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then shall the Triangle

be made as required.




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2. Let the Triangle be at the Periphery. Describe i the Primi

tive BDEG; set 36° 15' of Chords A off from

B to A, and draw the Right Circle AOF;


* then with the Half: Tangent of 61% 30', describe the Parallel


HCI, and thro’ the Point

of Intersection C, draw the Oblique Circle then is. the Triangle

made as required.


1. To find the Hypothenuje B C. Soc.

2. To find the Angle C., Both these Parts are found, as in the foregoing Cafes, and B C will i=1 442 52's and Ç = 56° 57.

3. To find the Angle B. In Fig. 1, EF = 42° 34' is the Measure thereof on the Chords ; and in Fig. 2, DK = to the Number of Degrees on the Tangents from 90°.

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Having described the

Primitive BEFG draw the Oblique Circle

BHF, to make

make an Angle with the Primitive

la of 42° 34'; then by (Problem 3,) : draw the Oblique Circle

CAD to make an Angle at the Primitive of

56° 57', So will the Triangle

be constructed as required.


1. To find the Hypotbenuse. BC sa o This measured on the Chords will be found to be 44° 52'.

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2. To find the Leg A B. A Ruler laid from the Pole a of the Oblique Circle B H F, and the Angular Point A, will cut the Primitive in d; then B d measured on the Chords, will be found 369 15'.

2. To find the Leg A C. Lay a Ruler from the Pole b of the Oblique Circle Call, and the Angular Point A, and it will cut the Primitive in the Point C; then C c measured on the Chords, will be found 288 30'.

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The ninth and tenth Methods of solving

Right-angled Spherical Triangles, by the Orthohraphic and Stereographic Planisphere.

HE Nature, and Manner of Construction of T both these Instruments hath been already fully

taught in former Chapters ; it now remains, that their Use be shewn in folving Right-angled Spherical Triangles, for which they are very apt (especially the Latter ) if made large enough for the Purpose.

In order to this, it must be understood, that each Planisphere is to be curiously graduated in its outer Limb, or Periphery ; that the Semidiameter of the Orthographic Planisphere, or Analemma, is to be graduated both ways from the Center, as a Line of Sines ; and that of the Stereographic Planisphere as a Line of Half-Tangents. Also, that on the Center of each, there must be placed a Label or Ruler

graduated as is the Semidiameter of the Instrument to which it belongs, in order to be set to contain any given Angle with the Semidiameter. Lastly, there must be all the Meridians, and Parallels described on each Planisphere for every Degree of the graduated Dia

or lome Method for describing such a Meridian and Parallel as is proper to any particular Occasion.

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Having given a Description of those two PlaniBheres ; I have thereto subjoin'd a Figure of each ; ii which, that I might represent the Triangle in the fregolrg Synopsis (which I have 'bitherto made ule 0?) I have placed the Label B L to 42° 34' on the Periphery, and described the Meridian which is 36° 15. diftant from the Center B, viz. DAG; and also the Parallel E F of 2° 30'; so that thereby on each Planisphere is constituted the Triangle aforesaid A B C ; those being sufficient to Mew the Use of these Instruments, I have omitted drawing the other Circles which fill up and compleat the fame for universal Ule.

As the Manner of using either of these Planispheres is the fame, so I have here treated of both those Methods together in one Chapter, the Directions which ferve to the one, serving also equally to the other.

And indeed were these Instruments made funciently large, viz. of 1, 2, or 3 Feet Diameter, and furnished with all its Circles, great and small, it would be a moft easy and expeditious Manner of solving Triangles ; and I question not but the Ingenious Student would soon be convinced, that the Use would abundantly compensate for the Pains and Time expended in making them thus large, especially the Stereographic Planisphere, which is pretty easily made of üny Size. Considering also that either of them may be contrived to answer diverse other Purposes in Astronomy, &c, at the same time.

But I proceed to their Use in folving the fix Cafes of Right-angled Triangles thereby; wherein the Reader must observe, that the Directions equally relate to each Planisphere, the faine Parts on each, being marked with the line Letters.


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