one above and the other below the horizon of the places of observation: and the double sign in the denominator indicating two different positions of the point E. It may be shown, that one point will lie within the triangle ABC, and the other without it, and thus all ambiguity will be removed from the solution *.

Further examples for exercise.

8. From the edge of a ditch, of 36 feet wide, surrounding a fort, the angle of elevation of the top of the wall was found to be 62° 40': required the height of the wall, and the length of a ladder to reach from my station to the top of it. Ans. height of wall = 69 649272, ladder 78'402942ft.

=

9. Required the length of a shoar, which strutting 11ft. from the upright of a building, will support a jamb 23ft 10in from the ground. Ans. 26-249ft.

10. A ladder, 40ft long, can be so placed, that it shall reach a window 33ft from the ground, on one side of the street; and by turning it over without moving the foot out of its place, it will do the same by a window 21ft high, on the other side: required the breadth of the street. Ans. 56 6493981ft.

11. A maypole, whose top was broken off by a blast of wind, struck the ground 15ft from the foot of the pole: what was the height of the whole maypole, the broken piece measuring 39ft in length? Ans. 75ft.

12. At 170ft distance from the bottom of a tower, the angle of its elevation was found to be 52° 30': required the altitude of the tower. Ans. 221 55ft.

13. From the top of a tower, by the sea-side, of 143ft high, it was observed that the angle of depression of a ship's bottom, then at anchor, was 35°: what was the ship's distance from the bottom of the wall? Ans. 204 2271ft.

14. What is the perpendicular height of a hill; its angle of elevation, taken at the bottom of it, being 46°, and 200yds farther off, on a level with the bottom, the angle being 31°? Ans. 286.2906yds.

15. Wanting to know the height of an inaccessible tower: at the least distance from it, on the same horizontal plane, I took its angle of elevation equal to 58°; then going 300ft directly from it, found the angle there to be only 32°: required its height, and my distance from it at the first station.

Ans. height = 307·5456, distance = 192·162.

16. Being on a horizontal plane, and wanting to know the height of a tower placed on the top of an inaccessible hill, I took the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measuring in a line directly from it to the distance of 200ft, I found the elevation to the top of the tower to be 33° 45': what is the height of the tower? Ans. 93 33149ft.

The fourth problem may be included in this, viz. when the three stations are in one line. For then bae and C = ; whence b2 — c2 — a2 = 2ac, c2 — a2 — b2 — — 2ab, and a2 — b2 — c2 —-2bc: and the fundamental equation becomes a complete square, and equivalent to {a cot2a (a + c) cot2ß + e cot2y} u2 = ac (a + c), as found at p. 459.

Again, if the distances a, c, be equal, we get {cota - 2 cot2ß + cot2y}u2 = 2a2, either by substituting in the last, or in the fundamental, equation.

The following is the process indicated by the investigation for the solution of the problem. 1. Find h and k from ha cot a tan ẞ, and ke cot y tan ẞ.

2. Find C and 0 from c2 a2-2ab cos C + b2, and k2 = h2 — 2ah cos 0 + a2.

3. Find e, from c2a2-2 ah cos (C0) + h2; and finally

Corresponding steps will be applicable to the case of the three stations in one line, though the solution will not be simpler than that already given, p. 459.

17. From a window near the bottom of a house, which was on a level with the bottom of a steeple, I observed the angle of elevation of the top of the steeple to be 40°; then from another window, 18ft above the former, the elevation was 37° 30' required the height and distance of the steeple, and a general formula of solution. Ans. height =210-436, distance = 250 792.

18. Wanting to know the height of, and my distance from, an object on the other side of a river, which was on a level with the place where I stood, close to the side of the river; and not having room to measure backward, in the same line, because of the immediate rise of the bank, I placed a mark where I stood, and measured, in a direction from the object, up the ascending ground, to the distance of 264ft, where it was evident that I was above the level of the top of the object; there the angles of depression were found to be, viz. of the mark left at the river's side 42°, of the bottom of the object 27°, and of its top 19°. Required the height of the object, and the distance of the mark from its bottom. Ans. height = 57.2734, distance 150.5058.

19. If the height of the Peak of Teneriffe be 2 miles, and the angle taken at the top of it, as formed between a plumb-line and a line conceived to touch the earth in the horizon, or farthest visible point, be 88° 2′; it is required from these measures to determine the magnitude of the whole earth, and the utmost distance that can be seen on its surface from the top of the mountain, supposing the form of the earth to be perfectly spherical.

Ans. greatest visible dist. = 135·943, diam. = 7917.85 miles. 20. Two ships of war, intending to cannonade a fort, are, by the shallowness of the water, kept so far from it, that they suspect their guns cannot reach it with effect. In order, therefore, to ascertain their distance, they separate from each other a quarter of a mile, or 440 yds; then each ship observes the angle which the other ship and the fort subtends, which angles are 83° 45′ and 85° 15′: what is the distance between each ship and the fort?

Ans. 2292-266 and 2298 051yds respectively.

21. Wanting to know the breadth of a river, I measured a base of 500yds in a straight line close by one side of it; and at each end of this line I found the angles subtended by the other end and a tree, close to the bank on the other side of the river, to be 53° and 79° 12': what was the perpendicular breadth of the river? Ans. 529-4847 yds.

22. Wanting to know the extent of a piece of water, or distance between two headlands, I measured from each of them to a certain point inland, and found the two distances to be respectively 735 and 840 yds; also the horizontal angle subtended between these two lines was 55° 40′: what was the distance of the two headlands? Ans. 741-2085yds.

23. A point of land was observed, by a ship at sea, to bear east-by-south; and after sailing north-east 12 miles, it was found to bear south-east-by-east: it is required to determine the position of that headland, and the ship's distance from it at the last observation. Ans. 26-07282 miles.

24. Wanting to know the distance between a house and a mill, which were seen at a distance on the other side of a river, I measured a base line along the side where I was, of 600 yds, and at each end of it took the angles subtended by the other end and the house and mill, which were as follow, viz. at one end the angles were 58° 20′ and 95° 20′, and at the other end the corresponding angles were 53° 30' and 98° 45': what was their distance? Ans. 959 604yds.

25. Wanting to know my distance from an inaccessible object O, on the other side of a river, and having only a chain for measuring distances, I chose two stations, A and B, 500yds asunder, and measured in the direction

from the object O, the lines AC and BD each equal to 100 yds; also the diagonals AD.BC equal to 550, 560 yds respectively: what was the distance of the object O from each station A and B ? Ans. AO 536 441, BO = 500·237.

26. In a besieged garrison are three remarkable objects, A, B, C, the distances of which from each other are discovered by means of a map of the place, to be as follow, AB = 266}, AC = 530, and BC = 327 yds. Now, having to erect a battery against it, at a certain spot without the place, and it being necessary to know whether my distances from the three objects be such, as that they may from thence be battered with effect, I observed the horizontal angles subtended by these objects from the station S, and found them to be ASB = 13° 30′, and BSC = 29° 50'. Required the three distances, SA, SB, SC; the object B being situated nearest to me, and between the two others A and C. Ans. SA 757·1407, SB = 537·1028, SC = 654·0996. 27. Required the distances as in the last example, when the object B is the farthest from my station, but still seen between the two others as to angular position; and those angles being ASB = 33° 45', and BSC = 22° 30', also the three distances, AB: 600, AC = 800, BC = 400 yds respectively. Ans. SA710-195, SB 1042.545, SC 934.29. 28. If CB (fig. 1. p. 422.) represent a portion of the earth's surface, and C the point where a levelling instrument is placed, then DG will be the difference between the true and the apparent level; and it is required to show that, for distances not exceeding 5 or 6 miles measured on the earth's surface, DG, estimated in feet, is nearly equal to a CD2, taken in miles.

=

29. On the opposite bank of a river to that on which I stood, is a tower, known to be 216 feet high, and with a pocket sextant I ascertained the vertical angle subtended by the tower's height to be 47° 56'. Required the distance, across the river, from the place where I stood, to the bottom of the tower; supposing my eye to be 5 feet above the horizontal plane which passes through it. Ans. 200 22ft.

30. In the valley of Chamouni three positions, A, B, C, were selected, in a straight horizontal line, such that AB = 80, and BC = 75yds. Three remarkable points, A', B', C', on the side of the Jura, were also chosen to be observed. The angles of elevation of A', as seen from A, B, and C, were 67° 10′, 68° 15', and 52° 18'; those of B', as seen from the same points, were 72° 18', 78° 15', and 70° 10′; and finally, those of C' were 60° 5', 61° 10, and 58° 5' respectively. It is required from these observations to find the heights of A', B', C', from the horizon of the line of observation.

Ans. A'A" = 159·8134, B'B" = 286·3938, C'C" = 323·3860. 31. (II. 1, p. 48.) An obstacle prevented my measuring the part BC of a line AD, and a point E was selected from which the angles subtended by the segments AB, BC, CD, were a, ß, y, respectively, and the two accessible segments AB and CD were found to be a and c respectively: from which data it is required to find the length of the line AD.

Ans. BC= x, is found from (x + a) (x + c) =

ac sin (a + ß) sin (3 + y) sin a sin y

32. (II. 2.) Being on the opposite side of a river from two steeples O and W, which I knew from a previous survey to be at the distance of 6954 yds, and wishing to know the distance between two other objects on the side on which I stood, but which the irregularity of the ground prevented my measuring, I took the following horizontal angles, OAW = 85° 46', BAW = 23° 56′, OBW = 31° 48', and OBA = 68° 2′. What was the length of AB?

VOL. I.

Hh

33. (II. 4.) A person walking from C to D on a straight horizontal road, can see a tower on the summit of the hill A at every point except E, where he can just see the top of the tower over the hill B. He then measures a base EC of 150 yds, and at C observes the elevation of A to be 59° 18' 15"; and he also finds that ACB = 10° 12′ 20′′, ACE = 69° 18′ 30′′, and AEC = 108° 12′ 15′′. From these observations, the horizontal distance of the hills from each other, and from the places of observation, C and E are to be found.

34. (II. 5.) From three positions A, B, C, in the same horizontal plane whose distances were AB = 150·25, BC = 179.69, AC = 205.36, the elevations of the top of a tower on a hill were observed to be 6° 10′ 55′′, 7° 18′ 3′′, and 6° 58′ 58′′ respectively: whilst the elevation of the bottom of the tower from A was 6° 2′ 58′′: and from these data the height of the tower is required.

35. (II. 6.) Four points A, B, C, D, are accessible, and three M, N, P, inaccessible, but are to be found from the following observations:

AB 815
BC = = 670
CD = 660

ABC 49° 54′
BCD= 73 57

AMB =
80° 8'
BMN= 24 55
CNM 124 16

CNP = 98° 44'
CPN = 29 13
CPD 51 19

36. A tree growing on the side of a hill which rises due north at an angle of 30°, had the upper part blown off 12 feet from the ground by a gale from W.S.W: now supposing the tree to stand perpendicularly to the horizon, and the top (before the other part was wholly separated from the tree) to strike the ground 40 feet from the bottom, what was its original height? Ans. 51.204ft.

37. Passing along a straight and level road, near a very lofty tower on the same horizontal plane with the road, I wished to know its height: but having no instrument for taking other than vertical angles, I proceeded thus: at a convenient point (A) on the road, I observed the angle of elevation of the top of the tower to be 30° 40′, and 60 yds farther on the road the elevation was found to be 40° 33'; at the end of another 60 yards, I was prevented, by a high wall, from taking the elevation, and therefore I measured 12 yds still farther, and found the angle of elevation to be 50° 23′. From these data it is required to find the height of the tower, and its horizontal distance from each of the stations. Ans. the height = 94.835, and the distances 159-087, 110 8414, and 78.507, from the stations.

38. A person in a balloon observed the angle between two places A, B, bearing N. and S. of each other, and a miles apart to be ao, and from B, which bore due east of him, to a point directly under him, to be 6o: show that his altitude is expressed by a cot a cos ẞ°.

39. A tower a feet high stands in the centre of a field whose form is an equilateral triangle, and each side subtends an angle of 2a: find the side of the field. 40. Walking along a horizontal road I observed the elevation of a tower to be 20°, and the angular distance of the top of the tower and an object on the road to be 30°; also the nearest distance of the tower from the road was 200ft : find its height. Ans. 187 57534.

41. From a station A, the angle subtended by two objects B and C was 2a, and at B and C the angles subtended by A and a fourth point D were right angles; also the distances AB and AC were b and c : show that if 20 be the dif

ference of the angles BAD, DAC, then tan 0 =

b

C

tan a, and the distance

b+c

42. AB is an obelisk on a hill BHG, and there is no horizontal ground in front of it: on the opposite hill, I therefore measured 160ft in the same vertical

plane with the castle from C upwards to E: the elevations of A and B from C were 47° 27′ and 45° 17′, and that of A from E was 46° 20′, whilst the inclination of CE to the horizon was 10° 10'. Find the height of the obelisk.

Ans. 367 851ft.

43. From a point on a level with the bottom of a flagstaff its elevation was 23° 8' 15", and from another point 18-2961ft higher, the angle subtended by the flag-staff was 23° 15′: required its height and distance.

44. A person on the mast-head at S, 105 6ft above the level of the sea, just sees over the earth's surface at P the top of a cliff T known to be 660 ft high: how far was the ship from the cliff, the earth's diameter being 7800 miles?

Ans. 35 051miles.

45. A church O is to be built for the accommodation of three villages A, B, C, whose distances asunder are BC= 2 26, CA = 1·14, and AB = 1.58 miles : but as they contribute unequally to the expense, their distances are to be to one another in the ratio AO BO : CO :: 5 : 129: what is its distance from each? Ans. If O be within ABC, AO = ·54699, BO= 1.31278, CO = 98459, .....without.... AO 99486, BO= 2·38768, CO = 1·79076.

=

46. Three stations of the trigonometrical survey of Britain can be seen from the Eddystone lighthouse, viz. Kit's Hill, Carraton Hill, and Butterton Hill, (which denote by A, B, C respectively, and the lighthouse by D): and it appears from the survey that AB = 33427 ft, BC = 131576ft, and CA = 100969ft; and likewise that from D, BC subtended an angle of 64° 1' 48", AC an angle of 48° 45′ 53′′, and AB an angle of 15° 15′ 55"; from which it is required to determine the distance of the Eddystone from each of the stations.

Ans. AD 123411ft, BD = 126896ft, and CD = 121123ft. 47. In the French trigonometrical survey, three stations, Villers Bretonneux (A), Vignacourt (B), and Sourdon (C), and a station (D) within the triangle, were taken, and the following data obtained: viz. log BC = 4·2734544, ABC = 49° 4′ 13′′, ACB = 31° 49′ 57′′ 8, ADC = 130°44′ 16′′ 5, and ADB=60° 31′ 53′′•8: to find the distances of D from each of the stations.

Ans. AD 8064 61, BD = 11124-25, CD = 7733:49. 48. In the trigonometrical survey of Scotland the three stations High Pike (A), Cross-fell (B), and Crif-fell (C) were observed to subtend angles from Helvellin (D) as follows; BDC = 100° 17′ 45′′ 25, ADC= 32° 39' 57'25, BDA= 67° 37′ 48′′; whilst the previously determined distances of the three stations were, BC= 255886 1ft, AC = 147733 5ft, and AB=120904 9ft: it is required to find their respective distances from Helvellin.

Ans. AD=65724-6, BD = 129531-8, and CD = 198738 6ft. 49. At the commencement of the trigonometrical survey, a base line BC of 27404 2 ft was measured on Hounslow Heath, between Hampton Poorhouse (C) and King's Arbour (B); and from both these stations, Hanger-Hill Tower (A) and St. Ann's Hill A' (on opposite sides of BC) were visible, and the following angles were observed: ABC 70° 1' 47", ACB = 67° 55′ 39′′, A'CB= 61°26′ 35′′ 5, and A'BC = 74° 14′ 35′′, from which to find the distance between Hanger Hill Tower and St. Ann's Hill. Ans. 68896ft.

=

50. To find the distance of Inchkeith Lighthouse (A) and the spire of North Leith Church (A'), the following observations were made at the Edinburgh Observatory (C) and Beincleuch (B); viz. BC=146314 ft, BCA = 73° 16′ 28′′·5, BCA' = 55° 38′ 41′′·1, CBA = 11° 53′ 56′′, CBA' = 2° 34′ 2′′2; to find AA', both points being on the same side of BC. Ans. 23045-53ft.