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 VARIETY OF SPHERICALL PROBLEMS Naturally arising out of every Sphericall Triangle, both Right and Oblique-angled, and that are resolvable thereby, described as they are perspicuous to the Eye in the Projection. The Fifth EXERCISE.

IN the foregoing Part of this Book you have the Doctrine of Plain and Sphericall Tri∣angles Geometrically performed. And in the Solution of Right-angled Sphericall Triangles there were 16 Cases; and in Ob∣lique-angled there were 12 Cases: but the 16 Cases of Right-angled Triangles will by this projective way be reduced to 5 Ca∣ses, and the 12 of Oblique-angled will be reduced to 6; so that in both there will be but 11 Cases, whereas before there were 28. That this may appear plain to the Reader, I will make use of two

Triangles in the Projection; one whereof shall be Right-angled, as the Triangle P O ☉, Right-angled at O; and the other shall be the Oblique-angled Triangle Z E P.—The Right-angled Triangle is constituted by the Intersection of three great Circles of the Sphere; namely, of P O, an Arch of the Meridian, ☉ O, an Arch of the Horizon, and P ☉, the Arch of an Hour-Circle. —The Oblique-angled Triangle, Z E P, is constituted also of three Arches of great Circles of the Sphere, (as all Sphericall Trian∣gles whatsoever are;) namely, of Z P, an Arch of the Meri∣dian, P E, an Arch of an Hour-Circle, and Z E, an Arch of an Azimuth Circle.

In the Right-angled Sphericall Triangle, P ☉ O,

• The Side P O is the Latitude of the Place.
• The Side ☉ O is the Sun's Amplitude from the North.
• The Side P ☉ is the Sun's distance from the Pole, or the Complement of his Declination.
• The Angle ☉ P O is the Hour from Midnight, or from the North part of the Meridian.
• The Angle P ☉ O is the Angle of the Sun's Position at the time of the Question.
• The Angle P O ☉ is the Right Angle.

The Parts of the Triangle being declared, and of what Cir∣cles of the Sphere the Sides do consist; I will now come to the Cases, which (as I said before) are 5 in every Right-angled Triangle. So that any two parts of the Triangle (besides the Right Angle) being given, I will shew in every of the 5 Cases what parts may be found.

I. In a Right-angled Sphericall Triangle.

CASE I. The Base and Perpendicular being given, to finde the other parts of the Triangle.

IN the Triangle P ☉ O here is given P O, the Latitude, and ☉ O, the Amplitude from the North part of the Meri∣dian; by which you may find 3.

• 1. P ☉, the Complement of the Sun's Declination.
• 2. ☉ P O, the Hour from Midnight.
• 3. P ☉ O, the Angle of the Sun's Position.

CASE II. The Hypotenuse and Perpendicular being given, to find the other parts.

IN the Triangle P ☉ O here is given P O, the Latitude, and P ☉, the Complement of the Sun's Declination; by which may be found 6.

• 1. ☉ O, the Amplitude from the North.
• 2. O P ☉, the Hour from Midnight.
• 3. P ☉ O, the Angle of the Sun's Position.
  • And if in stead of the Perpendicular there had been given the Base ☉ O, you might then find
• 4. P O, the Latitude.
• 5. ☉ P O, the Hour from Midnight.
• 6. P ☉ O, the Angle of the Sun's Position.

CASE III. The Hypotenuse and an Angle being given, to find the other Parts.

HERE is given in the Triangle P ☉ O, the Complement of the Sun's Declination, P ☉ and ☉ P O, the Hour from Midnight; by which may be found 6.

• 1. ☉ O, the Amplitude from the North.
• 2. P O, the Latitude.
• 3. C ☉ O, the Angle of the Sun's Position.
  • And if in lieu of the Angle at P, the Angle at ☉ P had been taken, then you might have found
• 4. ☉ O, the Amplitude from the North.
• 5. P O, the Latitude.
• 6. ☉ P A, the Hour from Midnight.

CASE IV. The Perpendicular, or Base, and either of the Angles given, to find the other Parts.

IN the Triangle P ☉ O let there be given the Amplitude, ☉ O, and the Angle of the Sun's Position, P ☉ O; by which you may find 12.

• 1. P O, the Latitude.
• 2. ☉ P O, the Hour from Midnight.
• 3. P ☉, the distance of the Sun from the Pole.
  • But if the Side given had been ☉ O, and the Angle given ☉ P O, then you might have found
• 4. P ☉, the Complement of the Sun's Declination.
• 5. P O, the Latitude.
• 6. P ☉ O, the Angle of the Sun's Position.
  • But again, if ☉ O and ☉ P O had been given, then might be found
• 7. ☉ P, the Complement of the Sun's Declination.
• 8. ☉ O, the Amplitude from the North.

• 9. P ☉ O, the Angle of the Sun's Position.
  • And again, if P O and O ☉ P had been given, we might then have also found
• 10. ☉ P, the distance of the Sun from the Pole.
• 11. ☉ O, the Amplitude from the North.
• 12. ☉ P O, the Hour from Midnight.

CASE V. The Angles being given, to find the other Parts.

IF the two Angles P ☉ O and O P ☉ be given, there may be found 3.

• 1. P O, the Latitude.
• 2. ☉ O, the Amplitude from the North.
• 3. P ☉, the Complement of the Sun's Declination.

Thus you see, that in this one Right-angled Sphericall Tri∣angle, by the several Parts given in these five Cases, there are 30 Sphericall Problems resolved; and so many are resolvable in every Right-angled Triangle.

II. In an Oblique Sphericall Triangle.

In the Oblique-angled Sphericall Triangle, Z P E,

• The Side Z P is the Complement of the Latitude.
• The Side P E is the Complement of the Sun's Declination, or his distance from the North Pole.
• The Side Z E is the Complement of the Sun's Altitude.
• The Angle E Z P is the Sun's Azimuth from the North part of the Meridian.
• The Angle Z P E is the Hour from Noon.
• The Angle Z E P is the Angle of the Sun's Position.

CASE I. The three Sides being given, to find an Angle.

IN the Triangle Z E P, if there be given the Side E Z, the Complement of the Sun's Altitude, Z P, the Complement of the Latitude, and E P, the Sun's distance from the Pole, or the Complement of his Declination, you may find 3.

• 1. E Z P, the Sun's Azimuth from the North.
• 2. Z E P, the Angle of the Sun's Position.
• 3. Z P E, the Hour from Noon.

CASE II. Two Sides and the Angle comprehended by them being given, to find the other Parts of the Triangle.

IF in the Triangle Z E P there be given the Side E Z and Z P, and the Angle between them E Z P, you may find 9.

• 1. Z E P, the Angle of the Sun's Position.
• 2. Z P E, the Hour from the South, or Noon.
• 3. E P, the Sun's distance from the Pole.
  • But if the Sides Z P and P E, and the Angle Z P E between them, had been given, then might have been found
• 4. P E Z, the Angle of the Sun's Position.
• 5. E Z, the Complement of the Sun's Altitude.
• 6. E Z P, the Azimuth of the Sun from the North.
  • And if the Sides Z E and P E, with the Angle Z E P contained by them, had been given, there might be found
• 7. E Z P, the Sun's Azimuth from the North.
• 8. Z P, the Complement of the Latitude.
• 9. Z P E, the Hour from Noon.

CASE III. Two Angles, and a Side contained by them, being given, to find the other Parts.

IF in the Triangle Z E P the Angle E Z P and the Angle Z P E, with the Side contained between them, Z P, be given, we may find 9.

• 1. Z E, the Complement of the Sun's Altitude.
• 2. Z E P, the Angle of the Sun's Position.
• 3. E P, the Complement of the Sun's Declination.
  • But if the Side E P, and the Angles Z E P & Z P O, had been given, then might be found
• 4. E Z, the Complement of the Sun's Altitude.
• 5. E Z P, the Azimuth from the North.
• 6. Z P, the Complement of the Latitude.
  • And if the Side Z E, and the Angles P Z E and P E Z, had been given, then you might find
• 7. Z P, the Complement of the Latitude.
• 8. Z P E, the Hour from Noon.
• 9. P E, the Sun's distance from the Pole.

CASE IV. Two Sides, with an Angle opposite to one of them, being given, to find the other Parts.

IF there be given the Side Z P, the Side E P, and the Angle Z E P, there may be found 18.

• 1. E Z, the Complement of the Sun's Altitude.
• 2. E Z P, the Sun's Azimuth from the North.
• 3. Z P E, the Hour from the South.
  • But if the Side Z P, and the Side E P, with the Angle E Z P, had been given, then might be found
• 4. E Z, the Complement of the Sun's Altitude.

• 5. Z E P, the Angle of the Sun's Position.
• 6. Z P E, the Hour from Noon.
  • And if there had been given E P, E Z, and E Z P, then might be found
• 7. Z P, the Complement of the Latitude.
• 8. Z P E, the Hour from Noon.
• 9. Z E P, the Angle of the Sun's Position.
  • In like manner, if the Sides P E and E Z, with the Angle Z P E, had been given, then might be found
• 10. Z P, the Complement of the Latitude.
• 11. E Z P, the Azimuth from the North.
• 12. Z E P, the Angle of the Sun's Position.
  • Again, if the Sides E Z and Z P, with the Angle Z P E, had been given, then would be found
• 13. P E, the Sun's distance from the Pole.
• 14. Z E P, the Angle of the Sun's Position.
• 15. E Z P, the Sun's Azimuth from the North.
  • Lastly, if the Sides E Z and Z P, with the Angle Z E P, had been given, then you might find
• 16. P E, the Complement of the Sun's Declination.
• 17. Z P E, the Hour from Noon.
• 18. E Z P, the Azimuth from the North.

CASE V. Two Angles, and a Side opposite to one of them, being given, to find the other Parts of the Triangle.

IN the Triangle Z E P, if there be given the Angles E Z P and Z P E, with the Side P E, there may be found 18.

• 1. Z P, the Complement of the Latitude.
• 2. Z E, the Complement of the Sun's Altitude.
• 3. Z E P, the Angle of the Sun's Position.
  • But if there were given E Z P and Z P E, with the Side Z E, then might be found

• 4. Z P, the Complement of the Latitude.
• 5. P E, the Sun's distance from the Pole.
• 6. Z E P, the Angle of the Sun's Position.
  • And if the Angles Z P E and Z E P, with the Side E P, were given, then might be found
• 7. Z P, the Complement of the Latitude.
• 8. P E, the Hour from Noon.
• 9. E Z P, the Sun's Azimuth from the North.
  • Again, if there were given the Angles Z P E and Z E P, with the Side Z P, you might then find
• 10. Z E, the Complement of the Sun's Altitude.
• 11. P E, the Complement of the Sun's Declination.
• 12. E Z P, the Sun's Azimuth from the North.
  • Also if there were given the Angles Z E P and E Z P, with the Side Z P, you might find
• 13. Z E, the Complement of the Sun's Altitude.
• 14. P E, the Complement of the Sun's Declination.
• 15. Z P E, the Hour from Noon.
  • And lastly, if there were given the Angles Z E P and E Z P, with the Side P E, then might be found
• 16. Z E, the Complement of the Sun's Altitude.
• 17. Z P, the Complement of the Latitude.
• 18. Z P E, the Hour from Noon.

CASE VI. The three Angles being given, to find the other Parts.

IN the Triangle Z E P, if the three Angles E Z P, Z P E, and P E Z, be given, there may be found 3.

• 1. Z P, the Complement of the Latitude.
• 2. P E, the Sun's distance from the Pole.
• 3. Z E, the Complement of the Sun's Altitude.

Thus have you in these six Cases all the Varieties that will arise out of an Oblique-angled Sphericall Triangle, in the Conversion of

which Cases you may observe 60 Questions of the Sphere to be re∣solved; and so many are resolvable in every Oblique-angled Sphe∣ricall Triangle, and 30 in every Right-angled: So that in these two Triangles 90 Questions are resolved. For,

In a Right-angled Sphericall Triangle,

By the	First	Case are resolved	3	Sphericall Questions.
Second	6
Third	6
Fourth	12
Fifth	3
		In all 30.

In an Oblique-angled Sphericall Triangle,

By the	First	Case are resolved	3	Sphericall Problems.
Second	9
Third	9
Fourth	18
Fifth	18
Sixth	3
		In all 60.