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Cafe 1.

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BC 44 52,

The Hypothenufe
and the Oblique Angle B = 42 34.

In order to folve this Cafe, move the Label B L to 42° 34' on the Limb, and there let it abide faft; then obferve what Meridian paffeth through the Point of 44° 52′ C on the Label, fuppofe DAG; then laftly obferve the Parallel paffing through the fame Point, as E CF.

1. Then will that Meridian interft the Diameter or Equinoctial in A at 36° 15' AB, the Bafe.

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2. The faid Parallel will cut the Limb in E in 289 30 AC, the Perpendicular.

3. To find the Angle C; you muft alternate the Legs A B and AC; making A C the Bafe, and AB the Perpendicular. And then apply the Label or Index to the End of that Perpendicular, and it will cut the Limb in 56°57′ = C.

Given {

Cafe 2.

The Hypothenuse
and the Leg

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, fhall alfo pass through the Point C of 449 521 on the Label, and there hold it faft. Then

1. The Label will cut 42° 34' on the Limb for the Angle B.

2. The Parallel F E paffing through C fhall be the Parallel of 289 30'



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3. The

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

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Place the Label to the Quantity of the given Angle on the Limb, viz. 42° 34'; and then to find Requifites,

1. Obferve the Point C in which the Meridian paffing through 36° 15' of the Equinoctial at A, interfects the Label, and you will find it to be in 44° 52′ BC, the Hypothenufe.

2. Obferve the Parallel which paffeth through that Point of Interfection C, and you will find it to be FE of 288 30′ = AC.

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

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Set the Label to the given Angle on the Limb, viz. 42 34', and there keep it faft; then to find the Requifites,

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′ B C, the Hypothenufe.

2. Obferve where the Meridian paffing through the Point of Interfection C, cuts the Diameter or Equinoctial; and you will fee 'tis in A, at 368 15′ BA, the Bafe.

3. The Angle C is found, as in Cafe 1.



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Move the Label upwards 'till fuch time as it meets with the Point C, where the Parallel F E of 28° 30′ interfects the Meridian D A G of 36° 15′, and there let it reft. Then

1. This Point of Interfection C of the faid Meridian and Parallel, will coincide with the 44° 52′ 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 Planifphere is the Angle B 42° 34'. 3. The Angle at C is again found as directed in Cafe 1.

Cafe 6.

Given both the Oblique Angles

Angles {


42 34, C = 56 57.

This Cafe not being conveniently wrote on thefe Planifpheres directly; must be reduced to Cafe 4 of a Side and its oppofite Angle; thus in the Triangle LDC, being oppofite or vertical to the Triangle ABC 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. Confequently DC, the Complement of AC; LC, the Complement of BC; and the Angle LDC, the Complement of AB, may all be known by the faid Cafe 4, if C L be made Bafe, and L D the Cathetus or Perpendicular Leg.


Thus have I fhewn the Ufe of Planifpheres in this refpect to be very eafy and facile both to be underftood, and to practice; and here I end the Ten various Methods of folving Right-angled Spherical Triangles; which may be alfo applied to Oblique-angled ones, when reduced to right ones as is taught in the next Chapter.


Of the Doctrine of Oblique-angled Spherical Triangles; a Synopfis of the Six Cafes thereof; 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 by those only who are well acquainted with the Circles of the Sphere, the Doctrine of Projection, and the Demonstration of the foregoing Trigonometrical Theorems; thefe Things are the Grounds and Rudiments of this Part of our Art; in all which the young Student is presupposed to be confiderably well verfed, 'ere he enters on the Doctrine of Oblique Spherics.

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


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