C

little by 5776 square links. This divided by 440 (half the guessline) gives 13 links, to be set off from m toward D; consequently, EF is the true line of division.

Again, draw the guess-line G n, which suppose = 878 links; then will the diagonal G F = 1017, the perpendicular E a = 430, and the perpendicular na = 385: hence, the area of the trapezium EF n G, is found =414427 square links, which is too little by 3203 square links. This divided by 439 (half the guess-line) gives 19 links, to be set off from n toward D; consequently, G H is the true line of division; and the trapezium A B C D is divided into three equal parts, as required.

=

Now, by the scale, we find B E = 450, E G 500, C F = 508, and FH 468 links, which distances must be measured in the field, in order to determine the situations of the division-fences.

NOTE. If we subtract the area of the triangle B C E, from the quantity to which each person is entitled, and divide the remainder by half the line C E, the quotient will be the perpendicular of the triangle C E F. By drawing a line parallel to C E, at the distance of this perpendicular, the point F may be determined. In a similar manner, may be parted off the trapezium E F H G.

2. Divide a field, whose dimensions are contained in the following notes, among three persons, A, B, and C, so that each partaking of a pond at P, A may have 3A., B 4A., and C the remainder.

Α

Begin at A. Range W

=

1025,

From the pond P, draw the line P D, and also the guess-line P m, which suppose 558 links; then will the diagonal D m be the perpendicular P a = 400, and C a =195 links; hence, the area of the trapezium Pm C D is found 304937 square links, which exceeds A's share by 4937 square links. This divided by 279 (half the guess-line) gives 17.7 links, to be set off from m toward C; consequently, P F is the true line of division, and the trapezium PF CD contains A's share.

Again, draw the guess-line P n, which suppose = 696 links; then, the diagonal PE will be = 848, the perpendicular n a = 247, and Da = 552 links: hence, the area of the trapezium P n E D, is found = 338776 square links, which is less than B's share, by 61224 square links. This divided by 348 (half the guess-line) gives 176 links, to be set off from the line P n perpendicularly toward A; consequently, P G is the true line of division, and the trapezium P GED

contains B's share.

Now, the irregular polygon A B F P G contains C's share, which will be found 4A. 3R. 19P.

By the scale, we find A G = 252, and B F 545 links, which distances must be measured in the field, in order to determine the situations of the division-fences.

NOTE 1. The foregoing example may also be performed by subtracting the area of the triangle C D P from A's share; and then laying out the remainder in the triangle C F P, as before directed.

In a similar manner, may be parted off B's share.

2. The division of the last, or any other figure, may be proved by finding the area of the whole figure, which, if equal, or nearly equal to the sum of the areas of the parts into which it has been divided, demonstrates the work to be right.

PROBLEM VI.

To divide a Common, or any Quantity of Land, of uniform Value, among any number of Proprietors, in the proportion of their respective interests.

In this case, the land to be divided must first be surveyed; and next, the estate of each proprietor, if its quantity be unknown. Then, if it be required to make the division according to the value of each person's estate, there must be proper persons appointed to value them, which, in this Problem, we will suppose, may be done at so much per acre, uniformly, throughout each estate.

NOTE 1. When the land to be divided is of uniform value, nothing more is wanted than its quantity.

2. It is immaterial whether the land be valued at 5s. or 51. per acre, if the same proportion, according to the quality of the land, &c. be observed in valuing each person's estate.

To determine each Person's Share.

RULE. As the number of acres, &c. contained in the sum of the estates, is to the whole quantity of land to be divided, so is each person's estate to his respective share. Or, as the sum of the values of all the estates, is to the whole quantity of land to be divided, so is the value of each person's estate to his respective share.

EXAMPLES.

1. Divide a common containing 56A. 2R. 16P. among three persons, A, B, and C, whose estates are 58, 96, and 128A. respectively.

Here 58 +96+128= 282, the number of acres contained in all the estates; and 56A. 2R. 16P. = 5660000 square links. Then,

Each person's share thus determined, the common may easily be divided by the methods already described.

2. Three gentlemen, A, B, and C, have each an estate consisting of 300A.; divide among them, according to the values of their estates, 75A. 3R. 32P.; A's estate being valued at 25s., B's at 32s., and C's at 40s. per acre, per annum.

And 75A. 3R. 32p. =7595000 square links. Then,

NOTE. It sometimes happens that two, three, or more persons join in taking a common pasture, and agree to pay in proportion to the number of cattle, with

share.

which each person depastures. In such cases, when the whole of the cattle graze an equal time, you must make use of the rule of Single Fellowship, by saying, as the whole of the cattle, is to the rent of the whole pasture, so is each person's cattle to his share of the rent. But when the cattle graze an unequal time, you must then have recourse to the rule of Double Fellowship, by saying, as the sum of the products of each person's cattle and time, is to the whole rent, so is each person's product to his share of the rent.

PROBLEM VII.

To divide a Common, &c. of variable Value, among any number of Proprietors, in the Proportion to their respective Interests.

In a work of this kind, the quantity of every different quality is required, not only of the land to be divided, but also of each proprietor's estate; consequently, the Surveyor, accompanied by the persons appointed to value, generally called "Commissioners," must examine each person's estate, and also the Common, previously to the survey being taken.

In doing this, they must stake out lines between the different qualities of the soil; and, in surveying, these lines (called by Surveyors, "Quality-lines ") must be considered as boundaries, and represented in the field-book, and upon the plan, by small dashes.

By way of distinction, there ought to be two stakes put down at each angle formed by the quality-lines; and also marks cut in the ground, pointing in the direction of these lines, so that if the stakes should be pulled up, these marks may serve as directors.

When the survey is finished and laid down, every different quality, represented upon the plan, must be successively numbered, 1, 2, 3, &c. The Surveyor must then require the Commissioners to put the different valuations upon the land; and, in doing this, he must accompany them with the plan, in order that both he and they may know the ground corresponding with each number. Surveyors generally use letters to represent the different values of land :