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Sine ADC 130°: s. CAD 11° :: AC: DC.

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8. At the top of a castle 54 feet high, erected on a hill near the sea shore, the angle of depression⇒ CAE of a ship at anchor was 4° 52'; and at the bottom of the castle the depression = DAE was 4o 2'. Required the horizontal distance of the vessel, and the height of the top of the castle above the sea. (See the last fig.) Answer. Distance AE= 3690 feet, and height EC = 314 feet.

9. Desirous to know the breadth of a river I measured100 yards in a straight line by the side of it, and at the ends

of this distance I took the an

gles formed by it and a tree on the opposite bank, which were



53° and 79° 12'. Required the breadth of the river.


BAC=180°-B-C-180°-132° 12′ = 47° 48′. Sine BAC 47° 48′: sine C 79° 12′ :: BC 100: AB.

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R or sine D 90° : sine B 53° :: AB : AD.

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10. Desirous to know the distance of a house, which stood on the opposite bank of a broad river, I measured 200 yards in a straight line by the side of the river, and took the angles formed at each end by the line and the house, which were 73° 15' and 68° 2. Required the distance of the house from each station. (See the last fig.) Answer. The distances are found as in the first analogy of the last problem, and are AB 306-19 yards, and AC = 296.54.

11. To find the distance between two places A, B, which are not accessible in a straight course, I seek a third place C, from which A and B can be seen; I then measure AC = 735 yards, and BC= 840 yards, and take the angle subtended by the two places ACB = 55° 40'.

BC+ AC 1575 BC-AC 105 :: tan.


62° 10'


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Hence the angle A = 69° 22′.

Sine A 69° 22′ : sine C 55° 40′ :: BC 840: AB.

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12. A man being desirous to know the distance between two inaccessible objects A, B, measured a base CD of 300 yards; at Ċ he found the angle BCD = 58° 20′, and ACB = 37°; at D he found the angle ADC= 53° 30′, and ADB = 45° 15'. Required the distance AB.

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The angle CDB = ADC + ADB = 98° 45'.

Sine CAD 31° 10' sine ADC 53° 30':: CD 300: AC.

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Sine CBD 22° 55′ : sine CDB 98° 45′ or its sup.

81° 15' :: CD 300: CB.

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Log. BC= 761-4655

BC+ AC 1227-4431: BC-AC 295.4879 ::

(CAB+ ABC) 71° 30′ or cot. ACB 18° 30′ :

tan. (CAB-ABC).

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Sine ABC 35° 46': sine ACB 37°:: AC 465. 9776: AB.

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90. In certain trigonometrical operations, when a base is measured on sloping ground, it is sometimes necessary to reduce it to the corresponding horizontal line. Let AB be the measured base, OB a theodolite, H and AR a staff equal to the height of OB. Let HOR be the angle of depression of the top of R below the horizontal line HO. If CO be Ri perpendicular to HO, then AC, which is parallel to HO, will be



the horizontal base corresponding to AB. Hence R: AB :: cos. HOR or BAC: AC.

If AB be 300 yards, and the angle of depression HOR 5°, the horizontal line AC will be 298-9 yards, which differs from the measured base AB by 11 yard only. Therefore when the measured base is inclined to the horizon in a small angle a reduction is not necessary, unless accuracy be required.

91. The distances of the most remarkable places in a town, or of several villages from one another, may be determined by the rules of trigonometry; also the plan of a camp, or of a country may be taken.




Let A, C, D, E, B, H, G, F, be several objects, whose situations are to be laid down in a map. Find a convenient position AB for the base, from which all the objects can be seen; and let the base be as long as possible, in proportion to the most distant object. From the extremity A of the base A take the angles EAB, DAB, CAB, HAB, GAB, FAB; and from the other extremity B take the angles





CBA, DBA, EBA, FBA, GBA, HBA. The common base AB and the angles of all the triangles being now known, the sides AC, AD, AE, &c., and consequently the points C, D, E, &c. may be determined by article 54.

To insure the accuracy of the operation all the objects C, D, E, &c. should be intersected from some third station O, in the base AB; otherwise the figure or plan of the survey may appear correct, when it is not so; and there will be no means of discovering whether the angles have been justly taken.


92. To carry on a Measurement by a series of Triangles.


A measurement may be carried on, or the distance of any two remote objects may be found, by means of a series of triangles, formed from a measured base. In this manner the trigonometrical survey of a country is generally performed. Let AB represent the measured E base, and C, D any two objects visible from the two stations A, B. If the angles CAB, CBA, DAB, DBA, be taken with a theodolite, or other instrument, we can find the sides BC, BD by article 54, and the angle BDA. The angles A DBA, CBA being known, their difference CBD is also known. From the two sides BC, BD, and the included angle CBD we can find the third side CD, and the two angles BCD, BDC.



If E, F be two other objects which are visible from the stations C, D, we can find the lengths of EF, DF in the same manner. And thus may the measurement be continued from one base to another to any distance.

To render the conclusion less inaccurate, the mensuration from one base to another may be carried on by different sets of triangles, leading to the same two objects. For instance, instead of C, D other two proper objects might have been chosen, by means of which the length of EF might have been found in the same manner. The mean result of several sets of triangles will probably approach nearer the truth than the result of a single set.

The distancc AF between the first station and the last may be also determined. For the two sides AB, BD, and the included angle ABD being known, the side AD may be found; therefore the two sides AD, DF, and the included angle ADF being known, the side AF may be found.

In surveys of this kind it is expedient to observe every angle of all the triangles, if the situations will permit; because the difference between 180° and the sum of the three angles of each triangle will enable us, in some measure, to judge of the accuracy of the operations.

All the principal distances ought to be laid down from a scale of equal parts, because a triangle can be protracted less inaccurately from the sides than from the angles.

After a series of triangles has been carried on to some distance, in this manner, it is expedient to measure the interval

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