that distance down by the edge of the scale parallel to it, to b; and there describe an arc on the point b; and if it just touches the ruler's edge, the point b is in the true place of the extended line. Lay then the fiducial edge of the scale from b to D, and take a distance from C, that will just touch the edge of the scale; carry that distance along the edge, till the point which was in C cuts the produced line in c; keep that foot in c, and describe an arc, and if it just touches the ruler's edge, the point c is in the true place of the extended line. Draw a line from c to D, and it will take in and leave out equally in like manner the other side of the figure may be balanced by the line e D.

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 right-lined figure ABCDEFGH, will be evident from problems 18, 19, geom. for thereby, if a line were drawn from b 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 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.

PROBLEM 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

Is to 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 scale.

.Root. 1600(40 16

8(00

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

PROBLEM XVIII.

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

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. 0In. 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 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 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 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 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 cs+qa=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 filth 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 Hypothe

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 out at

nuse 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.

Hence AC is the Departure and BC the Difference of Latitude, which corresponds 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.