Acd will be to the triangle A B C, as the square of A c is to square of A B.

Then, as the whole quantity is to the square of the side A B, so is the remaining quantity to the square of Ac; extract the square root, and the product will give the division required.

ЗАВС

1st. 10.002 x 1.433 = 143.300 ÷ 4.300 = 33.325

And √33.325 = 577 = A e

ЗАВС

2nd. 10.00 x 2.866 = 286.66000 ÷ 4.300 = 666514

And√666514816 = A c

Note. The same method must be repeated for any number of parts the triangle is required to be divided.

Fig. 8, Plate 27.

Problem 47.

Example 10. It is required to cut off a portion of land equal to two acres, in the direction shown by the dotted line A B.

Note. When a piece of land has to be cut off, and the boundary very crooked, and no fixed point to measure from, to fix the stakes of the true line of division, that portion must be accurately measured, and a chain line fixed as near to the division as can be guessed, as at AB, and at each end put in temporary stakes. The survey must then be plotted, and scaled to this assumed line, from which the correct line CD can be calculated, returning then to the field to make the amendment.

When calculated, it was found to be minus 1 rood 3 poles, or .268 decimals; therefore divide the decimals .26800 by the length of AB equal to 660 links, the quotient will be 40 links nearly.

Then set off 40 links at each end of the line AB, which will be the exact division required.

Example 11. It is required to cut off a portion of a field from a given point, A, to its opposite side, equal to 1 acre 1 rood.

Note.-Let this also be surveyed and the quantity calculated, as in the last example.

From the plan draw an assumed line as A B equal to 640 links; find the contents of the piece to the line A B equal to 1.887, or 1A. 3R. 22P., being minus the quantity required by .362 decimals, or OA. 1R. 18P.

Then divide 36200 by 640, the length of A B, the quotient will be 56 links, the width to be added if the amended line was parallel. As the increased quantity would form a triangle, this quotient must be doubled, or 112 links for the perpendicular to a, then will the line A a be the exact division required.

To set this out on the ground:

From the plan, scale the distance B a equal to 130 links, and at the point a put in a stake. Scale, 4 chains to the inch.

Example 12. A field containing 3A. OR. 6P. is to be divided into four equal portions, in such manner that access to the pond of water be given to each. By the Table, No. 13, 3 acres 6 poles = 3.03525 square links.

Thus: 3.035254.75881 to each share

No. 1. Set off the line A F as a fence; draw B F an assumed line; calculate the quantity A B F = .59040, which is less than the quantity required by .16841; divide this quantity by the length of B F = 380 equal to 44 links, being half the perpendicular; therefore set off 88 links, which will complete the first trapezium A B a F.

No. 2. Draw the guess line C F, and calculate the quantity of a CF equal to .49099, less than the quantity required by .26782; divide this by the length C F equal to 37871, the half; set off the perpendicular, 142 links, which completes the trapezium a Cb F.

No. 3. Draw the guess line EF; calculate the quantity A E F equal to .56121 less than the quantity required by .19760; which divide by E F = 350 links; the quotient 56 the half, or 112 the perpendicular to c, which completes the trapezium A E C F.

No. 4. The remaining trapezium is to be cast up according to the previous examples, and is found to be .75881, the quantity required.

Problem 50.

Example 13. To divide a common field amongst sundry claimants according to the value per acre of the various parts of the common, and in proportion to the sum of each proprietor's share.

Rule. Divide the yearly value of each person's estate by the value per acre; then, as the sum of all the quotients is to the quantity of the whole common, so is each particular quotient to the quantity of each particular share.

It is required to divide a common containing 500 acres amongst A B C D E, whose respective claims per annum are as follows:

Divide the yearly value of each person's estate by the value per acre, and find a factor, or common multiplier, by dividing the sum of the whole common by the sum of the quotient; then multiply each particular quotient by the common factor, will give the quantity to each share.

In dividing or allotting an extensive parish or common, the land will have various qualities or value, consequently it is impossible in all cases to put out allotments of one value; it becomes necessary on the part of the surveyor to calculate the quantities of each value; a line is therefore drawn on the plan to distinguish the changes.

An allotment will, in some cases, be made up in value by land of different qualities; the following system is then adopted:

Rule. Every portion of the common must be surveyed,

plotted, and computed, according to the several annual values that are set on them; they are then collected together to find the value of the whole common or parish.

Ascertain the annual values of each estate having rights of

common.

Then, as the annual value of all the estates is to the annual value of the whole common, so is each respective estate to its annual share of such common.

Or, as the price per acre of each portion of the common is to one acre, so is each annual share to the quantity due thereon. The quantity of each claimant being truly found, must be set out in such situations as may be considered beneficial to the property to which they belong.

Problem 52.

Fig. 11, Plate 27, represents the plan of a common containing 69 acres 25 poles, to be allotted in proportion to the value of the five claimants' estates.

The dotted lines on the plan divides the common into the several portions of value, which are numbered respectively, with their quantities and value per acre, in private characters,* viz. :

Annual value.

sh. dec. 393.148

Value per acre.

No. 1

28/

2

3

9.996

4

5

9.928

.

9.058

*When a parish or common has to be inclosed, commissioners are appointed under the act, whose duty it is to value the land. Letters of the alphabet to denote the value, to prevent confusion with other numbers, and also for secresy, are used; in this instance the word MAYFLOWERS, consisting of ten different letters, is adopted.

Note. The several quantities to make up the value of each share is entered on the plan, each of which must be accurately calculated and scaled, together making the sum required. A guess line must then be drawn as near to the division as can be, and adjusted in the manner shown by the first examples.

The rate to be raised to defray the expenses amounting to 1507.

Rule. First find the factor by reducing the amount into shillings; divide that product by the number of acres in the whole common; the quotient will be the factor; then multiply each person's allotment by the factor.