1000000030

698 / 1042

691- G E o M
flrument : L't M 'V Ih ~ pUl'enJiwiu be fu?pofed 10 CUI
off oa ,lhe ng!n tid~ K. So e<p .• lt 15 clear Ihal
L:(N, ACK. are :¡~li!.r Iriln;;ks ; for Ihe .ngb LKN,
ACK ¿fe ri~líl angl:~, a!ld Ihcréfore equ. l; m~reovcr
J..N anJ AC are parJlld. as bcing uo,h pcrp~n licular lO
t,he horizon; c 'nfequend)', (by art o60. curo 1. part 1.)'
~he angles KLN, KAC, are eq ,¡al; wherefore, (I,yan.
6:>. cor. 2. of par! l.) Ihe angb LNK, and AKC. are.
IjkewiCe e'pal: fo Ihal in Ih1: triangles NKL, K \ C,
(by art. 71 . of pan 1 ) as NK : KL :.: KC (i . e: BD)
;. e .o\.; Inat is, as 80 lO 100, fo is 96 fm to C:1.
Therefore, by lilo:. rule of time, eA will bc fouad 10
be 120 fm; and CS; which is 6 fm, being .addcd, Ihe
whole heiglll is I~6 fee!.
· BUI if Ihe di!laoce of Ihe ohfcrvator from Ihe heighl,.
~ HE, be fu ,h, 11m, when Ihe inOrumenr is dire8ed as
(ormerly lhe fummil A, rhe ptrp. ndicular falls on
Ihe angle P, oppolir~ lO H, Ihe cenue uf Ihe inllrumenl,
and BE or CG he given of 120 ftel; CA will alfo be
120 feet. ¡' or io Ihe triaogles HGP. ACG, equian–
guiar, as in Ihe preceoing cafe, as PG : GH: : GC :
CA. EUI PG is eqnl 10 GH j thercfore GC is like·
wiCe equal lO CA : Ihal is, CA wiil be 120 fCCl, and
the whole heighl 126·fm as before. .. •
· Let Ihe dillance BF be 300 fcet, and Ihe perpendicu.
lar or plumb lioe CUl off 40 equal parts from Ihe mli·
niog fide: Now, in lhis cafe, lhe angles ~C, «tI,
are equal, and Ihe angles «tI, ZIS, are equal; Ihere·
{ore lhe aogle ZIS is equal lO Ihe aogle ~C. BUl
the angles zsr QfA are equal, being right angles;
Iherefore, in Ihe equiaogular Iriangles AC~ szr, it
",iJl be, as ZS : SI : : CQ.; CA; thal is, as 100 1040,
fo is 300 lO CA. Wheretore, .by Ihe rule of e,
CA wilI be found lO be of 102 feet. And, by addlng
{he heighl of Ibe obfervalor, lhe whole HA wilI be 126
{eet. NOle, lhal Ihe height is grealer lhao Ihe diflance,
wbeo Ihe perpendicular CUIS Ihe rigbl fide, aod lefs if
il .CUI Ihe redined fide; and lbal Ihe heighl and diflance ·
are equal, ir lhe perpendicular faJl on the oppoGle
;'ngle. s e H 01. [U M.
· If lhe heighl of a lower, lO be'meafured as above, end
io a poinl, (as io fig. 3') lhe diOaoce oi Ihe obfervaror
oppoGle lO il, is nOI CO, bUI is 10 be ac,ounteo from
Ihe perpendicular 10 lhe poinl A ; lhal is, lO CO muíl
be addcd Ihe half of lhe IhickoeCs of lhe lower, lIiz.
RD : which mull likewife be underOood in Ihe fullowing
propOfilion!, when lhe cafe is Gmilar.
PRO P O S 1 T ION Ill.
FIG . 4. Fr071l Ihe Úighl o( R IfJ'IJur AS gillm, ID
find adiflallce 011 Ihe hprizMlalplalle BC, }¡y Ihe ,emlle·
Irical/quare ~Lel Ihe ioOrumenl be fo placed, as Ihat
the mark C io Ihe oppoGle plAnemay be feeoIhrough Ihe
Jighls ; and Jet il be obCmed how many pans are cut off
by Ibe perpendicular. Now, by whal halh beeo already
• demooílralcd, Ihe triao~le! AEF, ABe, are fimilar;
Iherefore, it wilJ be as EF 10 AE, Co AB (compofed of
lhe beighl of Ihe lower EG, aod of Ihe heighl of Ihe
centre of lhe inOrumeot A, abovelhe tower BG) lO Ihe
.ó.iflaocc BC. Whcrefore, if, by Ibe rule of Ihrce, you
Ji: T n y,
Cay, as El" 10 AE, Co is Aa 10 BC, it w;lI be Ihe aj.
r.¡¡¡;~ lo~glll.
y R O P O S [ T r O N IV.
FIG. 5· . 'To 1Iun/u/', nI/y JijJunc: ~lldl/Jor fla, b,
Iheg<MltlflCul jq//oN.-In tlm o~trauon, Ihe index IS
10 be apploed 10 rhe inOrutllcnr, as was (hown in Ihe d.!.
fcription; anJ, by Ihe help of a fupport, the in(!turne"l
is \O be plJced horizonlally al Ihe pOlnl A; Ihen lel il
be luroed ull ,he reruole poiol F, whofe diOance is 10 b~
mea~uJ'td , b~ feea Ihrouflh Ihe fixed (¡g~ls ; ao.I bringin~
the IOdex 'o be pmlld IVuh Ihe olher ltde of Ihe innru.
menl, o?C.rveby Ihe figh!s upon il any acceffible mark ll,
al a Ccnllble ddlance : Ihencan ying Ihe inflrumenl \O ,h.
poinl B, lel Ihe immove,Ible r, ' hls be dire8ed 10 Ihe lirll
flallon A, and Ihe :ir,hrs of tI~ index 10 Ihe puinr F. lf
lhe index CUl rhe rir,101 fide of Ihe fq uare, as in K, in 11..
~ lVolrrar,gl¡s BRK, anJ I3AF, which are 2 uldr ,
11 wdl be as SRlO RK, fo nA (lhcdilldnceof rhe lI.rioll$
10 be meaCured wi,h a 10 AF; and lhe (lIi!.nc~
AF foughl will be found by Ihe rule of Ihree. BUI ii
Ihe ind~x eUI Ihe reclined fiúa of Ihe fquare in any poinl
L, where Ihe diOance of a more remol~ poinl js 10u&bl ;
in lhe uiangles IlLS, BAG, Ihe fide LS (hall be 10 SU,
as /lA 10'AG, Ihe dillance fOllghl; which accordiogly
will be found by Ihe ru le of Ihree.
PRO P O S 1 T ION V.
FIG . 6. 70 1lun/un 011 am!Jiblt heighl by tIIeonl
if a plaill 1IIirr~r..-Lel AS be lhe heighl 10 be
meaCured; Jet Ihe mirror be plaeeq al C, in Ihe horizon–
tal plaoe BO, al a known diOaoce BC; let Ihe oufmer
go back lO O, lill he fee Ihe imag~ of Ihe Cummil in Ihe
mirror, al a cerlain poiO! of il, which he muO diligeody
mark; and let OE be Ihe heighl of Ihe obCervalor'seye.
The Iriangles ABC and EOC are equjang~lar; for Ihe
angb al D aod B 'ar~ right aogles; and ACB, ECO,
aré equal, being Ihe angles of incidence and reflexion of
l~e ray AC, as is demonOraled in oplics; wherefore Ihe
remaining angles al A alld E are alCo equal: Iherefore
il will be, as CO lO OE, fo. cn 10 BA j Ihal is, as Ihe
diOance of Ihe obCetvalor from Ihe poiQI of Ihe mirror
in Ihe righl lioe b~lwixI Ihe obfcrvaror and Ihe heighl, is
10 the hti;:hl of Ihe obfervalor's eye, Co is Ihe dilfance of
the lower from lhal poinl of Ihe mirror, lO lhe heighl of
Ihe 10IVer foughti which Iherefore will be fouod by Ihe
rule of Ihree.
NOle l. The obfermioo lVill be more exa8, if, al Ihe
poinl O, aOaff be placed in Ihe ground perpeodicularly,
over Ihe IOpof which ¡he obfemlor may f~e a poinl oi
the glaCs exa81y in a line.belwixl him ar.d lhe lower.
NOle 2. lo place of a mirror may be ufed rhe Cur·
face of waler conlai oed in a vclfel, which oaturaJly be·
comes raraJlel lO lhe horizon'.
PRO P O S 1 T ION VI.
flG . 7. 70 mea/lite 011 nm ffiUt height AB by mranl
if l 'lPO jJnffi .-Lel Ihm be placed pcrpendicularly in
Ihe ground a lonlÍcr lIalf OE, IrkelVife a OlOrter ooe FG,
fo as th: obfcrvator may fee A. ,he 10p of Ihe heighl 10
be mcafured, over lhe cnds O F of rhe IIVO OaU's ; lel
FH and OC, parnlkl 10 Ih~ horizon, mm DE and AS
io H ano e i Ihen Ihe Iri~ncl~s FHD, DCA, Olall be
equiaDguJar j