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. Two Places which differ onely in Longitude, to find their Distance.

IN this Proposition there are two Varieties also. For 1. The two Places may lie both under the Aequinoctial, and have no Latitude: in this Case the difference of their Lon∣gitudes (if it be less then 180 degr.) reduced into Miles is their Distance; but if their difference exceed 180 degr. take it out of 360 degr. the remaining Degrees turned into Miles will be the Distance of the two Places.

Example. The Island Sumatra and the Island of S. Thoma lie both under the Aequinoctial, the Island of S. Thoma having 33 d. 10 m. of Longitude, and the Island Sumatra 137 d. 10 m. The lesser Longitude taken from the greater leaves 104 d. 0 m.

which converted into Miles is 6240. And that is the Distance of the two Islands.

[illustration] geometrical diagram

2. But if the two Places differ onely in Longitude, and lie not under the Aequinoctial, but under some other intermediate Parallel of Latitude, between the Aequinoctial and one of the Poles, then to find their Distance, this is

The Analogie or Proportion.

As the Radius 90 degr. is to the Co-sine of the common Lati∣tude 47 degr.

So is the Sine of half the difference of Longitude 21 d. 37 m. to the Sine of half their Distance 15 degr. 38 m.

So let the two Places be the Cities Constantinople and Com∣postella, both lying in the Latitude of 43 degr. but they differ in Longitude 43 degr. 15 min. So that their Distance accor∣ding to the former Analogie will be found to be 31 degr. 16 m. which, converted into Miles, is 1876 miles, which is the Di∣stance between the two Cities Constantinople and Compostella.

To find the Distance of these two Places upon the Projection.

The two Places upon the Projection are noted with the Letters C and D, both lying in the Latitude of 43 degr. but Constantinople in the Longitude of 63 degr. and Compo∣stella in the Longitude of 106 d. 15 min. So that their diffe∣rence of Longitude is 43 degr. 15 min. Wherefore through the two Places C and D draw the Arch of a great Circle, and find the Pole thereof; (which to effect is already taught at the beginning of this Book:) which Pole will be at the Point M. Then laying a Ruler upon M and C, it will cut the first Meridian in the Point d; and laid from M to D, it will cut the first Meridian in e: the Distance between d and e, mea∣sured upon the Line of Chords, will be found to contain 31 degr. 16 min. which, converted into Miles, giveth 1876, the Distance as before.