MEASUREMENT OF HEIGHTS, ETC.] MATHEMATICS. MENSURATION. 



643 



(4). (riven, one side and two Angles; e.g., given, oB C, to 



find A. c, 6. 

 We have A = 180 (B + C). which gives A. 



6 sin. B c sin. C 



row, = . = - T- 



o sin. A a sin. A 



.'. log. ft = log. a -f- log. sin. B + ar. : comp. : log. 

 sin. A 10. and log. c = log. a + log- 81T *' "I" ar ' 

 comp. : log. sin. A 10. Whence we can immediately 

 calculate b, <fcc. 



Example : 

 Given B = 49. 



C = 29. 19' and a=93-4, we shall find 

 A = 101. 41' 



6 = 73-52255 



c = 47 -69952 



Several practical applications of the science of Trigo- 

 nometry will be found in the following chapter on 

 Mensuration. 



CHAPTER XIV. 



MENSURATION. 



THE object of the following pages is to show the prac- 

 tical application of the truths investigated in previous 

 chapters on Geometry and Trigonometry to solve a variety 

 of questions about measurements such as may frequently 

 occur in practical life. For instance, the determination 

 of the area of a field, the distance between two inacces- 

 sible points, the solid content of a pyramid, and others. 

 \\ 'i; shall endeavour in each instance to refer these ques- 

 tions to the principles on which they rest. By doing so, 

 the attentive reader will be able to apply the same prin- 

 ciples to cases that may occur practically, but which our 

 limits forbid us to treat of. We shall take in order a 

 variety of questions concerning the determination (1), 

 of 1 f eights and Distances ; (2), of Areas of Surfaces ; (3), 

 i tents of Solids. 



L ON HEIGHTS AND DISTANCES. 



In the following articles we are supposed to have the 

 means of measuring the angle subtended at the eye of 

 the observer by the line joining two points. The instru- 

 ment by which this can be done is called a sextant. For 

 measuring the angle subtended by two objects on a hori- 

 zontal plane, and for determining the vertical elevation 

 of one point above another, an instrument called a 

 theodolite can be used. 



(1). To determine the height of a Tower standing on a 

 Horizontal Plane, the bate of which is accessible. 



Let A P be the tower. B any L 



convenient place for the observer. 

 At B measure the angle A B P ; 

 also measure the line A B. Then, 

 if A P = x, 



x = a tan. B. 



.'. log. z= log. a + L. tan. B 10. 

 which gives x. 



N.B. If P be the top of a 



steeple, half the breadth of the tower must be added to 

 the measurement, which is made to the outside of the 

 tower. It will be observed, moreover, that B is the 

 place of the observer's eye, and .'. AP, or x, is the 

 height of P, above the horizontal plane passing through 

 the observer's eye. Hence, if h be the height of the eye 

 above the ground, x + h is the height of the summit of 

 the tower above the ground. 



(2). To determine the height of a Tower, the base of which 

 is inaccessible. 



Let A BN be the horizontal 

 plane ; P N the height re- 

 quired. At B measure angle 

 P B N ; move backward to 

 another point, A, taking care 

 that A and B are both on the 

 same vertical plane passing 

 through P, and measure A B, 

 and the angle PAN. 



Let P B N =/3. P A B = a 

 B P = x. P N = y. 



Now anglo A P B = (/3 - a) 

 tan i 

 .'.y m* a. 



Flg.J. 



AB=0. 



sin.(/3-a) 



and 



x = y sin. /J. 

 , _ sin. a. sin. /3 



.'. log. x = log. a + L. sin. a + L. sin. j3 + ar. comp. 

 L. sin. (p- a) -20. 



(3). To determine the height of the Tower in the last article, 

 when the nature of the ground does not admit of the 

 observer moving backward. 



Let P N be the vertical 

 height required ; place pickets 

 at two points, A and B, APN 

 and B P N being two dif- 

 ferent vertical planes. 



Measure A B = n. Let 

 PN = x. 



Measure angle P A N = a. 

 PAB = A PBA = B. 



Then x = A Psin. a. 

 NowAPB=180-(A + B) 



sin. B 

 . . AP=o. 



Fig. S. 



sin. (A + B) 

 sin. a. sin. B 

 >x= "" sin. (A + B)' 



.'. log. x = log. a + L. sin. a + L. sin. B -f- ar. comp. 

 L.sin. (A+B)-20. 



In the above case such an angle as P A N being the 

 vertical elevation of P above the horizon, is measured by 

 a theodolite. While the angle PAB, which is not in a 

 horizontal plane, being the angle subtended by the dis- 

 tance between two objects, is measured by a sextant. 



If the object be a long way off, we may measure the 

 angle subtended at the eye by the line joining the object 

 and its image in still water, by a sextant ; half that 

 angle will be the vertical elevation of the object. 



Example : We observe that the altitude of a hill is 

 3 15'. On proceeding 1J miles towards it, its altitude is 

 15 37'. Find the height of that hill, neglecting the 

 sphericity of the earth. 



Here (Fig. 2) A B = If miles. 



PAB = 3* 15'. 

 P B N = 15 37'. 

 .'. PBN-PAB = 1222'. 



Then by the formula proved in Article (2) 



sin. PEN sin. PAB 

 '" a " sin. (PBN-PAB) 



.'. log. P N = log. a + L. sin. P B N + L. sin. PAB 

 + ar.comp. L. sin. (PBN-P AB)-20. 



2430380 

 9-4300750 

 8-7535278 



^6692473-20 

 r2958881 

 2958748 



log. 1-75 



L. sin. 16 37' 



L.sin. 3 15' 



Ar. comp. L. sin. 12 2^ 



19764 

 6 



197646 of a mile 6 = 

 or \ of a mile very nearly. 



133 

 132 



(Answer.) 



