An acute-angled Triangle is that, which 'has all its angles acute. ' _ ' ;

N

23, Every lane Figure bounded by four' right: \,

i-lines, is calle a Wadrangle, or Wadrilateral:

24. Any Wadrangle, whoſe oppoſite ſides are parallel, is

25. A Parallelogram,whoſe an- r*fi'""'***'i' 1.

gles are all right-ones, is called a Rectangle, as E.

.-' 26. A Square is a parallelogram

*wh0ſe ſides are all equal, and its an- ſſ 7:

gles all'right-ones, as F. 'ſſ ' 27. A Rhombus is aſſparallelo

gram whoſe ſides are all equal, but its angles not right, as G. *

-,"

Booh the Firſt. ſi

' Or triangle.

29. All other four-ſided figures, beſides' theſe, i

are called Trapeziums. .

2 9. A right-linejoining any two oppoſite angles;

of a fourſiſided figure, is called a Diagonal.

30, That ſideAB upon ſſ C- i 1' i

which any parallelogram . _

AC'EB, or triangle ACB i 4 T

is ſuppoſed to ſtand, is . '

called 'the baſe ;.' and the' A I) , - B perpendicular CD falling thereon from the oppoſite;

angle C, is called the altitude'of the. parallelogram,

r

3 r. All plane figures contained under mOre than four ſides, are called Polygons; whereof thoſe hav ing 'five ſides, are called Pentagons 3 thoſe havmg ſix-ſides, Hexagons 3 and ſo On. "

1 \ i ſſ '

32. A Regular Polygon is one whoſe angles, as well as ſides, are all equal. ' . -

33. A Circle is a plant:

figure, btmnded by one

curve-line APCD, called its Circumference, every-i' where equally diſtantfrom a point E within the circle, 'called the center thereof.

3'4; The Radiusiofa'tir'clc, is the diſtance of the centeſitfrom the circctumference, or' a right-line EA drawn from the center to the' Circumference. '

- v ſ

' .B 3

\

A

AXIſi

6 Elements of Geoenetiy. _

A X I OJM S, or Selſ-evident Truths.

1. Things, equal to one and thevſame thing, are alſo equal to each other.

2. Every whole is greater than its part.

3. Every whole is equal to all its parts taken

together. \

4. Iſ to equal things, equal things be'added,

the whol'es will be equal. . i

5. Iſ from equal things, equal things be taken' away, the remainders will be equal.

\

6. If to, 'or'from unequal things, equal things v be added, or taken away, the ſums, 'or remainders, will have the ſame difference, as the unequal things

firſt propoſed. ' * * >

7. All right-angles are equal to one another. 7 -8. More than one right-line cannot be drawn from one given pOlnt A to an- A B other given pomt B.

. If two oints M 1) "

D,9.F, in a Eight- F -N

lineMN,arepoſited r

at unequal diſtances A c is; A

DC, hE, from am '

otherright-line AB in the ſame plane-ſurſace ; thoſe >

two hnes,_ being infinitely produced, on the ſide of . , the leaſt diſtance" EF, will meet each other. '

' * ' 10.If'

Y-'.- ſſ.

a __. LA_

l

i . 'Book the

10. If' two C

right-lines CA, ' CB, making an

angle C, be ree ſpectivelycqual

to two other .

right-lines FD, A B * E

FE, making an angle F," and the angles whiCh they make C, and F, be likewiſe equal 5 the right-lines AB, DE-joining their_cxtremes will be emial, and the two triangles ACB, DFE equal in 'all rcz

ſpects.

Ifthis ſhould not appear ſufficiently evident For an axiom ; conceive the triangleDFE to be remov ed, and lo applied to the triangle ABC, that the point F may coincide with'C, and the fide PD Fall

upon the ſide CA; then, becauſe FD is ſuppoſed ſi equal to CA, thepoint -D -will alſo fall upon A. '

And, 'the angle F being Equal to the angle C, the ſi ſide FE will fall upon CB; and conſequently the -

point E upon the point B, 'becauſe FE is ſuppoſed equal to CB. Therefore, ſeeing all the bounds of . the two 'triangles coincide, it is maniſeſt, that not

' only the' baſes AB, DE, but the angles oppoſite to the equal-ſides, are alſo equal.

When all the ſourlines CA, CBſFl), FE are >

equal; the triangleDFE, being contrariwffe applied to ACD ſo that FE may coincide with CA, will, ah'o, agree with the triangle ACB (as is manifeſt from the reaſoning above) : and ſo,l the angle E (as - D did before) now coinciding with the angle ſiA, the - V two angles E and D muſt neceſſarily be equal to

each, other, in this caſe,'.where 'the trianglc DFE

lS_ an iſoſceles one. '*

B4 POSTLLz

78 \. 'Elements of Geometry. '

ff____._, i

POSTULATES, or PETITIONS.

I. That, from any given point, to any other _ given point, a right-line may be drawn.

2. That, a right-line may be produced, or con tinued out, at pleaſure. \

3. That, from any point as a center, with ara dius. equal to any 'right-line aſiigned, a circle may '

be deſcribed.v \

4. That, a right-line may be drawn perpendi cular to another, at any point aſſigned ; and that it is alſo poffible for to make a right-line, or a right-lined angle, equal to any right-line, or right lined angle aſſigned, or to the half thereof.

This fourth Poſtulate is added, morefor theſake of making the proper references, than through ahſolute neeeffity; ſince, what is here harely afflumed as poffiſi , hle, is effected, and actually demonſtrated, in the hegz'n ning of the Fifth Booh, intirely independent of every ' thing t'ut Axioms and the other ſſPoſhulates, aho've laid down. It may < alſo he proper to note here, that.

though theſe Poſtulates are not always quoted, it will he aſſ to perceive where, and in what/erſt, they are to he underſtood.

NOTEs and OBSER'vATloNs, rwith the ſigni fication: of Sign: zg/Ed in this Tract.

A PROPOSlTION is, when ſomething is either

propoſed to be done, or to be demonſtrated, and is either a Problem,- or a Theorem.

A PROBLEM is, when ſomething is propoſed to

be done.

A THEO

I ,

BaoL't/Be Firſt. _ i

A THEOREM is, when ſomething is propoſed to be demonſtrated.

A LEMMA is, when ſome premiſe is demonſtrat ed, in order to render the thing in hand the more

eaſy. . ' , ' .

A'COROLLARY is, a conſequent truth, gained from ſome preceding truth, or demonſtration.ſi

A SCHOLXUM is, when remarks and obſerva

tionsare made upon ſomething going before.

The ſignz'flcatzſiaiz SIGNS,

The ſign :, denotes'that the quantities betwixt which it ſtands, are equal. * r ſi.

The ſign r', denotes that the quantity preceding it, is greater than that which comes after it.

The ſign -=, denotes that the quantity preced ing it, is leſs than that which comes after it.

The ſign +, denotes that the quantity which it precede-s, is to be added.

The ſign -, denotes that the quantity which it precedes, is to betaken away Or ſubtracted,

A figure, or number, prefixed to any quantity, ſhews how often that quantity is to be taken, or re peated; as 5 A ſhews, that the quantity repreſented by A, is to be taken '5 times. * '

D

When ſeveral angles are C formed about the ſame point ct >

(as at B), each particular angle _ is deſcribed by three lette'rs,' _

whereof the middle one ſhews A B " E the angular- point, and the

'gather two, the lines that form the angle: thus

'C BD or DBC ſignifies 'the angle formed by the

. ' When,

Io'

Elements of Geoenetry. -

7

ſ When, in any demonſtration, you meet with ſea veral quantities joined the one to the other conti nually by themark of equality (:), the concluſion drawn from thence, is always gathered ſrom the firſt and laſt oſ them ; which are equal to "each other, by virtue of the firſt axiom. Thus iſA:B:C:,D,

- then will thefirſt (A) and the laſt (D) be equal to

* Def. 8.

Þ Poſt. 4,

C Ax. 3.

* Ax. 4.

each other.

l Alſo, when in the quotations you meet with two

numbers, the firſt lhews the propoſition, and the ſecond the book. , Moreover, Ax. denotes axiom ;

Poſt. poſtulatum 5 Def. definition ; Hyp. hypothe

ſis. Note alſo, that, whenever the word Line oc curs, without the addition of either right, or carved,

a right-line is always underſtood: and that, when a line is ſaid to be drawn to, or from an angle,

the 'ſiangular point is meant.

T H E O R E M I.

11 line (AB) ſtanding upon another line (CD) makes with it two angles (ABC, ABD) which, taken to-o 4 gether, are equal to't-wo right-angles.

If the angles ABC, ABD are equal, it is plain they make two right-angles', if unequal, let BE be perpen dicular to,CD b, dividing *

the greater oſ them_ (ABC) B _ D

into the parts EBC, EBA 5 _

then the former part EBC being a right-angle _*, and the remaining part EBA together with the whole leſſer angle ABD, equal to another right

EA

angle EBD c5 the whole, of both the propoſed angles, taken together, muſt neceſſarily be equal' to two right-angles d.

C O-R OL

, * Baah the Firſt. ' II

, ſame fide of a right-line (CD)* are'equal to/two COROL'LARY."

Hence all the angles' at the ſame point (B) onthe

' right-angles *. \ = Ax. 3.

T H E O R E M II.

If one line (AB) meeting two other: (BC, BD) in the ſame point (B), make: two angles 'with them (ABC, ABD) which together are equal 't0_ two right-angles 5 theſe lines (BC, BD) willform one con

tinued rights/fire. .

For, poffihle, let BH, A /

and not BD, be the con- ' -'

tinuation of the right line If -

CB: then the angles ABC and' ABH being = two

' , right-anglese = ABC and C ' B

ABD ſ; iffrom theſe equal quantities, ABC, corn-raw, mon to both, be taken away, there will remain

ABH : ABD g s which is impoffihle h. ZZX- 5- t

- " x. 2,

THEORE/M III.

'A The oppoſite angle; (DEB,' AEC), made hy two

- line: (DC, BA) intetſſhcting each other, are equal.

For DEB + DEA = two is .

rightaanglesi 7- AEC + DEA; . a ,_ ,

whence, by taking away DEA, common, there remains DEB

: AEC ".

_ THEO

Denlf A i

/ a.

I 2 E/ements of Geonzetzzy.

,- TH'EOREM IV.

Two right-lines (A B, C perpendicular to one and theſame right-line (EF), are parallel to each other.

l If you ſay, they are not parallel; then let them, r

when produced out, meet in ſome pomt, as G.

InEA, pro- ſ C F' .D*

' duced (ifneceſ- .

ſary) let there - * a . '

be takenEH: > , B g

l Poſt- 4- EG. 1, and let H A " E

' m Poſt- 1- the right-line FH be drawn n'. The triangles EHF and EGF, having EH : EG, the angle HEF :

' DCF- 8- GEF ", and EF common, are therefore equal in all o Ax- 10- reſpects 0: and ſo, the angle EFH being =ſſ EFG' P Hyp. (EFD) :a right-angle 9, H FDG(aswell as HEG) I 2. 1. muſt be one continued right-line q: which is im

* Ax. 8. poffihle '. Therefore AB and CD. are parallels.

' S C H O L I U M.

In this theorem, the poffihilz'ty of parallel lines (or ſuch, which' being infinitely produced, in the ſame plane, can never meet) is demonſtrated: for' EF may be drawn perpendicular to AB 1; and CFD, again, perpendicular t'o EF ' 5 which laſt, it is de

\ monſtrated, will be parallel to AB. '

' THEOREM V.'

- p - 3 \l/Perpendieulars (EF, GH) to one (AB) of two pa rallel lines (AB, CD) terminated hy theſe lines, are equal to each otherjanel alſo perpendicular to the other

qf the two parallel: (CD). * * '

For, AB and CD being parallel to each other, ' Ax. 9. GH can neither be greater, nor leſs than EF ' ;

and Def. and therefore muſt be ' equal to EF. If you ſay, 13- that EF'is not perpendicular to. CD ; then let FM

'Pod- 4- be perpendicular to EF', meeting GH produced (Ft

nece -

I

a - Booh the - 33

neceſſary) in M: ſo ſhall . * M

ſſ H

_,F'Mbe parallel to AB"; C F' - D--4x and conſequently GM : Tſiſictſi"

EFWIGH; whiehisz'm- "5*1

poffihle x. Therefore EF is _ſi 2'

perpendicular to CD. And WO

by the ſame argument, GH E - ' B

is perpendicular to CD.

coROLLARY

Hence, thr0ugh the ſame point F, more than one parallel cannot be draWn to the ſame line given

AB.

'* SCHOLIUM." '7

From the preceding propoſition, the conſiſtence - -: -* -*

oſthe twenty-fiſth definition, or' the poſſibility, that * ' all the propertlies aſcribed to a G H_

rectangle, can ſubſiſt together in E E _ the ſame figure, will appear, V * together with the method oſ

conſtruction. For, at any two

points C, D in a right line RS, X _ ſ

two perpendiculars CG, DH R L D 5 r

may be erected 7 5 and a perpendicular to one oſ, poſh 4_

theſe, at any point E, meeting the other in F, may be drawn. The figure CþEFD thus conſtructed - will be a rectangle: ſor CE and DF are parallel z 3 z 4_ 1, as are alſo CD and EF z : therefore the angle F (as well as C, D, and is a right-angle '. If CE be a 5. 1.

'made : CD, then will the rectangle CEDF have

all its ſides equal b. Which' anſwers to the defini< b5_1_and

tion 'oſa ſquare. 4 ' ' * Ax. 1.

THEOREMXſſp

. Right-lines _(AB, EF) parallel to iheſome right Iine (CDJ are pored/e] to eath other.

. ' '_ A '_ ' For - ,

\

' Element: of Geometry.

14._

'5.1.

'441

l Ax. 7.

hAx. 10.

\

For let' the line HIG G,

be perpendicular to CD :_ A* l . B then, that line being alſo E I I,

perpendicular to both AB q t

and EFC, theſe laſt are C' 1Y__'*D\

Parallel to each other d.

THEOREM vn._

ſI line (AB) interjecting ter/o parallel lines (SR, (LP) make: the alternate angles (SDC, PCD) equal

to each other. B '

Let CF and DE be per

pendicular to V), and SR e; S F D\ R then theſe lines FC and DE

are likewiſe parallels ſ ; and

ſo the triangles CFD and Q C

CDE, having the ſideCF A/

:DE*,FD:CE®, and _

the angle F : E 3, they will alſo have the angle

FDC : ECD h. ' v -

COROLLARY I._ ,

Hence, a line interſecting two parallel lines, EP

A makes the angles (BDR, BCP) onthe ſame ſide, equal to each other: for BDR (:CDS i) :BCP*'.

COROLLARY II.

Hence, alſo, a line falling upon two parallel lines,

makes the ſum oſ the two internal angles (SDC + QCD) on the ſame ſide oſ it, equal to two right . angles: for the angle SDC being = PCD, and

PCD + QC'D = two right-angles'l z thEnce is -SDC + (LCD : alſo to two right-anglesm.

THEOREM VIII.

If a line (AB) interſſectlng two other lines (PQL JRS), make: the alternate angles (DCP, CDS) equal

to each other 3 then are theſe two lines parallel.

9 ' \ For,

'1.1.

"Ax. 4.

-l

__ t. hat-i"

'Book theFihſtſi , _ I 5

For; if poffible, let * ſbme other line DT,,.and

not DS, be parallel to' nsch. to

PQ; then muſt CDT 4. t.

: DCP0 :CDSF:' P;I7:i71.

which it impoffihle q. 1 A ,

C O R O L L A R Y.

Hence, if a line falling on two others, makes the anglects (BDR, BCP) above them, on the ſame fide, equal to each' other; then thoſe two lines

are parallels: becauſe SDC : BDR '. ' 3- '

THEOREM lX.

[fone/isle (AB) of a triangle (ABC) he produced, the external angle (CBD) will he equal to hath, the internal oppoſite angle: (A, C) take/'1 together. v

For, let BE be pa-' '

rallel to AC s3 then ' O , E ' Sch- 'w

will the angle C z - 1..

CBEK, and'the angle t 7.1. v

A = "3 tltere , o.- " COr. to

foreC+ AzCBE ſi B A D 7-1

+DBſiEx=CBD ſ. - - r 'Axſi.4.

* * . . . _ 7Ax. 3.

C O R O L L A R Y.

Hence the external angle of a triangle is greater ; than either of the internal, oppoſite angles. _

_ THEOREM X. >*£

7 The three angle: of any plmze triangle (ABC) taken - tOgether, are equal to two right-angles.

For, if AB be produced to C D,thcn C + A I CBD z,to which equal quantities let the

angle CBA be added, then \ * ,

willC + A'+ CBA ZCBD A 13 1)

+CBA®:tworight-anglesb. _ 2 Ax-A

16

Elements of Geometty. i _ſſ

*AX.5L

d 10. r.

e Ax. 4.

CQROLLARIES.

1. Iſ two angles in one triangle, be equal to two angles in ſſanother triangle, the remaining angles will alſo be equal c.

2.' Iſ 'one angle in one triangle, be equal to one _ angle in another, the ſums of the remaining angles

will be equal c. ' _'

3. Iſ one angle of a triangle be right, the other two taken together, will be equal to a right-angle.

4. The two leaſt angles, of every triangle, are

acute.

THEOREM XL

The four inward angles (ffi a quadrangle (ABCD) _ taken together, are equal to four right-angles.

Let the diagonal AC be drawn; then the three angles oſ the triangle ABC being : two right-angles d, and thoſe oſ t-he triangle ACD equal alſo to two right-an- _ gles "3 it follows that the ſum

oſ all the angles of both trian- *

gles, which make the ſour angles oſ the quadrangle, ' muſt be equal to ſour right-angles e. -

C O R O L L A R Y I.

Hence, iſ three oſ the angles be right ones,' the fourth will alſo be a right-angle.

"COROLLARY II.

Moreover, iſ two oſ the four angles, be equal to two right-angles, the remaining two together will likewiſe be equal to two right-angles. _ '

> - ' S C H O-ſſ

S C_ H O L I U M- .

Iſſrom any point P, within a polygon ABCDE, lines be drawn to all the angles, ſo as to divide the whole into as many tria'ngles " * ' _ APB,BPC,CPD,DPE,EPA,

as the polygon has ſides; the B

ſum of all the angles of theſe triangles, (which togetherſſma-ke

Up,_0r compoſe the angles oſ the - polygon,'over and above thoſe A- _ \ I:

about the point P) will be equal to twice as many right angles as the polygon has ſides (by IO 1.) Therefore, 'ſeeing all the angles about the point P, whereby the angles oſ all the'triangles exceed thoſe of the polygon," are equal to four right angles, it is manifeſt, that all the angles oſ the polygon, taken together, will be equal to twice as many right

anglesz wanting'ſour, as the polygon has ſides.

*THEOREM XII. K

The angles (A , 'B,) at 'the hoſe of an iſoſceZes trian gle (ABC)'ore equal to ezzeh other. \

For,-let 'the line CD biſect, U * _ ' or divide the angle ACB in

to two equal parts ACD, '

BCD, and meet ABinD:

then the trianglffgſifi'bl),

BCD, ha'vin AC : BC ſ, , r _

CD c'ommoZ, and the an- ' - D B Def' 18 _

*gle AACP : BCD 3, will alſo have the angle £Hyp. *

A : B . ' w

A h Ax. to?

-COR0LLARYI. .

Hence, the line'which biſectsſi the vertical angle oſ an iſoſceles triangle,, biſects the baſe, and is alſo perpendicular to it u. A _ '

C C O R O L