The Third Book of the Cross-Staff
CHAP. V.
To draw the Hour-lines in a Vertical Plane.
A vertical Plane is that which is parallel to the Prime Vertical Circle in the Fundamental Diagram, represented by EZW. It hath two Faces, the one to the North, the other to the South; in each of them the Substylar will be the same with the Meridian Line, and the Angle of the Style above the Plane will be equal to ZP, the Complement of the Latitude; and the Hour-lines here required may be supplied by imagining Right Lines drawn from the Center C to the Intersections of the Hour-circles EZW.
The Triangles here considered are made by the Vertical, the Meridian, and the Hour-circles, in which we know the Side ZP, the Angles at the Pole, and the right Angle at the Zenith, and therefore may find the Arks of the Vertical, between the Meridian and the Hour-circles, after this manner.
- As the Sine of 90 gr. is to the Co-sine of the Latitude:
- So the Tangent of the Hour, to the Tangent of the Hour-line from the Meridian.
Extend the Compasses from the Sine of 90 gr. to the Sine of the Complement of the Latitude, so the same extent shall reach from the Tangent of the Hour, to the Tangent of the Hour-line from the Meridian.
Thus in the Latitude of 51 gr. 30 m. I extend the Compasses from the Sine of 90 gr. to the Sine of 38 gr. 30 m. and find the same extent to reach from the tangent of 15 gr. to the Tangent of 9 gr. 28 m. for the distance of the first Hour from the Meridian; And from the tangent of 75 gr. unto the Tangent of 66 gr. 42 m. for the fifth Hour: and so the rest, as in the Table.
| Latit. | 51 | 30 | ||
| Ho. | Ang. Po. | Arc. Po. | ||
| Gr. | M. | Gr. | M | |
| 12 | 0 | 0 | 0 | 0 |
| 3 | 45 | 2 | 20 | |
| 7 | 30 | 4 | 41 | |
| 11 | 15 | 7 | 3 | |
| 1 | 15 | 0 | 9 | 28 |
| 18 | 45 | 11 | 56 | |
| 22 | 30 | 14 | 27 | |
| 26 | 15 | 17 | 4 | |
| 2 | 30 | 0 | 19 | 45 |
| 33 | 45 | 22 | 35 | |
| 37 | 30 | 25 | 32 | |
| 41 | 15 | 28 | 38 | |
| 3 | 45 | 0 | 31 | 54 |
| 48 | 45 | 35 | 22 | |
| 52 | 30 | 39 | 3 | |
| 56 | 15 | 42 | 58 | |
| 4 | 60 | 0 | 47 | 9 |
| 63 | 45 | 51 | 36 | |
| 67 | 30 | 56 | 20 | |
| 71 | 15 | 61 | 23 | |
| 5 | 75 | 0 | 66 | 42 |
| 78 | 45 | 72 | 17 | |
| 82 | 30 | 78 | 3 | |
| 86 | 15 | 84 | 0 | |
| 6 | 90 | 0 | 90 | 0 |
These arks being known, I may come to the Plane, and then by help of a Thread and Plummet draw a Vertical Line, serving both for the Meridian and the Hour of 12, and the Substylar; then may I draw an occult Vertical Circle, and therein inscribe the Chords of those former Arks, and draw the Hour-lines, and set up the Style, as before in the Horizontal Plane.
If it be the South face of the Plane, the Center will be upward, and the Style will point downward: If the North face, the Center must be in the lower part of the Meridian Line, and the Style point upward in all such Places as are to the Northward of the Equinoctial Line, as it may appear considering how the Line do fall in the Fundamental Diagram.
