18<19.] Scientific Intelligence. SSfe 



;y,\<\\-> -;<;" gfalOW 9-FF 



w\j\$ii\ The formula above alluded to is 



- '/ c . ,. T ~ 2 sin. A .sin. B . cos. S H . cos.. MR n 



cos. S A = cos. C CUj , mR . cos . aH =* ' CDS - C 



'2 M i 



Since cos. x S M= ^/ ; and cos. -iC = -y/ — , 



we have 

 cos. S M = 2 cos. 2 ± S M - 1, and cos. C = 2 jfias.'i C - 1, 



then by substitution, 

 2 cos. 9 4- SM- 1 =2 cos. 2 £ C — 1 — , from whence cos." 



.V S M = 

 cos. 9 ■ C - £ =: (l - ^ t -) cos. 9 * C. Put sin. 9 D = 



M 



¥T7o^r-c' 

 then will cos. 2 i S M = (1 - sin. 9 D) cos. 9 4 C = cos. 9 D . 



cos. 9 -1 C, therefore, 

 cos. {SM = cos. D . cos. i* C. But 2 log. sin. D = log. 



_ ,' ■. therefore, 



losf. sin. D = 4 I02;. --, — ■ — -^. Hence the following rule : 



° ' ° i\ , cos." |C ° 



- 



Lw. Sin. A - = 



_> 



sin. B = 



cos. * 's true alt — 



cos. J 's true alt = 



cos. ♦ 's app. alt. A.C = 



cos. _) 's app. alt. A.C = 



2 1o_-. cos. x C A.C = 



'O 



Sum of logarithms = 



. 8 .',- sum corresponding to sin. D. ....... . = 



cos. D = 



cos. J- C = 



______ 



cos. ,' r true distance = 



* 



H.>rda's formula, as given in Dr. Gregory's Trigonometry, 

 gives the sin. J- true distance, and has the same number df 

 operations as in the above rule, but the demonstrations are 

 different. 



