termed the departure, being the sum of all the meridional dis

tances passed over.

Again, in the triangle ABC: let A B represent the meridian distance (or departure), and the angle BAC be equal to the latitude, then AC, the hypothenuse, will be equal to the difference of longitude.

Also, if DB represent the nautical distance, and CD the difference of latitudes, then B C D will be a right angle, and B C, the departure, nearly equal to the meridian distance in the middle latitude. If, then, in the triangle ABC the angle ABC be measured by that middle latitude, AB, the hypothenuse, will be nearly equal to the difference of longitude between D and B.

For further information on this subject, no better work can be consulted than Riddle's

By the use of Mercator's "Projection," most of these questions can be solved without calculation. In this ingenious system the globe is conceived to be so projected on a plane that the meridians are all parallel lines, and the elementary parts of the meridians and parallels bear in all latitudes the same proportion to each other that they do upon the globe. The uses to which this species of projection can be applied, and the vast benefit its invention has proved to the navigator, will be evident by reference to any work on navigation.

N

The latitude and longitude of any place being known, that of any other station within a short distance can also be determined by plane trigonometry. Suppose the latitude and longitude of G for instance to be known, from whence that of O, an adjacent station, is to be determined; the distance O G must be measured, or obtained by triangulation, and the azimuth NOG observed; then the difference of longitude G L between the stations is the sine of the angle

L

LOG to radius OG; and OL, the difference of latitude, is the cosine to the same angle and radius. The following example will show the application of this simple method :—

The distance of a station O', 238 feet due south of the Rl. Engr. Observatory at Chatham from Gillingham Church, was ascertained to be 7547-4 feet, and the angle SOG, the supplement of the azimuth, 78° 55′ 55′′; Gillingham Church being situated in 51° 23′ 24′′ 12 north latitude, and 0° 33′ 49′′41 east longitude.

1448.9-3.161039 Diff. of latitude (north), in feet.

7407 -3.869642 Diff. of longitude (west), in feet.

The lengths of one second of latitude and longitude in latitude 51° 23′ are

= 16" 53. Difference of latitude in arc,

116"81'56" 8. Difference of longitude in arc.

Latitude.

Gillingham Church N. 51° 23′ 24′′-12 E.

Longitude. 0° 33′ 49′′-41 1'56.8

Observatory

51° 23′ 40 65

0° 31' 52 6

It is always necessary to ascertain the variation of the compass before plotting any survey, for the purpose of protracting such parts of the interior details as have been filled in by magnetic bearings, and also of marking the direction of the magnetic meridian upon detached plans. The laws of this variation are at present but little known; and it is only by accumulating a vast num

ber of observations at different places, and at different periods, that the position of the magnetic poles and the annual variation and dip can be ascertained with anything like certainty.

A meridian line being once marked on the ground, the bearing of this line by the compass is of course the variation east or west. It can be traced with an altitude and azimuth instrument, or even a good theodolite, by observing equal altitudes and azimuths of the sun, or a star, on different sides of the meridian. With the latter object no correction whatever is required: the cross hairs are made to thread the star exactly (by following its motion with the tangent screws) two or three hours before its culmination; the vertical arc is then clamped to this altitude, and the azimuth circle read off. On the star descending to the same altitude, at the same interval of time after its transit, it is again bisected by the cross hairs, and the mean between the two readings of the azimuth circle gives the direction of the true meridian, which being marked out on the ground, its bearing is then read with the compass.

When the sun is the object observed, the altitude taken may be that of either the upper or lower, and the azimuth that of the leading or following limb; the mean of the readings of the azimuth circle does not necessarily therefore in this case give the true meridian; a correction must also be applied for the change in the sun's declination during the interval of time between the observations.

If the sun's meridian altitude is increasing, as is the case from midwinter to midsummer, his lower limb when descending will have the same altitude at a greater distance from the meridian than before apparent noon, and the reverse when it is decreasing. The following formula for this correction is taken from Dr. Pear

son:

x= D x sect. lat. x cosect. T, where D is the change of declination* in the interval of time expressed by T.

Example:-In latitude 51° 23′ 40′′ N. on May 12, 1838, the upper limb of the sun had equal altitudes.

* The sun's change of declination is given for every hour in the first page of each month in the Nautical Almanac.

And the readings of the azimuth circle at these times were— 311° 47′ 20′′ morning observation.

The sun's change of declination in one hour of mean time on May 12 appears, by the Nautical Almanac, = 37′′-53, therefore for 2h. 5'6m., the half interval, it is = 78′′-5.

The magnetic bearing of the pole star, or of any circumpolar star at its upper or lower culmination, gives at once the variation of the compass; a meridian may likewise be traced by observing the azimuths of a star at its greatest elongations, and taking the

mean.

If only one elongation is observed, the sine of the angular sin polar distance of star which added to, or subdistance =

cosine latitude

tracted from, the observed azimuth, gives the direction of the meridian.

The time at which any star is at its greatest elongation is thus found. The cosine of the hour angle in space tan polar dist. x tan lat. This hour angle divided by 15 gives the interval in sidereal time.

The other methods of finding the variation of the compass by the amplitude of the sun at sunrise or sunset, and by his azimuth at any period of the day, requiring more calculation, will be found among the Astronomical Problems.

A meridian line can be marked on the ground, without the aid of any instrument, with sufficient accuracy to obtain the variation of the needle for common purposes, by driving a picket vertically into the ground on a perfectly level surface. At three or four hours before noon, measure the length of its shadow on the ground, and from the bottom of the picket, as a centre, describe an arc with this distance as radius. Observe, when the shadow intersects this arc about the same time in the afternoon; and the middle point between these, and the picket, gives the line of the meridian. It is of course better to have three or four observations at different periods before and after noon; and these several middle points afford means of laying out the line more correctly.

The method hitherto described of laying down stations by triangulation, or by means of distances calculated from astronomical observation, is, however, only applicable within certain limits; as, on account of the spherical figure of the earth, the relative positions of places on the globe cannot be represented by any projection in geographical maps embracing very large portions of its surface, except by altering more or less their real distances, the content of