1000000030

693 / 1042

,c: F. o 111
c,·. nf t1d~ t\\'o ares nD ud, and Cl) col l~.e ,,,¡ii
of lhe ,irdcs; lhen f,y, e l) : cJ :: J'L) : (d, énJ
<':D: etl :: BD: bJ, "nJ CD : ca :: HI~: L,c, ¿c.
Far finee lhe ~ 'cs Bl) bd are equal, the an~b BCD
ued lVill be equ,d; anJ FD fd, bcing langcn!' 10 llre
puinls 1) antl ti, lhe anglcs COF dfwil l be <qual, !>e·
illr caelr " " angle (an . 22.) 1'0 in lhe llVú Ir:ang!cs
Cl)F etl f, lire 111'0 ,ln~l<s reD COI' vf llr,· one, be·
¡!l~ equal 10 Ihe 1 \\'u angles fe d leJ ( of llre ollra, caeh
10 caelr, ll,e "nlJi"iug anglc CfO, lI'ill be equal 10 lhe
' ''llollnin~ an;;lc e¡'d, (by art o60.) ; lherefore llre lri–
l.n.;les CFl) .rfd are hmilar, anJ eonfequendy (byan .
i,) CD : cJ :: F D : fd . ¡n lhe f. me manner il may
1~ ~emonfrr.tlcd, lh.I C'D : ed :: BD : bd, anJ CD :
l J :: BE : be, te.
i l' Lel AB D (fig. ,.) be a quadrant of a eirde de·
f«ibed hy lile rad,us e D HD any are of il, and H ..I.
ilS.eomplrmenl, SG or Cf lhe fine , CG or 1)1' llre eo·
I'ne, DE lhe ungent, and CE lhe fecant of llral are RD.
Tiren finee lhe triangles CDE CGn are fimilar or equi·
:lIlgular, il will be (by .rt. 72 .) DE: EC :: GU : BC,
1. '. llre langen~ of any are, is 10 lhe fecam of lhe fan,e,
as llre fine cf il is 10 lhe radius Alfo fince DE : Ee ::
GB: BC; lherefore, by invening lhat proponiun, \Ve
Ira"e EC : DE :: HC : GB. i '. lhe fmm is 10 lhe
lJngent, as lhe radius is io Ihe fine of any are.
Again, lince lhe triangles e OE CGU are fimil",
therefore (by arto72 .) it \ViII be CD : CE :: ce; : C'S,
i (. as lhe radius is 10 the fecant of aoy are, fo is the eo–
fineof lhat are to lhe radius. And by invening lhe pro·
porti"n I'Ie hdve this, viz. As the feeamof any are is to
lhe radiu!, fo is llre radius 10 lhe eo·fine of lhal are.
75 . In all eireles lhe cbord of 60 is alIYays equal in
b glh 10 lhe rad,us. Thus in tire eirele AEBD, (fig. 8.)
if the are AEB be an are of 60 degrees. lhen uraw–
ing tire elrord AH, 1 AH nlall be equal 10 lhe
radius CB or AC; for in llre tri.ogl~ ACB, lhe angle
ACIJ is 60 degrees. oeing meafurcd by tI,e are AEU ;
therefor. lhe fum of lhe olher ,IYOanglcs is 120 uegrecs,
(by w. 1. of 60.) ; blll finee AC and Cl:! are equal to
lhe two 2ngl. 5 CAH, CHA will ~I fo be equal; eonfe·
queotlyeaeh of lhem half lheir fi,m 120, viz. 60 de&rets;
lherefore all lhe lhree . ngks are t'qua! la one anolher,
eonfeqtleotly all lhe kgs, lherefore Al:! is equal 10 CIl.e.,.lleoee lhe rdJ,US froOl whieh llre liots on aoy
fcaleare formed, is lhe dlOrJ of 60 GO llre line of ehor.!s.
Cco1l!ctricr./ Probl':lI/J .
PROBo l . From. poinl C (fig. 9.) in agiren linc ¡,U
to raife a perrend,,"I., 10 ,Ir. l I,nc
I/u/•. From tI,e poi nl C l.l!>e lhe equ~1 dia,toces ell,
CA 00 eaehlide uf il. T hcn lI,m!t lhe ~ompaffcs ter nny
dill.nee grealer lhan CB or CA, :tnd wrth ol1e f".1I d'
tbcm in li, Cwt:t p t!IC are EF \'Iith t!h! othl.'r; ,Ig:..in,
Wilh lhe f.l.lIe op(ning. and one fllOl io A, [-., ccplhe ."
GH Wilh lhe oll,er, an" llrel" llVO a" s wi ll ,olerll .'t
one anolher ,n lhe poinl 1) ; lhen joio lhe ~i"en ¡r,'"'' e
tlnd D wilh tI,e Ime CD, an,j lhat Ih.lIl" lh¡ p"l',n~l'
cu!;r rcqlli"d.
E T R Y.
Z. Tllei,·id" oI:il'cn ,:¡ 11 FlOr !, eJI'. l e. \ into t· ...oe.
(l:1al p:uIS ; 1::;, tu Ut; ... d ¡l. '
Jlu/,. .nyel,;l"ne,· \litl, )om ,o:nr,ffcs d,al )n,:
ale is g,al" lhan b :f lh.: ¡''''IO line ; ll,cn fClllns
(!le fUOl of lh('m in H, Wilh tI,e 1.:1"" r'Nctp the are
1) Fe; . nd wilh tI.e r.,me d:n:,ree . :tnd un( fOOl in A
,.ilh ,he o'her f"'l'ep lhe are ('] [J ; lWO a,e;
\" ill inltrf,(¡ en( "nolher in ll.e pOi01S CD, wl,i,h join.
cu hy lbe rigill I,nc De '1ill bifeél AIJ io G.
3. Froma giveo poinl D, (fig. 11.) to Itt f.1I a per–
pendicular en a ginn line 1'.13.
JIu". Sel One fuot of lhe ec.mpalTes in lhe point D,
and extend lhe othu 10 any diOan~e greater Ihan lhe
Jean diaanee belween the given poinl aod lhe line, and
\lilh lhal CXltnt fweep lhe are AEB, eultiog the
I,ne in lhe two points A and H, llreo (~)' lhe IJO prob. )
b,fcél lhe !ine AA in Ihe poinl C. 1.lHy join C :.nd D,
and lhal I,ne CD is lhe pe rpendieul.r ,equired.
4' (Fig. 12 .) Upon lhe elld J3 of agivcn rightlioe Bi\
to raile a perpendicular. '
11,,1.. ,¡'ake anyexleot io your eompaffes, "nd Wilh
one foot in H fix lhe other in any point C IYilhoUI lhe
given line; tl,en w'lh one poim of the eompaffes in C,
dcf~rrbe Wilh lhe oliler lhe eirele E13D, aod thro' E
and C dr.w tlie diammr EC'D meeling lhe eirele in D;
join D aod lJ , and lhe right !ine DB 's thal required'
Jur EllD is ti lighl angle (by coro4' vf 63.). '
5· (Fig. 13') To dfdW one Jine paraJleJ lU another gi–
"en Irne AH, lhal il,all be diOanl I, om one anolher by
any giren dir.lnce !J.
R,,". EXlcnd your eompaffes 10 lhe giren diOanee D·
lhen fett ing one fool of lhem in aoy poinl of the gil'e~
line (fuppole A), Wilh Ihe other fweep lhe are FCG; a–
gaio, al lhe fame extem, and one foot io any olher poine
üf lhe gil'en line B, fweep ll,e are HDK, and draIY lhe
I,ne Cl) lOucl,ing ¡hem, and lhal IYill oe pm llcJ 10 Ihe
gi"en Jine AB, alld dinanl from il by lhe Jinc D as was
,eqllired.
6. (f'ig. 14') To di"ide aci l'enline AR inlo l ny num–
ber of (qual pans, furpore i .
R,,/,. F~ um lhe .rO,nl A draw ~n)' line AD, I.laking
~n , ogle.'vllh lhe IrIlC AH, tI,en lhrough lhe point B,
,I,,,IV a l'll~ BC pwllel 10 AD;. and f,umA, Wilh aoy
111,,11 0remng of lhe eomranes, lel olr o numbcr of (qu,1
pans (IIn llre line AO,) Jd, I,y one lhan llre propofed
nu,"lre, (lru e 6 ); lhen frn.n R fel off lhe (,me oumber
of llre fjme p~' IS (oll lhe linc UC); IJUI)', join 6 onu 1,
2 anu 5. 3 , ou 4, 4 and; . 5 aoel 2, 6 aod 1, and
llrdc lines ~',II CIII lhe gilCn I",e .s req"iIlU.
7. (F,g..15·) 1'0 qUdIllr a giren errele, l.r 10 diriJe
it WI(I J~I\1r \.~p•.d pans.
/:''/,. T hr,,' lhe centre C' (lf tI,e gi" en !'irrle, d",v a
(¡i,l1lll!. r:\~1, t1u.:n\.1' 10 ¡he r pltll e ¡ ..¡fe ,( pn pc:ndi.:u-
111 !lL'C l(\ tI,,· lio, ,~Il; anJ 1:1,1~' 1\\'0 ¡Jrrnrelcrs :\Il
:,,,,1 J)!': Ih.,1I qllo1llcr lhl' ~i " k.
~ (f',!:. ll!) TIr:o' tI" "e I:i"(n ruints :l" B. and J)
ti) ,: , .: \\' .1 UI.:!t:. (,\','Ir, TLc: l hrc.'~ ¡'UlOb nll:lt nOl 1\;
il! dll': 11I1I~ ¡ll l1~ht 11I1l'.)
/: d. Jo"o 1\ alld H. al!'" B aod D, Wilh ll,e 1I" i.• ' 1
1.1.l~ ..\ t, j:1) ; l/HII lt.~ ¡ 'lltl . : .' b¡I~~: .\13 ,,"h '¡hl'
Loe