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SCHOLIUM.

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 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 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.

EXAMPLE.

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 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. and 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,

and the distance 79 four-pole chains. Under 35°. 15' or 35 degrees, and opposite 79, we find 64. 52 for the latitude, and 45. 59 the departure, which signify that the end of that station differs in latitude from the beginning 64. 52 chains, and in departure 44. 59 chains.

Note, We are to understand the same things if the distance is given in perches or any other measures, the method of proceeding being exactly the same in every case.

Again let the bearing be 543 degrees, and distance as before; then over said degrees we find the same numbers, only with this difference, that the lat. before found, will now be the dep. and the dep. the lat. because 54 is the complement of 35 degrees to 90, viz. lat. 45. 59. dep. 64. 52.

2. Suppose the same course, but the distance chains 90 links, or as many perches. Here we find the same numbers, but the decimal point must be removed one figure to the left.

Thus, under 351, and in a line with 79 or 7.9, are

Lat. 6. 45
Dep. 4. 56

the 5 in the dep. being increased by 1, because the 9 is rejected; but over 543 we get

Lat. 4. 56*
Dep. 6. 45

3. Let the course be as before, but the distance 7.

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THEO. II.

When the first Meridian passes through the Map. 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.

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Let the figures abkm 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.

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