METHOD OF DETERMINING THE AREAS OF RIGHT-LINED FIGURES UNIVERSALLY, OR BY CALCULATION.

Definitions.

PL. 8. fig. 7.

1. 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 NS 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 sta tionary 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. 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.

THEOREM 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 the westings.

PL. 9. fig. 1.

Let abcefgh 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, cs+ga=ob+de+if+rg+oh, the westings. For aq+yo (az) =de+if+rg+oh, and bo=cs-yo. If to the former part of this first equation cs-yo be added, and bo to the latter, then cs+aq=ob+de+if+rg+oh; that is, the sum of the eastings is equal to that of the westings. Q. E. D.

SCHOLIUM.

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 show how the difference of latitude and departure for any stationary line, whose course and distance are given, may be obtained by the table usually called the Traverse Table.*

* This table is calculated by the first case of right-angled plane trigonometry, taught in the fifth section of the first part of this book, where the hypothenuse and an acute angle are given, to find the legs.

In the right-angled triangle ABC (Pl. 8, fig. 7), given the distance or hypothenuse AB 91 chains, links, or perches, the course or one of the acute angles ABC 41°; it is required to find the legs, or the difference of latitude and departure.

To find the difference of latitude and departure by the Traverse Table.

This table is so contrived, that by finding therein the given course, and a distance not exceeding 120 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 left-hand margin of the page.

EXAMPLE.

In the use of this table, 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.

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

Hence AC is the departure and BC the difference of latitude which correspond to those in the table. In the same manner the difference of latitude and departure to every degree in the table is calculated, by which the practitioner can at any time prove the exactness of those in the table.

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, let us suppose a course or bearing to be S. 35° 15' E., and the distance 79 four-pole chains. Under 35° 15', or 35 degrees, and opposite 79, we find 64.51 for the latitude, and 45.59 the departure, which signify that the end of that station differ in latitude from the beginning 64.51 chains, and in departure 45.59 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 543 is the complement of 351 degrees to 90, viz. lat. 45.59, dep. 64.51.

2. Suppose the same course, but the distance 7 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 542 we get

Lat. 4.56
Dep. 6.45

3. Let the course be as before, but the distance 7.79, then opposite

Dep. 4.43

7

6

THEOREM 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.

PL. 10. fig. 1.

Let the figure 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.

These four areas am+ow+xp+gl will be the area of the whole figure cmswiprlc, 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 ox; the parallelograms xp and gl are composed of the west meridian distances ef and hh multiplied into the northings xg and ga (my): but these four parallelograms are equal to the area of the map; for if from them be taken the four triangles marked Z, 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. Q. E. D.

THEOREM 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.

Let abc be the map.

PL. 10. fig. 2.

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 xw, 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 bm, the remainder