811 / 868
'10
T R
G
o
N O M E T R Y.
TH Eo"M 11. In any righ,.angled fpheneal triangle
AIlC
(ibid.
nO
3,,)
it wil l be.
A.
radius i. tO lhe eo rone
of
one leg, fo is the co·fine of d,c other leg
lO
the co.fine of
'he,
hypothenofe,
THEO aE M V , In .ny fpheri ca l ,rriangle ABC
(ihid.
nO
-4
and
5.)
It will be,
as
the co.ungent of
h.lfthe fu m of
half Ihcir dlfferenee, fo i, the eo-tangen, of
h.lfthe b.felo
the
t.n~en\
of the diClance
(DE)
of the perpendicular
fr.omthe middlc of the baf•.
Henee, if ,wo righ' angled fpherieal triangleo ABC, CBn
(ibid,
nO
2 )
ha". the fame pe rpendicul.. BC. ,he co, rones
of ,heir hypothenufe. will be 'o
e.ehother, diretlly, a. the
eo .fines of their ba[(s.
Since the
I~(l
prOpOrtlon,
by
permufa.tion, becomcs co-
AC+IlC:
AC-llC:
tang ' - -.2-- eo.tang.
AE::
toog'--2-- tango
T u E
o
R" I
Ill.
In ony fpherieal tri.ngle i, will be, AH'·
(;Jiu, is to (he Cine of eitl'ler ;logle. fo is the
co-fine
of the
; dj"cent leg to
che
co-fine of
lne
oppoGte angle.
Hencc, in rigllt.aoglC'd fpherical tr¡:'Ingles. having the
r ..
me perpendicul:tT, (he
co-(j~,
of the
anglc, al
(he
bale
\Vdl
t,c:
16 c3ch other, direllly. as (he fines of
lhe
'fCrtic¡dangles.
DE, and as the tangen'ts of any
tWO
arches are,
inYCTfdy,
as thei r co-rangents ;
it
follow" therefo re, that
tango
AE :
AC+HC
AC-IlC
tan~.
--2 '- -:
:t3ng. ----;--: uná!.
DE;
or,
that
THEOREM IV. In any righ,.angled fpheric.1 " iangle i,
will be, Aocadius i. 'o Iheco,Gne of ,he hypothenufe , fo i.
thl.!
tang=nt of
either angle to
lhe
co·tangent
orlhe
othe rangle.
As the (um of the lines of
tWo
u"equal archesj s to
thelT
d:íferenec, fo i. the tangent o( half the fum of thofc "ches
10
the ungent of halr
tht:ir difference :
and, as the
fum
of
Ihe co·fine, is 10 their
difference,
fo is the co.tangent of
hd; f tl>e fu", of the arehes to the tangent of
h.lfthe diffe.
renel! of rhe {ame arches .
the
ta.~ent
of h. lf the
b.feis'
'o the tangent of
h.lf,he fu ..
of the fides ,
a.
the tangent of half the d, ffcrtnce of the fideo
to the tangent of the d,n.nce of the perpendicular from the
middle ot' the bafe.
TH
<ORE ..
VI. In .ny fphcri cll trnnglc ABC
(ibid..
oC>
4 )
it wlli
be,
as the co·tangent of hal ( the fum of the
ang,les
al
the baCe, is
10
the tangent
oC
hi lf
tbeir differencc,
fo
15
the tangent of h<llf the ver'lcO\)
~ ngle
to t·he tangent
Qf
the .ngle wltieh the perpendicular cn
m.k..
with tbe line
CF bifetling the .ertical ongle.
I
The Solu,ion of ,he C.fes of right.angkd fpheric.1 Tri.ngles,
(i'id,
nO
3')
\
C.feI
G ioen
1
Sought
I
Solution
Tite hyp,
A C
and
I
The oppofito leg
I
As rad,us : line hyp. ,\ C:: fi ne
A :
Gne
on< ,nple
'1.
R C
RC
(bv
the forOler ,ar' of lheor .
L)
l IJe: Il) p
A
e
Cilld
Il'he
dOJ .l t:tlt
1c:g
I
riS
radlus: co·Cine ot A : :
taog.
A C.
one
.n~l~
A
A B
..ng . ,AB by the I.tter
p.rtof tbeo. t ,
The hyp.
AC
. od
I
rhe olher .ngle Ás radius : co·fine of AC :: tang, A:
one .ngle A
C
eo
tanC'
C
(by
theorem
4')
'4
The hyp AC aod
l
Tb< o,her leg
As co.rone AB: radius :: eo, fin e AC :
one
Ic~
AB
BC
co·rone' Be (by theorem 2,)
I
The hyp AC .od
I
The oppofi ,e
0 0
I
As fi ne AC : "d,us :: fine AB: hne C
5
one leg AB
I
gle C
(by ,he forme,- part of ,heorem
1,)
I
The hyp. AC . ad
I
The .dj. coot.o
I
A,-;.n¡:-:-AC: tlQg-:-AB : : rad,us :
CU·
6
one leg AB
gle A
fin, A
( ~v
theorem
1,)
I
One leg AB
~nd
the
I
r he
o
l/h.
( I~¡
I
r\S I . OIUS :
hnc.
r\.,lj ::
'tllugCDl
A :
ta.n-
7
adjaeent . ngl. A
BC
gen. BC (by theorem 4,)
8
I
Une log AR .nd the
I
r lte
oppofi~
I
A.
~diüs:
fine
A;:
c·o: rone of AB: co
.....::....J
.dj.een,
an~le
A
gle C
ftne nf C (hv the..,,", 3' )
One leg AH
.M
th<
I
The hyp,
l A. eo.rone of A: ..dI", :: tongo
AB :
9
. dj.cent .ngle A
AC
tango AC (by ,heorem
L)
I
One leg BC anel ,he
I
The o,her leg
1
As ..ng, A : tango BC :: radlus : fi ne
le
onpofite .ngle A
AB
. ~<3':.-cthc:,e::::o,=r::..e_m-,,:4':")'-_-;'--;,...,._1
I
One leg
He
dnd Ihel'Che adJ.,'ent.o
I
A. eo.rone HC : radiu,:: eo·fine of A :
11
npporitc
~"gle
A
gl.
e
fin .
e
(8y
th"'orr m
3.)
\
One leR BC and the
I
The hyp,
I
As fi n. A: lin, Be:: " 'hus: ti". AC
t
2
oppnrote
.n~le
A
A C
U'y Ih<nrrnN t )
\
8 0th leg'
I
T he hyp,
lA. ..d,u.: <u.hne
A I.! ::
co.fine BC :
' 3
AIl . nd BC
AC
eo ron. :\C (by theorem
2 )
I I
BUlh leg'
I
An .nele, fuppofe
I,\,
rooe A B: ..diu, :: tang, -BC : !dng.
' 4
AH .nd BC
A
''\ (by thcorem
4. )
\ I
Both
dn~le,
I
A
leg. f.p pole
I
A<-fih~
A :
~';:fine·
C :: radiu,-;CO.fine
1
5
Á
and C
flB
Al!
(hy IIt<or.",
3 )
-61
Doth angles
I
T he h"l"
I
As IJn A: cu.tang. C : : radiu. : eo·
I _~_
.__
~
•.
nd
e
AC
fiot,· AC(bo Iheor<m
4)
N ote, T he
J
Oth,
1
Ilh, anel '2 th
~ar" 5
;trc .11l1blpUOUS; fin("c
It
cannOI
be
deh
rmlned
by thc c1ata, "hether A, B. C, . nd
A
C, be groater or lefs
,h, 90
dcgree, cach,
Th~