which distance being measured, the straight fence Wb may now be set out, and the other shares to the right of d Wb may be next proceeded with, according to the method given in this or the following cases.

THE ABOVE EXAMPLE IN NUMBERS.

WS = a = 460, SU = b = 400, U F = c = 420 links; the values are m = £30, n = £40, and o= £60 per acre; and the value V' £80: required the distance Fb or x by the preceding formula.

Here sa+b+c=460 + 400+ 420 1280 links, whence

2 X 1280 X 100000 X 80

4602 X 30+(920+400) × 400 × 40+ (920+800 +420) X 420 X 60 20480000-000

81396.000

= 251.6 links

F b,

which determines the position of the fence W b.

NOTE. This method possesses the advantage of being practicable on the ground without the help of either a map or guess-line; however, in the case of the division of commons a map is always necessary for the satisfaction of the several parties interested therein.-The investigation of formula here given, is founded on similar triangles combined with the same principle as that in Case I. If there be less than three different qualities of land to be laid out, the symbols referring to the additional qualities, must be made to vanish in the formula; and, if there be more than three qualities to be laid out the law for the extension of the formula is obvious.

CASE III. To set out from a field or common of any form, and of variable value, a quantity of land of given value by a straight fence in a given direction.

N

a

EXAMPLE.

Let CABD be a portion of a common, of variable value, from which it is required to set out a quantity of land of a given value V, by a straight fence ab parallel to B N, the quality lines PQ, RS being so adjusted as to be perpendicular to BN at p and r respectively, as per Note 1, Case I., and the value of the land between A C, P Q being m, between PQ, RS = n, and between RS, BD = 0.

R

B

First find the value of the land in the triangle A N B, and if the fence AB had been crooked, the offsets would have to be included; the value of the land to the left of BN being supposed to be less than

=

V',

is required to make up the given value V. Let the value of the land in the triangle AN B: v, then the value of the land still remaining to be set out will be V - v, which put let Np = a, pr= b, rB = c, p q = r s = x, cot. of the angle CNB: 2a, and cot. of the angle DBN: = 28, the symbols denoting the different values of the land being already given in the example, and 7 : =square links in an acre;

=

21 V'

=pqorrs;

am+b n+co√ (am+bn+co)2 — 4 (am+Bo)¡V' which distance may be set out at any two points perpendicular to NB; and, if the land in question be a common, requiring the shares of several claimants to be set out, the quality lines may be again adjusted to the right of a b, and the next share may then be set out as before.

NOTE 1. The investigation of the general formula, just given, is the same in principle as those in the preceding cases, being only a little more complex, on account of the land to be set out not being rectangular, and therefore involving the solution of a quadratic equation, thus giving an apparent complexity to the result which will doubtless startle the generality of surveyors, who are accustomed to use "guess lines" when even the most trifling difficulty occurs, thus making repeated approximations to obtain the position of the correct divisional fences, which may be obtained at once by these methods, with even less calculation than by guess lines, without naming the repeated measurements accompanying these bungling operations. It would, therefore, appear that neither the surveyors, nor the writers on this department of surveying, have had the least acquaintance with Mathematics, otherwise the results here given, which are comparatively void of extreme complexity, would have been known long ago.

NOTE 2. When one or both of the angles CNB, DBN are obtuse, their cotangents a and B will be one or both negative respectively; and when these angles are right ones, their cotangents vanish; and the formula for the perperdicular breadth p q or rs becomes.

IV'
am+bn+co

= pq = rs.

which is the same as the formula given in Case I., as it obviously ought to be. CASE IV. When the boundary of the land to be laid out is very irregular, also when part of the divisional fence is predetermined, either for the purpose of drainage or irrigation, and when the quality lines run irregularly.

EXAMPLE.

Let C ABD be the boundary of a portion of common, from which it is required to lay out a portion of land of variable value, part of the divisional fence of which it is desirable to have in the direction of a watercourse abcd, and the remainder of the divisional fence to run from d to f

A

P

Ъ

Join df, and find the value of the land included thereby and the other boundaries in question, and, this value being found less than the required value, the remaining land still required may be determined by assuming two lines, so as to form a triangle on df, as a base, till the correct position of the remaining part def of the divisional fence shall be ascertained, it being advisable to let the angle e, in the required fence, fall on the quality line PQ, for the purpose of more readily calculating the areas of the two triangles formed thereby, and from thence finding their values.

B

D

NOTE. It will here be proper to notice that surveyors who are unaccustomed to use mathematical formula, will more readily and accurately make the divisions of commons, &c., by means of assumed or guess lines, than by the methods I have given in the three first Cases of this Problem. Their not having had any practice beyond finding the areas of triangles, &c., gives them little confidence in more scientific methods, which they therefore view as chimeras, or at best as methods very difficult to be understood. It is to be hoped that the very general diffusion of science at the present day will root out this ignorant prejudice.

PROBLEM VI.

CASE I. To divide a common of uniform value among any number of proprietors, in proportion to the values of their respective estates.

In this case, the quantities and values per acre of each proprietor's estate must be determined, by survey if necessary, and the map, with the quantity of the common, must also be prepared; the value of the common, being uniform, is not required.

TO FIND EACH PROPRIETOR'S SHARE.

RULE.-As the sum of the values of each person's estate: the whole quantity of common to be divided :: the value of each person's estate to that person's share.

EXAMPLE.

Divide a common of 100 acres among three persons, A, B, and C; A's estate is 120 acres at £4 per acre, B's is 180 acres at £5 per acre, and C's 260 acres at £2 per acre.

The quantity of each person's share being now determined, the common may be divided by one or other of the methods already given.

CASE II.-To divide a common of variable value among any number of claimants, in proportion to the values of their respec

tive estates.

In this case, the map of the common with the quantity of every different quality of land thereon, must be prepared, as well as the quantities and qualities of the several parts of each claimant's estate, in order that the values of both the common and each person's estate may be properly ascertained; the quality lines on the common being marked out by the value or valuers, previous to the survey being made.

When the survey of the common is completed and mapped, with the quality lines laid down thereon, the several qualities must be number 1, 2, 3, &c., with the values of the land corresponding to each number, as determined by the valuers, who are usually called "Commissioners." Most surveyors use the letters of the alphabet to denote the different values of the land: thus a may represent five shillings, b six shillings, &c., per acre per annum. By using the letters in this manner a multiplicity of figures, which may be confounded with other numbers, is avoided.

ROADS, QUARRIES, WATERING PLACES, &c., REQUIRED TO BE SET OUT PREVIOUS TO ENCLOSING COMMONS, WASTES, &c.

Before the lands of a common or waste can be divided and allotted, both public and occupation roads must be set out in the most convenient manner; they should be straight and, as far as practicable, at right angles to one another, as this arrangement not only facilitates the division of the land, but contributes greatly to the economy of cultivation with the plough. All old roads that may be deemed unnecessary may be stopped up and

allotted to the different claimants, or diverted into more convenient directions, at the discretion of the Commissioners.

Portions of the common are now to be set apart for quarries, sand or gravel pits, if such abound in the common. The ground, thus set out, is considered as the common property of the several claimants, for the purpose of building, making roads, &c. Also, if there are any good springs or ponds on the common, they must be left unenclosed, in like manner, for common use; or the water must be conveyed from them by drains or channels to more convenient situations, previous to the enclosure of the common.

The lord of the manor in some places claims th of the common, in someth, &c. His claim, whatever it be, must be next set out, after its value has been determined from the whole value of the common. The lord of the manor will also be further entitled to his share of common, in proportion to his property, in the same manner as the other proprietors.

Lastly, when the roads, watering places, quarries, sand and gravel pits, and manorial rights have been set out, the remainder of the common must be divided equitably, as it respects quantity, quality, and situation, among the proprietors of lands, tenements, houses, cottages, &c., situated in the parish or township where the inclosure is to be made.

THE METHOD OF DIVIDING AND ALLOTTING THE REMAINING PART OF THE COMMON.

Having found the sum of the values of each proprietor's estate, and the whole value of the remaining part of the common to be divided, proceed to find the value of each proprietor's share by the following

RULE. As the sum of the value of each proprietor's estate, is to the whole value of the common remaining to be divided, so is the value of each proprietor's estate to his share of the value of the common.

It will be quite unnecessary to give an example in this case, as the laying out of the shares of the several claimants, after their respective values have been found by the above Rule, would only be a repetition of the methods already given, on a large scale; and after the work of laying out the several inclosures on the ground has been completed, the last enclosure or share of the common must be of the same value as that assigned by the rule, or so very near to it that the error is of no importance; otherwise a mistake has been made which must be immediately rectified.