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ó86 G E o 1\1
~ t._Tile par! of ,he radius comprcheoied betwceo Ihe
extrcmi,y of ,he "gilt line ¿od ,h~ 1 ..\Ver eod <lf ,he ar.:,
viz. OU, is calleJ Ihe verfed fin e of Ihe .re AB.
n . If tOaoy point io the circumfcreoee, !"iz. H, .here
be dra\Vn a diAm~l<r FCB, and from Ihe point B, perp<n–
dlcul., 10 thM di,mmr, there b~ dra",o the line BH;
that line il callcd alan'(r.1 tO th~ eirele io the poinl !:I ;
"'hic!! taogeot cao louch Ihe eirele 'ooly io ooe point B,
e1fe if it touched il in more, il would go within il, aod
fo nOI be a laogenl bUI a chord, (hyart. <7.)
23 . The tangent of aoy are AH, is a right lioe drawn
perpendicular 10 a di,mmr through the ooe eod of the
ore B, and termioateJ by a line CAH, drawo from Ihe
eeolre through the olhe .. eno A; Ihus J3H is the langen!
of Ihe m h AB .
24' Aod the lioe whieh terminates the taogenl, vi:.
CH, is ealled the fceaot of Ihe are AB .
25. Whal ao are waols of a quadllnl rs ealled Ihe
lomplmunl of Iha! are: bcio¡; \Yh" tI,e are AI3
wanlS of the quadranl EB, is eallcd the eomplemenl of
Ihe ¿re AB.
26. And whal .0 are \Van:s of a femieirele il eallcd
Ihe ¡u¡plufJ(n' of Ihal are; Ihus fine< AF is wh~1 Ihe
are AB wanls of Ihe femieirele l:!:\F, it is Ihe fupple–
menl of the are AB.
27. The fioe, taogenl, &,. of the eomplemeol of aoy
are, is ealled Ihe eo jin" (o.l.ngml, 6 e. of Ihat are;
Ihus the fioe, taogenl, &e. of the are AE is ealled Ihe
co·Gne, eo·tangent, be. of Ihe are AB.
,s. T he fine of Ihe fupplemeot of .n are il Ihe fame
with Ihe fine of the are itfelf; for, drawing Ihem aeeord·
íng 10 Ihe denoitioos, Ihm refults the fel f·fame lioe
29. A right.lioed angle is me.fured byao are of a
¿rele deferibed upon the angular poiot a~ a eeotre, eom–
prehended belween Ihe I\VO legs thal form Ihe angle ;
I~US (fig. 14') the aogleAnOi, meafured bythe are AO
of the eirde CAOE thal is deferibed upon the point D
as a centre; and the angle is faid 10 be of as mdny de·
&fCU as the are is ; fo if Ihe are AO be 45 degrees, Iheo
Ihe angle ABO is f,id tO be ao aogle of 45 degrccs.
Hence the aogles are gremr or Itfs, according as Ihe
are dtfcribed aboul ,he aogular poiot aod lermiomd by
Ihe lwo legs eoolaio a grmer or a lefs oumber of de–
¡ms.
30. When ooe line falls perpeodicularly 00 anoth~r,
as AS on CO, fie 15. Ihen the aogles are right (by the
10th der. ); ano deferibiog a eirele on the ceolre B, fince
Ihe angl.., ABC ABO are equll, their m:afures mun
be fo 100, i. (. the ares AC AO mun be equd; but
Ihe w~ole CAD is a femieirde, finee CD, a lioe palliog
Ihrough the ceotre S, is adi,meler; Iherefore eaeh of the
partS AC .'\.0 is a quadraot, i. (. 90 degrm j fa Ihe
meafure of a righl angle is always 90 degrees.
31 . If on~ lioe AH fal! any way upon aoother, CO,
Iheo the fum Qf the t\Vo ao¡¡les ABC ABO is .Iwa)'!
equal to the fum of I\VO right aoglcs; fig 16. For on
tite point I!, deferihiog the cirde CAD tt is plaio, Ihal
CAD IS a femicrrdc (b~ Ihe 14Ih); bu t CAD is equal
tu CA aod AO Ihe medfu re of the IWO an~leJ; therefore
tbe f"m of Ihe 11110 .ngles is equal I~ a femicir 'Ie, t1w
is, 10 I\VO righl aoSI s, (by the JaO).
E T l~ Y.
C. r. I Fro" wheoee il is pl..in, tita an Ihe an~lc.,
whie" eao be frol11 a point io an)' Iloe, tOlvlrJI u3e
[i,le of tile line, are equal tu t\YO right an~lts.
2. t'.od tlllt al!lh~ .ln~ks wl1lch can be auouI a
poiot, :!re cClu,1to four lI~hl oocs.
31. [1' OOe line AC crufs Aoolher nD io Ih poiol E,
Iheo Ih- nppofitc ansle, are equal, 1:. !lEA to CEO,
aod BEC eqlul 10 IIED; fig. 17. For upon Ihe poiot E,
as a centre, Jtbiblog the eirde ABCD, Il IS pl_io
ABC is a fc",ieirde, as alfo I3CD (by ¡he 14th); there.
fore the lre ABe is equ_1 tO th: are !:ICE; aod from
both takin~ Ihe eOIllOlOO are BC, there will ret:lain !\ B
equ.lllo CD, i. (. the Aogle BEA equ,¡J tO the angle
CED (by arto29.) ' Mler the r.IOle maoner we m.y
prove, tlldl the .ngle BEC is cqual 10 Ihe anele AE O.
33· Lines which are equally cliO,ol from one aoolher.
are eallcd /,a, . /I.-1 lillo ; as AB, CO, ng. IS.
H · lr a lio~ GH erof"wo pm llds AH, CO, (fig. I?)
Iheo the ex!Croal opp~file an~les are equal , .'Z. GEB
equallo CFH alld AEG equallo HFO. For fioee Afj
and CO are pMdlld tOone anolher, Ihey may be con–
as one broad lioe, .od GH erolling il; then Ihe
vertical or oppofile ao~les GEIl CFG are equal (byan,
32 .), as alfo AEG aod llFO O)' the fame.
35. If a Jioe GH erofs I\YO pmllds AH CD, theo Ihc
alteroale ancles, !liz . AEF and EFO, or CFE and FEIl,
are equ,l; thal is, the . 0r,le AEF is equal tO the anl:le
EFO, aod Ihe aogl~ cn: is equa.110 the angle FEB.
for GEB is equal tO AEF (by arto32)' and CFH i, e·
qual tO ErO (by the f. me); but GEIl is equal 10 CfH
(by the laH) ; thetefore AEF, is ¡qual tO EFO. The
f.m~ way we may rrove FEI3 equdllo EFC.
36. If alioe GH erofs t\Vo pmllcl lin~s AB, CD,
Ihen .he' eXleroal aogle GEB IS equ.1 to the inlcrnd
oppolite one EFO, or GEA equal lO CFE. For the
angle AEF is equal to Ihe ancle ErO (by the Idl); bUI
AEF is equal tO GEl! (by arto 32. ), thuerore GEB is
~quallo EfD. Th~ fame v:ay \Ve mayprore AEG equal
10 CFE.
37. If a lioe Gil erofs 111/0 pmllel lio~s AH CO,
Ihen Ihe fumof the I\VO inte.o.1 ansles, .iz. BEF and
OFE, or AEF aod CFE, are equal 10 IWO righl angles ¡
for fince Ihe angle GEH is equal to the angle EfD (by
3rt 36.), 10 bOlh add the aogle FEB, th, n GEH aod
l:!EF are equ,1110 BEF aod IJFE; but GEl:! aod BU'
are equa l 1011110 right angle. (by art. 31.) Ihmfore UF.f'
aod OFf are .Ifo .",uallo tlVOr¡ght angles. The fame
way we may prove Ihal AEF and CFE are cqual 10 tWO
right angb.
39. A figurc is .oy pan or fpace Uound~d by linn or
a lioe. If the bouodlOg lints be llrail, it IS e.lled am·
lilinMI g"", as A, fi . 20. if Ihry be cu"ed, il iscal·
Icd a (un 'in,al/gllrt, as II or C, fog. ? l . aod foe. 22 . ir
th~y be partly eUn'c Ilnes ar,d panly (Iran, It IS ,.U,d a
miol!f!un, 31 O, fig. ?J
35. '[11'! mon fi mplc rdlilinm figure is Iha! which i.
bOlloded by three rigl" lines, aod is calkod a Iri.ngl(, as
A, fig. 2).
40' T riaonles .r~ Ji.ideo iOlo rlill'acnt kinds, hOlh
with refpdl to lit", ro,J ao&lcs; lII"h rcfpt,1 10 IItetr
liJel Ihey re cunllnooly dmo,d inlo tllIee kinJs, ur:.
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