Note. When it is required to part from a trapezium, approaching very nearly to a rectangle, any number of acres, &c. by a line parallel to one of its sides; it may be done by this Problem.

PROBLEM X.

To part from a trapezium or any irregular polygon whatever, any proposed quantity of land, by a line drawn parallel to any of the sides, or by a line drawn from any of the angles, or from any assigned point in one of the sides, to any of the opposite sides.

RULE.

Having surveyed and laid down the whole figure, draw a guess line in the direction required, parting off, as nearly as can be judged, the proposed quantity; after which, by the scale, measure with the greatest accuracy, the guess line, and also the quantity thus parted off. Then, if the guess line or line of division be drawn from an angle, or from any assigned point in a side, divide the difference between the proposed quantity and the quantity parted off, by half the guess line; and the quotient will be the perpendicular to be set off, on one side, or the other, of the guess line, accordingly as the quantity parted off is more or less than the quantity proposed. To the end of this perpendicular, from the point assigned, draw a new line of di vision; and it wipart off the quantity required.

But if the guess line be drawn parallel to any of the sides, divide the difference before mentioned, by the whole guess line, and the quotient will be the perpendicular to be set off from each end of the guess line, on one side, or the other, as above.

EXAMPLES.

1. From a trapezium, whose dimensions are contained in the following notes, part off 2a. 2r, 24p. by a line parallel to the side AB.

Having laid down the figure, draw the line guess mn parallel to AB; and from n, let fall the perpendicular an; then suppose mn = 1058 links, an will be ≈ 230, and Aa ≈ 1052; therefore Ba = 1114 – 1052 =62 links.

15220 the difference between the quantity proposed, and the quantity parted off by the guess line, which being divided by 1058, we obtain 14.4 links, to be set off perpendicularly from m and n towards D and C. Hence EF is the true line of division; and the trapezium ABEF contains 2a. 2r. 24p.

As A is very nearly a right angle, measure, in the field, 230 + 14.4244.4 links, from A to F; and upon any part of the line AB (towards B) as at e, erect the perpendicular er, which make 244.4 links; stake out the line ErF, and the work will be completed.

=

Note. In order to prove the operation, find the area of DCEF; then if it be equal to the difference between the area of ABCD and the quantity parted off, the work is undoubtedly right.

2. From an irregular field, whose dimensions are contained in the following notes, part off 2a. 3r. 20p. towards the line AE, by a fence made from the angle D to the side AB.

R off A.

Having laid down the figure, draw the guess line Dn, which suppose 766 links; then the diagonal AD will be 824, the perpendicular Ea = 278, and the perpendicular ar = 372 links; also re will be 228, and rn = 52 links.

Square links.

267800 the area of the trapezium Ar DE. 12000 the area of the offsets

16000

DE.

taken on the line.

12348

Ar.

The sum is 2a. 3r. 20p.

308148 the area of AnDEA.

287500

20648 the difference between the quantity proposed, and the quantity parted off by the guess line, which being divided by 383 (half the guess line), gives 54 links, to be set off from n towards A. Hence, DF is the true line of division; and the irregular figure AFDE contains 2a. 3r. 20p.

Measure there

Now, by the scale, Ac = 377 links. fore, in the field, 377 links from A to c; stake out the line DCF, and the work will be completed.

Note 1. The Rules given in this Problem, for parting off land from irregular fields, are generally adopted by Practical Surveyors; because they may be applied to any irregular figure whatever. Land, however, may sometimes be parted off more directly: for instance; the last example may be performed by finding the area of the irregular figure ADE, and subtracting it from the quantity to be parted off, then, if the difference be divided by half the line AD, the quotient will be the perpendicular of the triangle ADF, the side AB being nearly straight from A to F.

Now, at the distance of this perpendicular, draw a line parallel to AD; and it will intersect AB in F, the point to which the division fence must be inade. 2. It is not absolutely necessary to survey and plan the whole figure, in order to part a portion from it, as the guess-line and portion parted off may he measured in the field; but, in my opinion, the former, in general, is a more eligible method than the latter.

3. In order to divide a trapezium or an irregular polygon, among any number of persons, by fences made in a given direction, proceed thus: Part off the first person's share; then from the remainder of the figure, part off the second person's share; and thus continue, till the whole field be divided.

4. Those who desire to see a greater Variety of Examples in surveying single Fields, and to make themselves fully acquainted with the Methods of Laying out, parting off, and Dividing Land also of Dividing a Common, &c. of variable Value, among any Number of Proprietors, in the Proportion of their respective Interests, may consult my Treatise on Practical Land-Surveying, Third Edition, in which I flatter myself they will find these subjects satisfactorily treated.