On the Center Z, with any Radius defcribe the Circle E NWS for the Horizon of London; on this Circle project the Sphere as has been taught in the Stereographic Projection; then from Z lay a Ruler to the Points 4, 5, 6, 7, 8, &c. ( where the Hour-Circles cut the Plane of the Horizon) and draw ftraight Lines to the Border of the Dial, and they will be the true Hour-Lines required to be drawn.

Then take 51° 32' (the Latitude of the Place) and fet from S to A, and draw Z A for the Stile of the Dial; the Subftile (or Line on which the Stile ftandeth) is the Meridian, or Hour-Line of XII; and thus your Dial is finished as in the Figure.

PROBLEM XV.

To make the foregoing Horizontal Dial by Trigonometrical Calculation only.

Practice.

Practice.

To do this, is only to find the Distance of the Points a, b, c, 4, 5, from N the Meridian in the Projection above. For which Purpose, in the several Right-angled Triangles NP a, NPb, NPc, NP4, NP 5, there is given the Side NP = 51° 32', the Pole's Elevation equal to the Latitude; and the Angle at P = 15°, 30°, 45°, &c. to find the Arches Na, Nb, Nc, &c. thus ;

To the Tangent of Na= 11° 51′ 9.3217977

And thus, by this Analogy, all the other Arches of Distance from the Meridian, for every Hour from Noon are found as in the Table annexed; and after the fame Manner you calculate the Arch of Distance for every Half and Quarter of an Hour; thus those Numbers in the third Column being fet from a Scale of Chords on any Circle, will give the Points, through

which ftraight Lines being drawn from the Center will be the Hour-Lines, as before.

PROBLEM XVI.

To defcribe Hour-Lines upon an erect direct South or North Plane.

Practice.

This Dial may be made two different Ways by Projection; For

1. A Projection of the Sphere on the Horizon of Latitude 38° 28', (as being the Complement of the Latitude of London) will produce an Horizontal Dial in the fame manner as in the laft Problem; which

Horizon

Horizontal Dial in that Latitude will be a South erect direct Dial here in London. Only the Stile in this erect South Dial points to the South Pole; and its Height above the Plane is S A= 38° 28' the Latitude or Elevation of the South Pole thereon. And the Dial made thus you have before,

2. This Dial following is alfo made by a Projection of the Sphere on the Horizon of London, thus ; let NWSE be the Plane of the Horizon on which the Sphere isprojected; let WE be the Interfection of the Plane of the Dial therewith; then will the Hour-Lines interfect it in the Points a, b, c, d, e; and fo za, zb, zc, &c. in this Projection will be equal to the Arches Na, Nb, Ne, &c. in the laft Projection above. Now take a Ruler and lay from N to the Points a; b, c, d, e, and it will cut the Plane of the Horizon

in the Points X, X, X, X, &c. then lay the Ruler on z the Center, and each of thofe Points x, X, X, X, &c. and draw ftraight Lines to the Border of the Dial, and they shall be the true Hour-Lines required.

3. This Dial is alfo to be made by Calculation, thus; In the first Projection for Latitude 38° 28' you have, in the Right-angled Triangle NP a, given the Side N P = 38° 28' the Latitude; and Angle NP a 15 oo'; to find Na, the Distance of the firft Hour from the Meridian. And in the fecond Projection on the Horizon of London, you have the fmall Right-angled Triangles Z P a, Z P b, &c. in which is given the Side Z P = 38° 28' the CoLatitude; and the Angle at the Pole z Pa = 15o ool; to find the Arches Z a, Zb, Z c, &c. So that in either Projection you find those Arches by the fame Analogy as before, thus ;

As Radius

10.0000000

Is to the Sine of P N, or Z P, = 38° 28′ 9.7938317 So the Tang. of NPa, or ZPa, = 15° 00′ 9.4280525 To the Tangent of Na, or Za, = 9° 27′ 9.2218842

And thus are all the other Arches of Distance from the Meridian found as in the Table following. Now by any Radius defcribe a Circle; and from a Scale of Chords belonging thereto lay off the Degrees of the feveral Arches za, zb, &c. (as in the Table) on the faid Circle each way from the Meridian S N, and they will give the Points X, X, X, X, X, &c. through which ftraight Lines drawn from the Center Z will be the Hour-Lines, as before by Projection.

Hours