116.5

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,

say,

As the Log. of the half Sum of CB and CD, 748.5

2.8741918

Is to the Log. of half their Difference, 116.5

2.0663259

So is the Tang. of half the Sum of B and D, 32.41

9.8026808

11.8690067

To the Tangent of 5.64 Deg.

8.9948149

This 5.64 Deg. is half the Difference between the Angle B and D,- Which added to the half Sum of those Angles 32.41 Deg. gives 38.05 Deg. for the Greater Angle at B. But being fubstracted 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. CB::SC: Log. DB. [By Axiom II.

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

CASE IV. Two Sides, DB and CB, 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.

To the refolving of this Cafe, Two Operations are required. The First, To find the Segments of the Base DG and GB, by these

PROPORTIONS.

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

Is to the Log. of the Sum of the Sides CD and CB:

So is the Log. of the Difference of those Sides DE,

To the Log. of DG. By Axiome 4.

And GD substracted from DB, shall be equal to GB,
and half G B, is equal to A Gor A B. Then,

(1.) As Log. DC,

Is to Radius:

So is Log. D A,

To the Sine of the Angle DCA. By Axiome 2.
And AG and AB together, are equal to GB; and therefore,
the Angle at B, may be found in the same manner.

OPERATION.

The Sum of the Sides CD and C B, is
Their Difference is

The Bafe BD is

Being thus prepared, say,

As the Log. of the Base DB, 1270

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

To the Log. of the Segment DG 274.65

H

Cafe V.

Fig. VI.

Substract D G 274.65, from the whole Base D B 1270, the Remainder will be G B 995.35; the half whereof, 497.675, is

equal to A Gor A B.

As the Logar. of D C, 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

10.

12.8877973

9.9507812

Whose Complement to 90 Deg. is 26.77 Deg. for the Angle CDB.

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

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

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JANUA