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Time sight of the sun, a navigator's morsel.

The formulae
Method 1 : cos t = (sin h - sin L sin d) / (cos L cos d)  (1)
Method 2 : hav t = cos S sin(S - h)/ sin p cos L   (2)
with : S = (h + p + L)/2.

Formula 1 can be solved by a semi-logarithmic computation
Formula 2 can be solved by a logarithmic computation and in fact formula (2) 
is derivated from formula (1) which is derivated from de fundamental formula 
:

sin h = sin L cos p + cos L sin p cos t
with : p = 90deg - d
Extracting t we have :
cos t = sin h - cos p sin L/ sin p cos L (1 bis)
Using the versine we have:
1 - cos t = (sin p cos L + cos p sin L - sin h) / sin p cos L
which gives:
1 - cos t = (sin(p + L) - sin h)/sin p cos L
and
2 hav t = 2 sin((p+L-h)/2) cos((h+p+L)/2)/sin p cos L

With:  p + L + h = 2S and : p + L - h = 2(S-h)
we get finaly the formula (2)

to avoid a substraction of log, formula (2) is rewrited  so :
hav t = cos S sin(S - h) csc p sec L   (2 bis)
csc = 1/sin, sec = 1/cos.
which become in casu :
log hav t = log cos S + log sin (S-h) + log csc p  + log sec L  (2 ter)

------------------------------------------------------- just for fun ------------
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