The Third Book of the Cross-Staff

CHAP. XI.
To describe the Tropicks and other Circles of Declination in an Equinoctial Plane.

Such Circles as are parallel to the Equinoctial, and yet fall within the Tropicks, may be described on any Plane by the help of these Lines of proportion, but after a different manner, according as the Style shall be either perpendicular of parallel to the Plane, or cut the Plane with Oblique Angles.

In an Equinoctial Plane, where the Style is perpendicular to the Plane, the Tropicks and other Circles of Declination will be perfect Circles: Wherefore consider the length of the Style in Inches and parts, and the Declination of the Circle which you intend to describe in Degrees and Minutes, the proportion will hold,

• As the Tangent of 45 gr. to the Length of the Style:
• So the Co-tangent of the Parallel, to the Semidiameter of his Circle.

Suppose the Length of the Style above the Plane to be 10 Inches, and that it were required to find the Semidiameter of the Tropick, whose Declination is known to be 23 gr. 30 m. Extend the Compasses from the Tangent of 45 gr. unto the Tangent of 66 gr. 30 m. the same extent will reach in the Line of Numbers from 10 unto 23, which shews the Semidiameter of the Tropick to be 23 Inches. So if the Declination be 20 gr. the Semidiameter will be 27 Inches 47 cent. If 15 gr. then 37.32; if 10 gr. then 56.71; if 5 gr. then 115.305: and so the rest.

Or if it were required to proportion the Style to the Plane,

• As the Tangent of 45 gr. to the Tangent of the Declination:
• So the Semidiameter of the Plane, to the Length of the Style.

As if the Semidiameter of the greatest Parallel upon the Plane were but six Inches, and that Parallel should be the fifth Degree of Declination; extend the Compasses from the Tangent of 45 gr. unto the Tangent of 5 gr. the same extent will reach in the Line of Numbers from 6.00 unto about 0.53, which shews that the length of the Style must be 53 parts of an Inch divided into 100: Then the length of the Style being known, the Semidiameter of the other Circles will be found as before.

I begin here with the fifth Parallel, and thence proceed unto the Tropick, because the Shadow of the rest near the Equinoctial would be over-long, and the Equinoctial it self cannot be described. The Parallels of North Declination are to be set on the North Face, and the Parallels of South Declination on the South Face of the Plane. Neither need these Parallel to be drawn in full Circles, but onely to the Horizontal Line, which shall be described in Chap. 18.

Having by these means set up the Style to its true Height, and drawn the Circles of Declination, if we shall place the Plane so as it shall make an Angle with the Horizon equal to the Complement of the Latitude, and then turn it until the top of the Style cast the Shadow of the Style upon the Parallel of Declination belonging to the Time, the meridian of the Plane will shew the Meridian of the Place, and the Shadow of the Style the Hour of the day, without the help of a Magnetical Needle.