For dividing of commons, &c. Surveyors generally a measure the land of different value in separate parcels, and find the separate value thereof, which added in different sums, gives the whole content, and wholes value.

By problem 1, they find each man's proportional share of the whole value, and then lay out for each person, a quantity of land equal in value to his share; this they effect by first laying out a quantity by guess, and then casting it up and finding its value; and if such value be equal to his share of the whole value, the dividing line is right; if otherwise, they shift the dividing line a little, till by trial they find a quantity just equal in value to the value of the required share.

If any single share contains land of several different values, each is measured separately, and their several values found; and if the sum of them be equal to the value sought, the division is right; if not, it must be altered till it is so.

OF LAYING OUT ANY GIVEN QUANTITY OF LAND.

As the quantity of land is generally given in acres, roods and perches, it is necessary, first, to reduce them to square links, which may be performed by the following rule.

To reduce acres, roods, and perches into square links.

Rule 1. To the acres annex five cyphers on the right hand, and the whole will be links. 2. Place five cyphers to the right of the roods, and divide this by 4, the quotient will be links. 3. Place four cyphers on the right hand of the perches, divide this by 16, the quotient will be links. 4. These sums added together, give the sum of square links in the given quantity.

PROBLEM 1. To lay out a piece of land, containing any given number of acres, in form of a square.

This is no other than to determine the side of a square that shall contain any desired number of acres; reduce, therefore, the given number of acres to square links, and the square root thereof will be the side of the square required.

PROBLEM 2. To lay out any desired quantity of land in form of a parallelogram, having either its base or altitude given.

Divide the content, or area, by the given base, and the quotient is the altitude; if divided by the altitude, the quotient is the base; by the same rule a rectangle may be laid out.

PROBLEM 3. To lay out any desired quantity of and in form of a parallelogram, whose base shall be 2, 3, 4, &c. times greater than its altitude.

Divide the area by the number of times the base is to be greater than the altitude, and extract the square root of the quotient; this square root will be the required altitude, whieh being multiplied by the number of times that the base is to be greater than the altitude, will give the length of the required base.

PROBLEM 4. To lay out a given quantity of land in form of a triangle, having either the base or the perpendicular given.

Divide the area by half the given base, if the base be given; or by half the given perpendicular, if the perpendicular be given; and the quotient will be the perpendicular or base required.

PROBLEM 5. To lay out any given quantity of land in a regular polygon.

1. Find in the following table, the area of a polygon of the same name with that required, the side of which is 1. 2. Divide the proposed area by that found in the table. 3. Extract the square root of the quotient, and the root is the side of the polygon required.

For dividing of commons, &c. Surveyors generally measure the land of different value in separate parcels, and find the separate value thereof, which added in different sums, gives the whole content, and whole value.

By problem 1, they find each man's proportional share of the whole value, and then lay out for each person, a quantity of land equal in value to his share; this they effect by first laying out a quantity 23 by guess, and then casting it up and finding its value; and if such value be equal to his share of the whole value, the dividing line is right; if otherwise, they shift the dividing line a little, till by trial they find a quantity just equal in value to the value of the required share.

If any single share contains land of several different values, each is measured separately, and their several values found; and if the sum of them be equal to the value sought, the division is right; if not, it must be altered till it is so.

OF LAYING OUT ANY GIVEN QUANTITY OF LAND.

As the quantity of land is generally given in acres, roods and perches, it is necessary, first, to reduce them to square links, which may be performed by the following rule.

To reduce acres, roods, and perches into square links.

Rule 1. To the acres annex five cyphers on the right hand, and the whole will be links. 2. Place five cyphers to the right of the roods, and divide this by 4, the quotient will be links. 3. Place four cyphers on the right hand of the perches, divide this by 16, the quotient will be links. 4. These sums added together, give the sum of square links in the given quantity.

PROBLEM 1. To lay out a piece of land, containing any given number of acres, in form of a square.

This is no other than to determine the side of a square that shall contain any desired number of acres; reduce, therefore, the given number of acres to square links, and the square root thereof will be the side of the square required.

PROBLEM 2. To lay out any desired quantity of and in form of a parallelogram, having either its base or altitude given.

Divide the content, or area, by the given base, and the quotient is the altitude; if divided by the altitude, the quotient is the base; by the same rule rectangle may be laid out.

PROBLEM 3. To lay out any desired quantity of and in form of a parallelogram, whose base shall be 2,3,4, &c. times greater than its altitude.

Divide the area by the number of times the base is to be greater than the altitude, and extract the square root of the quotient; this square root will be the required altitude, whieh being multiplied by the number of times that the base is to be greater than the altitude, will give the length of the required

base.

PROBLEM 4. To lay out a given quantity of land in form of a triangle, having either the base or the perpendicular given.

Divide the area by half the given base, if the base be given; or by half the given perpendicular, if the perpendicular be given; and the quotient will be the perpendicular or base required.

PROBLEM 5. To lay out any given quantity of land in a regular polygon.

1. Find in the following table, the area of a polygon of the same name with that required, the side of which is 1. 2. Divide the proposed area by that found in the table. 3. Extract the square root of the quotient, and the root is the side of the polygon required.

J.

PROBLEM 6. To lay out any quantity of land in

a circle.

1. Divide the area by 7854. 2. Extract the square root of the quotient, for the diameter required.

OF PLAIN TRIGONOMETRY.

Plain trigonometry is the art of measuring and computing the sides of plain triangles, or of such whose sides are right lines.

As this work is not intended to teach the elements of the mathematics, it will be sufficient for me just to point out a few of the principles, and give the rules of plain trigonometry, for those cases that occur in surveying. In most of those cases, it is required to find lines or angles, whose actual admeasurement is difficult or impracticable; they are discovered by the relation they bear to other given lines or angles, a calculation being instituted for that purpose; and as the comparison of one right line with another right line, is more convenient and easy, than the comparison of a right line to a curve ;