lel of the middle latitude; and, though these be not strictly equal, yet the error, which arises from this method, is of no consequence in a day's run, except in very high latitudes.

The middle latitude is found by taking half the sum of the two latitudes, if of the same name; or half their difference, if of contrary names; and the questions are solved by the help of the following proportions.

1. Cosine middle latitude

: Radius

:: Departure

: Difference of longitude.

This proportion is the inverse of the theorem of parallel sailing.

2. Cosine middle latitude
Tangent of course.

:: Difference of latitude

: Difference of longitude.*

For examples in Middle Latitude Sailing, see the examples in Mercator's Sailing.

MERCATOR's

* This proportion is deduced from the foregoing, as is here demonstrated.

DEMONSTRATION.

Cosine middle latitude : radius departure diff. long. And diff. lat. : radius :: departure tang. of course, by plane sailing.

Therefore, cosine middle lat. tang, of course :: diff. lat. : difference of longitude.

Which was to be proved.

MERCATOR's SAILING.

MERCATOR'S SAILING is the method of finding a ship's place on the globe by Mercator's chart, or equivalent

tables.

In Mercator's chart the meridians are drawn parallel to each other, as in the plane chart, and therefore the meridjonai distance is every where the same. But, in order to compensate for this error, parallels of latitude, which are equidistant on the globe, are not so on the chart, but are more distant the higher their latitudes. Or, in other words, the degrees of the meridian are not all equal, but increase in length the more remote they are from the equator. And this is so provided, that any very minute part of the artificial meridian bears the same proportion to a like part of the parallel of its latitude, as do the like parts of both on the globe itself.*

THEOREM

* This method of constructing a chart of the world, whose meridians and parallels are right lines, is hinted at in the writings of PTOLEMY. GERARD MERCATOR first published one of these charts, in 1556, with the theory of which he did not appear to be acquainted, for the parts of his meridian were not increased in the true ratio. In the year 1599 Mr. EDWARD WRIGHT рublished his Correction of Errors in Navigation, in which the theory is demonstrated, and the method of computation by a table of meridional parts explained. And Dr. HALLEY, in the Philo. sophical Transactions, first demonstrated, that the artificial me. ridian line is a scale of logarithmic tangents of half the colatitudes, beginning with radius.

THEOREM I

Radius

: Cosine of the latitude

:: Secant of the latitude
: Radius.*

THEOREM. II.

The distance of any parallel of latitude from the equator, on Mercator's chart, is as the sum of the secants of all the arcs of latitude, beginning at o, and increasing arithmetically, by an indefinitely small common difference, till the last arc be that of the given latitude.t

COR.

Let ED represent an arc of latitude. Then AF will be its cosine, and AG its secant. The triangles ADF, AGE, are similar, for they are right-angled, and

have a common angle at A.

Therefore AD:

radius: AF

cosine of lat.

1:: AG secant of lat. : AE radius. Which was to be proved.

COR. The nautical meridian may be divided practically, by assuming arcs of latitude, whose common difference is known and determined; and the graduations will be more accurate the smaller the common difference. Thus, if

'And radius: cosine lat. :: part of meridian : ́like part of parallel, because the meridian and equator on the globe are equalAgain, radius : cosine lat. ::

Theorem I.

secant lat.

radius, by

Therefore, secant lat. ; radius :: part of meridian : like part of parallel.

And on Mercator's chart, secant lat. : radius :; part of meridian lying in that latitude, (which part, consequently, must be indefinitely small) like part of parallel. ;

Whence

secant lat. part of Mer.

radius
part of parallel

But radius is a constant quantity, and so likewise is the part of the parallel, because all the parallels of latitude are equal on Mercator's chart. Assume, therefore, any number of arcs of latitude increasing arithmetically from o, by an indefinitely small common difference, and call the parts of Mercator's meridian, corresponding successively with the differences of latitude, by the letters a, b, c, &c. Then,

=

secant lat. 2

=

secant lat. 3

&c.

part of mer. b part of mer.c

respectively equal to the constant quantity

Whence sec. lat. 1: part of Mer. a :: secant lat. 2 : part of Mer. b :: secant lat. 3 : part of Mer. c, &c. And secant lat. I : part of Mer. a :: secant lat. '. secant lat. 2+ secant lat. 3, &c. : parts of Mer. a+b+c, &c.

Now

if the common difference be taken 1 minute, it will be secant im. meridional part im. :: secant im. + secant 2m.+secant 3m. &c. : meridional parts 1+2+3m. &c

And

mer. pt, Im. X sec. Im. + sec. 2m. + sec. 3m. &c. secant I min.

meridional parts 1+2+3m. &c.

=

That is to say, the sum, or aggregate of the secants of all the arcs, arithmetically increasing by the difference of 1 minute to any given latitude, being divided by the secant of 1 minute, will give the length of the nautical meridian to that latitude, in parts equal to the first meridional parts, namely, in equatorial miles, because the first meridional part, contained between the equator and the latitude of 1 minute, does not exceed an equatorial mile by any quantity, which need be considered in practice. By this method Mr. WRIGHT calculated the table of meridional parts.

SCHOLIUM. The line, marked mer. on Gunter's scale, is a nautical meridian, adapted to the scale of equal parts, or equatorial degrees, which is under it, and marked EP. By the help of these lines, and the instructions already given concerning the plane chart, it is easy to construct a Mercator's chart, whose degrees shall be of the particular magnitude exhibited on the scale. But the most convenient method of making a chart, whose degrees shall be of a required magnitude, is to graduate the equator into equal parts or degrees as required, and from it, as a scale, take the distances of the several parallels of latitude from the equator,

Now the sum of all the parts of the artificial meridian, regu larly taken, will be equal to the distance from the equator on Mercator's chart of the parallel of latitude, expressed by the highest secant.

Therefore, &c.

Which was to be proved.