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THese Dials tho' Vertical and Direct (as passing thro' our Zenith, and facing also two Cardinal Points or Quarters of the World) are very different from the former, nor has any body (I believe) taught yet their Description by the Globe.
To perform therefore this Operation, you must by the help* 1.1
of your String or Compasses describe on your Globe, with Chalk (or the like matter) an Arch (as in Sch. 11.) which having its Pole at K (the East-point, for examples sake, of the Aequinoctial) cuts somewhere or other the 11 a Clock Nor∣thern hour Circle, I mean the 11 a Clock hour Circle on the Northern, or black part of the Globe; and this Arch by reaching from the point C in the Aequinoctial Colure (or 6 a Clock Circle) to H in the Horizon on the said Northern side of the Globe, will be a piece of a little Circle parallel to the Meridian containing the Degrees of the Elevation of the Pole, and cutting all the Hour-Circles also from 6 to 11. But if this be thought too troublesom a work, the Globe-maker may avoid it by putting 6 Pricks or Asterisks upon the Globe, where the said Arch and Hour-Circles would intersect, as may be seen in the said 11 Scheme at C, O, S, T, V and Z; so that if beyond C he adds one prick more, viz. at R, to give you from H the Radius, or 60 Degrees of the said Arch, you need nothing else.
This being premis'd, describe on a sheet of paper (HR, or 60 degrees of the said Arch being Radius) a blind Circle* 1.2 as in Sch. 12, and drawing the Line H h how you please throu' K its Center to represent the intersection of the Horizon, open your Compasses to the said Arches full extent, to wit, from H to C, and putting one foot on the blind Circle at H, and the other marking there at C, draw the line PC π, throu' the Center K, and 'twill represent the intersection of the Ae∣quinoctial Colure (or 6 a Clock hour Circle) with your said blind Circle or Plane; so that if you take from off your Globe, the distances between the point C, and the several Intersecti∣ons of the Hour Circles with the said Arch CH, and place them on your blind Circle on the right hand side of PC π, as well below the Horizon H h, as above it, and draw lines thro' them (viz. O ο, S σ, T τ, V υ, and Z ζ) you will have a com∣pleat East Dial describ'd, after you have drawn 2 lines more on the left side of the said C π, to wit, the Line N ν distant from it as is O ο, and the Line M μ, as is S σ. As for the figuring each hour line, it must be according to the Figures of the corresponding Hour-Circles cut by the aforesaid Arch CH, and thus you will find them figured in the forementioned Scheme 12, which shews you too how the Borders or Parallels are drawn for the said Figures to lye in, as being only
double Lines (equidistant at pleasure) on both sides of the Ho∣rizon H h; and here also by the blind Lines, and by the fair ones, you have before your Eyes what is necessary to be exprest on your fair Plane, and what not.
Nor is there any difference in the Construction of a West-Dial,* 1.3 except it be in turning on your draught the Hour-Lines or Parallels the other way, to the end they may all point North∣wards on their respective Planes; for thus (in Sch. 11.) do the Prick Lines (m 8, n 7, c 6, o 5, s 4, t 3, u 2 and z 1.) which would truly represent this Dial, if they were produced in the said Scheme.
Now for the Substilar 'tis the 6 a Clock Hour Line, since that* 1.4 Hour Circle falls on the Plane at right Angles, and as for the Cock it may be a Gallows Stile (as in Scheme 13) or a Pin (as in Scheme 14) so it be plac't on the Substilar and perpendicular to it, having its height equal to the Distance between the Pricks or Asterisks C and P in the said 11 Scheme, or (which is all one) to the distance between K and X. viz. the nearest distance between the Substilar, and the 9 a Clock hour line in an East-Dial, and the Substilar and the 3 a Clock Line in a West Dial.
But here you are to remember, that when I say, that the height of the Stile is to be equal to the distance between C and P. I mean in rigour equal to the Sine, and not the Chord of that Arch; but seeing the Chord of 10 Degrees, differs not sensibly from the Sine (and by the way the Arch CP on the Globe will not be above 10 Degrees from the Meridian,) the interval be∣tween C and P will serve the Turn. But if you would be more exact take between your Compasses the distance of double CP, to wit the interval of (suppose) 20 degrees, and half of it is the required distance; for half the Chord of 20 Deg. is equal to the Sine of 10. Or if you, please you may erect a needle at C Paralel to P (the elevated Pole of the Globe) and the distance between them will be the true Height of your Stile. To Conclude, You may contract and enlarge these Dials as you please, by draw∣ing the hour-lines twice or thrice (or according to any other proportion) nearer or farther asunder, and so abateing or heightning in the like manner your Stile.
The Demonstration is obvious, for since the points M, N, C, O,* 1.5 S, T, V and Z in the upper part of the blindCircle or Plane, and the Points μ, ν, π, ο, σ, τ, ••, ζ, on the lower part of it are (by being equal in distance to those on the Arch) the intersections of
the morning hour Circles of 4, 5, 6, 7, 8, 9, 10, 11▪ with the edges of the said Plane, it follows that the Lines drawn from the corresponding Points, must be the true hour lines of this Dial, since the hour Lines (as we said) of all Dials, are only the Intersections of the respective hour Circles with the Plane. Again the shade of the Axis (the Axis being a part of all the hour Circles) falls ever on the Hour-Line or Interfection of this or that Hour Circle, as often as the Sun comes into the Plane of that Hour-Circle, therefore the Stile of this Dial re∣presenting truly the Axis▪ (since 'tis above the Plane, and di∣stant from it as 'tis on the Globe) will cast its Shade every hour on the corresponding hour Line▪ and as for the reason, why the height of the said Axis is equal to the distance between the 3 or 9 a Clock Lines and the Substilar, it shall be shown in the Demonstration of the next Operation.
Some few things to be premis'd.
The Constru∣ction of an East Dial.
Of a West Dial.
Of the Stile and Substilar.
The Demon∣stration.