The English globe being a stabil and immobil one, performing what the ordinary globes do, and much more / invented and described by the Right Honorable, the Earl of Castlemaine ; and now publish't by Joseph Moxon ...

OPERATION. XVI. How to describe Geometrically a Dial declining 40 Degrees Eastward, for the Elevation of London.

The second way.

HAving made an Horizontal Dial for this Elevation in the lower part of your Paper Plane, (as 'tis exprest by the prick lines in Scheme 18) and drawn from the Center A the several Hour-Lines upward as far as you think fit, and Figur'd them to show what Hour-Lines they are, chuse in AC (the 12 a clock line) any Point, suppose P, and draw throu' it the blind Line GD making with the said AC an Angle of 50 De∣grees or Complement of your Declension; then erect the Perpen∣dicular PB on the said blind line at P, and taking with your Compasses (AP being your Radius) the Tangent of 5•• Degrees and ½, or true Elevation of the Pole, put one foot on P, and where the other marks on the said Perpendicular (suppose at F) there will be the Center of your Declining Dial; so that having bordred your Plane with fitting Parallels, to contain the standing Figures of each hour, you have nothing more to do,

but to draw fair Lines from the said Center F, to your Border, throu' the Intersections of the Line GD with the several Hour∣lines of the Horizontal Dial; that is to say, you have nothing more to do, but to draw fair Lines throu' the Points KLMNO PQR which give the hours of 7, 8, 9, 10, 11, 12, 1 and 2; and by the way you may have as many other Morning or Evening hours as you please if you draw the said GD long enough for the other hour-lines of the Horizontal to meet with it. Nor is there more difficulty here about the Stile and Substilar than in any of the former Dials; for (AP being Radius) 'tis but taking the Sine of 40 Degrees (or Declination of the Plane) with your Compasses from the Sector, and putting one foot on your 12 a clock Line at P, the other foot will in the line GD (to wit, at M) give you the Point for to draw the Substilar FM, and the Sine Complement of the Declension, or Sine of 50 Degrees, will be XM the Stiles height. Nay, if (for want of a Sector or the like) you cannot conveniently find the Sine of the said Declen∣sion, do but observe where a Perpendicular from A falls on GD suppose at M, and PM will be the distance in the said GD be∣tween the 12 a Clock line of this Dial and its Substilar, and AM (equal to XM) the height of the Stile above it. Thus then we see that the Fabrique of a Declining Dial (which is wont to terrify young Students) is in a manner as quick and easy as that of the Horizontal, since two ordinary Lines more, viz. GD and BP give us all the Points necessary for its Description.

The Demonstration and Reason of this Dial is evident; for, the Horizontal being by construction true, any Erect Plane fa∣cing the South, that crosses its Meridian (or 12 a clock line AC) at right Angle•• will represent a Primary Vertical or Direct South Plane, and then the Center of the Dial described on it will be distant from P the intersection of the two Planes on the said AC) the Tangent of the Elevation, as I shew'd you before. Now since GD is (by Hypothesis) the Edge of a Vertical De∣clining Plane, and since (as we show'd you in the before cited place) that the 12 a Clock line, as well in a Declining as in a Primary Vertical Dial, is Perpendicular to the Horizon, con∣taining in it the Centers of the said Dials, it follows that FP (being the Tangent of the Elevation, and Perpendicular also to the said DG where it cuts the 12 a Clock line of the Horizon∣tal) must be the 12 a Clock line, and F the Center of our present

Dial, whose Declension is 40 Degrees Eastward, since FP de∣clines so many Degrees from CP toward the morning Hours; for the said CP and FP represent the 12 a Clock lines of a Direct, and of our thus Declining Vertical Plane, if you consider them flatted down, and lying in the Horizon. This being so, 'tis evident that the Lines drawn from F to KLMN, &c. are the true Hour lines of our Dial, as falling from its Center to the several Points made on its Horizontal edge, by the Hour Circles or (which is all one) by their Intersections with the Horizontal Dial. As for the Stile and Substilar, let us but consider the Triangle AMP, and we shall find that P is by construction the Angle of 50 Degrees, and A that of 40, as substended by the Sine of the Declension, so that A being a right Angle, AM must be a perpendicular; therefore the Hour Circle, whose intersection the said AM happens to be, falls at right Angles on our present Plane, and consequently gives the Substilar; Now since the Axis of the World passes through F and A, the Centers of the two Dials, when they are joyned (as we now suppose them) at GD the common Section of their Planes; I say, since the Axis passes throu' their Centers, its Elevation or Height above our Plane must be AM, as being the only Perpendicular that can fall from it upon the said Plane, and consequently its Mea∣sure; but AM you see is the Sine Complement of 40, since PM is the Sine of 40, Therefore in all Declining Dials, The Sine of the Declension (from their 12 a Clock Line) gives in their Horizontal Edge their Substilar, and the Sine Complement their Stile. Q. E. D.

Notes

• The Constru∣ction.
• A ready way to find the Stile and Sub∣stile of a de∣clining Dial.
• Demonstrati∣on.
• pag. 92.
• The Demon∣stration of the Stile and Sub∣stilar.