If the inclined side of the hill be a 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.

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

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 a 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, 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 shew 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 four-pole ones, to which annex the links given, thus,

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,

Ch. L.

2. In 17. 42. of two-pole chains, how many four

Ch. L.

Answer 8. 92.

'PROB. II.

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 1 to them, and take 50 from the links, and the remainder will be the links, thus,

Ch. L.

1. In 8. 37. of four-pole chains, how many twó pole ones?

2

16 37

Ch. L.

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

2 50

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 being the hundredth part of a chain;

therefore if the chains 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,

Ch. L

How many perches in 13. 64 of four-pole chains?

4

Answer 54. 56 perches.

PROB. IV.

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.

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

2.

Answer 34. 84 perches.

Ch. L.

2. In 15. 38. of two-pole chains, how many 2. 4 perches?