arc of a circle, equal to the radius in degrees be denoted by R°, &c., then

numbers of great use in various calculations, as is frequently shown in the course of this work.

From fig. 1 may be readily derived

the developed length on the mean arc of the sector.

But in the triangle SA C, fig. 1, SA = tan ACS = cot l,

therefore 180°: cot :: S:

T
× S cot AB
180

(2.)

Equating these two values, we have, after striking out the common

Thus we form an angle M S N, fig. 2, of as many degrees as there are of longitude required in the map, and we shall then have the angle S of the developed segment. Next S A must be taken equal to cot l, which will give the radius of the arc representing the mean parallel. Afterwards there must be set off in a straight line from A to M and m, the developed lengths of the meridian E A P, between the limits of the extreme latitudes. If the map should contain d degrees of latitude, Mm will be

1° being its assumed length in latitude on the scale selected for the map.

Now dividing A M and Am into as many parts in Ρ and q as may be required, we describe through these points of division from the centre S arcs forming the parallels of latitude, while the meridians are straight lines drawn from the centre S, and passing through the equal divisions on the arc A B.

For countries near the equator, however, the centre S becomes too distant to be practicable. In this case Flamsteed's projection may be employed, in which the central meridian is first drawn in a straight line. This meridian is next divided into degrees or less divisions, through which, perpendicularly, the parallels of latitude are drawn. On these parallels, divisions conformable to the convergence of the meridians, according to their respective latitude, are inserted. These operations may be very simply effected by the aid of Table XXV. of the general tables.

If the numbers in the column titled "Minute of Longitude" be divided by 100, or if the decimal point be moved two places to the left, the result will be a degree of longitude in geographical miles nearly.

For more precision, the line S A, in figure 1, is sometimes made to cut the meridian Mm in the middle of Am and A M, or at one fourth of the meridian distance from either of its extremes.

This

will give―

AS=

cos /
sin(+2)

(6.)

in which 7 and l' are the latitudes of the two points where S A cuts the meridian.

To apply these we shall select the British Isles, which extend from latitude 50° to 60° N., and therefore the middle latitude, or that of the mean parallel, A B, is 55° N. Also, including Ireland, they extend in longitude from 2° E. to 10° W., and therefore include 10° of latitude and 12° of longitude, that is, the arc Mm equals 10°, and A B 12°.

Hence, as formerly shown, A SAC cot 7 to radius A C=1. But the length of an arc equal to the radius is 57°.2957795, therefore AS 57°.2957795 cot 7 = R° cot l.

=

=

Hence, also, AS+d=40°.119± 5° 45°.119 and 35°.119, the lengths of SM and Sm respectively, therefore the radii of the mean and two extreme parallels are determined.

By formula (3) the angle A SB in figure second must also be found, that is, SD sin l.

Now D=12° of longitude 7 =55° N. the mean parallel.

These values of A S and S have been here computed in degrees, but they may be easily converted into minutes and seconds by multiplying successively by 60, or by using in the computation log R or log R" respectively.

To complete the construction, draw two straight lines, SA, SB, figure 2, making an angle of 9°.8298, or 9° 50', with each other ; then with S as a centre, and 40°.119 or 40° 7′ as radius, describe the arc A B, and divide it into twelve equal parts, each of these will be 1 degree of longitude. In like manner the other arcs M N, m n, &c., may be described. Divide MA, Am, each into five equal parts, each of these will be 1 degree of latitude, and it is obvious the subdivision in both cases may be carried to minutes, or even seconds, if required.

The lengths of the cords joining the extremities of these paral

lels of latitudes may also be computed, it thought desirable in peculiar cases, by the formula

in which is the cord or straight line, A B, fig. 1, a the length of the arc A a B, and r the radius S A. Hence such maps may be readily constructed with ease and accuracy. It was in this manner that the outline map exhibiting part of the triangulation of Scotland was constructed, and any portion of the whole may be readily selected or cut out, as the part defg, fig. 2, when, as in this case, it becomes necessary to retain a section only.

There are several other projections, besides the above, employed in forming the map of a country, particularly a modification of Flamsteed's, adopted for the national maps of France, whose principles are fully discussed in the Topographie of Puissant and the Geodesie of Francœur, which may be advantageously consulted. The tables of Plessis very readily compute the magnitude and position of each rectangular portion, but our limits will not permit of entering upon its discussion here.

TRIGONOMETRICAL SURVEY OF A BAY.

In order to fill in properly the coasts in marine surveys, the following example has been given, in which the base was measured with the chain, and the various angles measured with a theodolite or sextant, or a combination of both, assisted by the prismatic compass occasionally in sketching the contours. The whole is completed and delineated on Plate XXV., which will enable students of surveying to follow out all the details successfully.

The base AB was carefully measured with a hundred feet

chain, and found to be 6524.5 feet, bearing 15° 36′ 20′′ N.W. true. The variation of the compass was 28° 35′ W. in 1836. Latitude of B, 55° 31′ 56′′ N. Longitude 5° 7' 30" W.

If, therefore, the base be extended indefinitely, the perpendiculars falling upon it may be calculated and set off from an appropriate plane scale, to determine the points C, D, E, F, G, H, I, and K, as in Mr Gale's method of plotting, shown in a preceding part of this work.

Plate XXVII. will give the hydrographic engineer an example of a naval attack on a fortified town, with arrangements both for defence and attack. Plate XXVIII. indicates the arrangement followed in naval battles by one of the greatest of our national heroes, and may be submitted as an instructive example of that portion of a marine surveyor's duty.

Plate XXIV. is the plan of Kingston harbour, at Dublin Bay, and will show the method of planning a modern harbour, though the field-book is not given. The method of proceeding is exactly similar to that in Plate VII., which may be consulted.

Plate XXIII. is a chart of part of the sea-coast, laid down from a scale of one inch to a mile, and the bearings taken with a theodolite divided into twice 180°, and the distances measured with a Gunter's chain of 66 feet. This chart will give a surveyor who is employed in the survey of a county some idea of the labour in a work of that kind, as the whole coast must be measured and