CASE V. The Angles B and C, and the Leg. A C, given; Fig. XII.

To find the Hypotenuse B C.

Cafe V.

PROPORTION.

Ass B: Leg. A C:: Radius: Hypotenuse BC. [By Axi. II.

CASE VI. The Hypotenuse BC, and the Leg. A C, given, Fig. XII.

To find the other Leg. A B.

PROPORTIONS.

1.) As Leg. BC: Rad.:: Leg. AC:s B. [By Cafe I. 2.) As 1 B: Leg. AC:: Rad. : Leg. A B. [By Cafe L.

Cafe VI.

i

L

:

Fig. XII. CASE VII. The two Legs, B A and C A, given; To find the
Cafe VII. Hypotenuse B C.

PROPORTION S.

1.) As Leg. A B: Rad.:: Leg. AC:t B. [By Cafe I.
2.) Ass B: Leg. AC:: Rad. : Leg. BC. [By Cafe V.

The End of the Seven Cafes of Right-angled Plain Triangles.

CHAP. VI.

Of the Solution (or Mensuration) of Oblique-angled
Plain Triangles.

Fig. XII. T
HE Triangle which I shall make use of in the Solution of
Diagram the Five CASES of Oblique-angled Triangles, shall be
S. that noted with S; whose Sides and Angles are

:

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

PROPORTION.

is Log. CB: SD:: Log. CD, :s B. [By Axiom II.

But here it must be known whether the Angle fought be

Acute or Obtufe.

ASE II. Two Sides, DC, 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. abstract it from 180 Deg. the Remainder is 64.82 Deg. ✓hich must be made use of in the Calculation: And then, This the

PROPORTION.

is half the Sum of the given Sides DC and BC,

Is to half the Difference of those Sides;

o is the Tangent of half the Sum of the Angles at B and D,.

To the Tangent of half the Difference of those Angles.

By Axiom III. Then,

(

To the half Sum of these Angles D and B, if you Add the half

Difference thus found, you will have the Greater Angle; but eing substracted, it will give you the Leffer Angle.

Cafe II..

OPE

Fig. XII. CASE VII. The two Legs, B A and C A, given; To find th Cafe VII. Hypotenuse B C.

PROPORTION S.

1.) As Leg. A B: Rad. :: Leg. AC:t B. [By Cafe I.
2.) Ass B: Leg. A C :: Rad. : Leg. BC. [By Cafe V.

Is to Leg. A C, 142.72 Parts

12.15448

So is Rad. Sine 90 Deg.

To the Hypotenuse B C, 274.93 Parts

The End of the Seven Cafes of Right-angled Plain Triangles.

10.

243929

CHAP. VI.

Of the Solution (or Mensuration) of Oblique-angled
Plain Triangles.

HE Triangle which I shall make use of in the Solution

XII. TH

Diagram the Five CASES of Oblique-angled Triangles, shall S. that noted with S; whose Sides and Angles are

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

PROPORTION.

As Log. CB:SD :: Log. CD, :s B. [By Axiom II.

But here it must be known whether the Angle fought be

Acute or Obtufe.

CASE II. Two Sides, DC, 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. Substract it from 180 Deg. the Remainder is 64.82 Deg. Which must be made use 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 those Sides;

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

To the Tangent of half the Difference of those Angles.

By Axiom III. Then,

To the half Sum of these 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 Leffer Angle.

Cafe.II..

こ

OPE