The Third Book of the Cross-Staff
Those Planes wherein a Line may be drawn parallel to the Axis of the World are called Polar Planes, because that Line pointeth unto the Poles; and these Planes are always parallel to some one of the Hour-circles. If they be parallel to the Hour of 6, they are called Direct Polar Planes; if to the Hour of 12, they are called Meridian Planes; and both these are described before: if to any other of the Hour-circles, they are then called by the name of Polar Declining Planes, because their inclining to the Pole, and declining from the Vertical.
These kind of Planes may be known in this sort: First, consider the Inclination of the Plane to the Horizon, which in these parts of the World must always be Northward, and more than the Latitude of the Place. Then find the Declination from the Vertical. These two being known, if the proportion hold,
It is then a Polar declining Plane; otherwise not.
For example: In our Latitude of 51 gr. 30 m. a Plane is proposed declining from the Vertical 65 gr. 40 m. and inclining Northward 71 gr. 51 m. the upper Face being open to South-east, and the lower to the North-west. If I number those 65 gr. 40 m. in the Horizon of the Fundamental Diagram from E unto Q, and draw the Line HCQ, it shall represent the Horizontal Line of the Plane; then crossing it at Right Angles with the Plane BZD drawn through the Zenith, I number 71 gr. 51 m. for the Inclination from D unto R, and there draw the Circle HRQ: this Circle so drawn shall represent the Plane proposed; and because it also passeth through the Pole, it is therefore a Polar Plane. But for further trial, I extend the Compasses from the Sine of 90 gr. To the Sine of 24 gr. 20 m. the Complement of the Declination, and I find the same Extent to reach from the tangent of 71 gr.51 m. the Inclination proposed, unto the tangent of 51 gr. 30 m. which is the true Latitude of the Place; and therefore it is a Polar Plane.
Again, I number the Inclination of 71 gr. 51 m. in the Circle BZD from Z unto M, so this point M will fall at the meeting of BZD with the Equator, and being 90 gr. From the Plane R, it shall be the Pole of this Plane; and the Circle drawn though M and P will be the proper Meridian of this Plane. The Meridian MP here falling on the Hour of 8 doth give MPZ, the Angle of the Inclination of Meridians, to be 4 Hours, or 60 Degrees; then crossing the Plane at the point P, it shews that the Substylar should be CP, and be placed at the Hour of 8. But because P is the Pole, and CP the Axis of the World wherein all Hour-circles do meet, and so there would be no distinction between the Axis, the Substylar, and the Hour-lines, I now suppose the Plane in a parallel to the Circle HRQ, according to the distance that I would have between the Axis of the Style and the Substylar, then will the Style be parallel to the Plane, as appears in the Fundamental Diagram.
Here then the Style will be parallel to the Plane, and the Hour-lines parallel one to the other, as in the meridian and Direct Polar Planes. Yet that we may better know how to draw the Hour-lines, and where to place the Style, we are to consider,
In a Meridian Plane, the Ark between the Horizon and the Pole, which represents the Ark between the Horizon and the Hour-lines, is always equal to the latitude of the Place; in a direct Polar it is an Ark of 90 gr. In these Declining Polars it is greater than the latitude, and yet less than 90 gr. This Ark here is represented by PQ, and may be known by resolving the Triangle QNP, or PRZ.
Extend the Compasses from the Sine of 90 gr. Unto the Sine of 38 gr. 30 m. the Complement of the Latitude, the same extent will reach from the Sine of 65 gr. 40 m. the Declination proposed, unto the Sine of 34 gr. 34 m. whose Complement is 55 gr. 26 m. the Ark of the Plane required between the Horizon and the Pole.
And so extending the Compasses from the Sine of 18 gr. 9 m. the Complement of the Inclination to the Tangent of 24 gr. 20 m. the Complement of the Declination, the same extent doth reach from the Sine of 90 gr. unto the Tangent of 55 gr. 26 m. And such is QP the Ark of the Plane between the Horizon and the Pole, the measure of the Angle QCP between the Horizontal Line and the Substylar.
The Substylar in a Direct Polar Plane is always the same with the Hour-line of 12; in a Meridian Plane it is the same with the Hour-line of 6; in these Declining Polars it must be placed between 12 and 6, according to the Inclination of the Meridian of the Plane to the Meridian of the Place, which is here represented by MPZ, the Complement of the Angle RPZ, and thus known.
Extend the Compasses from the Sine of 90 gr. to the Sine of 51 gr. 30 m. the latitude of the Place, the same extent will reach from the Tangent of 65 gr. 40 m. the Declination proposed, unto the tangent of 60 gr. and such is the Angle of Inclination between the Meridian of the Place and the proper Meridian of the Plane, which resolved into Time, doth make four Hours; and so the Substylar must here be placed upon the Hour of 8 in the Morning.
This Angle being known, the rest of the Angles at the Pole are easily gathered: For if the Hour of 12 be 60 gr. distant from the meridian of the Plane, the Hour of 1 will be 75 gr. and the Hour of 11 will be 45 gr. distant, and the rest of the Hours, as in the Table following. Then coming to the Plane,
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