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Third Jethod for determining the Areas of right
lined Figures universally, or by Calculation.
1. Meridians are north and south lines, which are supposed to pass through every station of the survey.
2. The Bearing of a line, is the angle it makes with the meridian passing through either end of it.,
3. 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 :
· 4. 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.
Cor. Hence it appears that the stationary distance, difference of latitude, and departure, constitute the three sides of a right angled plane triangle': the distance being the hypothenuse, the difference of latitude and departures, the two legs, and the angle opposite the departure, the bearing or course; consequently, any two of these four parts being given, the other two can be found.
5. 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.
6. 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,
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 east or west lines, or departures.
The northings ei+go+hq =a0+68+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 eirk 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 cs+qarob+de+if+rstah, the westings. For aq -yo ( az)=de+if+rgtoh, and bo=C8—9o. If to the former part of this first equation, cs—yo be added, and bo to the latter, then es laq rob+de+if +rg toh ; 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 fieldwork 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 it is 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 and distance are given, may be obtained by the tables hereunto annexed.
To find the Difference of Latitude and Depar
ture, 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 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 gurveying) the latitude and departure are found at sight under the bearing or course, if less than 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. anil dep. will be found as above, either over or under the bearing; the decimal point or separatrix being removed one figure to the left, which leaves a figure to the right to spare.
3. If the distance be any number of chains or perches, and the decimals of a chain or perch, the lat. and dep. must be taken out at two or more operations, by taking out the lat. and dep. for the chains, or perches in the first place, and then for the decimal parts.
To save the repeated trouble of additions, a judicious surveyor will always limit his stations to whole chains, or perches and lengths, which can commonly be done at every station, save the last.
1. In order to illustrate the foregoing observations,