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George Huxtable wrote:
>
> Chuck Taylor has taken some of the mystery out of haversines and the
> cosine-haversine method, and shown how such calculations were made.
>
> I think there's a bit more to be add, if Chuck doesn't mind.
>
> ====================
>
> He wrote-
> >Haversines are merely a vehicle for simplifying the
> >computations.  While sines and cosines range from -1
> >to + 1, haversines range only from 0 to + 1, and the
> >haversine of a negative angle is the same as the
> >haversine of the absolute value of that angle.
>
> That's all correct, but Chuck has omitted the important reason WHY it was
> necessary to avoid negative values. The reason was the USE OF LOGS.
>
>

A lot of the "harry" formulas that were used in the past had to do with
avoiding negative values and making it easy to use logs.

I have a great book picked up at a used book store call H.O. No.200
"Altitude, Azimuth, and Line of Position Comprising Tables for Working
Sight of Heavenly Body Line of Position by the Cosine-Haversine Formula,
Marcq Saint Hilaire Method and also Aquino's Altitude and Azimuth Table
for Line of Position, Marcq Saint Hilaire Method. GPO 1918.

Notice also that in the older almanacs, before World War II, I am not
sure when they made the change, the Sun's Declination and Equation of
Time are given for every 2 hours NOT the Declination and GHA. Eq. of
Time
is in mins and secs. For the Moon the Right Ascension and Declination
were given for every 2 hours.


Example of a problem: At sea, January 15, 1916 8.30a.m., in Lat. by
D.R.,
30deg 10min N., Long. by D.R., 45deg 15min W., the sextant altitude of
the sun's lower limb, bearing southward and eastward, was 17deg 40mins
50 secs.
Index correction + 1min 00sec. Height of eye, 29 feet. Watch time, 8h
30min 10s; C.-W. 3h 02m 40s. Chronometer slow of G.M.T 7m 10s. Required:
The altitude difference and the azimuth for laying down the line of
position of the D.R. position.

 Formula: hav z = hav (L~d)+cos L cos d hav t
                = hav(L~d) + hav theta

Notice that logs of the haversines are used in the cos L cos d hav t
and this is then called log theta. It is then converted to natural
haversines and added to hav(L~d) and then the inverse haversine is
looked up.

z is the co-altitude. i.e. 90 - alt so the computed altitude is
90 - z.



Times:
W.T.  8h 30m 10s
C.-W. 3   02  40
    -------------
C.T.  11  32  50
C.C.       7  10
-----------------
G.M.T.
14th Jan.
or 15th Jan.

    23 40 00
 -  0  20 00
-------------
G.M.T. 23 40 00
Long W  3 01 00
---------------
L.M.T. 20 39 00
Eq. of T. - 9 11.1
------------------

L.A.T. 20 29 48.9
t =     3 30 11.1


Now Declination
G.M. noon, S. 21deg 18.9 mins
Jan 15.
 Corr. S.            0.1
----------------------------
 S. 21deg 19.0min

H.D.
N. 0.4min
G.M.T. - 1/3 hr
--------------
Corr. S. 0.1

Eq. of T.
(-) to M.T.

9 min 11.4 + 0.9 sec H.D.
Corr.       -.3  - 1/3h
-----------------------
9--11.1      Corr. 0.3


Altitude
By obs    17deg 40min 50sec
Index corr. +    1    00
Table I      +  8     09
---------------------------
h 17deg 49min 59sec


Solution
t = 3 h 30min 11.1 sec     log hav = 9.29213
L= N. 30deg 10.0min         log cos = 9.93680
d= S. 21deg 19.0min         log cos = 9.96922

                       theta log hav = 9.1981ta

                       theta nat hav = .15782
 L~d 51deg 29.0min       nat have = .18863

z = 72 06 55           nat hav = .34645
Calculated h = 17 53 05
Observed h   = 17 49 59
-----------------------------

Alt. Diff = 3min 06secs = 3.1mins

Azimuth S. 51deg E. from Table V

---------------------------------------------------------

Notice that the arguments are not LHA and Dec for the tables,
but LAT and Dec. Long was converted to time!

-- Gordon



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| Gordon Talge WB6YKK           e-mail: gtalge AT pe DOT net |
| Department of Mathematics      http://www.nlmusd.k12.ca.us |
| Norwalk High School                   Norwalk, CA          |
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