CASE I. Two Sides, D C, and C B, with the Angle at D, Fig. XII. oppofite to C B, given; To find the Angle at B, oppofite to CD. Cafe I.

PROPORTION.

As Log. CB:s D:: Log. C D, :s B. [By Axiom II.

But here it must be known whether the Angle fought be
Acute or Obtufe.

CASE II. Two Sides, D C, and BC, with the Angle C, com- Fig. XII. prehended by them, given, To find the other Angles B or D.

The given Angle C being above 90 Deg. viz. 115.18 Deg. Subftract it from 180 Deg. the Remainder is 64.82 Deg. Which must be made ufe of in the Calculation: And then, This is the

PROPORTION.

As half the Sum of the given Sides DC and BC,
Is to half the Difference of thofe Sides;

So is the Tangent of half the Sum of the Angles at B and D,.
To the Tangent of half the Difference of thofe Angles.

By Axiom III. Then,

To the half Sum of thefe Angles D and B, if you Add the half Difference thus found, you will have the Greater Angle, but being substracted, it will give you the Lesser Angle.

Cafe II.

O.P.E

The Sum of the Angles at B and D-64.82 Deg.

Their Difference

11.82 Deg.

The half Sum

32.41 Deg.

The half Difference

5.64 Deg.

Being thus prepared,

fay,

As the Log. of the half Sum of CB and C D, 748.5
Is to the Log. of half their Difference, 116.5
So is the Tang. of half the Sum of B and D, 32.41

2.8741918 2.0663259 9.8026808 11.8690067 8.9948149

To the Tangent of 5.64 Deg. This 5.64 Deg. is half the Difference between the Angle B and D- Which added to the half Sum of thofe Angles 32.41 Deg. gives 38.05 Deg. for the Greater Angle at B. But being fubftracted therefrom, it leaves 26.77 Deg. for the Leffer Angle at D.

Fig. XII. CASE III. Two Angles C and D, with the Side C B, oppoCafe III. fite to D, given; To find either of the other Angles.

PROPORTION.

Ass D: Log. CBC: Log. DB. [By Axiom II.

OPERATION.

As the Sine of the Angle at D, 26.77 Deg.

9.6536081

To the Logar of the Side C B, 632 Parts;

2.3007171

So is the Sine of the Angle C, 115.18 Deg. (or 64-82) 9.9566369

12.7573540

To the Log. of the Side DB, 1270 Parts

3.1037549

CASE

CASE IV. Two Sides, DB and C B, with the Angle B, com- Fig. XII. prehended by them, being given; To find the Side CB.

1.) Find the Angle C, by the Third Axiome.

2.) Find the Side C B, by the Second Axiome.

Cafe IV.

CASE V. The Three Sides DC, CB, and D B, being gi- Fig. XII. ven; To find the Angle at D.

Cafe V. To the refolving of this Cafe, Two Operations are required. The Firft, To find the Segments of the Bafe D G and GB, by these

PROPORTIONS.

(1.) As the Log. of the Bafe D B,

Is to the Log. of the Sum of the Sides CD and CB:
So is the Log. of the Difference of thofe Sides DE,
To the Log. of DG. By Axiome 4.

And GD fubftracted from DB, fhall be equal to G B,
and half G B, is equal to A G or A B. Then,

(1,) As Log. DC,

Is to Radius:

So is Log. DA,

To the Sine of the Angle DC A. By Axiome 2.
And A G and AB together, are equal to GB; and therefore,
the Angle at B, may be found in the fame manner.

As the Log. of the Bafe DB, 1270

Is to the Log. of the Sum of CD and C B, 1497 So is the Log. of the Difference of CD and CB, 233

To the Log, of the Segment DG 274.65

H

Fig. VI.

Subftra& D G 274.65, from the whole Bafe D B 1270, the Remainder will be G B 995.35; the half whereof, 497.675, is equal to A G or A B.

As the Logar. of DC, 865,

Then,

Is to the Radius, Sine 90 Deg.

So is the Log. of A D, 772.32

To the Sine of DCA, 63.23 Deg.

2.9370161

IO.

12.8877973

9.9507812

Whofe Complement to 90 Deg. is 26.77 Deg. for the Angle C D B.

In the fame manner; may the Angle at B be found, by, firft, finding the Angle A C B, which you will find to be 51.95 Deg. Whofe Complement to so Deg. 38.05 Deg. is the Quantity of the Angle at B.

The End of the Five Cafes of Oblique-angled Plain Triangles.

JANUA