Let the point of your compasses be kept to the last point of the extended line, till you lay your scale from it to the next station, to prevent mistakes from the number of points.

That the triangle c D e, is equal to the rightlined figure ABCDEFGH, will be evident from problems 18. 19. geom. for thereby, if a line were drawn from 6 to C, it will give and take equally, and then the figure b CDEFGH, will be equal to the map. Thus the figure is lessened by one side, and by the next balance line will lessen it by two, and so on, and will give and take equally. In the same manner an equality will arise on the the other side.

The area of the triangle is easily obtained, as before, and thus you have the area of the map.

It is best to extend one of the shortest lines of the polygon, because if a very long line be produced, the triangle will have one angle very obtuse, and consequently the other two very acute; in which case it will not be easy to determine exactly the length of the longest side, or the points where the balancing lines cut the extended one.

This method will be found very useful and ready in small enclosures, as well as very exact; it may be also used in large ones, but great care must be taken of the points on the extended line, which will be crowded, as well as of not missing a station.

PROB. XVII.

A map with its area being given, and its scale omitted to be either drawn or mentioned; to find the scale.

Cast up the map by any scale whatsoever, and

it will be

As the area found

Isto the square of the scale by which you cast up, :: The given area of the map

To the square of the scale by which it was laid down.

The square root of which will give the scale.

EXAMPLE.

A map whose area is 126A. 3R, 16P. being given; and the scale omitted to be either drawn or mentioned; to find the scale.

Suppose this map was cast up by a scale of 20 perches to an inch, and the content thereby produced be 31A. 2R. 34P.

As the area found, 31A. 2R. 34P. 5074P. Is to the square of the scale by which it was cast up, that is to 20×20=400. :: The given area of the map 126A, 3R, 16P. =20296P.

To the square of the scale by which it was laid down.

5074: 400 20296: 1600 the square of the required scule.

Answer. The map was laid down by a scale of 40 perches to an inch.

PROB. XVIII.

How to find the true content of a survey, though it be taken by a chain that is too long or too short.

Let the map be constructed and its area found, as if the chain were of the true length. And it will be,

As the square of the true chain

Is to the content of the map,

:: The square of the chain you surveyed by To the true content of the map.

EXAMPLE.

If a survey be taken with a chain which is 3 inches too long; or with one whose length is 42 feet 3 inches, and the map thereof be found to contain 920A. 2R. 20P. Required the true content.

As the square of 42F. OIn.=the square of 504 inches 254016.

Is to the content of the map 920A. 1R. 20P.147260P.

The square of 42F. 3In. the square of 507

inches 257049.

=

METHOD OF DETERMINING THE AREAS OF RIGHTLINED FIGURES UNIVERSALLY, OR BY CALCULATION.

DEFINITIONS.

1.

PL. 8. fig. 7.

MERIDIANS are north and south lines,

which are supposed to pass through every station of the survey.

2. The difference of latitude, or the northing or southing of any stationary line, is the distance that one end of the line is north or south from the other end; or it is the distance which is intercepted on the meridian, between the beginning of the stationary line and a perpendicular drawn from the other end to that meridian. Thus, if N. S. be a meridian line passing through the point A of the line AB, then is Ab the difference of latitude or southing of that line,

3. The departure of any stationary line, is the nearest distance from one end of the line to a meridian passing through the other end. Thus Bb is the departure or easting of the line AB: but if CB be a meridian, and the measure of the stationary distance be taken from B to A; then is BC the difference of latitude, or northing, and AC the departure or westing of the line BÄ.

4. That meridian which passes through the first station, is sometimes called the first meridian; and sometimes it is a meridian passing on the east or west side of the map, at the distance of the breadth thereof, from east to west, set off from the first station.

5. The meridian distance of any station is the distance thereof from the first meridian, whether it be supposed to pass through the first station, or on the east or west side of the map.

THEO. I.

In every survey which is truly taken, the sum of the northings will be equal to that of the southings; and the sum of the eastings equal to that of of the westings.

PL. 9. fig. 1.

Let a, b, c, e, f, g, h, represent a plot or parcel of land. Let a be the first station, b the second, c the third, &c. Let NS be a meridian line, then will all lines parallel thereto, which pass through the several stations, be meridians also; as ao, bs, cd, &c. and the lines bo, cs, de, &c. perpendicular to those, will be the east or west lines, or departures.

The northings, ei+go+hq=ao+bs+cd+fr the southings for let the figure be completed; then it is plain that go+hq+rk=ao+bs+cd, and ei→ rk-fr. If to the former part of this first equation ei―rk be added, and fr to the latter, then go+hq +ei=ao+bs+cd+fr; that is, the sum of the northings is equal to that of the southings

The eastings es+qa=ob+de+if+rg+oh,the west