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(491) Its nature. This operation is precisely the reverse of those of Surveying properly so called. The latter measures certain lines as they are; the former marks them out in the ground where they are required to be, in order to satisfy certain conditions. The same instruments, however, are used as in Surveying.

Perpendiculars and parallels are the lines most often employed. The Perpendiculars may be set out either with the chain alone, Arts. (140) to (159); still more easily with the Cross-staff, Art. (104), or the Optical-square, Art. (107); and most precisely with a Transit or Theodolite, Arts. (402) to (406). Parallels may also be set out with the chain alone, Arts. (160) to (166); or with Transit, &c., Arts. (407) and (408). The ranging out of lines by rods is described in Arts. (169) and (178), and with an Angular instrument, in Arts. (376), (409) and (415).

(492) To lay out squares. Reduce the desired content to square chains, and extract its square root. This will be the length of the required side, which is to be set out by one of the methods indicated in the preceding article.

An Acre, laid out in the form of a square, is frequently desired by farmers. Its side must be made 316 links of a Gunter's

*The Demonstrations of the Problems in this part, when required, will be found in Appendix B.

chain; or 2080 feet; or 695 yards. It is often taken at

70 paces.

The number of plants, hills of corn, loads of manure, &c., which an acre will contain at any uniform distance apart, can be at once found by dividing 209 by this distance in feet, and multiplying the quotient by itself; or by dividing 43560 by the square of the distance in feet. Thus, at 3 feet apart, an acre would contain 4840 plants, &c.; at 10 feet apart, 436; at a rod apart, 160; and so on. If the distances apart be unequal, divide 43560 by the product of these distances in feet; thus, if the plants were in rows 6 feet apart, and the plants in the rows were 3 feet apart, 2420 of them would grow on one acre.

(493) To lay out rectangles. The content and length being given, both as measured by the same unit, divide the former by the latter, and the quotient will be the required breadth. Thus, 1 acre or 10 square chains, if 5 chains long, must be 2 chains wide.

The content being given and the length to be a certain number of times the breadth. Divide the content in square chains, &c., by the ratio of the length to the breadth, and the square root of the quotient will be the shorter side desired, whence the longer side is also known. Thus, let it be required to lay out 30 acres in the form of a rectangle 3 times as long as broad. 30 acres = 300 square chains. The desired rectangle will contain 3 squares, each of 100 sq. chs., having sides of 10 chs. The rectangle will therefore be 10 chs. wide and 30 long.

An Acre laid out in a rectangle twice as long as broad, will be 224 links by 448 links, nearly; or 147 feet by 295 feet; or 49} yards by 983 yards. 50 paces by 100 is often used as an approximation, easy to be remembered.

The content being given, and the difference between the length and breadth. Let e represent this content, and d this difference, Then the longer sided + v(d2 +4 c).

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Example. Let the content be 6.4 acres, and the difference 12 chains. Then the sides of the rectangle will be respectively 16 chains and 4 chains.

The content being given, and the sum of the length and breadth. Let c represent this content, and s this sum. Then the longer side+√(82 —4c).

Example. Let the content be 6.4 acres, and the sum 20 chains. The above formula gives the sides of the rectangle 16 chains and 4 chains as before.

(494) To lay out triangles. The content and the base being given, divide the former by half the latter to get the height. At any point of the base erect a perpendicular of the length thus obtained, and it will be the vertex of the required triangle.

The content being given and the base having to be m times the height, the height will equal the square root of the quotient obtained by dividing twice the given area by m.

The content being given and the triangle to be equilateral, take the square root of the content and multiply it by 1.520. The product will be the length of the side required. This rule makes the sides of an equilateral triangle containing one acre to be 4801 links. A quarter of an acre laid out in the same form would have each side 240 links long. An equilateral triangle is very easily set out on the ground, as directed in Art. (90), under "Platting," using a rope or chain for compasses.

(495) The content and base being given, and one side having to make a given angle, as B, with the base

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Fig. 342.


Example. Eighty acres are to be laid out in the form of a triangle, on a base, AB, of sixty chains, bearing N. 80° W. the bearing of the side BC being N. 70° E. Here the angle B is found from the Bearings (by Art. (243), reversing one of them) to be 30°. Hence BC=53.33. The figure is on a scale of 50 chains to 1 inch 1:39600.

Any right-line figure may be laid out by analogous methods.

(496) To lay out circles. Multiply the given content by 7, divide the product by 22, and take the square root of the quotient.

This will give the radius, with which the circle can be described on the ground with a rope or chain. A circle containing one acre has a radius of 178 links. A circle containing a quarter of an acre will have a radius of 89 links.

(497) Town lots. House lots in cities are usually laid off as rectangles of 25 feet front and 100 feet depth, variously combined in blocks. Part of New-York is laid out in blocks 200 feet by 800, each containing 64 lots, and separated by streets, 60 feet wide, running along their long sides, and avenues, 100 feet wide, on their short sides. The eight lots on each short side of the block, front on the avenues, and the remaining forty-eight lots front on the streets. Such a block covers almost precisely 3 acres, and 171⁄2 such lots about make an acre. But, allowing for the streets, land laid out into lots, 25 by 100, arranged as above, would contain only 11.9, or not quite 12 lots per acre.

Lots in small towns and villages are laid out of greater size and less uniformity. 50 feet by 100 is a frequent size for new villages, the blocks being 200 feet by 500, each therefore containing 20 lots.

(498) Land sold for taxes. A case occurring in the State of New-York will serve as an application of the modes of laying out squares and rectangles. Land

on which taxes are unpaid is B

sold at auction to the lowest bidder; i. e. to him who will accept the smallest portion of it in return for paying the taxes on the whole. The lot in question was originally the east half of the square lot ABCD, containing 500 acres. At a sale for taxes in 1830, 70 acres were bid off, and this area was

Fig. 343.



set off to the purchaser in a square lot, from the north-east corner. Required the side of the square in links. Again, in 1834, 29 acres more were thus sold, to be set off in a strip of equal width

around the square previously sold. Required the width of this strip. Once more, in 1839, 42 acres more were sold, to be set off around the preceding piece. Required the dimensions of this third portion. The answer can be proved by calculating if the dimensions of the remaining rectangle will give the content which it should have, viz. 250-(70 +29 +42)=109 Acres.

The figure is on a scale of 40 chains to 1 inch =1:31680.

(499) New countries. The operations of laying out land for the purposes of settlers, are required on a large scale in new countries, in combination with their survey. There is great difficulty in uniting the necessary precision, rapidity and cheapness. "Triangular Surveying" will ensure the first of these qualities, but is deficient in the last two, and leaves the laying out of lots to be subsequently executed. "Compass Surveying" possesses the last two qualities, but not the first. The United States system for surveying and laying out the Public Lands admirably combines an accurate determination of standard lines (Meridians and Parallels) with a cheap and rapid subdivision by compass. The subject is so important and extensive that it will be explained by itself in Part XII.



(500) It is often required to part off from a field, or from an indefinite space, a certain number of acres by a fence or other boundary line, which is also required to run in a particular direc tion, to start from a certain point, or to fulfil some other condition. The various cases most likely to occur will be here arranged according to these conditions. Both graphical and numerical methods will generally be given.*

The given lines will be represented by fine full lines; the lines of construction by broken lines, and the lines of the result by heavy full lines.

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