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.

Title

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.

Author

Leybourn, William, 1626-1716.

Publication

London :: printed by James Flesher, for George Sawbridge, living upon Clerken-well-green,

anno Dom. 1669.

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"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." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A48344.0001.001. University of Michigan Library Digital Collections. Accessed May 4, 2024. content_copy

Contents

• frontispiece
• title page
• THE CONTENTS Of the severall EXERCISES.
• To the Reader.
• Arts and Sciences MATHEMATICALL Professed and Taught by the Authour.
  • ADVERTISEMENT.
• insert
• mathematical treatise
  • GEOMETRICALL PROPOSITIONS and THEOREMS, Necessary to be known and practised for the more easie understanding of the subsequent EXERCISES. The First EXERCISE.
    • THE ARGUMENT.
    • GEOMETRICALL PROPOSITIONS. PROP. I. A right Line being given, to divide the same into two equal parts at right Angles.
    • PROP. II. Ʋpon a right Line given, to erect a Perpendicular upon any part thereof.
    • PROP. III. From a Point above, to let fall a Perpendicular upon a right Line given.
    • PROP. IV. A right Line being given, to draw another right Line which shall be parallel thereto at any distance required.
    • PROP. V. A right Line being given, to draw another right Line parallel thereunto, which shall pass through a given Point.
    • PROP. VI. Three right Lines being given, to make a Triangle, whose three Sides shall be equal to the three given Lines.
    • PROP. VII. Three Points (which lie not in a straight Line) be∣ing given, to finde the Centre of a Circle, which being described shall pass through the three given Points.
    • PROP. VIII. Two Points within any Circle being given, how to de∣scribe the Arch of another great Circle which shall pass through those two given Points, and also divide the Circumference of the given Circle into two equal parts.
    • How to make a Line of Chords.
      • The Uses of the Line of Chords.
        • I. How to protract (or lay down upon Paper) an Angle containing any number of Degrees and Minutes by the Line of Chords.
        • II. An Angle that is already protracted, how to finde the quantity of Degrees it containeth.
    • Its Description.
      • Its Ʋse.
  • Trigonometricall Theorems.
  • THE SOLƲTION Of Right-lined TRIANGLES By the LINES of EQƲAL PARTS, & CHORDS. The Second EXERCISE.
    • I. Of Right-angled plain Triangles.
      • CASE I. The Base B A 180, and the Perpendicular C A 135, being given, to finde the Angles B and C.
      • CASE II. The Hypotenuse C B 225, and the Base A B 180, being given, to finde the Angles B and C.
      • CASE III. The Base A B 180, the Angle C 53 degr. 7 min. and the An∣gle B 36 deg. 53 min. being given, to finde the Perpendicular C A.
      • CASE IV. The Hypotenuse C B 225, the Angle C 53 degr. 7 min. and the Angle at B 36 degr. 53 min. given, to finde the Base B A, and the Perpendicular C A.
      • CASE V. The Hypotenuse C B 225, and the Base A B 180, being given, to finde the Perpendicular C A.
      • CASE VI. The Base A B 180, the Angle C 53 degr. 7 min. and the Angle B 36 degr. 53 min. being given, to finde the Hypotenuse C B.
      • CASE VII. The Base A B 180, and the Perpendicular C A 135, being gi∣ven, to finde the Hypotenuse C A.
    • II. Of Oblique-angled plain Triangles.
      • CASE I. Two Sides, as the Base D B 335, and the Side C B 271, and the Angle D 43 degr. 20 min. opposite to C B, to finde the Angle at C, opposite to the Base D B.
      • CASE II. The Base D B 335, and the Side D C 100, with the Angle D, 43 degr. 20 min. contained between them, to finde either of the other Angles at B and C.
      • CASE III. The three Sides, D B 335, C B 271, and D C 100, being given, to finde any of the Angles, as B.
      • CASE IV. The three Angles, C 122 degr. D 43 degr. 20 min. and B 14 deg. 40 min. being given, to finde any of the Sides, as B C.
      • CASE V. The two Sides D C 100, and C B 271, with the Angle at C 122 degr. being given, to finde the Base D B.
  • THE DIMENSION Of Sphericall TRIANGLES BY A LINE OF CHORDS. The Third EXERCISE.
    • I. Of Right-angled Sphericall Triangles.
      • CASE I. The Base A C 27 degr. 54 min. and the Perpendicular C B 11 degr. 30 min. being given, to finde the Hypotenuse A B.
      • CASE II. The Hypotenuse A B 30 degr. and the Angle at the Base A 23 degr. 30 min. being given, to find the Perpendicular B C.
      • CASE III. The Base A C 27 degr. 54 min. and the Angle at the Base A 23 d. 30 min. being given, to find the Angle at the Perpendicular B.
      • CASE IV. The Perpendicular B C 11 degr. 30 min. and the Angle at the Base A 23 degr. 30 min. to finde the Hypotenuse A. B.
      • CASE V. The Perpendicular B C 11 degr. 30 min. and the Angle at the Base A 23 degr. 30 min. being given, to finde the Angle at the Perpendicular B.
      • CASE VI. The Angle at the Base A 23 degr. 30 min. and the Angle at the Perpendicular B 69 degr. 22 min. being given, to finde the Base A C.
      • CASE VII. The Base A C 27 degr. 54 min. and the Hypotenuse A B 30 degr. being given, to finde the Angle at the Perpendicular B.
      • CASE VIII. The Base A C 27 degr. 54 min. and the Hypotenuse A B 30 degr. being given, to finde the Perpendicular B C.
      • CASE IX. The Hypotenuse A B 30 degr. and the Angle at the Base A 23 d. 30 min. being given, to finde the Angle at the Perpendicular B.
      • CASE X. The Hypotenuse A B 30 degr. and the Angle at the Base A 23 d. 30 min. being given, to finde the Base A C.
      • CASE XI. The Base A C 27 degr. 54 min. and the Angle at the Base A 23 degr. 30 min. being given, to finde the Perpendicular B C.
      • CASE XII. The Base A C 27 degr. 54 min. and the Perpendicular B C 11 d. 30 min. being given, to finde the Angle at the Base A.
      • CASE XIII. The Base A C 27 degr. 54 min. and the Angle at the Base A 23 d. 30 min. being given, to finde the Hypotenuse A. B.
      • CASE XIV. The Perpendicular B C 11 degr. 30 min. and the Angle at the Base A 23 degr. 30 min. being given, to finde the Base A C.
      • CASE XV. The Base A C 27 degr. 54 min. and the Hypotenuse A B 30 degr. being given, to finde the Angle at the Base A.
      • CASE XVI. The Angle B at the Perpendicular 69 degr. 22 min. and the An∣gle at the Base A 23 degr. 30 min. being given, to finde the Hy∣potenuse A B.
    • II. Of Oblique-angled Sphericall Triangles.
      • CASE I. The Angle at E 38 degr. 15 min. the Angle at A 23 degr. 30 m. and the Side A B 30 degr. being given, to finde the Base B E.
      • CASE II. The Side A B 30 degr. the Angle at E 38 degr. 15 min. and the Side B E 18 degr. 47 m. being given, to finde the Angle at A.
      • CASE III. The Side A B 30 degr. and the Side B E 18 degr. 57 min. and the Angle at A 23 degr. 30 min. being given, to finde the Base A E.
      • CASE IV. The Angle at 23 degr. 30 min. the Angle at B 122 degr. 36 m. and the Side A B 30 degr. given, to finde the Angle at E.
      • CASE V. The Angle at A 23 degr. 30 min. the Angle at E 38 degr. 15 m. and the Side A B 30 degr. to finde the Angle A B E.
      • CASE VI. The Side A B 30 degr. the Base A E 42 degr. 51 min. and the Angle at A 23 degr. 30 min. being given, to find the Angle at E.
      • CASE VII. The Angle at A 23 degr. 30 min. the Angle at B 122 degr. 36 m. and the Side A B 30 degr. being given, to find the Side B E.
      • CASE. VIII. The Side A B 30 degr. the Side B E 18 degr. 47 min. and the Angle at A 23 degr. 30 min. being given, to finde the Angle at B.
      • CASE IX. The Angle at A 23 degr. 30 min. the Angle at E 38 degr. 15 m. and the Side B A 30 degr. being given, to find the Side E A.
      • CASE X. The Side A B 30 degr. the Side B E 18 d. 47 m. and the Angle at B 122 d. 36 m. contained by them being given, to find the Base.
      • CASE XI. The three Sides A B 30 degr. B E 18 degr. 47 min. and A E 42 d. 51 min. being given, to finde the Angle at E.
      • CASE XII. The three Angles A 23 degr. 30 min. B 122 degr. 36 min. and the Angle E 38 degr. 15 min. being given, to finde any of the Sides.
    • Postscript.
  • To project the SPHERE Upon the Plain of the MERIDIAN BY A LINE OF CHORDS. Whereby the Sides and Angles of Sphericall Triangles are naturally laid down in Plano, as they are in the Sphere it self; By which the nature of them is dis∣covered, and their Sides and Angles measured with speed and exactness. The Fourth EXERCISE.
    • I. Of the MERIDIAN.
    • II. Of the HORIZON.
    • III. Of the AEQƲINOCTIAL.
    • IV. Of the ECLIPTICK.
    • V. Of the PRIME VERTICALL.
    • VI. Of the HOƲR-CIRCLES.
    • VII. Of the AZIMƲTH CIRCLES.
    • VIII. Of the TROPICKS.
    • IX. Of the CIRCLES or PARALLELS of DECLINATION.
    • X. Of the CIRCLES or PARALLELS of ALTITƲDE.
    • How to project the Sphere upon the Plain of the Meridian.
      • Diversion.
    • I. A Sphericall Triangle being projected, how to find the quantity of any Angle thereof.
      • Example I.
      • Example II.
    • II. A Sphericall Triangle being projected, to find the quantity of any Side thereof.
      • Example I.
      • Example II.
  • 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.
    • I. In a Right-angled Sphericall Triangle.
      • CASE I. The Base and Perpendicular being given, to finde the other parts of the Triangle.
      • CASE II. The Hypotenuse and Perpendicular being given, to find the other parts.
      • CASE III. The Hypotenuse and an Angle being given, to find the other Parts.
      • CASE IV. The Perpendicular, or Base, and either of the Angles given, to find the other Parts.
      • CASE V. The Angles being given, to find the other Parts.
    • II. In an Oblique Sphericall Triangle.
      • CASE I. The three Sides being given, to find an Angle.
      • CASE II. Two Sides and the Angle comprehended by them being given, to find the other Parts of the Triangle.
      • CASE III. Two Angles, and a Side contained by them, being given, to find the other Parts.
      • CASE IV. Two Sides, with an Angle opposite to one of them, being given, to find the other Parts.
      • CASE V. Two Angles, and a Side opposite to one of them, being given, to find the other Parts of the Triangle.
      • CASE VI. The three Angles being given, to find the other Parts.
  • PROPOSITIONS ASTRONOMICALL, Usefull in the Practice of NAVIGATION: Performed by the resolving of severall Sphericall Tri∣angles upon the Projection. The Sixth EXERCISE.
    • PROP. I. The distance of the Sun from the nearest Aequinoctial Point (either Aries or Libra) given, to find his Declination.
      • The Analogie or Proportion.
      • To resolve the Triangle upon the Projection,
    • PROP. II. The Latitude of the Place, and the Declination of the Sun, being given, to find the Ascensional Diffe∣rence.
      • The Analogie or Proportion.
      • To resolve the Triangle upon the Projection,
    • PROP. III. The Latitude of the Place, and the Declination of the Sun, being given, to find his Amplitude.
      • The Analogie or Proportion.
      • To work the Proposition upon the Projection,
    • PROP. IV. The Latitude of the Place, and the Declination of the Sun, being given, to find the Angle of the Sun's Position at the time of his rising.
      • Analogie or Proportion.
      • By the Projection.
    • PROP. V. The Sun's Declination, and his Amplitude from the North part of the Horizon, being given, to find the Latitude.
      • The Analogie or Proportion.
      • By the Projection.
    • PROP. VI. The Sun's greatest Declination, with his Distance from the next Aequinoctial Point (Aries or Libra,) being given, to find his right Ascension.
      • Analogie or Proportion.
      • By the Projection.
    • PROP. VII. The Latitude of the Place and the Sun's Declination being given, to find at what Hour the Sun will be upon the true East or West Points.
      • The Analogie or Proportion.
      • To resolve the Proposition by the Projection,
    • PROP. VIII. Having the Latitude of the Place, and the Sun's Decli∣nation, given, to find what Altitude the Sun shall have when he is upon the true East or West Points.
      • The Analogie or Proportion.
      • To resolve the Proposition by the Projection,
    • PROP. IX. The Latitude of the Place, and the Declination of the Sun, being given, to find what Altitude the Sun shall have at Six of the Clock.
      • The Analogie or Proportion.
      • To resolve the Proposition by the Projection,
    • PROP. X. The Latitude of the Place and the Declination of the Sun being given, to find what Azimuth the Sun shall have at Six a Clock.
      • The Analogie or Proportion.
      • To resolve the Proposition upon the Projection,
    • PROP. XI. The Latitude of the Place, the Declination of the Sun, and the Sun's Altitude, being given, to find the Sun's Azimuth either from the East, North or South Points of the Horizon.
      • Analogie or Proportion.
      • To resolve this Probleme by the Projection.
    • PROP. XII. The Latitude of the Place, the Sun's Declination, and the Sun's Altitude, being given, to find the Hour of the Day.
      • Analogie or Proportion.
      • To resolve the Proposition by the Projection.
    • PROP. XIII. The Declination, Altitude, and Azimuth of the Sun, be∣ing given, to find the Hour of the Day.
      • Analogie or Proportion.
      • By the Projection.
    • PROP. XIV. The Sun's Declination, his Altitude, and the Hour from Noon, being given, to find the Sun's Azimuth from the North part of the Meridian.
      • Analogie or Proportion.
      • By the Projection.
    • PROP. XV. The Hour from Noon, the Latitude of the Place, and the Altitude of the Sun, being given, to find the An∣gle of the Sun's Position.
      • The Analogie or Proportion.
      • By the Projection.
    • PROP. XVI. The Sun's Altitude, his Declination, and Azimuth from the North, being given, to find the Latitude.
      • Analogie or Proportion.
      • By the Projection.
  • 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.
    • I. Propositions assistent to find the Latitude.
    • II. Propositions assistent to find the time of the Sun's rising and setting.
    • III. Propositions assistent to find the Hour of the Day and Night.
      • PROBL. I. How to find at what time any of the Stars in the Table of the Sixth Proposition 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.
    • IV. Propositions assistent to the finding of the Variation of the Compass.
      • I. To find the Variation by the Amplitude.
      • II. To find the Variation by the Azimuth.
  • PROPOSITIONS GEOGRAPHICALL, Shewing how the Distance of any two Places upon the Terrestriall Globe may be found, both by Trigono∣metricall Calculation and Geometricall Projection. The Eighth EXERCISE.
    • PROP. I. Two Places which differ onely in Latitude, to find their Distance.
      • To find the distance of these Places upon the Projection.
    • PROP. II. Two Places which differ onely in Longitude, to find their Distance.
      • The Analogie or Proportion.
      • To find the Distance of these two Places upon the Projection.
    • PROP. III. Two Places differing both in Longitude and Latitude being proposed, to find their Distance.
      • The Analogie or Proportion.
      • By the Projection.
      • Analogie or Proportion.
      • By the Projection.
      • Analogie or Proportion.
      • By the Projection.
  • The Doctrine of RIGHT-LINED TRIANGLES applied to Practice in NAVIGATION: Whereby Sundry Nauticall Questions are resolved; and many Problems of Sailing, both by the Plain and Merca∣tor's Chart, performed by Protraction, by Calculati∣on, and also wrought upon the Chart it self. The Ninth EXERCISE.
    • SECTION I.
      • QUESTION I. There are two Ships set sail from the Port A, the one saileth di∣rectly North 24 Centesms, (or 4 Leagues and ⅘ parts of a League,) and the other directly East 32 Centesms, (or 6 ⅖ Leagues;) I demand how the two Ships bear one from the other, and also how far they are asunder.
      • QUEST. II. A Ship at A discovers an Island at C, lying from her directly East, but she sails from A towards B 32 Cent. or 6 ⅖ Leagues directly South; but her Compass coming to some mischance, that use cannot be made of it, she again at B discovers the same Island, and sails upon an unknown Point of the Compass directly upon the Island, and touches upon it, having sailed 8 Leagues.—I demand upon what Point of the Compass the Ship sailed from B to C, and also how far off the Island was from A, where it was first discovered.
      • QUEST. III. There are two Ports at A and B which are distant 6 ⅖ Leagues, and lie directly North and South of each other; from whence two Ships set sail, both for the Port C: the Ship at B sails away upon a South-W. by South Point; and the Ship at A sails directly West.—I demand how many Leagues either of the Ships had sailed when they met at the Port C, and also how the Port C did bear from that at B.
      • QUEST. IV. A Ship at C discovers a Point of Land at A bearing from her S. S. E. but she shapes a Course E. by S. and sails away 8 Leagues to B, and at B she discovers the same Point of Land bearing from her W. S. W.—I demand how far the Ship was from Land being at C and B.
      • QUEST. V. A Ship being at A discovers two other Ships at C and B; the Ship at C bears from her directly East, and the other Ship at B bears from her directly South. The Ship at A sails directly South 32 Cent. to B, and being at B, steers away upon an un∣known Course to C 40 Cent. or 8 Leagues.—I demand upon what Point the Ship failed from B to C,—and also how far C is distant from A.
      • QUEST. VI. Two Islands at A and C are discovered by a Ship at B, the Island A bears from the Ship at B N. N. W. and the Island at C bears N. by E. from B; the Ship being at B sails away N. N. W. to the Island A, and having sailed 32 Cent. touches upon the Island, and being there findes that the Island C bears from the Island A E. N. E.—I demand how far the Ship at B was from the Island C, and also how far the two Islands were asunder.
      • QUEST. VII. Two Ships set out from one and the same Port A; the Ship C sails 24 Cent. or 4 ⅘ Leagues directly East, and the Ship B sails away 32 Cent. or 6 ⅖ Leagues directly South.—When they have thus sailed, I demand how far the two Ships are from each other.
      • QUEST. VIII. Two Ships set sail from the Port at K; the one sails 3 77/100 Leagues upon the S. W. Point towards M, the other sails 8 Leagues upon the West Point towards L.—I demand how many Leagues the Ships at M and L are asunder, and also how the Ship at M bears from the Port K, and the other Ship at L.
      • QUEST. IX. There are three Ships, K, L, and M: the Ship K is distant from the Ship L 8 Leagues; the Ship at L is distant from that at M 6 62/100 Leagues; and the Ship at M is distant from that at K 3 77/100 Leagues; and they lie directly North and South.—I demand how the Ship at M bears to that at L, and how that at L bears to that at K.
    • PROBLEMS Of Sailing by the Plain Sea-Chart. SECTION II.
      • The Making of the Plain Sea-Chart.
        • Some Uses of the Plain Sea-Chart.
      • PROBL. I. How to set any Place upon your Chart according to its Longitude and Latitude.
      • PROBL. II. Any Places being set upon the Chart, to find in what La∣titudes they are, and also how they differ in Longitude.
      • PROBL. III. Having the Rhumb, and the Distance that the Ship hath run upon that Rhumb, to find the Difference of Lon∣gitude and Latitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. IV. The Difference of Latitude and the Rhumb being given, to find the Distance run and the Difference of Lon∣gitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. V. Having the Difference of Longitude and the Rhumb gi∣ven, to find the Distance run and Difference of Latitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VI. The Distance that the Ship hath run, and the Difference of Latitude, given, to find the Rhumb and Difference of Longitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VII. The Distance that the Ship hath run, and the Difference of Longitude, being given, to find the Rhumb and Difference of Latitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VIII. The Difference of Longitude and Difference of Latitude being given, to find the Rhumb and the Distance run.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. IX. The Rhumb that a Ship hath sailed upon, and the num∣ber of Leagues she hath sailed upon that Rhumb, being given, to know how much she hath raised or depressed the Pole.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. X. The Longitude and Latitude of the Place from whence you came, with the Rhumb and Distance sailed, be∣ing given, to find the Longitude and Latitude of the Place to which you are come.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. XI. The Longitude and Latitude of the Place from whence you came, the Rhumb upon which you sailed, and the Latitude of the Place to which you are come, being given, to find the Distance and Difference of Lon∣gitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. XII. The Latitude of two Places, and the Difference of Longi∣tude between them, being known, to find what Rhumb leadeth from one to the other, and how many Leagues distant they are asunder.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
    • PROBLEMS Of Sailing by Mercator's Chart. SECTION III.
      • PROBL. I. How to make a Sea-Chart according to MERCATOR's Projection, by your Line of Chords.
        • To divide the Meridian Line of a Sea-Chart.
      • PROBL. II. To find how many Leagues do answer to one Degree of Longitude in every severall Latitude.
        • The Analogie or Proportion.
      • PROBL. III. By the Latitude of two Places and their Distance, to find the Rhumb.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. IV. The Longitude and Latitude of two Places being gi∣ven, to find the Rhumb.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. V. The Latitude of two Places and the Rhumb being given, to find the Difference of Longitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VI. The Difference of Longitude of two Places, the Latitude of one of them, and the Rhumb leading from one to the other, given, to find the Latitude of the other Place.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VII. Having the Latitude of one Place, the Rhumb leading from that Place to another unknown, and the Distance upon the Rhumb from the first to the second Place, to find the Difference of Longitude of the two Places.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. VIII. The Difference of Longitude between two Places, the Rhumb leading from one Place to the other, and the Latitude of one of the Places, being given, to find their Distance.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. IX. The Difference of Longitude, and Distance of two Places, with the Latitude of one of the Places, being given, to find the Rhumb that leads from one to the other.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. X. The Longitude and Latitude of two Places being given, to find the Distance upon the Rhumb.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. XI. The Latitude of two Places and their Distance upon the Rhumb being given, to find their Difference of Lon∣gitude.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. XII. The Difference of Longitude of two Places, their Distance upon the Rhumb, and the Latitude of one of the Pla∣ces, being given, to find the Difference of Latitudes.
        • The Analogie or Proportion.
        • Ʋpon the Chart.
      • PROBL. XIII. The Latitude of two Places and their Difference of Lon∣gitudes being given, to find the Rhumb leading from one to the other, and also how many Degrees distant they are asunder.
        • Ʋpon the Chart.
      • PROBL. XIV. A Ship set sail from the Latitude of 50 degr. upon the fifth Rhumb N. E. by E. after that she had made 36 Leagues of way upon that Rhumb, the wind changing, she was constrained to sail 50 Leagues upon the 7th Rhumb E. by N. I would know in what Longitude and Latitude the Ship is.
        • Ʋpon the Chart.
• chart