portion between themselves, yet as the Degrees of Longitude are every where the fame, it will follow, that the Meridians, and confequently the Distances, are every where (except under the Equinoctial) distorted, and the more fo as they come nearer the Pole, infomuch that the fame extent of the Compafs, which contains 60 Miles under the Equinoctial, contains but 20 Miles, or lefs, between Latitude 70 and 71, and nearer the Poles.

BUT as it was pity fuch an excellent Invention fhould be loft, for want of a way to measure Distances upon it, we have likewise subjoined the Method of making a Scale, whereby at one Extent to measure a Distance upon Mercator's Chart: The Scale is formed from the fame mathematical Principles with the Chart itfelf; we fhall first give the Conftruction of one, and then of the other.

WE have obfeved above, that by reafon of the Difficulty of truly drawing upon a plain, any confiderably part of the Surface of the Globe, with inclining Meridians, and all the other properties of the Globes, they endeavoured to find out a right lined Projection, which might be equally true, and yet easier in Practice, in order to which it is premised, as a Foundation,

1. THAT the Peripheries, or Circumferences, of all Circles, are in a direct proportion to their Diameters; and confequently to their Semidiameters or Radiufes. And,

2. As the Periphery of one Circle, to the Periphery of another, of a different Magnitude, So a Half or a Quarter, or any other part of the firft Circle's Periphery, to the like part of the other Circle's Periphery.

3. The Semidiameter or Radius of any Parallel of Latitude, is the Sine Complement of that Latitude; thefe confidered it will be then,

2

1. AS

1. AS Radius: To Sine Complement of any given Latitude, So 60, the Miles in a Degree of Longitude in the Equator, to the number of Miles contained in a Degree of Longitude, in the Latitude propofed.

2. BUT it is alfo demonftrable, that As Sine Complement of any Arch to Radius, So is Radius to the Secant of the fame Arch.

IN the Scheme let C be the Center of the Earth, and ABD be a Quarter of a Meridian, let A be the Pole, and CD the Semidiameter, or Radius of the Equinoctial, draw the Tangent DE, and the Secant CBE, draw alfo the perpendicular BG (the right Sine of the Arch DB) and FB equal to CG, the Sine Complement of the fame Arch, then is BD the Latitude or Distance of B from the Equinoctial: A B the Complement of Latitude: BG is the Sine of the Latitude, and CG equal to FB the Sine Complement of Latitude. CB equal to CD is Radius, DE the Tangent, and CE the Secant of the Latitude.

HENCE, for the firft, it will be, As CD, (Plate XI. Fig. B.) the Radius or Semidiameter of the Equinoctial, To CG equal to F B, So one Degree, or any other part of the Equinoctial to one Degree, or the like part of the Parallel paffing through B, that is as Radius to the Sine Complement of Latitude, So 60, the Miles in a Degree of the Equinoctial, to the Miles in one Degree of Longitude, in the Parallel of B.

Secondly, As CG (equal to FB) Sine Complement of Latitude, to CB (equal to CD) Radius, So CD Radius to C E the Secant of Latitude Euclid, Lib. vi. Prop. iv.

THIS being granted, it may thus be applied to a Demonftration of the truth of the Conftru&tion of Mercator's, or, more juftly, Mr Wright's Chart.

IN this Projection the Meridians are proposed parallel to each other, and confequently the Diameter of any or every Parallel, is likewise supposed equal to that of the Equator, and hence the Radius of any Parallel is equal to the Radius of the Equator; and therefore, in the last mentioned Scheme, FB or C G will be every where equal to CD, but when CG is become equal to CD, it is evident CB will become equal to CE, and confequently the Radius of the Meridian, at any Parallel, will be equal to the Secant of the Latitude of that Parallel, alfo as a Degree, or any small Arch upon the Equator, is equal to a Degree, or the like Arch upon the Meridian, upon the Globe, it will be as the Secant of any Parallel is to Radius, fo is the length of a Degree, or any small Arch upon the Meridian, to the Length of a Degree, or the like Arch upon that Parallel, and therefore a Degree on any Parallel, will be increased in the fame Proportion, as the Secant of Latitude is greater than Radius; as for Inftance, the Secant of 60 is juft twice Radius, therefore a Degree of Longitude which should be but half as much as a Degree of Longitude in the Equinoctial, is there, and all over the Projection, equal to a Degree in the Equinoctial, and therefore in Latitude 60 juft twice as large as it fhould be, and the Degrees of Latitude which are every where equal upon the Globe, are double in Latitude 6a, to what they are in the Equinoctial, by which means the proportion in northing and fouthing, is the fame to that of the eafting and wefting upon this Projection, as it is upon the Globe, and the Length of a Degree, or any fmall Arch upon the enlarged Meridian, muft every where be to a De gree, or like Arch of the Meridian on the Globe, as the Secant of the Latitude is to Radius,

HENCE

HENCE fuppofing any fmall Arch of the Meridian to be Radius, the following Corollaries may be deduced.

1. THAT the Length of a Degree, or any fmall Arch upon the enlarged Meridian, is every where equal to the Secant of the Arch contained between it and the Equator.

2. THE distance of any Point in the enlarged Meridian from the Equator, is equal to the Sum of all the Secants contained between it and the Equator.

3. THE Distance between any two Parallels, on the enlarged Meridian (being both on the fame Side of the Equator) is equal to the Difference of the Sums of all the Secants of every Minute, contained between each Parallel and the Equator.

4. THE Distance between any two Parallels upon the enlarged Meridian (being on contrary fides of the Equator) is equal to the Sum of the Sums of all the Secants contained between each Parallel and the Equator.

UPON the Principle of this fecond Corollary it is that the Table of meridional Parts is formed viz. That the Distance of any Point, upon the enlarged Meridian, from the Equator, is equal to the Sum of all the Secants contained between it and the Equator. It is plain that if to o0000 (which if the expreffion be proper, may be faid to be the meridional Parts of oo deg. oo min.) we add the natural Secant of oo deg. o min. viz. 10000, the Sum will be 10000 the meridional Parts for o deg. 1 min. to which if we add the natural Secant of o deg. 2 min. which is again 10000 (because the Secants of fuch small Arches, where they fall upon the tangent Line, at fo near a right Angle, increase infenfibly) the Sum is 20000, the meridional Parts for 2 deg. oo min. &c. But this will appear more confpicuous towards the latter part of the Table, Z 4

where

where the natural Secants, and confequently the meridional Parts, increase much fafter; and to make it more intelligible, we fhall have recourfe to that excellent Book, entituled Correction of Errors in Navigation, written by the ingenious Mr Edward Wright, the first Inventor of the Projection we are now treating of, and of the Table of meridonal Parts, by which it is projected, and the Operations therein performed. According to his Table of Latitudes (being of the fame Conftruction and Ufe, with those we now call Tables of meridional Parts) and Mr Sherwin's or any correct Table of natural Secants, we fhall find, by the continual Addition of the Secant of the fucceeding Minute, to the meridional Parts of the Minute propofed, the Sum is the meridional Parts of the faid fucceeding Minute, &c. as above, and as in the following Inftances in finding the meridional Parts to higher Latitudes.

IN this manner (not to proceed any further by way of Example) are all the meridional Parts, from oo deg. oo min. to 89 deg. 59 min. produced,

it