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Case 1.

BC = 44 52,
Given {

The Hypothenuse
and the Oblique Angle B = 42 34.

In order to folve this case, move the Label B L to 42° 34' on the Limb, and there let it abide fast ; then observe what Meridian passeth through the Poirt of 44° 521 C on the Label, fuppose D AG; then lastly observe the Parallel passing through the same Point, as ECF.

I. Then will that Meridian interf-et the Diameter or Equinoctial in A at 36° 15' = AB, the Base. .

2. The said Parallel will cut the Limb in E in 299 30 = AC, the Perpendicular,

3. To find the Angle C ; you must alternate the Legs A B and AC; making A C the Base, and AB the Perpendicular. And then apply the Label or Index to the End of that Perpendicular, and it will cut the Limb in 560.57' = C.

Cafe 2.

Given { The Hypothenuse

BC= 44 52,
AB = 36 15.


Move the Label upwards and downwards, till that Meridian or great Circle D. AG, interfecting the Equinoctial at Right Angles in A; shall also pass through the Point C of 440 521 on the Label, and there hold it fast. . Then

1. The Label will cut 420 34' on the Limb for the Angle B.

2. The Parallel F E passing through G shall be the Parallel of 280-301 AC.


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3. The

3. The Angle C is found by alternating the Legs AC, AB, as before,

Case 3:


The Leg
and adjacent Angle

AB = 36 153

B = 42 34.

Place the Label to the Quantity of the given Angle on the Limb, viz. 429 34'; and then to find Requisites,

1. Observe the Point C in which the Meridian pafsing through 36° 15' of the Equinoctial at A, intersects the Label, and you will find it to be in 44° 52' = BC, the Hypothenuse.

2. Observe the Parallel which passeth through that Point of Intersection C, and you will find it to be FE of 288 30' = AC.

3. The Angle C is found as in Case 1.

Cafe 4.

Given {

The Leg
and Opposite Angle

AC= 28 30,

B= 42 34

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Set the Label to the given Angle on the Limb, viz. 422 34', and there keep it fast ; then to find the Requisites,

1. See where the Parallel of 28 30' touches the graduated Edge of the Label, which you'l find to be in C at 44° 52' = BC, the Hypothenufe.

2. Observe where the Meridian passing through the Point of Interfection C, cuts the Diameter or Equinoctial ; and

you will see 'tis in A, at 368 15' =B A, the Base. 3. The Angle C is found, as in Cafe 1.


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Move the Label upwards ’till such time as it meets with the Point C, where the Parallel FE of 280 30' intersects the Meridian D A G of 36° 15', and there let it rest. Then

1. This point of Intersection C of the said Meridian and Parallel, will coincide with the 4452' on the Edge of the Label, and is the Quantity of B C, the Hypothenuse.

2. The Degrees and Minutes cut by the Label on the Limb of the Planisphere is the Angle B=42° 34'.

3. The Angle at C is again found as directed in Cafe 1.

Cafe, 6.

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This Case not being conveniently wrote on these Planispheres directly; must be reduced to Cafe 4 of a Side and its opposite Angle ; thus in the Triangle LDC, being opposite or vertical to the Triangle A B C of which the Angles are known, we shall have the Side L D, and the Angle LCD also known ; for the Angle LCD= ACB = 56° 57'; and the Side LD = 47° 26' the Complement of the Angle B, or Arch which is the Measure thereof. Consequently D C, the Complement of AC; LC, the Complement of BC; and the Angle LDC, the Complement of A B, may all be known by the faid Cafe 4, if C L be made Base, and L D the Cathetus or Perpendicular Leg.


Thus have I shewn the Use of Planispheres in this respect to be very easy and facile both to be understood, and to practice ; and here I end the Ten various Methods of folving Right-angled Spherical Triangles ; which may be also applied to Oblique-angled ones, when reduced to right ones as is taught in the next Chapter.


Of the Doctrine of Oblique-angled Sphe

rical Triangles ; a Synopsis of the Six Cases thereof i The Method of Resolving them by means of a Perpendicular; and Rules for the Solution of all Ambiguities attending them in any Cafe.

HE Nature or Doctrine of Oblique Spherical
Triangles is to be well understood

only who are well acquainted with the Circles of the Sphere, the Doctrine of Projection, and the Demonstration of the foregoing Trigonometrical Theorems ; these Things are the Grounds and Rudiments of this Part of our Art ; in all which the young Student is presupposed to be considerably well versed, 'ere he enters on the Doctrine of Oblique Spherics.

In Oblique Trigonometry we labour under the same Misfortune, with regard to the uncertain, promiscuous, and undue Limitation of the Number of Cafes, as


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