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I noticed a small typo in my previous post. On the form I put "inv hav" 
when is should have been simply "hav", taking the natural haversine for 
the value of the previous log haversine. This is the same as taking the 
antilog of the previous log haversine. Depending upon the arrangment of 
the trig tables you are using, it might be easier to take the antilog 
here instead of the natural haversine, your choice because you get the 
same value and the same result. This typo was also in the attached text 
document. The typo is not in the example using Chichester's numbers. I 
have attached a corrected text document.

Logarithms of trig functions are still found in Bowditch but the 
Haversine table, which had been table 34 until at least the 1975 
edition, is not in the online edition of Bowditch.

Incorrect form:

                            Hc                                      Az

LHA ___________        log hav LHA  ___________        log sin LHA  ______________          
Lat ___________        log cos Lat   ____________ 
Dec  __________        log cos Dec + ___________       log cos Dec  ______________                   
                                    ______________               

                       _*inv *_hav    _______________  <<<<<<<<<<<<<<    incorrect version
(L ~ D) ___________    hav (L~D)  + ____________ 

        89-60

ZD -  ___________<<<<  inv hav  _____________>>> >>>>>>log csec ZD  
+___________  <<<<<<< this line is correct
                                               
 
Hc ____________           
                   
Ho-_____________
A______________                     Z ______________<<<>> >>>>>>log csec ZD  
+___________    <<<<<<< this line is correct  
                                               
 
Hc ____________           
                   
Ho-_____________
A______________                     Z ______________<<< Now we will look at Chichester's illustration of the old method on page 
> 234 to compare it with the short method already discussed.
>
> What he is doing with this example is using the traditional Haversine- 
> Cosine method of calculating Hc and azimuth. The formulas used for this 
> were derived from the standard Sine - Cosine formulas and, in fact, uses 
> the same method and formula for calculating azimuth.
>
> The formula for calculating Hc is:
>
> hav ZD = hav LHA cos Lat cos Dec + hav (Lat ~ Dec)
>
> (Lat ~ Dec means the difference between latitude and declination, 
> subtracting the smaller from the larger if of the same name and adding 
> if of different names)
>
> (ZD is zenith distance)
>
> so Hc  = 90º - ZD
>
> For calculating azimuth we use
>
> sin Z = (sin LHA cos dec ) / cos Hc
>
> usually rearranged into the more convenient form of
>
> sin Z = sin LHA cos dec sec Hc
>
> Since csec ZD is the same as sec Hc
>
> we can rearrange this formula to
>
>  sin Z = sin LHA cos dec csec ZD
>
>
> Chichester used these formulas and solved them using logarithms by using 
> this format:
>
>                                                 
> Hc                                                      Az
>
> LHA ___________     log hav LHA  ___________        log sin LHA  
> ______________          
> Lat ___________        log cos Lat   ____________ 
> Dec  __________        log cos Dec + ___________       log cos Dec  
> ______________                   
>                                                       
> ______________               
>
>                                     inv hav    _______________  
> (L ~ D) ___________ hav (L~D)  + ____________ 
>
>              
> 89-60                                                                        
                                               
> (89-60 is a convenient way to write 90º when you will be subtracting)
>
> ZD -  ___________<<<<  inv hav  _____________>>> log csec ZD  
> +___________      
>                                                
>  
> Hc ____________           
>                    ..
> Ho-_____________
> A______________                     Z ______________<<<...inv log sin 
> _______________
>
>
> In contrast to the previous example using H.O 249, using the haversine - 
> cosine allows the use of the DR position and does not require the 
> selection of an AP that produces whole degrees of latitude and whole 
> degrees of LHA.
>
> I have attached a marked up version of page 234. "A" shows the 
> computation of Hc and "B" the computation of azimuth. Chichester uses 
> the DR latitude and determines LHA from the DR longitude. He transforms 
> the usual LHA into a value less than 180º by subtracting from 360º. This 
> has also been called hour angle, H.A., and angle "t".
>
> Using Chichester's numbers:
>
>
>                                                 
> Hc                                                      Az
>
> LHA _35-48.5_E___   log hav LHA _  8.97548____          log sin LHA    
> 9.76716_______          
> Lat ___37-08.5_N_    log cos Lat   ___9.90154____ 
> Dec  __08-02_N____log cos Dec + __9.99572___            log cos Dec  
> _9.99572_______                   
>                                                       
> =__8.87274___               
>
>                                                 hav    __.07459____  
> (L ~ D) 29-06.5_____ hav (L~D)  + __.06315_____
>
>           89-60       
>  
> ZD -  43-34______<<<<  inv hav  =__.13774___>>>  log csec ZD  
> +_10.16166____      
>                                                
>  
> Hc __46-26_____           
>                  
> Ho-__46-23___________
>
> A____3 away___                     Z ___57_________<<<...inv log sin 
> ___9.92454____________
>
>
> This method required referring to nine different pages in the log tables 
> including interpolating three times (I used Nories Air Tables). It 
> required subtracting Dec from Lat;  adding three five digit numbers 
> twice; adding two five digit numbers one time; and finally subtracting 
> ZD from 90º.
>
> This compares to two pages of H.O. 249 and one addition of the two digit 
> minutes correction.
>
> So take your choice.
>
>
> I have attached these nine pages from Norie's so you can follow along 
> with the computation. I have also attached a text document of this 
> explanation since the column format of emails usually gets distorted.
>
>
>
> gl
>
>
>
> Gary J. LaPook wrote:
>   
>> I am attaching a marked a up version of page 235, page 29 from Volume 3 
>> of H.O. 249 and the correction table from that volume. To make it easier 
>> to follow this explanation I have marked up page 235 with red boxes 
>> labeled A through G.
>>
>> "A" show his computation of GMT or Zulu time for entering the Almanac. 
>> 12:11:22 is his watch time and the watch is obviously set to GMT. 02:30 
>> + 1/2 is the correction for his watch error which is running that many 
>> minutes _slow_ on GMT, Adding these two numbers produces the GMT of the 
>> observation of 12:13:52 (he dropped the extra half second.)
>>
>> "B" shows the computation of LHA _for _entry into H.O. 249. Taking the 
>> entry of the sun's GHA (Greenwich Hour Angle) for 12:00:00 GMT he takes 
>> out the GHA of 000º 53.1'. since the sight was taken 13 minutes and 52 
>> seconds after 12:00 o'clock you look in the increments table in the 
>> Nautical Almanac or in the Air Almanac and take out 3º 28' additional 
>> for that extra time which you add to the GHA at 12:00: to find the GHA 
>> of the Sun at the time of the observation to be 004º 21'. Chichester 
>> automatically added 360º to this and wrote down 364º21' to make it 
>> easier for the next step of subtraction his assumed longitude. Since the 
>> DR position is 43N -  25W ("C") he choses an assumed longitude of 25º 
>> 21' so when he subtracts this from the GHA he ends up with a whole 
>> number of degrees of LHA which is 339º.
>>
>> "C" shows the DR position.
>>
>> "F" shows the declination of the sun taken out of the Almanac at the 
>> same time that the value of GHA was obtained.
>>
>> "D" show the computation of Hc using H.O. 249. Looking on page 29 of 
>> volume 3 for latitude 43 and declinations 15-29 same name, we go down 
>> the column for 20º declination and come across from 339º LHA and we take 
>> out the tabulated Hc of 61º 02', the "d" correction value of +50 and the 
>> azimuth of 136. The tabulated Hc is for 20º declination exactly. Since 
>> the declination of the sun was actually 6 minutes more at the time of 
>> observation we must correct this tabulated Hc for this difference. We go 
>> to the correction table and under the 50 column (the "d" value) we read 
>> down to the 6 minutes of extra declination line and extract 5' which we 
>> add to the tabulated Hc of 61º 02' to determine the actual Hc of 61º 07'.
>>
>> "E" shows the computation of Ho. Starting with the HS (sextant altitude) 
>> of 60º 50' we add the correction  for semi diameter (this is obviously a 
>> lower limb shot), subtract refraction and subtract dip which Chichester 
>> has combined into one correction of + 13'. (The SD alone is + 16 and 
>> refraction for this Hs is - 1' making a +15'. Since Chichester uses +13' 
>> he is including a dip correction of -2'.) On the next line he applied 
>> the Index correction (IC) of + 1 to arrive at the Ho of 61º 04'.
>>
>> "G" shows him subtracting the Ho from Hc to arrive at the intercept of 3 
>> away since Ho was less than Hc.
>>
>>
>> The other examples he gives are done the same way although it is 
>> interesting that in three of the examples he also combines the IC with 
>> the other corrections to Hs writing down +14 total correction.
>>
>> I will get to page 234 tomorrow.
>>
>> gl
>>
>>
>>
>>
>>
>>
>>
>> Beverley Maxwell wrote:
>>   
>>     
>>> Gary,
>>> Thank you.  I am sending page 235, which Chichester refers to as the 
>>> short method, and page 234 he calls the long old method.
>>>  
>>> Frank M.
>>>
>>> ------------------------------------------------------------------------
>>> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
>>> 
.
>>>
>>> ------------------------------------------------------------------------
>>> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
>>> 
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>>>
>>> ------------------------------------------------------------------------
>>>
>>>     
>>>       
>>
>>     
>>   
>>     
>
>
> >
>   


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Now we will look at Chichester's illustration of the old method on page 234 to 
compare it with the short method already discussed.

What he is doing with this example is using the traditional Haversine- Cosine 
method of calculating Hc and azimuth. The formulas used for this were derived 
from the standard Sine - Cosine formulas and, in fact, uses the same method 
and formula for calculating azimuth.

The formula for calculating Hc is:

hav ZD = hav LHA cos Lat cos Dec + hav (Lat ~ Dec)

(Lat ~ Dec means the difference between latitude and declination, subtracting the 
smaller from the larger if of the same name and adding if of different names)

(ZD is zenith distance)

so Hc  = 90� - ZD

For calculating azimuth we use

sin Z = (sin LHA cos dec ) / cos Hc

usually rearranged into the more convenient form of

sin Z = sin LHA cos dec sec Hc

Since csec ZD is the same as sec Hc

we can rearrange this formula to

 sin Z = sin LHA cos dec csec ZD


Chichester used these formulas and solved them using logarithms by using this format:

                            Hc                                      Az

LHA ___________        log hav LHA  ___________        log sin LHA  ______________ 
         
Lat ___________        log cos Lat   ____________ 
Dec  __________        log cos Dec + ___________       log cos Dec  ______________ 
                  
                                    ______________               

                               hav _______________  
(L ~ D) ___________    hav (L~D)  + ____________ 

        89-60

ZD -  ___________<<<<  inv hav  _____________>>> >>>>>>log csec ZD  +___________  
    
                                               
 
Hc ____________           
                   
Ho-_____________
A______________                     Z ______________<<<>>>>>>>> log csec ZD 
+_10.16166____      
                                               
 
Hc __46-26_____           
                 
Ho-__46-23___________

A____3 away___                     Z ___57_________<<<<<

   

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