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Prob. 47. To make an Angle of 36°. 40′ at the Point B in the primitive Circle.

From the Point B draw the Diameter B D, and at right Angles to it draw A C; from C fet off 36°. 40', by the Line of Chords, to r, and draw Br, which will interfect A Cine; thro' Be D, by Prob. 17, draw Be D, making an Angle of 36°.40 with the primitive Circle in the Point B. But here it will be proper to obferve, that, when an Angle is fignified by three Letters, the middle Letter is that at the angular Point; as, here it is the Angle BC that is the Angle required, for the Angle O Be is 53°. 20', being the Complement of the other to go°.

Prob. 48. To draw a right Circle perpendicular to the primitive Circle.

This is no more than to draw a Diameter, as AC in the laft Figure, thro' O, the Center or Pole of the primitive Circle.

Prob. 49. To draw a right Circle perpendicular to the right Circle A C.

Find B and D, the Poles of A C, by Prob. 38; thro' which drawing the right Circle BD, it will be perpendicular to A C, and pass thro' the Pole of the primitive Circle.

Prob. 50. To draw an oblique Circle perpendicular to the right Circle A C.

Having drawn the primitive Circle, and the right Circle A C, draw BD perpendicular to A C, by Prob. 49; fet one Foot of the Compaffes any where in A C, except in O, extend

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the other Foot to B or D, and draw BxD, which is perpendicular to A C, because it paffes thro' its Poles, B and D. But, when it is required to draw a Perpendicular to a right Circle, to make a given Angle with the primitive Circle, as of 49°, in this Cafe, take 41°, the Complement of the given Angle, from the Scale of Half-Tangents, and lay from O to E; then, by Prob. 17, draw

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an oblique Circle thro' the three Points B, E, and D, which will be perpendicular to A C, and make an Angle, CBE, of 49° with the primitive Circle.

Prob. 51. To draw an oblique Circle, m pn, perpendicular to another oblique Circle, P QR.

By Prob. 43, find p, the Pole of the oblique Circle PQR; then taking any two oppofite Points in the primitive Circle not in a right Line with p, as m and n, draw a Circle, by Prob. 17, to pass thro' the three Points m, p, and n, which will be perpendicular to the given oblique Circle PQR.

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P

R

Prob. 52. To draw an oblique Circle, reps, perpendicular to the oblique Circle PQR, to pass thro' e, any given Point in the oblique Circle PQR.

Draw the Diameter PR, alfo DE perpendicular to it; find P, the Pole of the oblique Circle, as before, and draw Rp; at R erect q R perpendicular to Rp, and extend it 'till it interfects the right Circle D E produced in 9; thro' q, e, and p, a Circle being drawn, by Prob. 17, will be perpendicular to PQR, and pass thro' the given Point e.

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Prob. 54. To measure a spherical Angle, SA E, when the angular Point A is in the Periphery of the primitive Circle.

Draw the right Circle A O B paffing thro' the Pole of the primitive Circle, alfo SOT perpendicular to AOB; then measure OE on the Scale of Half-Tangents 41°.28", which, being fubtracted from 90°, leaves SE 48°. 32', the Measure of the Angle S AE.

Prob. 55. To measure a spherical Angle, BIE, made by the Interfection of the oblique Circle EIF with the right Circle BD at the Point I,

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Prob. 56. To measure a spherical Angle, Em B, made by the Interfection of two oblique Circles, Em F and Bm D, in

the Point m.

Find x, the Pole of the oblique Circle B m D, and L, the Pole of Em F, as before; then draw m x G and mL H; and GH, being measured on the Line of Chords, is 48°. 30', the Measure of the given Angle Em B.

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But this Rule holds only when the two oblique Circles, Em F and Bm D, have their convex Parts the fame Way: For, if it were required to meafure the Angle Sa N, made by the Interfection of the two oblique Circles PaS and Za N; having, by Prob. 43, found p, the Pole of the oblique Circle Z a N, by the fame Prob. find x, the Pole of the oblique Circle Pa S,

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which is on the fame Side the Center of the primitive Circle with p, the before-found Pole of Za N; a Ruler, being laid from the angular Point a to p, will cut the primitive Circle in F, and, being laid from the fame Point a to x, will cut the primitive Circle in E; then the Distance E F, being meatured on the Line of Chords, is 72°. 23, the Measure of the given Angle Sa N.

Prob. 57. To draw an oblique Circle, qas, to make an Angle, A a q, of 39° with a right Circle, A C, and to pass thro' a given Point, a, in any Part of the right Circle.

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C

Draw BD perpendicular to A C,. thro' O, the Center of the primitive Circle; lay a Ruler from D to a, which will cut the primitive Circle in E, and draw OE to a convenient Length; at C erect Ct perpendicular to A C, 'till it meets EO in t; take Ct in the Compaffes, and fet it from the A Center O to e; at e erect cec perpendicular to A C, and extend it both Ways to cc; take 60° from the Line of Chords, fet one Foot of the Compaffes in a, and with the other draw an Arch, wxz; then take 51°, the Complement of the given 39°, fet it from x to r, and draw a r, 'till it meets cec in c; on the Point c, with the Radius ca, draw the oblique Circle qa s, and the Angle A a q will be 39°, as was required.

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D

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Prob. 58. To draw an oblique Circle, es I, to make an Angle of 80° with another oblique Circle, B c D, in c, any given Point in Bc D.

Find p, the Pole of the oblique Circle Be D, by Prob. 43; lay a Ruler from p to c, which will cut the primitive Circle in m, and from m fet off 90° to n; a Ruler, being laid from p to n, will cut the oblique Circle Bc D in x; a Ruler, being laid from to x, will cut the primitive Circle in y, from whence lay the given 80° to z; a Ruler, being laid from c to 2, will cut the right Circle A C in s; thro' either of the Points

G

E B

m

H

C

DF

e or s, as c, draw a right Circle, EF, which cross at right Angles with the right Circle G H; then draw the Line H, and at H erect HI perpendicular to H, which continue 'till it meets E F produced in I; laftly, thro' the three Points c, s, and I, by Prob. 17, draw the oblique Circle cs I, paffing thro' c, and making an Angle, se D, of 80° with the oblique Circle Bc D, as was required.

Prob. 59. To draw an oblique Circle, F a E, thro' a given Point, a, fo as to make an Angle of 40° with the primitive Circle.

Take the Tangent of 40° in the Compaffes, fet one Foot in the Center of the primitive Circle, and draw an Arch, e; then, with the Secant of the fame Angle of 40°, fet one Foot of the Compaffes in the given Point a, and describe an Arch interfecting the former in e, which is the Center of the required Circle; upon which, with the Extent ea, describe E a F, which will

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e

B

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meet the primitive Circle in E, and make an Angle, a E B, of 40°, as required.

Prob.

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