807 / 868
~
.
9<>6
T R
G O N
i. lO Ihe lancenl of C
53°,
3)'
IO.q212
fa
is .'\C
146
~ . I 61H
lO
Al:!, 17.5
2°7°00
C~
..
IV The two legs being glven, to fiad the .ng1<•.
E XA"PL' . In Ihe "i.ngle A
I:!
C,
(ib:d
nO 5 ) fuppofe
A
B 94 .nd B (' 56, required Ihc .nsle, A . nd C.
I
C~()mclr"(I/,, :
Draw A B equal co 94- (rom any Jioe
of ec¡u.d p"" ; Iheo frnm th< poinl
B
r,ife B C perpendi.
cul., lO A
n,
and takc B C froOl Ihe fo rmer Ji.e of equ.l
pans equal
to
56; Jaflly, joio Ih< points A aod C wilh lhe
llraish t Jine A C:
fa
Ihe Iriangle i. co. llruaed. aod the
aD~le.
may 'be 'meafured by • Itne of chord•.
11.
oy
calcula/ion:
t'tdl,
fuppofing Al:! ,hc radius. we
have this analogy,
v;z.
AH
:B C:: R : T.
A.
i
e.
as A B
94
1.9"
q
i, to BC
56
1.74819
fo
lS
lhe
radius .
90°
10.00000
'o ,he tangeot of
A.
30° 47'
9· 77
506
Seeoodly. making BC the radius. we h ave thi, propor–
tioo,
viz.
BC : BA :: R : T.C.
i.
e.
as BC
56
1.74819
iSlO
AB
94
1.973 13
fa is the radius
90°
lo.cooeo
tO Ihe tangent of C
59° 13
10 . 22494
CASE
V.
The hypotheoufc and ooeoftheleg' giyeo. tO
find the aogles.
EXAMPLE. lo the
triaogl~
DEF.
(ihid.
nO 6.) (.ppofe
.he leg DE=83. and the hypothenu(e DF=I,6; required
,he
an~les
D and F.
1.
Ceometr;c411j:
Dra.,." the line DE=83 (rom any line
of equal pa", ; and from the point E raiCe ,he perpeo–
dicular EF: then take the length of DF=126, from the
fame lioo of equal por,,: a.d (etting one foot of your com–
pa/Tes io D . with the o.her croCs the perpeodicular EF io
E:
lallly. join D and F; aod the triansle being thus coo–
Jlrutled. the aogles may be meafured' by a liBe of chord•.
U .
oy
calcul./ion
:
Fidl. makiog DF che radius. we
lhall baye thi. proportioo,
viz.
D F : D E:: R: S. F.
;.
e.
a.
DF
126
í••
0
DE
83
lo is radiut
90°
(O
,he fine of F
41° 12'
S..ondly. by CuppoGng D E lhe radiu,.
Jowing analogy,
viz .
DE : D
F ::
R : See. D.
2.,H,037
.. 9
1
9°8
10 .00000
9 . 818 7 1
we have the
foJ ·
i . '
as
DE
83
1' 919°8
j,
.0
DF
126
2.10037
fo is radius
'0°
10.00000
'0
the feeant of D
48° 48'
10 .18129
Thi, may be donc without .he help of reeants: for fince
R :
Cee. :: Ca·S. :
R• •
he foregoing anology will become
chis,
(liz .
D F: DE :: R : Co-S, D.
which gives Ihe
r.meanCwer with .hal deduced from the
fi rll (uppofi.ion.
CUE
VI.
The 'wo leg' beins given. to 6nd the hypo.
Ihenufe .
EXAMPLE:
In
the triangle ABD.
(ibiá.
0° 7.) CuppoCe
,he Jeg A B;:; 64. and
B
D;:;
56 :
required the
~ypo.he(lufe.
o
M E T
R.
Y.
l.•
Ce.,,;etrical":
The conílruélioó of tbi• .care
i.
per–
rorm~d
the (ame wayas in
(he
fOllrth
care. and the length
of ,he hy po.henuCe is found by taking
i.
in your eompa/Te.,
and applying i. to the Came line of equal
parto
that .be tWQ
legs wcre takcn
fram.
11.
O, calcula/ion:
Tbi. caCe beiag
a
compound of .he
fourth and recond cafe,.
we
mu!!
6dl
fi ad .he .ogle. by ,he
founh, thus:
.
AB:DIl::R:T.A.
i.
e.
as
Ihe leg AB
64
1.80618
is to lhe leg DB
56
' -74819
fo is the radius
90
10.0000'0
'0
Ihe tangen. of
A .
410 11'
9.9420.
T hen by .he Cecond caCe we fi od .he hypotheauCc. requí–
red .hus :
S.
A : R :: B D:A
D.
i . e.asth.fineofA
41 °. 11'
9.8'8H
¡,
to the radiul
90°
10.0000<)
Co
i. ,he leg B
ti
56
1.74819
" 0
the hypo.h Al}
SS.05
1.92965
Thi, e•
.re
may alfo be
Col
ved after .be following a..n–
ner.
viz.
From ,wice .he log..ilhm of Ibe greater
fide AB
3.6123'6
fubtratl .he
logari.hmof the le/Ter
fideBD
1.7481?
and there remains
1.86417
.he logarilhm of 73. 15; to 'which .dding the le/Ter fide
BD, we {hall have 189. 1S. whoC.
logari.hmi,
~ . 1I 09J
to which add the logari,hm
of
tbe lelfer
lide
B
DI. 748 19
and !he Cum will
b.
the half of which i,
lhe loga rithm of the hypotheoufe requi,ed.
Or it may be done by adding the fquuc of the t"o fideo
. ogether, aod .akiog the 10garithIU of ,ha. CUID, .he half
of which is .be logarithm of che hypocheauCe required:
thus, in
,he
prerent
care,
the Cquare of A
B
(61) i, 4096
che Cquare of B D (56) i. 3136
Ihe Cum of theCe Cquare, i, 72 32
the
Jo~arilblU
of "hich i, 3.85926
the half of whieh
¡,
"92962~
(O
.he logarithm of 85 ..°5. ,he length of ,he bypotheD"re
required.
.
C~
.. VIL Tbe
hypotheo~fe
.nd one of .he legs peinc
giveo, to 6nd the o.her leg.
EXHIPL6. In .he triangle BGD.
(ibiá.
nO 8.) ruppo!e
the lec BG=87••nd the hypo!henuCe
BD=14~ ;
requirod
the leg DG.
1.
Geonut,ical" :
The conllruétion here
iJ.
the rame
&1
in care V .•he Carne .biog' beiog giyen: and the leg DG i,
fouDd by taking i" leog.h in your eompa/Tes. aod applying
,hat to the .(ame li.e of equal
pUtl
.he o.heu were
tak""
from.
1I.
oy
calcul.t;on :
The Colutioo of this cafe dependo up–
on the
,{t
and 5th; and b,ll we llluílDod .he oblíque angles
by caCe 5th ,hu. :
D B : BG : : R :
S.
D.
i. (,
as
Ihe hypolh. D B
14'
:")229
js