3. Ascensional Difference, is an arch of the equinoctial, between the sun's meridian, and that point of the equinoctial that rises with him, or it is the angle at the pole between the sun's and the six o'clock meridian.

6. Oblique Ascension or Descension, is the sum or difference of the right ascension and the ascensional difference.

7. Sun's Longitude, is an arch of the ecliptic, between the sun and first part of Aries, as TO.

8. Declination, is an arch of the meridian between the equinoctial and the sun, as OK.

9. Latitude of a Star, is an arch of a circle of longitude between the star and ecliptic.

10. Latitude of a Plane, is an arch of the meridian, between the equinoctial and the place.

11. Longitude of a place on the earth is an arch of the equinoctial, between the first meridian (Isle of Ferro), and the meridian of the place. And diff. longitude, is an arch of the equator, between the meridians of the two places, or the angle at the pole.

12. Hour of the Day, is an arch of the equinoctial, between the meridian of the place and the sun's meridian, as EK; or it is the angle they make at the pole, as EPO.

Ex. 1.-To project the sphere upon the plane of the meridian, for May 12, 1767. Latitude 54° north, at a quarter past 9 o'clock be fore noon.

I. By the Orthographic Projection.-Plate A, fig. 1.

Here we will project the convex side of the eastern hemisphere. With the chord of 60 degrees describe the primitive circle, or meridian of the place HZON. Through the centre C draw the horizon HO; set the latitude 54 from 0 to P, and from H to p, and draw Pp the six o'clock meridian. Through C draw EQ perpendicular to Pp for the equinoctial. Make ED, Qd 18° 5′ the declination May 12, and draw Dd the sun's parallel for that day. By Prop. 11, make OG (3 hours or) 48° 45′ the sun's distance from the hour of 6, then

is the sun's place. Through O, by Prop. 5, draw AL parallel to HO for the sun's parallel of altitude. By Prop. 7, draw the meridian POP and the azimuth ZON. Also the ecliptic will be a6 ellipsis passing through O, which cannot be conveniently drawn in this projection. Also draw the parallel Ss 18 degrees below the horizon, and where it intersects Dd is the point of day-break, if there is any. Now the sun is at d at 12 o'clock at night, and rises at R, at 6 o'clock is at G, due east at F, at a quarter past 9, and is at D in the meridian at 12 o'clock.

Draw GI parallel to HO. Then GR, measured by Prop. 10, is 27° 14', and turned into time (allowing fifteen degrees for an hour), shows how long the sun rises before 6, to be 1 49"; GI, measured by Prop. 10, gives the azimuth at 6, 79° 16. CR, by Cor. Prop. 10, gives the amplitude 32° 19′, and CF gives his altitude when east 22° 25. FG 13° 28′ (turned into time) is 54", and shows how long after

6 he is due east. IO is his altitude at 6, 14° 38'. AH 41° 53' is his altitude at O, or a quarter past 9; and OL, measured by Prop. 10 is his azimuth from the north at the same time, 120° 40. And thus the place of the moon or a star being given, it may be put into the projection, as at . And its altitude, azimuth, amplitude, time of rising, &c. may all be found, as before for the sun.

II. Stereographically-Fig. 2.

To project the sphere on the plane of the meridian, the projecting point in the western point of the horizon; with cord of 60, draw the primitive circle HZON, and through C draw HO for the horizon, and ZN perpendicular thereto for the prime vertical. Set the latitude from O to P, and from H to p, and draw Pp the 6 o'clock meridian, and EQ perpendicular thereto for the equinoctial. Make ED, Qd the declination, and by Prop. 12, draw DGd, the sun's parallel for the day. Draw the meridian Pop by Prop. 17, making an angle of 41° 15' with the primitive, to intersect the sun's parallel in O, the sun's place at 9'4. Through O, by Prop. 12, draw the parallel of altitude AOL; through draw, by Prop. 17, the azimuth ZON. And, by Prop. 12, draw the parallel Ssd 18° below the horizon, if it cut Rd, gives the point of day-break. And through G draw the parallel of altitude GI. Lastly, by Prop. 20, through O draw the great circle cutting the equinoctial EQ at an angle of 23° 30', and this is the ecliptic, the first point of Aries, and that of Libra. This done, dR, measured by Prop. 23, is 62o 46', shows the time of sun rising; by CR, Prop. 22, is the amplitude 32° 19′. GI 79° 16′, by Prop. 23, the sun's azimuth at 6. IO 14° 38′ his altitude at 6. CF 220 25', by Prop. 22, his altitude when east. GF 13° 28′ the time when he is due east. OB 41° 53', by Prop. 22, his altitude at a quarter past 9; the 4OZP 122° 40', by Prop. 24, his azimuth at that time. Also rO, by Prop. 22, is his longitude, 51° 7'. YK his right ascension, 41° 40'.

And the place of the moon or a star being given, it may be put into the scheme as at ; and its time of rising, amplitude, azimuth, &c.

found as before.

III. Gnomonically-Fig. 3.

To project the eastern hemisphere upon a plane parallel to the me ridian. About the centre of projection C, describe the circle HON with the tangent of 45 the radius of projection, for the primitive. Through C draw the horizon HO, and the prime vertical ZN perpendicular thereto. Set the latitude 54 from H to a, and draw the 6 o'clock meridian Pp, and the equinoctial EQ perpendicular to it. Set the tangent of 48° 45' (equal to 3 hours) from C to E, and, by Prop. 10, draw the meridian EL parallel to Pp. Make Ee=Ea, and 4 Ee=18° 5' the sun's declination, then, by Prop. 11, is the sun's place. Through draw the hyperbola DOd, by Prop. 14, for the sun's parallel of declination; and draw OB perpendicular to HO, for his azimuth circle. And draw GI perpendicular to HO, and RM,

FT, || Pp. Also the ecliptic is a right line passing through O, and cutting EQ at an angle of 23° 30', which is difficult to draw in this projection.

Also, by Prop. 14. Draw the parallel Ss 18° below the horizon, and if it intersects Dd, it gives the point of sun-rise.

Then, if by Prop. 17 or 11, you measure GR, or rather CM, 27° '4', you have the time of sun-rising; GF to CT 13° 28′, the time when he is due east. Also, by Prop. 11, if you measure CR you have the amplitude 32° 19. CI the comp. of this azimuth at six, 10° 44'. IG, by Prop. 12, his altitude as 6, 14° 38'. CF his altitude when east, 22° 25'. And, by Prop. 11, OB=41° 53′, his altitude a quarter past 9. CB the complement of his azimuth at that time, 32° 40'.

And the place of the noon or a star being given, its place in the projection may be determined as before, and all the requisites found.

Ex. 2.-To project the sphere upon the solstitial colure for the latitude 541 N. May 23, 1767, at 10 o'clock in the morning.

Stereographically.-Fig. 4.

The projection of the western hemisphere, the first point of Libra. the projecting point. Describe the solstitial colure PEPQ, and the equinoctial colure Pp perpendicular to it; and through C draw the equinoctial EQ perpendicular to Pp. Set 23° 30' from E to, and from Q to W, and draw the ecliptic. Set the sun's longitude 61° 42′ from C to O, and through draw POKp for the 10 o'clock meridian. Make KA (two hours or) 30°, and draw PAp for the meridian of the place. Set the latitude of the place 544 from A to Z, and Z is the zenith. About the pole Z describe the great circle BHS for the horizon of the place. Through Z and draw an azimuth circle ZOB.

Then you have OK the sun's declination 20° 33'. CK his right ascension 59° 35'. OB his altitude at 10 o'clock 49° 10′; the 4 AZO or PZO his azimuth at 10=HB, 45° 44'. H the south point of the horizon. I the point of the ecliptic that is in the meridian. T the point of the ecliptic that is setting in the horizon.

Ex. 3.-To project the sphere on the plane of the horizon, Lat. 35, N. July 31, 1767, at 10 o'clock.

Gnomonically.-Fig. 5.

To project the upper hemisphere on a plane parallel to the horizon With the radius of projection and centre C, describe the primitive circle ADB. Through C draw the meridian PE, and AS perpendicular to it for the prime vertical. Set off CP 35 the latitude, and P is the N. Pole, and perpendicular to CP draw Pp the 6 o'clock meridian. Set the complement of the latitude from C to E; and draw EQ perpendicular to CE for the equinoctial. Make EB 30° (or two hours) and draw the 10 o'clock meridian PB. Set the sun's declination 18° 27′ from B to. And is the place of the sun at 10

o'clock. Through draw the azimuth circle CQ; likewise through O a parallel to the equinoctial EQ may easily be described, by Prop. 15, for the sun's parallel that day.

Then CO, measured by Prop. 11, is 31° 30′ the complement of the altitude. And the angle ECO, measured by Cor. Prop. 12, is his azimuth, 65° 10′.

Scholium. After this manner may any Problems of the Sphere be solved by any of these projections, or upon any planes, but upon some more commodiously than upon others. And if in a spherical triangle any sides or angles be required, they may be projected according to what is given therein, according to any of these kinds of projection before delivered; and it will be most easily done, when you chuse such a plane to project on, that some given side may be in the primitive, or a given angle at the centre; and then you need draw no more lines or circles than what are immediately concerned in that problem. But always chuse such a plane to project on, where the lines and circles are most easily drawn, and so that none of them run out of the scheme.