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Chufe two ftations C and D, from each of which, the places PART I. A and B may be feen; and let the angles ACB, BCD, and BDA, ADC be taken with the theodolite, and measure the distance CD, or find it by fome of the preceding problems. Then, in the triangle ACD, are given the angles, and the fide CD, to find AD, therefore fin. CAD fin. ACD:: CD: DA. In like manner, in the triangle BCD, all the angles are given, and the fide CD, to find DB; therefore fin. CBD : fin. BCD :: CD : DB. Then in the triangle ADB, are given AD and DB, and the angle ADB; therefore, by Prop. 4. Pl. Trig. the fum of AD, DB is to their difference, as the tangent of the fum of DAB, DBA to the tangent of their difference; whence these angles may be found, by Lem. 4. of Pl. Trig. and then fin. ABD: fin. ADB:: DA: AB, the distance to be found.

be

Note, It is not neceffary that the points A, B, C, and D, in one plane, but if they be not, the angles ACD, CDB must be taken with the theodolite.

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'O measure a height AF placed on a fleep, fo that Pl. II. one can neither go near it, on a horizontal plane, Fig. 26. nor recede from it.

Chufe any two ftations C and D, and find the angles ACD, and ADC, by the theodolite, and measure the distance CD: also with the quadrant, find the angles ACB and FCB. Then in the triangle ACD, the angles are known, and the fide CD; therefore fin. CAD : fin. ADČ : : DC: CA: and because ACB is known, its complement CAB is known; and FCB is known, therefore ACF is known: Wherefore, all the angles of the triangle ACF are given, and the fide CA, to find AF; and fin. AFC: fin. ACF :: CA: AF, the height required.

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O find how much one place is more elevated than Pl. II.
Fig. 27.

Tanother.

Here a level and poles are ufed. Let the poles ab, cd, ef be erected perpendicular to the horizon. Then let the level be placed horizontally between the poles ab, and cd, and mark the points b and c of the poles, which are feen in a straight line through the fights of the level. Place in the fame manner the level between the poles cd, and ef, and mark the points d and e upon

Ss 2

PART I.

upon them; and fo on, until you arrive at the top A. Then measure ab, cd, ef, and add them together, and the sum is the elevation of A above the place a; that is, it is equal to the height AB. Likewife the distances bc, de, fA, added together, give the horizontal diftance of A from a; that is, they give

aB.

COR. Hence the angle of elevation BaA, and the hypotenuse aA may be found by Trigonometry.

Note 1. Sometimes the angles of elevation are taken from ftation to ftation, and the distances measured on the ground, by which means, between each station there is formed a triangle, of which the hypotenuse and the angle at the bafe are given, to find the bafe and perpendicular by Trigonometry.

Note 2. The level ought to be very accurate, and the fights fitted with a telescope: and if the distance be great, an allowance must be made for the rotundity of the earth, at the rate of 7.96 inches for every mile measured on the earth; that is, the level at a mile's distance is 7.96 inches lower than what is found by the level.

Note 3. It is not neceffary to go in a straight line from one of the places to the other, but every two fucceffive flations may be taken in the moft convenient direction.

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Pl. II.

Fig. 28.

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O find the diameter of the earth, from one obfervation,

Let a high hill AB be chosen near the fea; and let the height of it be found as exactly as poffible, by fome of the foregoing methods. Then with a very exact quadrant, that can take an angle to feconds, find the angle ABE contained by the perpendicular AB, and the visual line BE, which touches the earth at E: and let AF be perpendicular to AB. Then, in the triangle ABF, right angled at A, are given AB, and the angle ABF, to find AF and FB; and R: tan. B:: BA: AF, and R : fec. B: : a Cor. to AB: BF; therefore AF and FB are found: and AF is equal 37. 3. E. to FE, therefore BE is known. But the fquare of BE is equal b b 36. 3. to the rectangle AB, BD; and AB is given, therefore BD can be found; from which, fubtracting BA, the remainder AD is the diameter of the earth.

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Note, There are many other methods of measuring the diameter of the earth, but they do not belong to this place: the latest attempts make the mean diameter to be 7913 English miles, though it was formerly determined, to be about 7970 miles.

OF

OF SURVEYING LAND.

IN furveying land, we use the chain for measuring lines; and PARTI.

in using it, there are 10 arrows, or long pins; and when the chain is ftretched, the follower directs the leader into the ftraight line which is to be meafured, and there the leader fixes an arrow in the ground, and goes forward, and when the follower comes to the arrow, he again directs the leader into the ftraight line to be measured, who then fixes another arrow in the ground, and proceeds, by which means they are enabled to measure any diftance, without danger of mistaking the number of chains. There is also an offset-ftaff, 10 links long, and divided into 10 equal parts, for meafuring the diftances of the hedges, and other things from the ftraight line measured by the chain; but care must be taken, to enter thefe fmall diftances in the field-book, with fuch remarks as fhall readily point out their fituation, and diftinguish them from one another.

The angles are to be taken with the theodolite; and likewife the elevation of the lines measured, when they are not parallel to the horizon, and then they ought to be reduced to horizontal lines, before they be entered in the field-bock; or else their elevations ought to be fo entered, that they may be easily known, and reduced afterwards, by the following

TABLE

For reducing Inclined Lines to Horizontal ones.

Deg. oLin Deg. ofLi Deg, ofLin.Deg. oi Lin. Deg. Lin.
elev. Sub elev.Sub. elev. Sub

elev. Snb

elev. Sub

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Suppofe the inclined line really to measure 1107 links, and the angle of elevation to be 19°. 95. Then looking in the Table against 19o. 95, I find 6 links, which multiplied by 11, is 66; and this fubtracted from 1107, leaves 1041, the true horizontal length to be laid down in the plan.

After

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