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Question regarding mathematics of Captain Thomas Sumner's calculations:
From: Russell Sher
Date: 2006 Jun 3, 22:10 +0200
From: Russell Sher
Date: 2006 Jun 3, 22:10 +0200
For those of you who enjoy a bit of maths, perhaps
you could take a look and explain this to me - I hope I've laid out
the question clearly, but if I haven't, let me know and I'll send further
info - thanks.
I am currently reading the book 'Line of
position navigation' which discusses Sumner's method of calculating a line
of position. (I think several list members mentioned the book previously). The
book includes actual workings of Sumner's calculations. I worked through some of
the examples but cannot quite understand Sumners application of
logarithms as used here. I can use a calculator to get the result using
the formula, but am trying to use his method with logarithms. Perhaps some of
you can shed some light on this...
The formula to solve would seem to be the
familiar LHA = arccos{ [ sin(Ho) - ( sin(Lat.) * sin(Dec) ) ] / [ cos(lat)
* cos(Dec) ] }
(Ho here is the corrected altitude of the body)
(remember to enter South declination as a negative quantity if you want to
confirm the workings)
I agreed with the result obtained by using a
calculator, but I would like to understand the use of logarithms. (Obviously
since calculators weren't around then, logs were used).
In the book in appendix A, page 15, the
following figures are given:
Latitude 51 degr. N, Decl. 23 degr. 23'
South
Sun's altitude (with all corrections applied) 12
degr. 10'
The solution begins as:
Latitude 51 degr. N --- sec.
0.20113 --I'll call this [value 1]
Dec. 23 degr. 23' S. -- sec
0.03722 -- I'll call this [value 2]
I managed to work that the reference above to 'sec'
means the natural logarithm of the secant of each of the angles. (remember
that secant(x) = 1/cos(x) , then take the logarithm of the
secant to get each of the two above values)
Now I understand that since the denominator of the
formula for LHA given above is [cos(lat) * cos (Dec)], the two 'sec' values when
added would appear to be equivalent of this denominator (since adding
logs is equivalent to multiplying the original numbers) and since sec is 1/cos,
this forms the denominator - correct?
Now I get lost...
Then Sumner adds latitude 51 degr. +
Decl. 23 degr. 23' - he writes:
Sum: 74 degr.
23' Nat. cos 26920 --I'll call this [value 3]
Alt. 12 degr. 10'
Nat.sin 21076 --
I'll call this [value 4]
Question 1. Why is value of declination added to
the value of latitude at this point?
continuing...
I calculate the cosine of 74 degr. 23' =
0.269199
and the sine of 12 degr. 10' = 0.2107561
Question 2. So where do [value 3] and [value 4]
come from? - they appear to be a scaled up (by 100000) equivalent of the the two
values just above for cos(lat. + dec) and sine(altitude). (Ok his values are
rounded off - so there are slight differences.). I assume Nat. cos and Nat. sin
means natural cosine and natural sine (as opposed to log. cos or log. sin
??)
Then sumner subtracts: [value 3] - [value 4] = 5844
and then he takes the logarithm of this: log(5844) = 3.6671 - call this
[value 5]
Then sumner takes [value 1] + [value 2] + [value 5]
= 4.00506. and he writes:
1 hour 43 min 59 sec from noon = log rising =
4.00506 (the value just calculated).
(now, if I use a calculator I obtain LHA = 25 degr.
59 min which agrees (in time) with 1 hour 43 min 59 sec) -- using 15 degr. = 1
hour)
Question 3. But how does Sumner associate 1
hour 43 min 59 sec. with log 4.00506 ??
Question 4. Although I agree with his final result
from the formula, how does it fit together to give the correct answer from the
use of logs as in the example?
regards
Russell
