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To draw a great Circle perpendicular to another given great Circle.

In this Problem are four Cases.
Cafe 1. To draw a Circle at
Right Angles to the Primi-

H tive

AEBF. Thro the Pole (or Center) C, draw any Right Circle EF;

c and it is done as required.

B Case 2. To draw a Circle per

P pendicular to a Right Cir

I cle

E F.


A, B, Through the Poles draw the Diameter

and it is the Circle required. Cafe 3. To draw an Oblique Circle perpendicular to a Right Circle, as

EF. Thro' the Poles thereof

A, B, describe the Oblique Circle

ADB; and it is done as required. Case 4. To draw an Oblique Circle perpendicular to an Oblique Circle, as

ADB. Find the Pole thereof then draw any Diameter

HII describe a Circle thro' the three Points H, P, I,

and that is the Oblique Circle required.

A B;

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P ;

Note, If the Point of Intersection, e, be limited, then

draw a Circle through e and a, by Prob. 4.


H 2



To graduate any leffer Oblique Circle.


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Let the Primitive

R Circle be ABCD, and the Circle IKLM be the Projection

of the lesser Circle FG,

P which is Oblique to

D the Circle AC let the Pole of the Oblique Circle 2,

P be projected into P; let it be distant from its nearest Pole


N T by OZ = 50°; and from its farthest Pole by OH = 130 with the Half-Tangent of (that is, with the Tangent of

65° ;) describe the Prick'd Circle

QRST, divide this Circle into any Number of equal Parts,

in and laying a Ruler from

P, to the Points

T, a, b, c, it will intersect the Oblique Circle in the Points 0, P,9,r; then will the Arches op = Ta, pq = ab, qr = bc, &c. And thus may the whole Oblique Circle I K L M,

be graduated as required.

1309 ;

a, b, c, &c.

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: C.


0; Basi

c H A P. VI. of the Stereographic Projection of the

Sphere on the Plane of the General Meridian ;* of the Horizon of the Ecliptica

of the Equinoctial ; of a Small Circle ; of ... an Oblique Circle. 1. To project the Sphere on the Plane of the

General Meridian.
N the Center E, describe the Primitive

Circle A B C D, representing the General Diameters AC, and BD; the first of which, viz. A C, represents the Equator ; which, because it is ac Right Angles with the Meridian, is projected into a Right Line. The other Diameter, B D, is the Axis of the Sphere, and its Extremities B, D, the Poles thereof, North and South.

2. If the Primitive Circle be made to represent the Solftitial Colure, then B E D shall represent the Equinoctial Colure ; and the Right Line & Eve is the Projection of the Ecliptic ; which is thus divided ; Take the Half-Tangent of 30° ( or one Sign ) and set it both ways from the Center É, and it will cut the Northern Signs 8 and 1R ; and the Southern Signs # and m : Thus the Half-Tangent of 60', shews the Signs A and ; and image and ; and thus may each

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Sign be also divided into its proper Degrees ; the Points of Cancer S, and Capricorn W, being 902 distant from r and in the Equinoctial Colure, are projected into the Primitive, 23° 30' above A, and belo7 Ç.

3. The Hour Cirdes, or Intermediate Meridians, as B 15 D, B 30, D, &c. are all drawn by Problem 8, Case 3, and Theorem 7 ;. That is, by setting the Half-Tangents of 15°; 30°, 45°, &c. both ways from the Center on the Equinoctial, which gives the Points of Intersection therewith, and the Secant Complement fet from those Points of Intersection (or the Tangent Complement set from the Center E) upon the Equinoctial, extended when there is occasion, will find the Centers, on which these Circles are described, passing thro' the Intersection and the two Poles B, D,

4. The Parallels of Latitude, ( among which the dotted Lines denote the Tropics and Polar Circles, on each side the Equator, ) are all drawn by Pros blem 7, Cafe 2, and Theorem 8; That is, by cutting off the Half-Tangents of 10', 20', 30°, &c. from E the Center, towards either Pole B, and D; and thus you will have the Intersection ; the Secant's Complement set from the Center E on the Diameter BD extended, (or the Tangent's Complement set from the Interfections) will find the Centers of those Parallels, on which they are to be described.

5. This Projection may be made for any Latitude, and then the Prime Vertical, aud Azimuch Circles ; as also the Horzson of the Place, with the


10 60 50 40 30

30 40 50 60 70 80 B



The Projection of the Sphere on the Plane of the general Meridian

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Circles of Position ; also the Circles of Longitude and Latitude for the Stars ; but they would entirely perplex the Scheme, and confound the Eye ; and therefore are omitted ; they being drawn in the same Manner as those above described.

6. By this Projection all the Parts of the Globe may be laid down, and thus a general Map of the whole World is constructed ; but because the projected

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