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.

I. Propositions assistent to find the Latitude.

THE Propositions which may be applied to the finding of the Latitude are the First and the Sixteenth.

The first Proposition is to find The Sun's Declination, which being obtained, and the Sun's Meridian Altitude observed at Sea or Land in any part of the World, the Latitude of that Place, by help of them, may be known; in which there are severall Cases, according as the Sun hath either North or South Declination, and as the Sun is situate, he being either upon the North or South-side of the Meridian.—The severall Varie∣ties are these which follow.

When the Sun is in the Aequinoctial, ha∣ving no Declination, and the Meridian Altitude is observed on the

• SOUTH-side of the Meridian,
  • The Meridian Alti∣tude taken from 90 degr. leaves the Elevation of the North-pole.
• NORTH-side of the Meridian,
  • The Meridian Alti∣tude taken from 90 degr. leaves the Elevation of the South-pole.

When the Sun's Declination is

• NORTH,
  • If the Meridian Altitude be less then 90 d. and the Sun upon the South-side of the Meridian; the Sun's Decli∣nation, being taken from the Meridian Altitude, leaves the height of the Aequinoctial, which taken from 90 d. gives the Latitude North.
• SOUTH,
  • If the Meridian Altitude be less then 90 d. and the Sun upon the South-side of the Meridian, adde the Meridi∣an Altitude and Declination together; their Sum is the height of the Aequinoctial, which taken from 90 degr. leaves the Latitude North. But if the Sum of the De∣clination and Altitude exceed 90 degr. take 90 degr. therefrom, the remainer is the Latitude South.

When the Sun's Declination is

• NORTH,
  • If the Meridian Altitude be less then 90 d. and the Sun upon the North-side of the Meridian, adde the Alti∣tude and Declination together; their Sum is the height of the Aequinoctial, which taken from 90 d. leaves the Latitude South.—But if the Sum be above 90 d. take 90 d. therefrom, the remainer is the Latitude North.
• SOUTH,
  • If the Meridian Altitude be less then 90 d. and the Sun upon the North-side of the Meridian, subtract the De∣clination from the Meridian Altitude; the remainer is the height of the Aequinoctial, which taken from 90 degr. leaves the Latitude South.

When the Sun's Decli∣nation is

• NORTH, If the Meridian Altitude be just 90 degr. the Sun's Declination is the Latitude North.
• SOUTH, If the Meridian Altitude be just 90 degr. the Sun's Declination is the Latitude South.

If the Meridian Altitude be observed under the Pole, within the bounds of the Polar Circles, in such case the Sun's Decli∣nation must be taken from 90 degr. and what remains is his distance from the Pole; which being added to the Meridian Altitude, the Sum is the Latitude of the Place.

The other Proposition which will be assistent to find the La∣titude is the Sixteenth. If you set your Compass to the given Azimuth, and when the Sun is upon that Azimuth, if you take his height, you may find your Latitude either by Trigonometri∣call Calculation or by Projection. As in the Sixteenth Propo∣sition.