The Third Book of the Cross-Staff
All those Planes that have their Horizontal Line lying East and West, are in that respect said to be Vertical; if they be also upright and pass through the Zenith, they are direct Verticals; if they incline to the Pole, they are direct Polar; if to the Equinoctial, they are properly called Equinoctial Planes, and are described before: if to none of these three Points, they are then called by the general name of Inclining Planes.
These may incline either to the North parts of the Horizon, or to the South; and each of them hath two Faces, one to the Zenith, the other to the Nadir, in which we are first to consider the height of the Pole above the Plane, by comparing the Inclination of the Plane to the Horizon with the Latitude of the Place.
As in our Latitude of 51 gr. 30 m. if the declination of the Plane EIW in the Fundamental Diagram shall be 13 gr. Northward, that is, if IN, the Ark of the Meridian between the Plane and the North part of the Horizon, shall be 13 gr. we may take these 13 gr. out of PN 51 gr. 30 m. the Elevation of the Pole above the Horizon, and there will remain PI 38 gr. 30 m. for the Elevation of the North Pole above the upper Face of the Plane, and therefore 38 gr. 30 m. for the Height of the South Pole above the lower Face of the Plane.
Or if the Inclination of the Plane shall be found to be 62 gr. to the Southward, we may number them in the Meridian from S the South part of the Horizon unto L, and there draw the Ark ELW representing this Plane; so the Ark of the Meridian PL shall give the Height of the North Pole above the upper face of this Plane to be 66 gr. 30 m. and therefore the height of the South Pole above the lower Face of the Plane is also 66 gr. 30 m.
In like manner, if the Inclination of the Plane EYW shall be 15 gr. Southward, that is, if SY the Ark of the Meridian between the South part of the Horizon and the Plane shall be 15 gr. the height of the North Pole above the upper Face of the Plane, and the height of the South Pole above the lower Face of the Plane, will also be found to be 66 gr. 30 m.
But if the Plane shall fall between the Zenith and the North Pole, then will the North Pole be elevated above the lower Face, and the South Pole above the upper face of the Plane, as may appear by the Projection of the Sphere in the Fundamental Diagram.
Xxxx in the Triangles made by the Plane, the Meridian, and the Hour-circles, we have the side which is the height of the Pole above xxxxx together with the Angles at the Pole, and the Right Angle at the Intersection of the Meridian with the Plane, by which we xxxxx Arks of the Plane between the meridian and the Hour-points in like manner.
- As the Sine of 90 gr. Is to the Sine of the Pole above the Plane:
- So the Tangent of the Hour, To the Tangent of the Hour-line from the Meridian.
Thus in the former Example, where PI the height of the Pole above the Plane was found to be 38 gr. 30 m. if you shall extend the Compasses from the Sine of 90 gr. to the Sine of 38 gr. 30 m. the same extent will reach from the tangent of 15 gr. unto the Tangent of 9 gr. 28 m. for the distance of the first Hour from the Meridian, and from 30 gr. unto 19 gr. 46 m. for the second Hour, and so forward, as in the direct Vertical.
And for the two last Examples, you may extend the Compasses from the Sine of 90 gr. unto the Sine of 66 gr. 30 m. so the same extent shall reach in the Line of Tangents from 15 gr. unto 13 gr. 48 m. for the first Hour, from 75 gr. unto 73 gr. 43 m. for the fifth Hour, from 30 gr. unto 27 gr. 54 m. for the second Hour, from 60 gr. unto 57 gr. 48 m. for the fourth Hour, and from 45 gr. unto 42 gr. 31 m. for the third Hour from the Meridian.
These Arks being known, you may first draw the Horizontal Line, and cross it in the middle with a perpendicular, that may serve both for the Meridian and the Hour of 12, and the Substylar; then knowing which Pole is elevated above the Plane, you may accordingly make choice of a fit Point in the meridian for the Center of your Hour-lines, and thence describe an occult Ark of a Circle, inscribe the Cords of those former Arks, and draw the Hour-lines, and set up the Style, as I shewed before in the Horizontal Plane.
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