C44 



MATHEMAT1CS.-MENSURATION. 



[MEASUKEMKNT or AREAS. 



(4). To find the datanee Ixiuttn (wo point*, one of whit* 

 it acttttibU and the other inaccttMU. 



Let P be the inaccessible point 

 A the accessible. 

 Drive a picket in at A, and 

 drive in another at any convenient 



Measure the angles P A B and 

 P B A, also measure the line A B. 



Th.n by the fourth case of tlio 

 solution of triangles 



AP-AB 8 > A1 ' U 

 " sin, ABP 



sin. (PAB + ABP) 



sin. ABP 

 .-.calling A B-o. PAB- A. ABP-B. 



we have 



log. A P - log. o 4- L. sin. (A 4- B) 4- ar. comp. L. sin. 

 B-10. 



(6). To find the distance between two points neither of 

 u-hich is accessible. 



Let A, B, be the two rig. S. 



points ; place on the 

 ground two pickets, C 

 and D, such that the 

 distance between* C and 

 D can be measured, and 

 that from each of C and 

 D, the two inaccessible 

 points and the other 

 picket may be visible. 

 Measure CD = p ACB = C BCD 

 A D C = D' 



Then in triangle A C D we know one side and two 

 angles, and therefore can calculate A C. 



In triangle B C D we know one side and two angles, 

 and therefore can calculate B C. 



And finally in triangle A C B we have already calcu- 

 lated AC, B C, and have measured the angle C, and 

 therefore can determine A B. 



Of course, from the first two triangles we can also 

 determine A D and B D, and then in triangle A B D we 

 know two sides, and the included angle, and hence can 

 determine A B, and can use these two calculations for 

 checks upon each other. 



The calculation is performed as follows : 



Call the sides and angles of ABC a. b. c. A. B. C. 

 and observe that C A D = 180 (C 4- C' + D') and 

 C B D= 180 (C' 4- D 4- D'). Then from triangle A C D 



sin. CDA 

 .'. log. A C=log. C D + L. sin. C AD + ar. comp. log 



sin. CDA-10. 

 or 



log. b - log. p + L. sin. (C -f- C' + D') + ar.: comp. : 

 log. sin. D' 10 (1) 



Similarly 



log. o = log p. + L. sin. (C' -|- D + D') + ar - : comp. : 

 log. sin. CK 10 (2) 



From triangle A C B we have s = 00 _- 



** 



A B a 6 



A+B 



- 



.. 

 a a + 6 1 + tan. 



.'. L. tan. 0- log. b log. a + 10 



log. 6 4- ar. : comp. : log. a 



tan. (45<>-0) 

 (3) 



and L. 



.-L. tau. (45-0) 



.=-! 10 



(4) 



-n 



whence we obtain and .'. A and B 



A + B 

 since we know g 



Finally from triangle ABC 



c-a 8i ": 

 sin. A 



.'. log. c log. o + L. sin. C + ar. : comp. : 



L. sin. A 10 (6) 



It will be observed that in tho above calculation we do 



not require a and 6, but only log. a and log. 6, which 



are given by equations (1) and (2). (Compare 1'lane 



Trig., Art. 40). 



Example: Given C D = 372.5 yds. C - 123. 15'. 

 C / =13.42 / . D=129.ll'. D'-19.13'. 

 (1) Log. 6 -log. 372.5 =2.5711203 



+ L. sin. 156. 10/ 9.6064647 



+ ar. comp. L. sin. 



19. 13' 10 .4820176 10. 



(2) Log. a = log. 372.5 



+ L. sin. 162. 6' 

 + ar. comp. L. sin. 

 13. 4^ 10 



(3) L. tan. = log. b 



+ ar. comp. log. a 



= 43 24' 



= 43 24' 37* 8 

 45 



2.6602086 

 2.6711263 



.6255483 10. 

 2.6843172 



7.3160828 



9.9768914 

 9.9767318 



4218)159600(37-8 

 12654 



:;:;nr,o 

 29526 



45 0= 135'22"2 



L. tan. = L tan. (45 0) 

 2 



.. --2. 10. 



= L. tan. 1 35' 22*. 2- 

 L. tan. 28 22' 30* 



A_-B 

 2 



51' 



30* 

 2 



_ 

 61' 32* 



:" = 28 22' 30* 



= 8-4432492 

 9-7326024-10 

 Mr 58516 

 8-1756658 

 1408)2858(2 

 2816 

 42 



(5). 



A = 29 14' 2* 

 B = 27 30' 58' 

 log. c = log. a -f L. sin. C + ar. comp. L. sin. A - 10. 



log. o. 

 L. sin. 56 45' 

 ar. coinp. L. sin. 29 14' 2" 



827.786yds. 



827.78 - 



6 = 



26843172 

 99223549 



3112458-10 

 29179179 

 9179149 

 30 

 32 

 A nsu.tr. 



827.786 



II. THE MENSURATION OF AKEAS. 

 (1). To find ike Area, of a Rectangle. 

 Let A B C D abed be two rectangles. Then (Euclid, 

 VI., 23) the areas of these rectangles are to each other 



in tho ratio compounded 

 of the ratios of the sides, 

 i. e., in the ratio com- 

 poundi'd of the two, 

 ab : A B and be : B C ; 

 and if we suppose these 

 lines to be represented 

 by numbcrsj this com- 

 pounded ratio iaabxbe: 



