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.

The foregoing PROPOSITIONS applied to Practice: By which the Ingenious young Sea-man may make them serviceable to him at Sea, to severall good and usefull Purposes. The Seventh EXERCISE.

OF the foregoing Astronomicall Propositions the ingenious Sea-man may make good and profitable use at Sea. For some of them will be assistent to him in the finding of the Lati∣tude of the Place he is in.—Some of them will help him to find the time of the Sun's rising and setting in any place, and at any time of the Year.—Some will help him to the Hour of the Day,—Some to the Hour of the Night, at any time and in any place.—And divers of them to find the Varia∣tion of the Compass. Examples of all which I will instance in, so that he may put them in practice at Sea.

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.

II. Propositions assistent to find the time of the Sun's rising and setting.

THE principal Proposition for this purpose is the Se∣cond, which is to find the Ascensionall Difference, from which the time of the Sun's rising and setting, the Semidiurnall and Seminocturnall Arches, may be gathered; and from thence the length of the Day and Night: all which are plainly shew∣ed in the Proposition it self.

III. Propositions assistent to find the Hour of the Day and Night.

THE Twefth and Thirteenth Propositions will be ser∣viceable to find the Hour of the Day. The Twelfth giving the Hour at any time of the Day by the Work of that Proposition it self.—The Thirteenth findeth the Hour upon a given Azimuth and Altitude. Wherefore set your Compass to the given Azimuth, and observe his Altitude when he cometh upon that Azimuth; the Sun's Declination (or time of the year) being known, you may then find the Hour by the Work of the Thirteenth Proposition.

The Proposition that will be assistent to you in finding the Hour of the Night is chiefly the Sixth, it shewing how to find the Sun's right Ascension, which, with the assistence of those other Tables which follow that Proposition, will help you to the Hour of the Night, and also to find at what time any of the Stars there inserted in the Table will be upon the Meridian. The manner how to effect either shall be shewed in these two following Problems.

PROBL. I. How to find at what time any of the Stars in the Table of the Sixth Proposition will be upon the Meridian.

SUbstract the Right Ascension of the Sun from the Right As∣cension of the Star, the remainer is the time of the Star's coming to the Meridian after noon. But if the right Ascension of the Star be less then the Right Ascension of the Sun, adde 360 degr. thereto, and substract the Right Ascension of the Sun from the Sum, and the Remainer is the time of the Star's coming to the Meridian.

Example. Upon the fourth of October 1667, the Sun being in 21. degr. of Libra, I would know at what time Sirius (or the Great Dog) will be upon the Meridian.

	d.	m.
The Right Ascension of Sirius is	97	27
The Right Ascension of the Sun that day is	199	23
Because Substraction cannot be made, adde 360 d. to the Right Ascension of the Star	360	00
The Sum is	457	27
The Sun's Right Ascension substracted from 458 degr. 4 min. leaves the time of the Star's coming to the Meridian	258	04

Which 258 degr. 4 min. being converted into Time make 17 hours almost, that is, at 5 of the Clock the next Morning Sirius will be upon the Meridian.

PROBL. II. To find the Hour of the Night by any of the Stars that are in the Table of the Sixth Proposition.

TAKE the Altitude of the Star, and by his Declination and Altitude find the Hour (by the Twelfth Proposition) as if it were by the Sun, which I call the Star's Hour. Then comparing the Right Ascension of the Sun with the Right Ascen∣sion of the Star, you may come to find the Hour of the Night.

Example. Upon the 16. day of November, in the Morning, I took the Altitude of Arcturus, finding it to be 27 degr. 12 m. and his Declination (by the Table) I find to be 20 degr. 58 m. By help of these two and the Latitude I find the Star's Hour to be 72 degr. Then compare the Sun's Right Ascension with the Star's Right Ascension, and find his time of coming to the Meridian, as in the former Probl. the difference between the Star's Hour and his coming to the Meridian is the Hour of the Night. See the manner of the Operation.

	d.	m.
The Right Ascension of Arcturus	210	13
The Right Ascension of the Sun	242	00
Adde to make Subt.	360	00
The Sum is	570	13
The Sun's Right Ascension substracted, rests	328	13
From which take 180 degr. or 12 hours	180	00
Rests	148	13
The Star's Hour substracted	72	00
Leaves the Hour of the Night	76	13

Which converted into Time is 5 h. 5 m. and that is the Hour in the Morning.

IV. Propositions assistent to the finding of the Variation of the Compass.

THE Propositions that will be serviceable herein are the 3. 7. 8. 9. 10. 11. and 14. but more especially the Third and the Eleventh: and those I shall here illustrate by Exam∣ple, though all the rest (as occasion may fall out) will be also usefull thereunto. By the Third Proposition you may find the Amplitude of the Sun's Rising and Setting.—By the Eleventh you may find the Azimuth at any time of the day.— By either of which the Variation of the Compass may be found, and also which way it varieth.

I. To find the Variation by the Amplitude.

BY the Third Proposition you found the Sun's Amplitude at his rising or setting to be 33 degr. 20 min. from the true East or West Points of the Horizon towards the North. Having thus before-hand found the Amplitude, in the Morning I set my Compass to the Sun at his Rising; and if I find that the Sun by my Compass do rise 33 d. 20 m. from the West-point thereof towards the North, then may I be ascertain'd that my Compass hath no Variation, but that the Fly or Wires do point directly North and South.—But finding before-hand the Amplitude to be 33 d. 20 m. and I should find the Sun to rise but 28 degr. from the East-point of my Compass, then substracting 28 degr. from 33 degr. 20 min. the difference is 5 degr. 20 min. and so much doth my Compass varie from the true East-point, and con∣sequently all the other Points of the Compass as much.

Now to find which way the Compass varieth, you must ob∣serve whether your Amplitude, found by your Calculation, be to the Right or Left-hand of the Sun's rising or setting. And

if it be on the Right-hand, you may conclude the Variation to be Easterly; but if on the Left-hand, Westerly.

As for Example; Finding by the Amplitude that the Sun should rise 33 d. 20 min. from the East Northerly, when I come to set my Compass to the Sun at his rising, I find that the Sun riseth but 28 degr. from the East Northerly; wherefore the Am∣plitude found is on the Left-hand, and so I conclude the Varia∣tion to be 5 d. 20 min. Westerly.

II. To find the Variation by the Azimuth.

SUppose the Sun's Azimuth found by the Eleventh Propositi∣on to be 107 degr. 30 min. from the North, and when I set the Compass, I find the Magneticall Azimuth to be 102, the dif∣ference between the true and the Magneticall Azimuth being 5 d. 30 m. which is the Variation.

Now to know whether this Variation be towards the East or towards the West: seeing by the Azimuth found the Sun should have been 107 d. 30 min. from the North, which is 17 degr. 30 min. from the East; but setting of the Sun with my Compass, I find that it was from the East to the Southward onely 12 degr. so that the Degree upon which the Sun should have been was more towards the Right-hand then the Degree on which it was; therefore I conclude the Variation to be 5 degr. 30 min. Easterly.