The Third Book of the Cross-Staff
A Direct Polar Plane is that which is parallel to the Hour of 6, and here represented by EPW, wherein the Style will be parallel to the Plane, and the Hour-lines parallel one to the other; and therefore may be best drawn by that which I have shewed in the Use of the Sector. They may be also drawn by the help of these Lines of Proportion, in this manner.
First draw a Right Line WE for the Horizon and the equator, and cross it at Point C, about the middle of the Line, with CB another Right Line, which may serve for the Meridian and the Hour of 12, and must also be the Substylar Line, wherein the Style shall stand. Then, to proportion the Style unto the Plane, consider the length of the Horizontal Line, and what Hour-lines you would have to fall on your Plane.
For the distance of any one Hour-line from the Meridian being known, we may find both the length of the Style, and the distance of the rest: because,
Suppose the length of the Horizontal Line to be 12 Inches, and that it were required to put on all Hour-lines from 7 in the Morning unto 5 in the Evening. Here we have 5 Hours and 6 Inches in either side of the Meridian: Wherefore I allow 15 gr. for an Hour, and extending the Compasses from the Tangent of 75 degrees, I find the same extent to reach in the Line of Numbers from 600 to about 161. This shews both the height of the Style, and the distance of the Hour-points of 9 and 3 from the Meridian, to be 1 Inch and 61 parts.
Thus having found the length of the Style in our Example to be 1.61, if I extend the Compasses from the Tangent of 45 gr. unto the Tangent of 15 gr. the measure of the first Hour from the Substylar, I shall find the same extent to reach in the Line of Numbers from 1.61 unto 0.43, for the Length of the Tangent between the Substylar and the Hour-points of 11 and 1. If I extend them from the Tangent of 45 gr. unto the Tangent of 75 gr. the measure of the fifth Hour, I shall find them to reach in thze Line of Numbers from 1.61 unto 6.00, for the length of the Tangent from the Substylar to the Hour-points of 7 and 5. For howsoever it be the same distance in the Line of Tangents from 45 to 75, as from 45 unto 15; yet because 75 are more, and 15 less than 45, the Tangent Lines that answer to them will be accordingly more or less than the length of the Style.
Again, if I extend them from 45 gr. in the Tangents unto 30 gr. the measure of the second Hour, I shall find them to reach in the Line of Numbers from 1.61 unto 0,93 for the Hour of 10 and 2: If I extend them from the Tangent of 45 gr. unto the Tangent of 60 gr. for the fourth Hour, I shall find them to reach in the Line of Numbers from 1.61 unto 2.79, and such is the length of the Tangent Line from the Substylar unto the Hour of 8 and 4. And the like Reason holdeth for the inscribing of all other Tangent Lines in the Propositions following.
But for such Tangents as fall under 45 gr. I may better use cross Work, and extend the Compasses from the Tangent of 45 gr. unto 1.61 in the Line of Numbers, so shall I find the same extent to reach from 30 gr. in the tangents, to 93 parts in the Line of Numbers, for the distance of the second Hour; and from 15 gr. in the tangents, to 43 parts for the distance of the first Hour from the Meridian.
Or if this extent from 45 gr. backward to 1.61 be too large for the Compasses, I may extend them forward from the Tangent of 5 gr. 43 m. to 1.61 parts in the Line of Numbers, and the same extent shall reach from 15 gr. in the Tangents, to 43 parts in the Line of Numbers, for the distance of the first Hour; and from 30 gr. to 93 parts, for the distance of the second Hour, as before.
Having found the length of the Tangent Lines in Inches and parts of Inches, and pricked them in the Equator on both sides of the Meridian, from the Center C; if we draw Right Lines through each of those Points, crossing the Equator at right Angles, they shall be the Hour-lines required; and if we set a Style over the Meridian, sa as the edge of it be parallel to the Plane, and the height of it be as much above the Meridian, as the distance between the meridian and the Hour-points of 3 and 9, it shall represent the Axis of the World, and be truly placed for the casting of the Shadow upon the Hour lines in a Polar Plane.
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