1000000030

G E

eqfli3n~ul.r;

fN tite

:In~l,::s

2n:i·H :ilre

ri,!ht

on('s ;

Io~<w;(c

111<

.nJIe

i, c.I....

tile angl:

1"1):1;

IYhorc·

ture

tll~ r('miUlI!tl~ 41t~1

.. )

üFH.

and A!)C,

:tre

alfo

t–

1":

\,.'I·,c«f"r<,

Fd,

til. Jill.nce of lhe

n,rrs,

hO,

t!,e

<\Cc:,

t,f

t!,~

looger llar. t.líove lhe Ihoncr; fo

De,

lhe tli"Joce uf lhé longcr

fl.ff

froOl lh. lowcr,

CA,

lhe ex.:d;; of lhe IYight o: lhe 10IYcr above

lh~

J.lOgcr

1¡,IIf.

AnJ

lh,oCé CA will be founo by ¡he rule of

tun.'C.

To wloich if lhe length DE be added, you will have

the \\ hole he'gill of lhe ¡ower HA.

. S

/'.1.

F,c.

Maoy OtltH Olelhods

I11'Y

be occafiooally coo·

" ived for Ole••furiog

accelüble height. For eX'lIlple,

fr"m lhe gil'en lenglh of

l~e

fl'Jdow

liD,

te>

find out lh.

hcight AH,. thos: Lel lloe:e be ertlled a fI,ff CE pte·

pendiculdrly, p:cducing tI.e Ibadow EF: The " iangles

ABO, CEt',

are equiaogular;

f~r

lbe angles at

and

E are righl; . od the aogles ADB aod CfE

>re

equal,

eaeh b:iog equal 10 Ihe aogle of tbe fun's eievalion above

the horizoo: There(ore, as EF, the Ibadow of Ihe

1l.1f,

10 EC, Ihe nar!' itU f; fo

BD,

Ihe Ibadow of the tower,

to HA, thé heigbt of the lower. Though Ihe plane

whic Ihe

ntaUOW

of lhe lower falls be 001 parallci 10

tbe bonzon, ir ,he Il.ñ be ercaed io Ihe (.mc

pl.oe,

the

rule will

the (ame.

O P O S 1 T ION VII.

m:.!",.,

an inaccrffibl(

hdghl hy

,"(a/JI o[ 1'lJJ0

jlaji

-¡{ilher,o we have (uprored. the

heigh~

10 be ac·

celJible, or Ihat we can come at the lower eod o( it ; nolV

if, becaufe

fome impcdimenl¡ we cannot gel 10 a tower,

or if the poin, IVho(e height is 10 be fouod OUI be the

(ummi, of a hill, fo lhat tbe perpendicular be hid witl¡io

lhe hill; if, for IVanl of bmer inUruments, fuch an in·

acce/lible heigo, is 10

mea(ured by meaos of IWO llafTs,

lhe fidl cb(ervalion be madt IVilh the /1>ffs DE and

FG,

(as in prop.

6.);

tben the ob(trvator is to go off in

a direa lioe íro01 lhe heigiu !nd fidlllation, till he come

to Ihe (econd Ilation; \liBere

(fi~ .

J.)

he is tO

pl.c~

the

lon~er

Ilaff pelPendictllarly

RN,

and ¡be Iboner Ildrr

at KO, fo that the (ummil

may be fcen alonr. lhtl<

io?s; Ihal is, (o Ihat Ihe points KNA m.y be in the

(ame right line. Through Ihe poin t N, let ,here be

~rlwn

toe righl line N

parallci 10 FA: Wherel1Jre in

IJ.e

trian~les

KNP,. KAF, the angles KNP KAF ale

eqoal, al(o lhe angle AKF is comOlon to bOlh; con(e.

quendy Ihe remaining angle KPN is equal 10 the remain·

iog angle

KFA.

Ancl therefore, PN : FA :: KP :

KF.

HUI Ihe t{¡annle! I'NL, FAS are r,mílar; Iherefore,

PN : fA :: NL :

SAo

Thm[ore, (by thc 11 .

Ellcl )

KP : KF ::

L :

SAo

1'hence, alternately. it wdl be,

as KP (tbe exce(s of the greatcr diUance e,[ Ihe Ibon

lIafT from the long ooe abuve its leO!.r dillancc from i:)

lONL, the exccís of Ihe longer

Ha[

abol'c lhe OlOrter ;

KF, the dillaace of lhe tIVO lIations of the OlOmr

naff to

the excefs of

lh~

Ireighl [ought above

th~

height of the OlOrter llaIF Whercfore

S.'\.

lVill be found

by lhe rule of Ibree. To IVhich let the heigh, of

lh~

!horter llafT be .dded, and the (U:11 will givt the whoJc

inaccelliblc hcighl HA.

NOle

lo thc fame maoner nuy an

inlcceOí~le

h,iZht bc founrl by a

r,eom~tl ic~1 [~nr.r',

or l.)' a piolO

(Ilclulllnt. But we ¡:lJlllcd\'c

ti,"

r",,, lu Le fuuoJ u;¡t

by Ihe lIu.l:oI, for his oIVn c,trClle.

NUlc~ .

Tl,al by lhe hci;:ill uf

d,lll.1r

we nnderllar.d

ils height above the grouod In II'hieh il i, fixed.

NOlc 3 Henced. pwds tllC me,hod

ufing olher in-

Ilrulllenls in'<nled by gtomelnciar.s; fOí c;<amplr, uf the

geometric::l crors: And if aJl Ihings be jtt{lly wtighed,

• likc rt:lc wdl fcn'e for it as here. BUI IVe incline to

touch only upon \'Ihat is moll maleriJ!.

R O P O S 1 T ION VII!.

F,c.

9·

1Iudflm

1;" dl}/anceAlI, 10

O11t

,",hol'

t ,Ylroll;l", 1U(

b.ll

accrji,

Ih.

h Ip

Jour

jlaji.–

Let there be a IIJff tix

;lt

lhe point A; theo going

bl, k at (ome (enfible oillance in the fa me right lioe, let

anolher bc fixed in

fo as that both the poinls

Rnd

be comed and hid by lhe

lI.tf

C: likewile gaing off in

perpeodicular (roOl the right lioe CH, at toe point

(the method of doing whlch rn,1I be Ibown io Ihe follo

IV–

ingfcho/iu11I),

let Ihere be placed aoolher UlíF al

and in ,be righl line CKG (perrennicuJar 10 ,he (ame CB,

at lite point

13),

and al tbe poio, of ,il K, (uch Iha¡ Ihe

pointS

J..:,

aod

may be in the [ame right line, Jet

Ihae bc fixed a (ounlr Uaff. Lel Ihere be dnwn, or

1" lhere be (uppofed tO be drawn, a right line

pa–

lit!

10 CA. The

triangl~s

KGH, HAll, \ViII be equi.

anglllar; for the anr,les H!\

KGH are rigilt aogles.

Alt~

the anglts .AHH, KHG are equal ; whereforc, as

KG (lhe exce(s of CK above

AH)

10 GH. or 10 CA,

the din.oce betwixI the fitll and (econd naff; (o is

AH,

,he díl!ance betlvixl Ibe firU and tbird llaIF, tO AB me

the di/lance foughl.

/11.

FIC .

10.

draw

a plane a right line

perpen–

dicular tO CH, from a given poin, A; take the right ·

lioes AH, AD, on eaeh r,de equal; and io lhe points

B·

and

let there be fixed Il:kes, tO IVhich lel Ihere be

lied IWO equal ropés BE, DE, or ooe having a mark io ·

Ihe middle, and holding in your hano their eXlrcOlilies

joincd, (or lhe mark in Ihe middle, if il be bUI one),

dra\V OUI lhe ropes ooe ¡he ground; ann then, \IIhere the

til'O ropes meel, or at lhe mar

when by it Ihe rope is

(ully f1relchcd, ltl Ihere be placed

third

U.ke

at E;

lhe righl linc AE will be perpendicnlar 10 CH iD

Ine puinl

(prob.

of parto i.). In a muner not un–

Itke 10 Ihis, m.y any problems that are rcfolved by lhe

(quare and comva{fcs, be dune by ropts and a cord

turned round as a radius.

PRO P

O S 1 T

ION IX.

F,c

12.

I:¡,n/I/N

Iht dijl.J"u

\.3,

611(

whojt

txlr:III;liu

occrftblr.-From

the roint A, let the

rtghl line

of a kno\Vo Icnglh be

Ola

de rerpendicular

to AH, (by the prccedingfchd;1I1I/): likewi(e draw the

rig!lt line

CD.

perpeodicular 10 CH,

mwin~

the ,ip,ht

lill~

J) :

lhen as DA:

AC::

,\e :

.'lB.

Whm·

fOlc, whcn

and AC are given,

¡lB

wiU be found by

lhc rul< of lhree.

/I'l.

AII the preceding

~pcralions d~rend

onthe cquality of

(,'me angles of triancles, anel on lhe liOlilarity of the tri·

anzl,s

Jlilin~

froOl that ,q\lc!ily. .'1od

the (.me prin.

clplo