the area of the figure dfohikd, which is equal to the area of the map.

Let bou=Y, urih=L, ric=O, wrc= 2, akw=K, and efb B, ade-A. I say, that Y+Z+B=K+ L+A.

Y=L+O, add Z to both, then Y+Z=L+O+ Z; but Z+0=K, put K instead of Z+O; then Y+Z=L+K, add to both sides the equal triangles B and A, then Y+Z+B=L+K+Â. If therefore B+Y+Z be taken from abc, and in lieu thereof we put L+K+A,we shall have the figuredfohikd

abc, but that figure is made up of the meridian distance when east, multiplied into the southing, and the meridian distance, when west, multiplied into the northing less by the meridian distance, when west, multiplied into the southing. Q. E. D.

COROLLARY.

Since the meridian distance (when west) multiplied into the southing, is to be subtracted, by the same reasoning the meridian distance when east, multiplied into the northing, must be also subtracted.

SCHOLIUM.

From the two preceding theorems we learn how to find the area of the map, when the first meridian passes through it; that is, when one part of the map lies on the east and the other on the west side of that meridian. Thus,

RULE.

The merid.east

Dist. when west into the

their sum is the area of the map.

{

multiplied

southings.
northings

But,

The merid.east multiplied northings
Dist. when west into the

southings

the sum of these products taken from the former gives the area of the map.

These theorems are true, when the surveyor keeps the land he surveys, on his right hand, which we suppose through the whole to be done; but if he goes the contrary way, call the southings northings, and the northings southings, and the same rule will hold good.

General Rule for finding the Meridian distances.

1. The meridian distance and departure, both east, or both west, their sum is the meridian distance of the same name.

2. The meridian distance and departure of different names; that is, one east and the other west, their difference is the meridian distance of the same name with the greater.

Thus in the first metood of finding the area, as in the following field-book.

The first departure is put opposite the northing or southing of the first station, and is the first meridian distance of the same name. Thus if the first departure be east, the first meridian distance will be the same as the departure, and east also and if west, it will be the same way.

At station 11, the meridian distance 0.12 W,

The next departure

The next meridian distance

5.84 E.

5.72 E.

PL. 10. fig. 3.

In the 5th and 11th stations, the meridian distance being less than the departures, and of a contrary name, the map will cross the first meridian, and will pass as in the 5th line, from the east to the west line of the meridian; and in the 11th line it will again cross from the west to the east side, which will evidently appear, if the fieldwork be protracted, and the meridian line passing through the first station, be drawn through the map.

The field-book cast up by the first method, will be evident from the two foregoing theorems, and therefore requires no furthur explanation; but to find the area, by the second method, take this

RULE.

When the meridian distances are east, put the products of north and south areas in their proper columns; but when west, in their contrary columns; that is, in the column of south area, when the difference of latitude is north; and in north when south: the reason of which is plain, from the two last theorems. The difference of these two columns will be the area of the map.