reg. b c

E cample.

ac, 407.6 In the A abc, given

ab,168.612

Proportion.
The Angles being found by the 6th Cafe.

S. L b: Rad, :: ac : b c.
S, 67'. 30': S. 90° :: 407, 6:44.6.

Operation.
To the Ar.co, SL b 67o. 30' 0. 034385
add Logar. ac, 407. 6 2. 609659

.

Sum = Logar. b c 446.

2. 64.4044.
By. Gunter's Scale.
The Extent from S, 67° 30' on the Line of
To the Sine

90,00 $ Sines.
Reaches froin

487: 6 2 on the Line • To

446. o Sof Nuinbers.

of Oblique-Angled Plane Triangles.

Wherein there are 6 Cases, and all rcsolved by the three following Axioms.

Α Χ Ο Μ ΑΙ.

In all Plane Triangles, the sides are preportional to the Sines of their opposite Angles.

A XIOM III.

In all Plane Triangles, As the Sum of the Legs of any Angle, is to their Difference;

So is the Tangent of half the Sum of their opposite Angles,

To the Tangent of half their Difference.

A X IOM IV.

As the Pale, or longeft Side, is to the Sun of the other Sides;

So is the Difference of those Sides,

To the Difference of the Segments of the Bale.

CASE T

Fig. 61. Two Sides, and an Angle opposite to one of them given; the Angle opposite to the other required.

Example.
In the 4 bca, Scd=139
(c b=64

requir'd the given,

Acute L <d=22° 309

d

Proportion. cb:$,_d::cd : S,L b. per-Ax, 2. 64: S, 22°, 300 :: 139: S, 56151

Opera.

:

add the {5,24 22:30

.

Operation.
To the Ar. co, Logar. cb 64 8.193820
S . 139

2. 143142

9. 582840 Sum-Radius,=S. Lb, 56° 151 9.919 802.

By Ganters Scale, The extent from 64 Z on the Line of Num. Το

139 S bers. Reaches from S, 22° 30' on the Line of To S,

56-15 S Sines.

In the a bed, given cb=64 , }requir'd b. .

CASE II. Two sides and an Angle opposite to one of them given, the third S de reqnired.

Example
cd=139

=

Ld22°, 30'

Preparatione
By Case the first find the Angle b, and
knowing the Angle d, you will find the
Angle c to be icio 151.

Proportion, per Ax, 2:
S, L b: cd :: S, Lcib d.
S, 56. 15 : 139:: S, 101: 164.

Op'ration.
Ar.co.S, Lb 56° 15' 0.080554

2. 143142 S. LC

9.991594. Sim-Radius=L-g.bd, 164. 12,214870.

add the long cd, 139

By Guters Scule.
The extent from the Sine of 56° 151 to
TOI' 15' on the Line of Sines, reaches from
139 to 164 on the Lines of Numbers.

Note, That the Sine of 101' 15'; is the
Sine of the Complement to 180, d. grees

Sine of 71° 45'.

CASE III.

Za b,

cd

Two Argles and a Side given ; the other
Side sequired.

Example.
In Obl. a bed Sc=ior':159)

Sod
L d = 22. 30' requ.
given,

= 64 5
Proportion. per Ax 2.
Sild:cb:: SLcibd.
S. 22° 30':64:: S, CI 15:164.

Operation.
To Ar. co, SL d 22° 301 0.417160
add the SLC101°; 159
CIO

9. 991574 Log, cb 64

1.806 180

<

Sum-Radius= Log. bd. 164. 12. 214914

By Gunters Scale.
The extent from the S, 22° 33' to the Sine
101° 15' on the Line of Sines, reaches from
6+ to 164 on the Line of Numbere.

Proportion.
S.Ld:S, Lb :: cb: cd
S, 22. 30:9,56.15 :: 64:139.

CASE

C:A SE IV.

Two Sides and an Argle comprehended by them, given; the other Angles required.

Example
In obl. a bed, Sac=101',1517

sld
cb = 64
given,

Freq.2.6 : Preparation by the third Axiom.

Scd= 139

{

L.

cd =139
cb = 64

Sam=cb + cd
=

203.
Differ, cd-cb = 75.
From 180' 0" te the 3 Angles of a Triangle.
Substract:07, 15 to the given Angle.
Remain 78, 45 = to the Sum of the opposiee Angles.
Remain=3922=to half the Sum of the opposite
Angles.

PROPORTION. cb+cd :cdcb::T, TLL:T, ! X 4,4,.

1
203 : 75:: T, 3022':T, 16'52'

OP ER AT IO N.
To Ar, co Log. cd tcb 203.

77. 692504 cd-75

1.875061 39o. 22' 9.914172

I 2

Adding {1,7 Sum

Sum--Rad.=T; Diff.

16. 52,+ 9.481 737.

.}

}uba{ {} ={

+

56 15 3S

Sum

}-{