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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.
As Log. CB:s D:: Log. C D, :s B. [By Axiom II.
But here it must be known whether the Angle fought be
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
As half the Sum of the given Sides DC and BC,
So is the Tangent of half the Sum of the Angles at B and D,.
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.
The Sum of the Angles at B and D-64.82 Deg.
The half Sum
The half Difference
Being thus prepared,
As the Log. of the half Sum of CB and C D, 748.5
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.
Ass D: Log. CBC: Log. DB. [By Axiom II.
As the Sine of the Angle at D, 26.77 Deg.
To the Logar of the Side C B, 632 Parts;
So is the Sine of the Angle C, 115.18 Deg. (or 64-82) 9.9566369
To the Log. of the Side DB, 1270 Parts
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.
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
(1.) As the Log. of the Bafe D B,
Is to the Log. of the Sum of the Sides CD and CB:
And GD fubftracted from DB, fhall be equal to G B,
(1,) As Log. DC,
Is to Radius:
So is Log. DA,
To the Sine of the Angle DC A. By Axiome 2.
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
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,
Is to the Radius, Sine 90 Deg.
So is the Log. of A D, 772.32
To the Sine of DCA, 63.23 Deg.
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.