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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. PR. 20P. Required the true content.
As the square of 42F. 3In.=the square of 504
inches=254016P. Is to the content of the map 920A. IR. 20P.
=147260P. : : The square of 42F. 3In.=the square of 507
inches_257049 To the true content.
A. R. P.
The third Method
for determining the Areas ofrightlined Figures universally, or by calculation.
Plate VIII. fig. 7.
ERIDIANS 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 outhing of any stationary line, is the distance that ne end of the line is north or south from the her end; or it is the distance which is intercept
on the meridian, between the beginning of the tionary line and a perpendicular drawn from the aser end to that meridian. Thus, if N. S. be a wridian line passing through the point A of the
AB, then is Ab the difference of latitude or ing 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 BA.
4. The 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.
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 the westings. Plate IX. fig. 1.
Let a, b, c, e, f, g, h, represent a plot or parcel of land. Let a be the 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 de partures.
The northings, ei+go+hq=20+65 +cd+fr the southings: for let the figure be completed; then it is plain that go+ha+rk-20+68+cd, and eimrk= fr. If to the former part of this first equation ei mrk be added, and fr to the latter, then go+hq+ ei=ao+bs+od+fr; that is, the suin of the northings is equal to that of the southings.
The eastings cs+qasob+de+if+rg+ah, the westings. For aq+yo (az) =detif+rg+oh, and bo=c-yo. If to the former part of this first equation, cs—yo be added, and bo to the latter, then cstaq=ob+defiftrg zoh; that is the sum of the eastings is equal to that of the westings. Q.E. D.
This theorem is of use to prove whether the field-work be truly taken, or not; for if the sum of the northings be equal to that of the southings, and the sum of the eastings to that of the westings, the field-work is right, otherwise not.
Since the proof and certainty of a survey depend on this truth, it will be necessary to shew how the difference of latitude and departure for any stationary line, whose course or distance are given, may be obtained by the tables hereunto annexed.
TO FIND THE DIFFERENCE OF LATITUDE AND DEPARTURE, BY THE HELP OF THE ANNEXED TABLE.
This table is so contrived, that by finding therein the given course, and a distance not exceeding 100 miles, chains, perches, or feet, the difference of latitude and departure is had by inspection : the course is to be found at the top of the table when under 45 degrees; but at the bottom of the table when above 45 degrees. Each column signed with a course consists of two parts, one for the difference of latitude, marked Lat. the other for the departure, marked Dep. which names are both at the top and bottom of these columns. The distance is to be found in the column marked Dist. next the margin of the page.
In the use of those tables, a few observations only are necessary.
1. If a station consist of any number of even chains or perches (which are almost the only measures used in surveying) the latitude and departure are found at sight under the bearing or course, if less that 45 degrees; or over it if more, and in a line with the distance.
2. If a station consist of any number of chains and perches, and decimals of a chain or perch, under the distance 10, the lat. and dep. will be found as above, either over or under the bearing :