3. Let the course be as before, but the distance

If the east meridian distances in the middle of each line be multiplied into the particular southing, and the west meridian distances into the particular northing, the sum of these products will be the area of the map.

PL. 10. fig. 1.

Let the figure ahkm be a map, the lines, ab bk to the southward, and km ma to the northward, NS the first meridian line passing through the first

station a.

These four areas am+on+tp+gl will be the area of the whole figure cmswiprle, which is equal to the area of the map abkm. Complete the figure.

The parallelograms am and ow, are made of the east meridian distances dz and tu, multiplied into the southings ao and or. The parallelograms rp and gl are composed of the west meridian distances ef and hh, multiplied into the northings g and ga (my) but these four par llelograms are equal to the area of the map; for if from them be taken the four triangles marked 2, and in the place of those be substituted the four triangles marked O which are equal to the former; then it is plain the area of the map will be equal to the four parallelograms. 2. E. D.

THEO. III.

If the meridian distance when east, be multiplied into the southings, and the meridian distance when west be multiplied into the northings, the sum of these less by the meridian distance when west, multiplied into the southings, is the area of the survey.

PL. 10. fig. 2.

Let abc be the map.

The figure being completed, the rectangle af is made of the meridian distance eq when east, multiplied into the southing an; the rectangle yk is made of the meridian distance rw, multiplied into the northings cz or ya. These two rectangles, or parallelograms, af+yk, make the area of the figure dfnyikd, from which taking the rectangle oy, made of the meridian distance tu when west, into the southings oh or bin, the remainder is the area of the figure dfohikd, which is equal to the area of the map.

Let bou-Y, urih=L, ric=0, wrc=Z=, 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+0+%; but 2+0=K, put K instead of +0; then 1+%= L+K, add to both sides the equal triangles B and A, then Y+Z+B=L+K+A. If therefore B+Y+ Z be taken from abc, and in lieu thereof we put L+K+A, we shall have the figure dfohikd=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,

But,

into the

The merid east multiplied Dist when west into the the sum of these products taken gives the area of the map.

northings southings from the former

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 method 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 Thus if the meridian distance of the same name. 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.

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 east to the west side, which will evidently appear, if the field-work 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 further explanation; but to find the area, by the second method, take this

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