Or the Tangent of the Diftance of the Center of Capricorn CE from the Center of the Primitive; but 162173 is the Natural Tangent of 58° 20, the Numbers in the Table. 8. Wherefore the Half-Tangent of 75° fet off from the Center Z (in the Projection) to the Point VS, gives there the Point of Interfection; and the Tangent of 58° 20', fet off from Z the other way, will find the Center, on which to defcribe the Tropic of Capricorn BA; and thus by the Half-Tangents of the Numbers in the Table of Interfections, and Tangents of the Numbers in the Table of Centers, you may speedily draw any Parallel of Declination, in this Projection. 9. In this Projection the Parallels of Altitude, or Almicanthars, are all Concentric Circles, or Parallel to the Primitive. And the Azimuths are all straight Lines, or Radii, drawn from the Center Z to the Divifions of the Primitive. The Circles of Longitude are all Oblique Great Circles (paffing thro' the Poles of the Ecliptic) in this Projection; but thofe Circles, with others, are all omitted, to prevent Confufion, and as being lefs neceffary than others. 10. This Projection of the Sphere on the Plane of the Horizon, is of the laft Importance in Dialling; the whole Theory thereof depends intirely hereon, and none can give the Rationale of his Calculations, in this moft curious Art, without duly understanding the Doctrine of Projection; as will appear further on. III. To III. To project the Sphere on the Plane of the Ecliptic. 1. Draw the Primitive Circle WNES, to repre fent the Plane of the Ecliptic, and divide it into its Signs, and each Sign into its proper Degrees. Then crofs it with two Diameters WE, NS, which will be the Projection of two Circles of Longitude; for thofe Circles will here be Right Lines, or Radii, paffing thro' the Center C, the Pole of the Ecliptic. Note, N is the North, S the South, E the Eaft, and W the West Points of the Ecliptic. 2. The Equinoctial will here fall towards the South, for 'tis the Northern Half of the Ecliptic that is elevated above the Equinoctial; and of confequence, the Southern Part of the Equinoctial (with regard to this Pofition of the Sphere) that is elevated above the Plane of the Ecliptic: Wherefore as it is elevated but 23° 30', the Half-Tangent of its Complement 66° 30' fet from the Center C fouthward to Æ, will there, give the Point of Interfection; and the Tangent of 23° 30' fet Northward from C to e, will find the Center e, on which it is defcribed, as WEE. 3. The Pole of the World in this Projection is elevated 66° 30′, above the North Point N of the Ecliptic; therefore the Half-Tangent of 23° 30' fet from C to P, will there fhew the Pole. Now fince the Hour Line of 6, paffeth thro' the Poles, and also interfects the Ecliptic, in the two oppofite Points V and, therefore the Tangent of 66° 30' (the Complement of C P = 23° 30′) fet from C fouthwards to R, will there give the Center, on which to defcribe the Hour Circle, or Meridian WP E; and by drawing the Line DRF normal to NS R, and fetting off the Tangents of 15o, 30′, &c. from R towards D, and F, continued out each way, all the other Hour Lines, or Meridians, may be drawn juft as in the last Projection on the Plane of the Horizon. 4. After the fame Manner as the Equinoctial was defcribed, may be described the Horizon of London WOE; for the Northern Part of the Horizon being elevated above the Southern Part of the Ecliptic 62°; therefore the Half-Tangent of its Complement 280 being fet from C to O, will there give the Point of Interfection; and the Tangent of 62° being fet Northward from C, in the Line S N continued, will find the Center, on which is defcribed the Horizon WOE. 5. The Tropics of Cancer, and Capricorn; the Polar Circles, and all other Parallels of Declination, are to be drawn by Problem 7, Cafe 3; as being all of them Parallels to the Oblique Equinoctial Circle WEE. By the fame Problem, may all the Almacanthers be drawn. 6. If you find the Pole of the Horizon, you may thro' it draw all the Azimuths, in the fame manner as the Meridians were defcribed. 7. The Circles of Longitude will here be all Right Lines, or Diameters paffing thro' the Center; and the Circles of Latitude, will be all parallel to the Primitive, and Concentric therewith. Hence the Stars and Conftellations, by their Latitude and Longitude, may very conveniently be reprefented in this Projection. 5. The IV. The Projection of the Sphere on the Plane of the Equinoctial. 1. Defcribe the Primitive Circle WNES, to reprefent the Plane of the Equinoctial; and crofs it with the two Diameters WE, NS, which are here the Projection of those two Hour Lines or Meridians, which we call the Equinoctial and Solftitial Colures. 2. All the other Hour Circles, or Meridians, are here projected into Right Lines alfo, or Diameters, paffing thro' the Center P, which is the Pole of the Equinoctial, and interfecting the Plane thereof at Right Angles in the Primitive; where they there point out the Hours, 1, 2, 3, 4, &c. Thus this will be an Horizontal Dial under the Poles of the World. 3. All Parallels of Declination are here defcribed by the Half-Tangents of their Distance from the Pole, or of the Complement of their Distance from the Equinoctial And thus with the Half-Tangent of 23° 30′ you describe the Polar Circle Leow; and with the Half-Tangent of 66° 30' is defcribed the Tropic of Cancer AS B. 4. As the Northern Part of the Ecliptic is elevated above the Equinoctial 23° 30; and alfo interfects it in the two oppofite Points W, E; therefore this fhall be a Great Oblique Circle in this Projection; and will fall between P, S, (in the Northern Hemisphere) as the Circle WE; and is projected by Theorem 7. Problem 8. VOL. II. K 5. The |
