Nine geometricall exercises, for young sea-men and others that are studious in mathematicall practices: containing IX particular treatises, whose contents follow in the next pages. All which exercises are geometrically performed, by a line of chords and equal parts, by waies not usually known or practised. Unto which the analogies or proportions are added, whereby they may be applied to the chiliads of logarithms, and canons of artificiall sines and tangents. By William Leybourn, philomath.

PROP. II. The Latitude of the Place, and the Declination of the Sun, being given, to find the Ascensional Diffe∣rence.

UPON the Projection this Proposition is to be resolved by finding the Side A B of the Right-angled Sphericall Triangle A ☉ B, Right-angled at B; which Triangle is com∣pounded of three Arches of great Circles, namely, of A ☉, an Arch of the Horizon, A B, an Arch of the Aequinoctial, and ☉ B, an Arch of an Hour-Circle.

In this Triangle you have given (1.) the Side ☉ B, the Sun's Declination 20 d. (2.) the Angle ☉ A B, the Complement of the Latitude 38 d. 30 m. and the right Angle at B. In this Triangle therefore you have given ☉ B, the Perpendicular, and ☉ A B, the Angle at the Base, to find the Base A B, which you may doe by the 14. Case of Right-angled Sphericall Triangles. For which this is

The Analogie or Proportion.

As the Co-tangent of the Latitude 38 degr. 30 min. is to the Tangent of the Sun's Declination 20 degr.

So is the Radius 90 degr. to the Sine of the Ascensional Dif∣ference 27 degr. 14 min.

To resolve the Triangle upon the Projection,

Lay a Ruler to P, the Pole of the World, (and also of the Aequinoctial,) and the Point B, it will cut the Meridian Circle in the Point b; the distance b S, being taken in your Compasses and measured upon your Line of Chords, will reach from the beginning thereof to 27 degr. 14 min. the Ascensional Dif∣ference; which is so much as the Sun riseth or setteth before or after Six a Clock. So these 27 degr. 14 min. being turned

into Time (by allowing 15 deg. for one Hour, and one Degree for 4 minutes of Time) is 1 Hour and 49 min. and so much doth the Sun rise or set before or after the hour of Six, ac∣cording to the time or season of the Year: for if the Sun hath North Declination, then he riseth before Six, and sets after; but if the Sun have South Declination, then doth he rise after, and set before Six.

This Ascensional Difference being added to 6 Hours will give you the Semidiurnall Arch or Half-length of the Day; and being taken from six Hours, will leave the Seminocturnall Arch or Half-length of the Night.

The Semidiurnall Arch, when the Sun hath 20 degr. of North Declination, is represented in the Projection by the Arch ♊ a ☉, and the Seminocturnall Arch by ☉ ♌. The Semidiurnall Arch, when the Sun hath 20 degr. of South Declination, is represented by the Arch ♒ E r, and the Seminocturnall by the Arch r c ♐.

Though I have shewed how these may be found by adding and subtracting the Ascensionall Difference; yet they may be found by the Projection, for the Arches are measured upon the Aequino∣ctial. Wherefore lay a Ruler to P, the Pole of the World, and the Point B, it will cut the Meridian Circle in b: So the distance b AE, being measured by your Chord, will be 117 degr. 14 min. the Semidiurnall Arch, and b ae measured will be 62 degr. 11 min. for the Seminocturnall Arch.