758 CHAP VI. of DIALLING. తి. SECT. I. How to project a DIAL Stereogra phically upon any Plain HE Projections of any Plain Dial, is no more but the Projection of the Sphere, upon the Plain of that great Circle, to which the Dial is Parallel, the Meridians at every 15Degrees Diftance reprefenting the Hour Lines, and the Axis of the World (terminating at the two Poles) reprefenting the Stile; as the following Examples will make evident. SECT. SECT. II. How to project an Horizontal DIAL. A N Horizontal Dial, is fo called, because it is parallel to the Horizon, and therefore is projected upon the Plain thereof, to the Latitude given. EXAMPLE. It is required to draw an Horizontal Dial for the Latitude of London, 51 Degrees 32 Minutes. First, Draw the Primitive Circle N WS E, to reprefent the Horizon, and cross it at Rightangles, with the Lines N S, and WE, then the Center Z is the Zenith of the Place (because it is the Pole of the Horizon) and the Points N, S, E, W, are the North, South, Eaft and Weft Points of the Projection. Then because the Pole of the World is diftant from the Zenith 38° 28′ (equal to the Complement of Latitude) fet of 380 281 upon the Right Circle S N, by Prob. 9. from z to P; then is P the Pole: Alfo because the Equator is distant from the Zenith $1° 32' the La titude given, fet off 51° 32' from z to Q, by Prob. 9. of Spherical Geometry; and draw the Oblique Circle WQ E, to represent the Equator; then from Q to a, fet off 15 Degrees each Way upon the Equator, by Prob. 9. and from 2 to b, 30 Degrees each Way, and from Qto c, 45 Degrees, and to d, 60 Degrees, and to e, 75, as in the Figure; then through P, and the feveral Points a be, &c. draw the Meridians 1 Pa 1, and 2 Pb 2, c. by Prob. 5; which done, draw Lines from the Center Z, to the Points 1, 2, 3, &c. these are the Hour Lines required. Then for the Stile's Height, measure the Arch P N, of the Right Circle S N, by Prob.. 8. the Quantity of that Arch is the Height of the Stile required. NOTE, If draw you your Projection upon the fame Plain (whether Wood, Stone, &c.) that you intend to make your Dial upon, 'tis beft to do it obfcurely, that when you have found the Hours Diftances upon the Periphery, the other Circles, may be done out at Pleafure, and the Stile may be laid down, by feting 519 320of the Chords from N to B; then draw the Line z R, and RN; but you may make the Lines z R, and Z N, longer, or fhorter, at Pleafure; only the Angle at z is to be always equal to the Latitude of the Place:Thus a Piece of thin Plate, c. of the Form of the Triangle z R N, fhall be the Stile required to much d To To find the Hours Diftances by Trigonometrical Calculation, the Proportion is, As Radius, To Sine of Latiude; So Tangent of 15 Degrees, To Tangent of the Hours Distance of 11 and 1 from the Meridian : And fo Tangent of 30 Degrees, To Tang. of the Hours Diftance of 10 and 23r And fo of 45 Degrees, For 9 and 3. exc.. This Proportion you have in many Books of Dialling; but no Reafon given for it, nor Demonftration of it, as you have from this Projection, in which, in the Triangle P N 1, you have given P N, the Latitude 51 32; and the Angle at P, 15 Degrees, to find the Side N1, the Distance upon the Primitive Circle, between the Hours of 12 and 1 (or 12 and 11, which is the fame ;) and you may find it by the Globical Projection, taught in the former Part of this Book; for Plate 8. fuppofing the Triangle P N 1, to re- Fig. 3. prefent the Triangle P N 1 in the Projection, you will find by the Rule laid down in Globical Projection, that the Propor tion lies in the Manfion P bc, and is, As As (Pb) Radius, 90.00 10.00000. To (Pn) the Sine of Latitude; 51 32 9.89374. So (bc) the Tangent of 15 Degrees? 9.42805; the Angle at P, To (N) the Distance upon the 9.3.2179: By the fame Proportion with Tangent of 30 Degrees, you find the Hour Diftances of 10 and 2, and by 45 Degrees, the Hour of 9 and 3, c. which is an intelligible Demons ftration of that common given Proportion a bove mentioned.. SECT. III. How to project an Erect Direct THI HIS Dial is parallel to (and therefore projected upon the Plain of) the Prime Vertical, or Eaft and Weft Azimuth. |