These two methods, when accurately performed, are both very juft, at least when the ftations are at no great distance from each other; yet the former is more to be depended on than the latter, in as much as the observations required by it, are less fufceptible of error; for befides the inaccuracy of the measured hypotenufes, on account of the unevennefs of the ground, the angles themselves can hardly be taken, with the generality of inftruments for this purpofe, to lefs than quarters of degrees.

But when the places, of which we would know whether of them is the higher, are far diftant from each other, instruments of greater accuracy are to be ufed, fuch as very exact levels with telescopes; but in fuch cafes an allowance must be made for the rotundity of the earth; for the true water-level course is determined by a line of which every part is at the fame distance from the center of the earth, and which is, therefore, an arc of a great circle of the earth, confidering it not as an oblate fpheroid, but as a sphere, its difference from that form having, in this cafe, no fenfible effect; and the allowance neceffary to be made in the level, is at the rate of 8 inches nearly to a mile measured upon the earth; for the visual line, when the level is properly fixed, being a tangent to the earth, or at least parallel to one, that line at the distance of one mile from the station is 7.96 inches above the water-level line, or fo much farther from the center of the earth than the inftrument is. And for other distances, above or below a mile, the allowance for the level varies in proportion as the fquare of the distance.

It may be neceffary farther to obferve, that in taking, at feveral ftations, the difference of level between two places, it is not neceffary to go in a line from the one to the other, but every two fucceffive ftations may be taken in any direction that seem moft convenient.

G 3

may

PRAC

PRACTICAL QUESTIONS IN TRIGONOMETRY,

&c.

QUESTION I. A may-pole, 50 feet 11 inches high, at a certain time will caft a fhadow 98 feet 6 inches long; what then is the breadth of a river, which, running within 20 feet 6 inches of the foot of a fteeple, 300 feet 8 inches high, will, at the fame time, throw the extremity of its fhadow 30 feet 9 inches beyond the stream? Anf. 530 feet 5 inches.

3

QUEST. 2. Required the length of a fhoar, which being to ftrut 11 feet from the upright of a building, will fupport a jamb 23 feet 10 inches from the ground. Anf. 26 ft 3 inc. QUEST. 3. A line 27 yards long will exactly reach from the top of a fort, to the oppofite bank of a river, known to be 23 yards broad: what is the height of the wall? Anf. 42 ft 5 inc. QUEST. 4. Two fhips fet fail from the fame port, one of them goes 50 leagues due caft, and the other 84 leagues due north: how far are they then asunder? Anf. 97 leagues.

QUEST. 5. The height of an elm, growing in the center of a circular ifland, 30 feet in diameter, plumbs 53 feet; and a line of 112 feet long ftretched from the top of the tree ftraight to the nearer edge of the water required the breadth of the moat, fuppofing the land on either fide of the water, to be level. Anf. 83 feet.

QUEST. 6. Suppofe a light-house, built on the top of a rock, the distance between the place of obfervation, and that part of the rock level with the eye, and directly under the building, is given 310 fathoms; the distance from the top of the rock, to the place of obfervation, is 423 fathoms; and from the top of the building 425 required the height of the edifice. Anf. 17 ft 7 inc.

QUEST,

QUEST. 7. A ladder, 40 feet long, may be fo planted, that it fhall reach a window 33 feet from the ground, on one fide of the street; and by turning it over, without moving the foot out of its place, it will do the fame by a window 21 feet high, on the other fide required the breadth of the street.

Anf. 56.649 feet. QUEST. 8. There are two columns left ftanding upright in the ruins of Persepolis; the one is 64 feet above the plane, and the other 50: In a right line between these ftands an antient ftatue, the head of which is 97 feet from the fummit of the higher, and 86 from that of the lower column; the bafe of which measures juft 76 feet to the center of the figure's bafe required the diftance between the tops of the two columns. Anf. 157 feet.

QUEST. 9. A may-pole, whofe top was broken off by a blast of wind, ftruck the ground at 15 feet distance from the foot of the pole: what was the height of the whole may-pole, fuppofing the length of the broken piece to be 39 feet? Anf. 75 feet.

QUEST. 10. Suppofe the breadth of a well at the top be 6 fect, and the angle formed by its fide and a visual diagonal line from the edge at top to the oppofite fide at the bottom, 18° 30': required the depth of the well. Anf. 17.89 feet.

QUEST. II. At 85 feet distance from the bottom of a tower, the angle of its elevation was found to be 52° 30' required the altitude of the tower. Anf. 110 feet. QUEST. 12. At a certain place the angle of elevation of an inacceffible tower was 26° 30'; then measuring 75 in a direct line towards it, the angle was then found to be 51° 30': required the height of the tower, and its diftance from the laft ftation.

height 62, distance 49.

QUEST.

QUEST. 13. To find the distance of an inacceffible caftle 'gate, I measured a line of 73 yards, and at each end of it took the angle of pofition of the object and the other end, and found the one to be 90°, and the other 61° 45' required the distance of the castle from each station.

Anf.

{

1358,

154°2.

QUEST. 14. From the top of a tower by the fea fide of 143 feet high, I obferved that the angle of depreffion of a fhip's bottom, then at anchor, was 55°; what was its diftance from the bottom of the wall?

Anf. 20456 feet.

QUEST. 15. How far at fea can the pike of Teneriff be feen, its height being 4 miles, and the radius of the earth 39784 miles? Anf. 178 miles.

Anf.

{

QUEST. 16. If a fhip, in the latitude of 50° north, fail 52 miles in the direction s wby s: what latitude is the arrived in, and how much farther to the weft? lat. 49° 16.8', weft 28.9 miles. QUEST. 17. Sailing w sw, I faw, at fome diftance, a point of land, which I fet, and found its bearing w by N; and after failing 6 leagues farther, I fet it again, and found its bearing N w by w. Required its diftance. Anf. 26.13 miles.

QUEST. 18. Obferving three fteeples, A, B, C, in a town at a diftance, whofe diftances afunder are known to be as follows, namely, AB 106, AC 202, and BC 131 fathoms, I took their angles of pofition from the place where I stood D, which was nearest the steeple в, and found the angle ADB 13° 30', and the angle CDB 29° 50'. Required my distance from each of the three fteeples. DA 302.8 Anf. DB 214.8 DC 262.0

QUEST. 19. Suppofing my ftation to be fartheft from the fteeple B, required to find the distances from it, when the diftance AB is 9 furlongs, AC 12,

and BC 6 furlongs; alfo the angle ADB 33° 45', and

the angle CDB 22° 30'.

QUEST. 20.

DA 10.64 Anf. DB 15.64

DC 14.01

Two ships fail from the fame port; the one fails ENE 85 miles, the other fails E by s till the first ship bears N w by w: what is the diftance of the fecond fhip from the port, and alfo from the first ship? from the port 1847 Anf.from the 1ft fhip 1234

QUEST. 21.

Two ports lie east and weft of each other a fhip fails from each, namely, the fhip from the weft port fails NE 89 leagues, and the other fails 80 leagues, when the meets the former: required the latter fhip's courfe, and the distance between the two ports. course 51° 52′ distance 112.3

Anf. {

QUEST. 22. Two ships fail from a certain port; the one fails s by E 45 leagues, and the other s s w 64 leagues. What then are their bearings and diftance afunder?

Anf.

bearing 43° 14' 14 distance 36.5 QUEST. 23. A fhip failing N w, two islands appear in fight, of which the one bore N, and the other w NW; but after failing 20 leagues, the former bore N E, and the latter w by s. What is the distance afunder of the two iflands?

Anf. 32.38 leagues.

PART