Of thefe direct South and North erect Dials it must be obferved, that the Sun never fhineth on the South Dial before fix in the Morning, nor after fix at Night; therefore Hour-Lines on it are to be drawn only from XII to VI, each way. Alfo, that on a direct North Dial, the Sun never fhineth at all, the Winter half Year; that the Sun never fhineth on it after 7 a-Clock 20 Minutes, 48 Seconds in the Morning, and before 4 Hours 39 Minutes 12 Seconds in the Afternoon, (in the Latitude of London) on the longest Day; and therefore Hour-Lines of VIII in the Morning, and IV in the Afternoon, are useless on this Dial, though always drawn by the Dialist.

Note, The Hours belonging to the North Dial, in the two laft Schemes, are marked with Figures, as 8, 7, 6, 5, 4 ; and 4, 5, 6, 7, 8; and thofe of the South Dial, with Numerical Letters; as VI, VII, &c.

PROBLEM XVII.

To defcribe Hour-Lines upon an Erect South, or North Plane declining Eaft or Weft.

VOL. II.

Y y 2

Practice.

Practice.

As the Reafon or Theory of this fort of Dials is not altogether fo obvious as of the foregoing, and the Practice of making them fomewhat different and more difficult; I have (the better to illuftrate the Reafon, and facilitate the Practice) fubjoined the following Scheme.

In it I have made a Dial-Plane A D C L decline from the South S towards the Weft W, 30° co' = SD, in the Latitude of London 51° 32′ = EZ or P N. Now in order to defcribe Hour-Lines on this

the perpendicular Line of the Plane.

D the Pole

of the Plane.

Meridian of

P the Pole of the World.
P the Pole of the World.

LPD the the Plane. WE E the Equinoctial. Therefore PH is the Height or Elevation of the Pole above the Plane, and fo the Stile's Height. ZH the Distance of the Subftile from the Meridian MÆ = HP Z the Inclination of Meridians, or the Plane's Difference of Longitude weftward.

Now the three laft Particulars, viz. The Height of the Stile HP, the Diftance of the Subftile from the

Meridian Z H, and the Plane's Difference of Longitude or Angle HP Z, muft all be found before a Dial of this Sort can be made.

Now all thefe are to be calculated from the little Triangle Z HP, Right-angled at H; for therein is given the Side Z P = 38° 28', the Co-Latitude; and the Angle PZ H = 60° 00', the Co-Declination. Wherefore we fhall thus find

1. The Stile's Height P H.

As Radius
Is to the Sine of ZP 38° 28'
So is the Sine of P ZH 60° 00'

=

10.0000000

9.7938317

9.9375306

To Sine of the Stile's Height HP=32° 36′ 9.7313623

II. The Distance of the Subftile Z H.

As Radius

To the Co-fine of P ZH 60° 00'
So is the Tangent of Z P = 38° 281

10.0000000

9.6989700 9.9000865

9.8937452

So is the Tangent of P Z H= 60° 00' 10.2385606

To the Co-tangent of the
Plane's Differ. of Long.
HP Z, or MÆ

}

360

24'= 10.1323058

The

The next thing is to find the Distance of the HourLines on the Plane, from the Perpendicular Line of the Plane L D, or Hour-Line of XII. And this may be done either by Projection or Calculation, as in the former Dials.

First, Let the Circles of the Sphere be projected as hath been taught, and you will find they will cut the Plane AC in the Points 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7; then lay a Ruler on D to each of those Points, and make Marks XX XXXX, &c. in the upper Semicircle of the Projection CLA, where the Ruler interfects it. Laftly, By a Ruler laid on Z to each of thofe Marks XXXX, &c. you draw ftraight Lines from the lower Semicircle CDA, to the Border of the Plane, and they fhall be the true Hour-Lines required.

Secondly,

Secondly, This Matter may be attained by Trigonometry, thus; Let there be drawn the dotted Meridian of the Plane D HPL; then we have before found the Angle Z PH = 36° 24, and becaufe the Hour-Lines include Angles at the Pole P of 15o each, therefore is the Angle HP 1 = 21° 24′; and HP 26° 24', and HP3 = 8° 36', and HP 4

23° 36′, and fo for the reft; whence in thofe Right-angled Triangles HP 1, HP 2, &c. there is given the Perpendicular PH = 32° 36' the Stile's Height, and the Vertical Angles at the Pole P = 21° 24', 6° 24', &c. as before; to find the several Bafes H2, H1, H3, H4, &c. which are the Dif tances of thofe Hour-Lines from the Subitile H. Now they are found by this Analogy.

As Radius

To the Sine of PH 32° 36′

10.0000000 9.7314040

So is the Tangent of HP 2 = 6° 24′ 9.0498689

To the Tangent of H 2 = 39 281

8.7812729

the Diftance of the Hour-Line of 2 from the Subftile, and in like manner are all the other Distances found as in the following Table. Wherein the first Column contains the Hours; the fecond Column contains the Angles at the Pole; and the third Column contains the feveral Diftances of the Hour-Lines from the Subftile, on the Plane.

Q

Now in order to fix the Stile of the Dial in its due Place on the Plane, do thus; take the Distance of the Subftile from the Meridian of the Plane, HZ = 21° 401, from the Chords, and fet from D to T, and draw Z T for the Subftile; then take the Height of the Stile P H = 32° 36′, and fet it from T to I, and draw Z1 for the Stile.

The