any latitude, is to the difference of longitude between these meridians, as the cosine of that latitude, is to the radius. (F. 359.) But, on a Mercator's chart, the distance between any two meridians is made equal to their difference of longitude, and consequently the parallels of latitude are enlarged, in the ratio of the cosine of the latitude to the radius: now, in order that the angle which the rhumb line makes with the several meridians on the chart, may correspond with the same angle on the globe, it will be necessary to increase the meridians in the same ratio, by which the parallels of latitude are increased, viz. as the cosine of the latitude is to the radius, or which is the same thing, as the radius is to the secant of the latitude. Hence the degrees of the meridians on a Mercator's chart increase towards the poles, as the secant of the latitude increases; likewise all the parallels of latitude, and every part of them, are larger than they are on the globe in the ratio of the radius to the secant of the latitude. Hence, though the latitudes, longitudes, and bearings of places are truly represented on a Mercator's chart, the distances are distorted in various proportions. To render these observations more clear, let us suppose WELL (Plate VI. Fig. 24.) a Mercator's chart, constructed to represent the spherical surface WELL (Plate III. Fig. 2). Then wl; 10,10; 20,20; &c. are meridians; ABCzeul a rhumb line; IB, HC, KZ, Te, YU, &c. parallels of latitude, each straight line on the chart representing its corresponding arc on the spherical surface.

The meridians in this projection being parallel to each other, IB is equal to w, 10; hence IB is the difference of longitude on the chart, corresponding with ib on the sphere. Therefore cos. wi: radius :: 26: IB (E. 359.).

Now to make the elementary triangle AIB on the chart, equiangular with Aib on the sphere, the difference of latitude ai must be increased in the same ratio with the departure ib; hence cosine wi: radius :: Ai: AI. But the cosine of any arc is to the radius, as the radius is to the secant; therefore radius : secant of wi:: Ai: AI.

(R) But A is an, indefinitely small portion of the sphere, which is increased to AI on the chart. Let us suppose the radius of the sphere an unit, and the difference of latitude Ai 1 minute *; then the last proportion will be 1: secant wi :: 1': AI; consequently in this case, the increased minute will be equal to the secant of the latitude.

(S) Hence it follows that the natural secant of any lati

* It is evident that if Ai were to represent a smaller portion of the meridian than one minute, the conclusions derived from such a supposition would be more

accurate.

tude,

tude, will be equal to the increase of the next minute on the chart, nearly.

(T) The meridional line on a Mercator's chart (constructed on the above principles) is equal in length to the sum of all the natural secants of every minute of latitude contained in it.

Thus suppose the difference of latitude Al on the sphere to be 6 minutes, then the enlarged meridian Al on the chart is equal to AI+BH+CK+ZT+CŸ+u, 50; and these are the secants of wa; 10, в; 20, c; 30, z; &c.

Hence is derived Mr. Wright's method of constructing a table of meridional parts.

(U) The difference of latitude (in geographical miles) between two places on the globe, is generally called the proper difference of latitude, to distinguish it from the difference of latitude (in geographical miles) on the chart, which is called the meridional difference of latitude.

(W) A plane right-angled triangle formed by the ship's course, distance, difference of latitude, and departure (Z. 357.), will be similar to a plane triangle formed upon a Mercator's chart, by the enlarged distance, meridional difference of latitude, and difference of longitude.

For all the elementary right-angled triangles on the sphere, (Plate III. Fig. 2.) viz. Aib, bhc, ckz, zte, &c. are equiangular and similar (U. 355.); and from the very nature of the chart, the elementary triangles (Plate VI. Fig. 24.) AIB, BHC, CKZ, ZTE, &c. are likewise equiangular and similar, not only to each other, but to the elementary triangles Aib, bhc, &c. on the sphere.

Now the elementary triangles on the sphere, may be truly represented by a plane triangle (Z. 357.); therefore the elementary triangles on the chart may likewise be truly represented by a plane triangle; equiangular and similar.

(X) I. MR. WRIGHT's method of constructing a table of meridional parts.

Meridional parts of 1'-Natural secant of l'

M. P. of 2'Nat. secants of l'+2′

M. P. of 3'Nat. secants of ' + 2′+ 3′

And by adding together the natural secants of a few pages, in either of the above mentioned books, you will find

The meridional parts of 1 degree= 60·0031231.

(Y) In this manner (by a continual addition of the secants) Mr. Edward Wright formed the FIRST table of meridional parts which is contained in his celebrated work intitled, Certain Errors in Navigation Detected and Corrected. This book was first published in 1599. Mr. William Oughtred was the next who constructed a table of meridional parts, by the continual addition of the intermediate secants of, 14, 21, &c. minutes; and these tables were again corrected, and extended, by Sir Jonas Moor.

(Z) It must be admitted that Wright's method of constructing a table of meridional parts is not strictly geometrical; for the cosines of any two parallels of latitude are not precisely the same, even if you suppose the common difference between these parallels, to be less than one minute of a degree, and the secants of such parallels do not increase in any regular ratio. A table constructed on Wright's principles, in high latitudes, will exceed the truth, a small matter. This, the learned author was sufficiently aware of; for he says he rejected some of the decimal parts of the secants in making his table, "Because that indeed,

at every point of latitude, a minute of the meridian in this "nautical planisphere [Mercator's chart] hath somewhat lesse "proportion to a minute of the parallel adjoyning towards the "Equinoctial, then the secans of that parallels latitude hath to "the whole sine. But in this table it was thought sufficient to "use such exactness as that thereby (in drawing the lineaments ❝ of the nautical planisphære) sensible error might be avoided. "He that listeth to be more precise may make the like tables "to decades or tennes of seconds, out of Ioachimus Rhæticus "his Canon magnus triangulorum. Notwithstanding the Geo"metrician that desireth exact truth, cannot be so satisfied "neither."

(A) II. ANOTHER Method of constructing a table of meri dional parts from the secant of an arc.

It is shewn (G. 123.) that if a the length of any arc, the radius being 1, the secant will be

a2 5a4

61a6 277a8 50521a1o 540553a12

1+ + + + +

+

2 24 720 8064 3628800 95800320

multiplied bya, the fluxion of the arc, will produce

a2a 5a à 61a 277a a 50521a1à

a+

+ +

2

24 720

+
8064 3628800

+&c. This

&c. the fluxion of

the sum of the secants in the arc a, the fluent of which is

Let it be required to find the meridional parts correspondent to 5 degrees. The length of the arc of one minute being ⚫000290888208665, &c. *, supposing the radius an unit.

First, 5° x 60 x 00029088820866.087266462599, &c.= a the length of an arc of 5 degrees.

This result, divided by ⚫000290888208665, &c. the length of the arc of 1' will give the meridional parts of 5°=300·381498, true to the last place of decimals.

By a similar process you will find the meridional parts of 10° to be 603 0695795. Proceed in the same manner to find the meridional parts of any other degree of latitude.

(B) III. The method of constructing a table of meridional parts, by a table of logarithmical tangents.

PROPOSITION I. (Plate VI. Fig. 25.)

If the logarithmical tangent of half the complement of any degree of latitude be subtracted from 10, and the remainder be multiplied by 7915.7044679, &c.; the product will give the meridional parts (in minutes), correspondent to that latitude.

* Gardiner's Edition of Sherwin's Logarithms, page 44 (1742.). Traité de Trigonométrie, par Cagnoli (Table AA), page 474.

DEMON

DEMONSTRATION. Let p be the pole, EQ a portion of the equator; o the latitude, ov its sine and OE=vw its cosine; and EU the radius of the sphere.

Put qy the latitude, m=the length of qu, on a Mercator's chart, called the meridional parts; y=ov=Ew the sine of the latitude, and r=EV the radius of the sphere. Then will E0= wo-y2, and because the degrees of latitude on a Mercator's chart increase as the cosine of latitude is to radius, we have

√ r2—y2 : r :: x : m, hence m=

rx

rj

But, ✔r-y3 : r :: ÿ:, conseq. *=√ — y2

r+y

m=rx 2.30258509299404, &c. ×× log.; + the correc

r-y

tion,=rx 2-302585, &c. x log. ++the correction.

r-y

But by Plane Trigonometry, in the triangle vws, vw: rad :: ws: tang wvs, or co-tang wsv.

the co-tangent of wsv, measured by half the arc

Po, the complement of the latitude. Hence we have m=

rx 2.302585, &c.x log.

co-tang.comp. lat.

+the correction.

But when m=o the co-tangent of half the complement of latitude=r (L. 31.); and hence the correction is nothing, thereforem rx 2.302585, &c. x log. co-tang.comp.lat. =

; for, the co-tangent of

an arc divided by the radius, is equal to the radius divided by the tangent (Z. 97.).

But the tables of meridional parts are generally expressed in geographical miles, therefore we must express the radius of the sphere in geographical miles. If the diameter of the earth be 1, the circumference will be 3.141592653589, &c., hence

3.141592653589