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no such person, should chain himself. The inaccuracies of most surveys arise from bad chaining, that is, from straying out of the right line, as well as from other omissions of the hinder chainman : no person, therefore, should be admitted at the hinder end of the chain, of whose abilities in this respect, the surveyor was not previously convinced; since the success of the survey, in a great measure, depends on his care and skill.

In setting out to measure any stationary distance, the foreman of the chain carries with him 10 iron pegs pointed, each about ten inches long ; and when he has stretched the chain to its full length, he at the extremity thereof sticks one of those pegs perpendicularly in the ground; and leaving it there, he draws on the chain till the hinder man checks him when he arrives at that peg: the chain being again stretched, the fore man sticks down another peg, and the hind man takes up the former; and thus they proceed at every chain's length contained in the line to be measured, counting the surplus links contained between the last peg, and the object at the termination of the line, as before: so that, the number of pegs taken up by the hinder chainman, expresses the number of chains; to which, if the odd links be annexed, the distance line required in chains and links is obtained, which must be registered in the field-book, as will hereafter be shown. T

If the distance exceeds 10, 20, 30, &c. chains, when the leader's pegs are all exhausted, the hinder chainman, at the extremity of the 10 chains, delivers him all the pegs; from whence they proceed to measure as before, till the leader's pegs are again exhausted, and the hinder chainman at the extremity of these 10 chains again delivers him the pegs, from whence they proceed to measure the whole distance line in the like manner; then it is plain, that the number of pegs the hinder chainman has, being added to 10, if he had delivered all the pegs once to the leader, or to 20 if twice, or to 30 if thrice, &c. will give the number of chains in that distance; to which if the surplus links be added, the length of the stationary distance is known in chains and links.

It is customary, and indeed necessary, to have red, or other coloured cloth, fixed to the top of each peg, that the hinder man at the chain may the more readily find them; otherwise, in chaining through corn, high grass, briars, rushes, &c. it would be extremely difficult to find the pegs which the leader puts down: by this means no time is lost, which otherwise must be, if no cloths are fixed to the pegs, as before.

It will be necessary here to observe, that all slant, or inclined surfaces, as sides of hills, are measured horizontally, and not on the plane or surface of the hill, and is thus effected.

PL. 8. fig. 4.

Let ABC be a hill, the hindmost chainman is to

hold the end of the chain perpendicularly over the point A (which he can the bettereffect with a plummet and line, than by letting a stone drop, which is most usual) as dis over A, while the leader puts down his peg at e : the eye can direct the horizontal position near enough, but if greater accuracy were required, a quadrant applied to the chain, would settle that. In the same manner the rest may be chained up and down; but in going down, it is plain the leader of the chain must hold up the

end thereof, and the plummet thence suspended, will mark the point where he is to stick his peg. The figure is sufficienttorender the whole evident; and to show that the sum of the chains will be the horizontal measure of the base of the hill; for de=40, fg=op, hi-pg, &c. therefore de-Hsg+hi, &c. = 40+op-Hpq, &c. =.AC,” the base of the hill. If a whole chain cannot be carried horizontally, half a chain, or less, may, and the sum of these half chains, or links, will give the base, as before.

If the inclined side of the hill be the plane surface, the angle of the hill's inclination may be taken, and the slant height may be measured on the surface; and thence (by case 1. of right angled trigonometry) the horizontal line answering to the top, may be found; and if we have the angle of inclination given on the other side, with those already given; we can find the horizontal distance across the hill, by case 2 of oblique tri

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* The number of chains, taken down in the Field-Book, is longer than the lines Mo, op, pg, &c. because the chain being elevated above the surface of the earth, (though stretched with a force at both ends,) forms a curve, which approaches a right line, according as the force is more or less applied; but does not coincide with it: as for example. Let the chain be stretched from d to e (Pl. 8. Fig. 4.), it does not coincide with de, but forms a curve line, which must be longer than de or its equal Ao, and so is so or op shorter than the chain, and in like manner with all the rest. And de, fg, &c. =.Mo, op, &c. ==-4C, consequently the number of chains being greater than de, ef, &c. or -ào, op, &c. is greater than MC, therefore the horizontal line AC (by surveyors in general) is made too long, therefore a deduction must be made for every chain in the Field-Book, the sum to be taken from AC, may be found by making an experiment on a two-pole chain (when extended above the surface of the earth by a force at both extremities), and measuring the distance from its middle point to the middle of the right line which would join its extremities,

which call a, and call the length of the chain b, then vös as -* de, or half the right line, therefore 2 Vä2 a.2 =de, or the right line, from whence 202 V82 age= the excess for every chain which is measured or taken down in

the Field-Book, calling the number of chains c, then c. 2b - 2 vb2 a” = the whole excess on the horizontal line.AC; from what is here demonstrated, the Practitioner will be able to find the sum to be taken from every horizontal line in surveying hills, &c.

All inclined surfaces are considered as horizontal ones; for all trees which grow upon any inclined surface, do not grow perpendicular thereto, but to the plane of the horizon: thus if Ad, ef, gh, &c. were trees on the side of a hill, they grow perpendicular to the horizontal base AC, and not to the surface A B: hence the base will be capable to contain as many trees as are on the surface of the hill, which is manifest from the continuation of them thereto. And this is the reason that the area of the base of a hill, is considered to be equal in value to the hill itself. - * *

Besides, the irregularities of the surfaces of hills in general are such, that they would be found impossible to be determined by the most able mathematicians Certain regular curve surfaces have been investigated with no small pains, by the most eminent; therefore an attempt to determine in general the infinity of irregular surfaces which offer themselves to our view, to any degree of certainty, would be idle and ridiculous, and for this reason also, the horizontal area is only attempted.

Again, if the circumjacent lands of a hill be planned or mapped, it is evident we shall have a plan of the hill's base in the middle: but were it possible to put the hill's surface in lieu thereof, it would extend itself into the circumjacent lands, and render the whole an heap of confusion: so that if the surfaces of hills could be determined, no more than the base could be mapped.

Roads are usually measured by a wheel for that purpose, called the Perambulator, to which there is fixed a machine, at the end whereof there is a spring, which is struck by a peg in the wheel,

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once in every rotation; by this means the number

of rotations is known; if such a wheel were 3

feet 4 inches in diameter, one rotation would be

10; feet, which is half a plantation perch; and be- .

cause 320 perches make a mile, therefore 640 rotations will be a mile also ; and the machinery is so contrived, that by means of a hand, which is carried round by the work, it points out the miles, quarters, and perches, or sometimes the miles, fur

longs, and perches.

Or roads may be measured by a chain more accurately; for 80 four-pole, 160 two-pole chains, or 320 perches, make a mile as before : and if roads are measured by a statute-chain, it will give you the miles English, but if by a plantation chain, the miles will be Irish. Hence an English mile contains 1760, and an Irish mile 2240 yards; and because 14 half yards is an Irish, and 11 half yards is an English perch, therefore 11 Irish perches, or Irish miles, are equal to 14 English OneS.

Since some surveys are taken by a four-pole,

and others by a two-pole chain; and as ground

for houses is measured by feet, we will show how to reduce one to the other; in the following problems.

PROB. I.

To reduce two-hole chains and links to four-hole ones.

If the number of chains be even, the half of

them will be the four-pole ones, to which annex

the given links, thus,

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