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.

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.

UPON a piece of thick and smooth Paper (or ra∣ther Past-board) draw a right Line, as A B, and upon the Point A, with 60 degr. of your Line of Chords, describe the Quadrant A C D, which divide into 90 equal Parts or Degrees, as here in this Figure there is onely every fifth Degree. —This done, upon the Point D erect the Perpendicular D F, (in which you must be very exact.) And from the Point A (through each Degree of the Quadrant) draw right Lines, as A 10, 10; A 20, 20; A 30, 30; A 40, 40; &c. to∣wards 90. till they touch the Line D F.—Then with your

Compasses, one foot being placed in A, extend the other to 60 degr. in the Line D F, and with that Distance describe the Arch F H G B. Again, extend the Compasses from A to 55 in the Line F D, and keeping one foot in A, with the other describe 55 K; so shall the Point K be the Point of 55 degr. in the Line A B.—Also, extend the Compasses from A to 50 in the Line F D, and draw the Arch 50, 50. Doe so with 45, 40, 35, &c. till you come to the beginning of the Degrees at D. So shall these Arch-lines, by meeting with the Line A B, divide that part of it D B into unequal parts, at 10, 20, 30, 40, 50, 60, and so forward. But this Figure is sufficient for Example.

[illustration] geometrical diagram

Now from this Line A B, being thus unequally divided, you may divide the Meridian-line of a Sea-Chart according to Mercator's Projection of any bigness, so that the Distance be∣tween Degree and Degree in the Aequinoctial be less then the Distance A D, which is here two Inches. And if a Chart were made that the Aequinoctial Degrees were two Inches di∣stant, and it passed upon a smooth Board, many Nauticall Con∣clusions might be wrought upon it very exactly. Being thus far prepared, I will now shew you how, from the Line A B,

To divide the Meridian Line of a Sea-Chart.

A Sea-Chart, according to this Projection, may be made either General, or Particular. I call that a General Sea-Chart, whose Line E H, in the following Figure, represents the Aequinoctial, as the Line E H there doth the Parallel of 49 degr. and so I will make the Chart following to contain all Latitudes between 49 degr. and 57 degr. whose Difference of Longitude exceedeth not 8 degr.

Now to project such a Chart, having drawn the Line E F for the Meridian, and crossed it at right Angles with another Line representing the Parallel of 49 d. parallel thereto draw another Line F G, representing the Parallel of 57 degr. and another Meridian G H, parallel to F E. So shall you have made the Parallelogram E F G H.

This done, consider how far distant you would have your Degrees of Longitude upon the Aequinoctial each from other, as suppose (and as in this Chart I have made them to be) half an Inch. Take half an Inch out of a Line of Inches, and run that Distance along the Line E H from E to 1, from 1 to 2, from 2 to 3, &c. And also doe the like upon the Line F G, at the top of the Chart, drawing the Lines 1, 1; 2, 2; 3, 3; &c.

Now for the Dividing of the Meridians E F and H G, re∣pair to the foregoing Figure, taking in your Compasses the Distance that is between Degree and Degree of the Aequino∣ctial,

which in our Example is half an Inch. With this Di∣stance, set one foot of the Compasses in the Point D, and with the other describe the Arch m m; by the very Edge whereof draw the Line A G: so is your Figure prepared to divide the Meridian-line of a Sea-Chart whose Degrees of Longitude are half an Inch distant.

Now in respect that your first Parallel of Latitude E H in your Chart is drawn for 49 degr. your next Parallel must be 50 degr. Wherefore set one foot of your Compasses upon 50 degr. in the Line A B, and with the other take the nearest Distance to the Line A G: that is done by turning the Com∣passes about till the moveable foot do onely touch the Line A G; which when it so doth, that Distance at which your Com∣passes then are, being set upon the Meridian of your Chart, will reach from 49 degr. to 50, which being set upon your Chart, on both sides thereof, from 49 draw the Line 50. 50 will give you the Parallel of 50 d. of Latitude. In like man∣ner for the Parallel of 51 degr. Set one foot of the Compasses in 51 degr. upon the Line A B of the former Figure, and with the other take the least Distance to the Line A G: this Distance set upon the Meridian of your Sea-Chart, on both sides thereof, will reach from 50 to 51; and there draw the Parallel 51, 51.—Likewise for the Parallel of 52 degr. Set one foot of the Compasses in 52 degr. in the Line A B, taking the nearest Distance to the Line A G: that Distance set upon the Meridian of your Sea-Chart, on both sides thereof, will reach from 51 to 52; and there draw the Parallel of 52, 52. Doe thus with all the Degrees, as 53, 54, 55, 56, and 57. So shall the Meridians of your Chart E F and H G be divided into whole Degrees.

For the Sub-divisions of these Degrees, they may be divi∣ded each of them into equal parts, as the Divisions at the top and bottome of the Chart ought to be; but the Degrees of the Meridian, as they grow higher, they ought still to grow greater. But the Difference is so small, that it cannot produce

any considerable Errour, though the Sub-divisions be all made equal between Degree and Degree. You may therefore di∣vide them either into 60 Minutes or English Miles, or into 20 Leagues, or into 100 parts of Degrees, as you shall best like of.

But if you would make a Chart that the Distance between De∣gree and Degree upon the Aequinoctial should be an Inch, or any other Distance less then A D in the foregoing Figure; take that Di∣stance (as suppose an Inch) in your Compasses, and setting one foot in D, with the other describe the Arch o o, and draw the Line A H onely to touch the Arch o o. The least Distance taken from each Degree to this Line A H shall give you the Distance of the Degrees upon the Meridian of a Sea-Chart, whose Distance of Degrees up∣on the Aequinoctial are an Inch from each other.

Your Chart being thus prepared, I will now come to shew you how to resolve severall Problems upon it.

PROBL. II. To find how many Leagues do answer to one Degree of Longitude in every severall Latitude.

UPON the two edges of your Protracting Quadrant there are two Lines, the one divided into 20, the other into 60 equal parts.

Take therefore the least Distance from the Complement of the Parallel's distance from the Aequator, (or the Complement of the given Latitude:) this Distance, being measured upon the edge that is divided into 20, shall shew you what number of Leagues make one Degree of Longitude in that Parallel of Latitude. And the same Distance, being measured upon the other edge that is divided into 60, will give so many of our Miles, or so many Minutes of the Aequinoctial, or any other

great Circle, as are answerable to one Degree of Longitude in that Latitude.

Example. Let it be required to find how many Leagues do answer to one Degree of Longitude in the Latitude of 18 d. 12 min.

Set one foot of your Compasses in 71 degr. 48 min. the Complement of the given Latitude, and with the other take the nearest Distance to the side of the Quadrant which is di∣vided into 20: that Distance, measured upon the Line 20, will reach from the beginning thereof to 19: and so many Leagues do answer to one Degree of Longitude in the Lati∣tude of 18 degr. 12 min.

Or, If you take the least Distance from 18 degr. 12 m. the Latitude it self, in the Limb of the Quadrant, to that edge which is divided into 60, that Distance will also reach to 19 upon the Line 20, as before.

And the same Distance, being measured upon the Line 60 of the Quadrant, will give you 57 parts: and so many Mi∣nutes of the Aequator are answerable to one Degree of Lon∣gitude in the Parallel of 18 degr. 12 min. of Latitude.

So likewise in the Latitude of 25 degr. 15 min. if you take the least Distance from the Complement thereof, or from the Latitude it self, to the edges of the Quadrant, you shall find that Distance to reach 18 in the Line of 20: and so many Leagues do answer to one Degree of Longitude in the Lati∣tude of 25 degr. 15 min. or unto 54 in the Line of 60: and so many Minutes of the Aequator do answer to one Degree of Longitude in that Parallel of Latitude.

The Analogie or Proportion.

As the Radius is to the Co-sine of the Latitude;

So is

• 20 Leagues to the num∣ber of Leagues
• 60 Minutes to the num∣ber of Minutes

which answer to one Degree of Longitude in that Lati∣tude.

PROBL. III. By the Latitude of two Places and their Distance, to find the Rhumb.

The Analogie or Proportion.

As the Distance upon the Rhumb is to the Radius;

So is the Difference of Latitudes to the Co-sine of the Rhumb from the Meridian.

Thus if the Places given were one in the Latitude of 50 d. and the other in the Latitude of 55 degr. and the Distance up∣on the Rhumb 6 degr. or 120 Leagues; the Rhumb leading from one to the other will be found to be the third from the Meridian, namely, N. E. by N. 33 degr. 45 min.

Ʋpon the Chart.

LET A represent the Place in the Latitude of 50 degr. and C that in 55 degr. whose Distance from A to C is 6 degr. Take 6 degr. out of the Meridian-line, by setting one foot as much below the lesser Latitude as above the grea∣ter, which will be from K in the Latitude of 49 ½ degr. to L in the Latitude of 55 ½; either of which are half a Degree above and under the two given Latitudes. Take this Distance K L in your Compasses, and setting one foot in A, (the lesser Latitude) with the other cross the Parallel of the greater La∣titude 55 degr. in the Point C, and draw a right Line from A to C. So shall the quantity of the Angle B A C, being found (either by your Chord or Quadrant,) shew you the Inclina∣tion of the Rhumb to the Meridian to be 33 degr. 45 min. the N. E. by N. Point.

Note, That in the Propositions following, the Difference of Lon∣gitude must always be taken out of the Aequator, and measured

thereupon also. But the Difference of Longitude and Distance upon the Rhumb must alwaies be measured upon, and taken out of, the Meridian Line of your Chart. And hereafter I shall call them the proper Difference, and proper Distance.

PROBL. IV. The Longitude and Latitude of two Places being gi∣ven, to find the Rhumb.

The Analogie or Proportion.

As the proper Difference of Latitude is to the Radius;

So is the Difference of Longitude to the Tangent of the Rhumb from the Meridian.

Thus if the Places should lie one in the Latitude of 50 deg. and the other in the Latitude of 55 degr. and the Difference of Longitude between them were 5 degr. 30 min. the Rhumb leading from one Place to the other will be found to be the third from the Meridian N. E. by N. 33 degr. 45 min.

Ʋpon the Chart.

THE Meridians and Parallels being drawn through the two Places at A and C, and a straight Line from A to C, for the Rhumb, by your Chord or Quadrant find the quanti∣ty of the Angle B A C, which you will find to be 33 d. 45 m. or the third Rhumb from the Meridian N. E. by N.

But if this Rhumb were to be found by the Common Sea-Chart, it would be found to be above 47 degr. that is, N. E. 2 degr. Easterly, that is, one whole Point and 2 degr. more Easterly then it should be.

PROBL. V. The Latitude of two Places and the Rhumb being given, to find the Difference of Longitude.

The Analogie or Proportion.

As the Radius is to the Tangent of the Rhumb from the Meri∣dian;

So is the proper Difference of Latitudes to the Difference of Longitude.

Thus the Latitude of one Place being 50 degr. and the other 55 degr. and the Rhumb leading from one to the other being the third from the Meridian, the Difference of Longi∣tude will be found to be 5 ½ degr.

Ʋpon the Chart.

LET a Meridian be drawn through A, and a Parallel of Latitude through C. Then upon the Angle A protract the Angle of the Rhumb 33 degr. 45 min. So the Distance B C upon the Parallel, being measured upon the bottome of the Chart, will be found to contain 6 degr. 30 min.

But if this Difference of Longitude were to be found by the Plain Sea-Chart, the Difference of Longitude would be found to be but 3 degr. 20 min. which is more then 3 degr. less then the truth; a vast Difference. And yet this Errour would be yet greater, if either the Latitude be greater, or the Rhumb farther from the Me∣ridian.

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.

As the Radius is to the Co-tangent of the Rhumb from the Meridian;

So is the Difference of Longitude to the proper Difference of Latitude.

Thus if the Latitude of one of the Places were 50 degr. the Rhumb leading from that to the other N. E. by N. 33 d. 45 min. and the Difference of Longitude between the two Places were 5 degr. 30 min. the Latitude of the other Place will be found to be in 55 degr.

Ʋpon the Chart.

LET A B and D C be two Meridians drawn through A and C, at 5 ½ d. the Difference of Longitude, and a Par∣allel of Latitude through A, crossing the Meridian C D in D. Then upon the Point A protract an Angle equal to the Rhumb from the Meridian given 33 degr. 45 min. So the Line C D, being measured upon the Meridian from A, the given Latitude, 50 degr. will reach to 56 degr. the proper Difference of Latitude. So that the other Place lies in the Latitude of 56 degr.

But if this Difference of Latitude were to be found by the Plain Sea-Chart, this Difference of Latitude would be found to be 8 d. 13 min. and the Latitude sought would be found to be 58 degr.

13 min. above three Degrees more then the truth. As by the Tri∣angle for that purpose drawn upon the Plain Sea-Chart, marked with T V E, may appear.

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.

As the Radius is to the Sine of the Rhumb from the Meridian;

So is the proper Distance upon the Rhumb to the Difference of Longitude.

Thus if the two Places were one in the Latitude of 50 degr. and the other in a greater Latitude, but unknown; the proper Distance upon the Rhumb leading from one place to the other being 6 degr. and the Rhumb N. E. by N. 33 degr. 45 min. the Difference of Longitude will be found to be 5 ½ degr.

Ʋpon the Chart.

THrough the Point A in the Latitude of 50 degr. let be drawn a Meridian A B, and a Parallell A D; and upon the Point A protract an Angle equal to the Rhumb from the Meridian 33 degr. 45 min. Then take with the Compasses 6 degres, the proper Distance upon the Rhumb, out of the Meridian-line, (having respect to the Latitude of the Places) as from K to L, and set that Distance upon the Rhumb from A to C. Then through C draw another Meridian C D, cros∣sing the Parallel drawn through A in the Point D. So the Line A D, being measured at the bottom of the Chart, will be found to contain 5 ½ d. the Difference of Longitude sought.

But if this Difference of Longitude had been to be found by the Common Sea-Chart, it would be found to have been onely 3 d. 20 min. which is 2 degr. 10 min. less then the truth; as in the Plain Chart may be seen, where the third Rhumb from the Me∣ridian cuts the Parallel of 55 degr. of Latitude in 3 degr. 20 m. of Longitude at the Point X.

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.

As the Sine of the Rhumb from the Meridian is to the Diffe∣rence of Longitudes;

So is the Radius to the proper Distance of the two Places up∣on the Rhumb.

Thus, if the Latitude of one Place were in 50 degr. the other in a greater Latitude unknown, the Difference of Lon∣gitude between the two Places 5 ½ degr. and the Rhumb N. E. by N. 33 degr. 45 min. from the Meridian; the proper Distance upon the Rhumb will be found to be 6 degrees.

Ʋpon the Chart.

LET two Meridians, A B and C D, be drawn through A and C, according to the Difference of Longitude, and a Parallel of Latitude through A, crossing the Meridian C D in the Point D. Then upon the Point A protract an Angle of 33 degr. 45 min. the quantity of the Rhumb from the Meri∣dian, and draw the Line A C crossing the Meridian C D in C. So the Distance C D, being taken in the Compasses, and

measured upon the Meridian-line of the Chart, (respect be∣ing had to the Latitude of the Places) that is, so much above the greater Latitude as below the lesser Latitude, you will find it to contain 6 degr.

But if this settting of the Compasses so much above one La∣titude as below another seem difficult, it may be thus other∣wise done.—For, the Rhumb Line being drawn, it will cut the Meridian C D in C: so a Parallel drawn through C will cut the Meridian A B in B: so is B the Latitude of the second Place, viz. 55 degr. Then divide the Distance be∣tween the two Latitudes A and B in two equal parts in the Point M; also divide the Rhumb-Line A C in two equal parts in N: then take the Distance N C or N A, and setting one foot of the Compasses in M, the other will reach to L above the greater Latitude, and from M to K as much below the lesser Latitude, namely, 30 min. or half a Degree on either side; so that between K and L are contained 6 degr. and that is the proper Distance upon the Rhumb.

But if this Distance were to be found by the Plain Chart, it would be almost 10 degr. or 197 Leagues, which is 77 Leagues more then in truth it should be. As may appear, if you measure the Line A L in the Plain Chart, upon the Side thereof.

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.

As the proper Distance upon the Rhumb is to the Difference of Longitude;

So is the Radius to the Sine of the Rhumb from the Meridian.

Thus, if one of the Places lay in the Latitude of 50 degr. and the other in a greater Latitude, but unknown; the Dif∣ference of Longitude between them 5 ½ degr. and their pro∣per Distance upon the Rhumb 6 degr. the Inclination of the Rhumb to the Meridian which leadeth from one Place to the other will be found to be 33 degr. 45 min. that is the N. E. by N. Point.

Ʋpon the Chart.

LET the Meridians A B and D C be drawn through A and C, and through A a Parallel of Latitude A D. Then open the Compasses (having respect to the Latitudes) from K to L, the quantity of 6 degr. in the Meridian; and setting one foot of that Extent in A, with the other foot cross the Meridian C D in C, and draw the right Line A C for the Rhumb. Lastly, by your Chord or Quadrant find the quantity of the Angle B A C, 33 degr. 45 min. and that is the Rhumb required N. E. by N.

But if you were to find this Rhumb by the Plain Sea-Chart, it would be found almost the E. N. E. Point within 1 degr. 30 min. differing from truth very near 3 whole Points to the Eastward.

PROBL. X. The Longitude and Latitude of two Places being given, to find the Distance upon the Rhumb.

The Analogie or Proportion.

As the proper Difference of Latitudes is to the Radius;

So is the Difference of Longitudes to the Tangent of the Rhumb from the Meridian:

And

As the Sine of the Rhumb from the Meridian is to the Diffe∣rence of Longitude;

So is the Radius to the proper Distance upon the Rhumb.

Thus, the two Places being one in the Latitude of 50 degr. the other in the Latitude of 55 degr. and the Difference of Longitude between them being 5 ½ degr. the proper Distance upon the Rhumb will be found to be 6 degr.

Ʋpon the Chart.

DRAW the Meridians A B and C D, the Difference of Longitude between them being 5 ½ degr. and through A and B draw two Parallels B C and A D, and then the Line for the Rhumb leading from the one to the other A C. So A C, being taken in the Compasses, and measured upon the Meridian-line of the Chart, with this Condition, that at the resting of the Compasses upon the Meridian-line, one foot be so many Degrees above the greater Latitude as the other foot is below the lesser Latitude; so will the feet of the Compasses rest in the Points K and L, one being 30 min. below the lesser Latitude, and the other 30 min. above the greater.

But if this Distance upon the Rhumb were to be found by the Plain Chart, it would be found to be almost 7 degr. 15 min. or 245 Leagues, which is 25 Leagues more then it should be.

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.

As the proper Distance upon the Rhumb is to the Radius;

So is the proper Difference of Latitudes to the Co-sine of the Rhumb from the Meridian:

And

So is the Sine of the Rhumb from the Meridian to the Diffe∣rence of Longitude.

Thus, if one of the Places be in the Latitude of 50 degr. and the other in 55 degr. and their proper Distance upon the Rhumb 6 degr. or 120 Leagues; their Difference of Longi∣tude will be found to be 5 degr. 30 min.

Ʋpon the Chart.

DRaw A D and B C, two Parallels of Latitude, through 50 degr. and 55 degr. which were the two given Lati∣tudes. Then out of the Meridian Line take the proper Distance upon the Rhumb (having respect to both Lati∣tudes) from K to L: the Compasses being opened to this Distance, one foot being set in A, the lesser Latitude, the other will cross the Parallel of the greater Latitude in C. So the Distance B C, being measured at the bottome of the Chart from E, will reach to 5 degr. 30 min. And such is the Diffe∣rence of Longitude between the two Places.

But if this Difference of Longitude were to be found by the Plain Chart, it would be but 3 degr. 20 min. which is no less then 2 d. 10 min. less then the truth; as by the Triangle T V. E drawn upon the Plain Chart may appear.

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.

As the proper Distance of the two Places upon the Rhumb is to the Radius;

So is the Difference of Longitudes to the Inclination of the Rhumb to the Meridian:

And

So is the Co-sine of the Rhumb from the Meridian to the Dif∣ference of Latitudes.

Thus, the Difference of Longitudes being 5 ½ degr. their pro∣per Distance upon the Rhumb 6 degr. and the Latitude of one of the Places 50 d. the Difference of Latitudes will be found to be 5 d.

Ʋpon the Chart.

THrough the given Latitude A draw a Meridian AB, and a Parallel A D, and upon the Parallel set the Difference of Longitude 5 ½ d. taken from the bottom of the Chart, from A to D, and through D draw the Meridian D C. Then out of the Meridian-line take the proper Distance upon the Rhumb, 6 d. from K to L, and setting one foot of the Compasses in A, with the other cross the Meridian C D in C: so a Parallel of Latitude drawn through C will be the Parallel of 55 d. So is 55 d. the Latitude of the other Place, and 50 being taken from 55, leaves 5 d. for the Difference of Latitudes required.

Which Difference, had it been to be found by the Plain Chart, would have been but 2 d. 25 m. that is, 2 d. 35 m. less then the truth; as by the Triangle T V E upon the Plain Chart may appear.

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.

THIS Proposition is already performed in the Example of the two Places A and B; but for Variety I will take two other Places, and onely shew the manner of working upon the Chart.

Suppose then two Places, one (as before) in the Latitude of 50 d. the other in the Latitude of 52 degr. 30 min. whose Dif∣ference of Longitudes is 6 degr.

Ʋpon the Chart.

Through the two given Latitudes 50 d. and 52 ½, at A and O draw two Parallels, O P and A D, upon which set the Dif∣ference of Longitudes from O to P, and from A to Q, 6 degr. Then draw the Line A P, which shall be the Line of the Rhumb leading from one Place to the other: wherefore, by your Chord or Protracting Quadrant find the quantity of the Angle O A P, which shall be the Inclination of the Rhumb to the Me∣ridian, and will be found to be 56 d. 15 m. that is the N. E. by E. Point; which was the First thing that was required.

Then to find the proper Distance upon the Rhumb; Take the Line A P in your Compasses, and measure it upon the Me∣ridian-line, so that one foot may be above the greater Latitude so much as the other is below the lesser; and you will find the Compass-points to rest in E and S, E being one whole Degree below the lesser Latitude, and S one Degree above the grea∣ter. So that there is intercepted between E and S 4 ½ degr. And that is the proper Distance upon the Rhumb; which was the Second thing required.

But if this Problem had been wrought upon the Plain Chart, the Rhumb from the Meridian would be found to be 67 d. 23 m. that is, within 7 m. of the 6^th Rhumb; which is more then the truth by 11 d. 8 m.

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 7^th Rhumb E. by N. I would know in what Longitude and Latitude the Ship is.

Ʋpon the Chart.

THE Rhumb-Line A P being drawn, set off thereupon 36 Leagues (which was the way that the Ship made upon the fifth Rhumb before the Wind changed) from A to T, (which Distance must be taken out of the Meridian-line by opening the Compasses from 50 d. to 51, 48. or better, to as much below 50 d. as above 51 d.) So shall the Point T be the Place that the Ship was in when the Wind altered. So a Paral∣lel drawn through T upon the Chart will cut the Meridian at V in 51 d. and in that Latitude the Ship was. Now to find in what Longitude she was; Take in your Compasses the Line T V, and measure it at the bottom of the Chart, you shall find it will reach from E to 2 d. 21 m. And in that Longitude the Ship then was.

This done, upon the Point T (where the Wind changed, and drove the Ship 2 Points more Eastwardly, namely, upon the E. by N. Point) protract an Angle of 22 d. 30 m. namely, the An∣gle P T X, which is the Rhumb upon which the Ship sailed 50 Leagues after the Wind changed. Therefore take 50 Leagues out of the Meridian-line, and set them from T to X. So shall X be the Place that the Ship was in after she had sailed 50 Leagues upon the E. by N. Point; which, by drawing a Parallel through K, will be found in the Latitude of 51 d. 30 m. and by drawing of a Meridian through K also, it will be found to be in the Lon∣gitude of 6 degr. 16 min.

But if these Courses had been protracted according to the Plain Sea-Chart, the Point T would fall in the Latitude of 51 degr. and the Point X in the Latitude of 51 degr. 30 m. But the Longitude of T would be onely 1 d. 30 m. and the Longitude of X in 3 d. 57 min. Both these Longitudes being added, make but 5 d. 27 m. for the Difference of Longitude between X and the first Meridian; whereas by the other Chart it is 6 d. 16 m. So that the Ship at X is 33 m. Westward of the Place to which she was bound.

These Differences, which I have observed to be between the Plain and Mercator's Chart, may be seen by comparing the Scheme of the two Charts together.