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From that Semidiameter
238905 cf = 36732 CE = 162173
There will remain
Or the Tangent of the Distance of the Center of Capricorn C È 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 750 fet off from the Center 2 ( in the Projection ) to the Point vs, gives there the Point of Intersection ; and the Tangent of 58° 20', fet off from 2 the other way, will find the Center, on which to describe the Tropic of Capricorn B V A ; 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 Déclination, in this Projection.
3. In this Projectior. 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 2 to the Divisions of the Primitive. The Circles of Longitude are all Oblique Great Circles (paling thro' the Poles of the Ecliptic) in this Projection ; but those Circles, with others, are all omitted, to prevent Confusion, and as being less necessary than others.
10. This Projection of the Sphere on the Plane of the Horizon, is of the last Importance in Dialling ; the whole Theory thereof depends intirely hereon, and none can give the Rationale of his Calculations, in this most curious Art, without duly understanding the Doctrine of Projection ; as will appear furcher on.
III. To project the Sphere on the Plane of the
1. Draw the Primitive Circle WNES, to repre. sent the Plane of the Ecliptic, and divide it into its Signs, and each Sign into its proper Degrees. . Then cross it with two Diameters WĒ, NS, which will be the Projection of two Circles of Longitude ; for those Circles will here be Right Lines, or Radii, passing thro' the Center C, the Pole of the Ecliptic. Note, N is the North, S the South, E the East, 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 consequence, the Southern Part of the Equinoctial ( with regard to this Position 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' set from the Center C southward to Æ, will there, give the Point of Intersection ; and the Tangent of 23° 30' set Northward from C to e, will find the Center e, on which it is described, as W Æ E.
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' set from C to P, will there shew the Pole. Now since the Hour Line of 6, pafseth thro' the Poles, and also intersects the Ecliptic, in the two opposite Points r and , therefore the Tangent of 66° 30' (the Complement of CP= 23° 30') set from C fouthwards to R, will there give the Center, on which to describe the Hour Circle, or Meridian WPE; and by drawing the Line DRF normal to NSR, and setting off the Tangents of 15°, 30', &c. from R towards D, and F, continued out each way, all the other Hour
me 20 10 or 20 10 09 2040_1 20 10
20 m, 10 20
Projection of the Sphere on the
of the ECLIPTIC .
Lines, or Meridians, may be drawn just as in the last Projection on the Plane of the Horizon.
4. After the same Manner as the Equinoctial was described, may be described the Horizon of London WOE ; for the Northern Part of the Horizon being elevated above the Southern Part of the Ecliptic 620; therefore the Half. Tangent of its Complement 288 being fer from C to 0, will there give the Point of Interiction ; and the Tangent of 629 being set Northward from C, in the Line S N continued, will find the Center, on which is described the Horizon WO E.
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 WÆ E. By the same 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 same manner as the Meridiairs were described.
7. The Circles of Longitude will here be all Right Lines, or Diameters passing thro' the Center ; and the Circles of Latitude, will be all parallel to the Pri. mitive, and Concentric therewith. Hence the Stars and Constellations, by their Latitude and Longitude, may very conveniently be represented in this Projection.
IV. The Projection of the Sphere on the Plane
of the Equinoctial.
1. Describe the Primitive Circle W NES, to represent the Plane of the Equinoctial ; and cross it with the two Diameters WĖ, NS, which are here the Projection of those two Hour Lines or Meridians, which we call the Equinoctial and Solstitial Colures.
2. All the other Hour Circles, or Meridians, are here projected into Right Lines also, .or Diameters, passing thro' the Center P, which is the Pole of the Equinoctial, and intersecting 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 described 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-Tangerit of 23° 301 you describe the Polar Circle Leow, and with the Half-Tangent of 66° 30' is described the Tropic of Cancer 8 AT B.
4. As the Northern Part of the Ecliptic is elevated above the Equinoctial 23.0 30%; and also intersects it in the two opposite Points W, E ; therefore this shall be a Great Oblique Circle in this Projection ; and will fall between P, S, ( in the Northern Hemisphere) as the Circle WO E ; and is projected by Theorem 7. Problem 8.