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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=Ao, fg = op, hi=pq, &c. therefore de+fg+hi, &c. = Ao+op+pq, &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 trigonometry.

* The number of chains, taken down in the Field-Book, is longer than the lines Ao, op, pq, &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 fg or op shorter than the chain, and in like manner with all the rest. And de, fg, &c. Ao, op, &c. AC, consequently the number of chains being greater than de, ef, &c. or Ao, op, &c, is greater than AC, therefore the horizontal line AC (by survey. ors 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 ✅6a u2 ⇒ de, or half the right line, therefore 2h2 a2 =de, or the right line, from whence 25 262 a2 the excess for every chain which is measured or taken down in the Field-Book, calling the number of chains c, then c. 2b 2b2 a2 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 AB: 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,

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 because 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, furlongs, 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-pole chains and links to four-pole 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,

Ch. L.

1. In 16. 37 of two-pole chains, how many fourpole ones?

Ch. L.

Answer 8. 37.

But if the number of chains be odd, take the half of them for chains, and add 50 to the links, and they will be four-pole chains and links, thus,

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To reduce four-pole chains and links, to two-pole ones.

Double the chains, to which annex the links, if they be less than 50; but if they exceed 50, double the chains, add one to them, and take 50 from the links, and the remainder will be the links, thus, Ch. L.

1. In 8.

2.

37 of four-pole chains, how many two-pole ones?

16. 37

Ch. L.

2. In 8. 82 of four-pole chains, how many

2. 50 two-pole ones?

17. 32 Answer.

PROB. III.

To reduce four-pole chains and links, to perches and decimals of a perch.

The links of a four-pole chain are decimal parts of it, each link before the hundreth part of a chain; therefore if the chain and links be multiplied by 4, (for 4 perches are a chain) the product will be the perches and decimal parts of a perch. Thus,

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To reduce two-pole chains and links, to perches and decimals of a perch.

They may be reduced to four-pole ones (by prob. 1.) and thence to perches and decimals (by the last,) or,

If the links be multiplied by 4, carrying one to the chains, when the links are, or exceed 25; and the chains by 2, adding one, if occasion be; the product will be perches, and decimals of a perch. Thus,

Ch. L.

1. In 17. 21 of two-pole chains, how many 2. 4 perches.

Answer, 34. 84 perches.

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