OF DIALING. 383 



Let the altitude be 48 ; aud because, in this case, LECT. 

 H- A ~ LD > and A (the natural sine of 48*)=743145, aud ^^ 



Id 



L D=.267664, A L D will be 0.475481, 



whose logarithmic sine is .... 1.6771331 

 from which taking the logarithmic sine of 



lxd= 1.7671354 



Remains ........... 1.9099977 



the logarithmic sine of the hour-distance sought, viz. of 

 54 22* : which, reduced to time, is 3 hours 37j min. that 

 is, IX h. 37i min. in the forenoon, or II h. 22i m. in the 

 afternoon. 



Put the altitude =18, whose natural sine is .3090170 ; 

 and thence A L D will be =.0491953 ; which divided 

 by Ixd, gives .0717179, the sine of 4 6T, in time 164 

 minutes nearly, before VI in the morning, or after VI in 

 the evening, when the sun's altitude is 18. 



And, if the declination 20 had been towards the south 

 pole, the sun would have been depressed 18 below the 

 horizon at 16j minutes after VI in the evening : at which 

 time, the twilight would end ; which happens about the 

 22d of November, and 19th of January, in the latitude 

 of 51J north. The same way may the end of twilight, 

 or beginning of dawn, be found for any time of the 

 year. 



NOTE 1. If in theorems 2 and 3 (page 363) A is put 

 =0, and the value of H is computed, we have the hour 

 of sun-rising and setting for any latitude, and time 01 

 th" 'ear. And if we put HzzO, and compute A, we have 

 the sun's altitude or depression at the hour of VI. And 

 lastly, if H, A and D are given, the latitude may be 

 found by the resolution of a quadratic equation ; for 



NOTE 2. When A is equal 0, H is equal - 



id- 



TLxTD, the tangent of the latitude multiplied by the 

 tangent of the declination. 



