2. The Distance of any Point upon the inlarg'd Meridian from the Equator, is equal to the Sum of all the Secants contain'd between it and the Equa

tor.

3. The Distance between any two Parallels on the fame fide of the Equator, is equal to the difference of the Sums of all the Secants contain'd between the Equator and each of the Parallels.

4. The Distance between any two Parallels on contrary fides of the Equator, is equal to the Sum of the Sums of all the Secants contain'd between the Equator and each Parallel.

4. Now fince it has been fhewn, that in this Projection the Distance of each point of the Meridian from the Equator, is equal to the Sum of all the Secants contain'd between it and the Equator; 'tis plain that by a continual Addition of the Secants, beginning at the Equator, we fhall have the Diftance of every particular Point in the Meridian from the Equator, which Distances collected together form the Table, commonly call'd A Table of Meridional Parts, which is annex'd to the End of this Section, and in which you may observe that the top Column contains, the Degrees, and the left-hand fide Column the Minutes; the other Columns contain the meridional Parts answering to these Degrees and Minutes. There is alfo upon Gunter's Scale, a Line of meridional Parts, mark'd Mer. which shows the distance of each Point of the Meridian from the Equator.

5. By either of these, viz. the Table of meridional Parts, or the meridian Line upon Gunter's Scale, may a Mercator's Chart be conftructed. Thus for Example, let it be required to make a Chart that fhall commence at the Equator, and reach to the parallel of 60 Degrees, and fhall contain 80 Degrees of Longitude.

Draw

Draw the Line EQ reprefenting the Equator; (fee Plate 1.) then take from any convenient Line of equal Parts, 4800 (the number of Minutes contain'd in 80 Degrees) which fet off from E to Q, and this will determine the Breadth of the Chart.

Divide the Line EQ into eight equal parts, in the Points 10, 20, 30, &c. each containing 10 Degrees, and each of these divided into 10 equal parts will give the fingle Degrees upon the Equator; then thro' the points E, 10, 20, &c. drawing Lines perpendicular to EQ, thefe fhall be Meridians.

From the fcale of equal parts take 4527.4 (the meridional parts anfwering to 60 Degrees) and fet that off from E to A and from Q to B, and join AB; then this Line will reprefent the Parallel of 60, and will determine the length of the Chart.

Again from the fcale of equal parts take 603.1, (the meridional parts answering to 10 Degrees) and fet that off from E to 10 on the line E A, and thro' the point 10 draw 10, 10, parallel to EQ, and this will be the Parallel of 10 Degrees. The fame way fetting off from E on the line EA, the meridional parts anfwering to each Degree, &c. of Latitude, and thro' the feveral points drawing lines parallel to EQ, we fhall have the feveral Parallels of Lati tude.

If the Chart does not commence from the Equator, but is only to ferve for a certain diftance on the Meridian between two given Parallels on the fame fide of the Equator; then the Meridians are to be drawn as in the last Example, and for the Parallels of Latitude you are to proceed thus; viz. from the meridional parts answering to each point of Latitude in your Chart, fubtract the meridional parts anfwering to the leaft Latitude, and fet off the differences feverally, from the Parallel of leaft Latitude, upon the two extream Meridians, and the lines joining these points of the Meridians shall represent the several Parallels upon your Chart.

Thus let it be required to draw a Chart that fhall ferve from the Latitude of 20 Degrees North, to 60 Degrees North, and that shall contain 80 Degrees of Longitude.

ཟ་

Having drawn the Line DC to represent the Parallel of 20 Degrees (fee Plate 1.) and the Meridians to it, as in the foregoing Example; fet off 663.3 (the difference between the meridional Parts anfwering to 30 Degrees, and thofe of 20 Degrees) from D to 30, and from C to 30; then join the points 30 and 30 with a right Line, and that fhall be the Parallel of 30. Alfo fet off 1397.6 (the difference between the meridional Parts answering to 40 Degrees, and thofe of 20 Degrees (from D to 40, and from C to 40, and joining the points 40, and 40 with a right Line, that fhall be the Parallel of 40. And proceeding after the fame Way, we may draw as many of the intermediate Parallels as we fhall have occafion for.

But if the two Parallels of Latitude that bounds the Chart, are on the contrary fides of the Equator; then draw a Line reprefenting the Equator, and Meridians to it, as in the first Example; and from the Equator fet off on each fide of it the feveral Parallels contained between it and the given Parallels as above, and your Chart is finished.

N. B. Here you must notice; that in all Charts, the upper part is the North Side, and the lower part or bottom is the South Side; alfo that part of it towards the right Hand is the Eaft, and that towards the left Hand the Weft Side of the Chart.

6. Since according to this Projection, the Meridians are parallel right Lines; 'tis plain, that the Rumbs which form always equal Angles with the Meridians, will be ftreight Lines; which Property renders this Projection of the Earth's furface much more eafy and proper for Ufe, than any other.

7. This method of projecting the Earth's furface upon a Plain, was firft invented by Mr. Edward Wright, but first publifhed by Mercator; and hence the failing by the Chart, was called Mercator's failing. 8. In the annexed Scheme, let A and D reprefent two places upon the furface of the Globe, A C the Meridian of A, and AD the Rumb Line between the two places; thro' D draw DB perpendicular to AC, and this will be the Parallel of Lati tude of the place D; from A fet off upon the

Meridian, the length A C, equal to the Meridional or inlarg'd Difference of Latitude, and thro' C draw CE parallel BD meeting AD produced in E; then AB will be the proper Difference of Latitude, and AC the inlarg'd Difference of Latitude, or the Difference of Latitude according to Mercator's Chart, between the places A and D: CE will be the Difference of Longitude, and BD the Departure, alfo AD will be the proper Distance, and AE the inlarg'd, or according to Mercator's Chart, and the Angle BAD will be the Course.

9. Now fince in the Triangle ACE, BD is parallel to one of it's fides CE; 'tis plain the Triangles ACE, ABD will be fimilar, and confequently the fides proportional (by Art. 74. Sect. 1.) Hence arifes the Solutions of the feveral Cafes in this failing, which are as follows.

CASE I.

The Latitudes of two Places given, to find the meridional or inlarg'd Difference of Latitude between them.

Of this Cafe there are three Varieties, viz. either one of the places lies on the Equator, or both on the fame fide of it; or laftly on different fides.

1. If one of the propofed places lies on the Equator, then the meridional difference of Latitude, is the fame with the Latitude of the other place, taken from the Table of meridional Parts.

Example.

Required the meridional difference of Latitude between St. Thomas, lying on the Equator and St. Antonio in the Latitude of 17°, 20' North. I look in the following Table for the meridional Parts anfwering to 179, 20, and find it to be 1056.2, the inlarg'd difference of Latitude required.

2. If the two propofed places be on the fame fide of the Equator, then the meridional difference of Latitude is found by fubtracting the meridional Parts answering to the leaft Latitude, from those answering to the greateft, and the difference is that required,