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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 infini!y of irregular surfaces which offer theinselves 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 inapped.
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 wil 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 chams, or 320 perches, make a mile as beiore: and if roads
are measured by a statute chain, it will give you the miles Fnglish, but if by a plantation chain, the miles will be Irish. Hence an English mile con tains 1760, and an Irish mile 2940 yards; and because 14 half yards is an Irish, and 11 half yards is at English perch, therefore 11 Trish 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.
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,
L. 1. In 16. 37 of two-pole chains, how many four-pole ones?
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, thuss
Ch. L. 2. In 17. 42 of two-pole chains, how many four-pole ones ?
To reduce four-pole chains and links, to two-pole ones.
Double the chains, to which annex the links, if y be less than 50 ; but if they exceed 50, douthe chains, add one to them, and take 50 from links, and the remainder will be the links, thus,
Ch. 2. In 8.
L. 82 of four-pole chains, how many 50 two-pole ones?
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 oi a chain; therefore if he chain and links be multipled by 4, (for 4 perches are a chain) the product will be the perches and decimal parts of a perch. Thus,
Ch. How many perches in 13. chains,
64 of four-pole 4
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. 1. In 17.
4 perches ?
To reduce perches, and decimals of a perch, co four-pole chains
Divide by 4, 80 as to have two decimal places in the quotient, and that will be four-pole chains and links. Thus,
In 31. 52 perches, how many four-pole chains and links?
To reduce perches and decimals of a perch, to two-pole chains
The perches may be reduced to four-pole chains (by the last) and from thence to two-pole chains (by prob. 2.) or,
Divide the whole number by 2, the quotient will be chains; to the remainder annex the given decimals, and divide by 4, the last quotient will be the links. Thus,
In 31.52 perches, how many two-pole chains and }inks?