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    Re: Fw: Chichester's Calculations ( Sine-Cosine method)
    From: Gary LaPook
    Date: 2008 Dec 10, 02:47 -0800

    
    
    I decided to work the same example using the standard Sine- Cosine 
    method to compare it to the Haversine-Cosine method. I discovered that 
    it takes fewer table entries and additions than does the Haversine 
    method so I do not know why that method became the prefered method, any 
    ideas?
    
    I am posting and example of this computation. I am also attaching a PDF 
    of it to ensure that the format doesn't become corrupted.
    
    The normal formulas for computing Hc and azimuth are:
    
    sin Hc = sin Lat  sin Dec + cos Lat cos Dec cos LHA
    
    and
    
    sin Z = (sin LHA cos Dec)/ cos Hc
    
    rearranged to the more convenient
    sin Z = sin LHA cos Dec sec Hc
    
    These can be solved using logarithms  using this format:
    
    
                                                   
    Hc                                                                          
    Az
    
    Lat _________  log sin Lat   ________  log cos Lat  __________          
    Dec _________ log sin Dec  + ______  log cos Dec  _________ log cos Dec  
    ________ 
    LHA  _______ >>>>>>>>>>>>>>>   log cos LHA + _______  log sin LHA  
    _______               
       
                                               
    _________                         _________
    
                                inv log    ________         inv log     
    __________
                                                                     +
                                                     
    .........................>>> ___________
    
     
    Hc _______________<<<<<<<..................inv sin    _________           
    
                       
    ......................................................>>>>>>>>> 
     >>>>>>log sec Hc +  _______
    
    Z 
    _________________<<<<<<<<<..................................................inv 
    log sin __________
    
    
    
    
    
    
    ----------------------------------------------------------------------------------------------------------
    
       
    Using Chichester's numbers:
    
    
                                        
    Hc                                                      Az
    
    Lat 37-08.5 N_  log sin Lat    9.78088____ log cos Lat   
    9.90154______         
    Dec 08-02 N__ log sin Dec  +9.14535___ log cos Dec   9.99572______ log 
    cos Dec 9.99572  
    LHA 35-48.5__  >>>>>>>>>>>>>>>>> log cos LHA +9.90901_____log sin LHA  
    9.76716
                                                 
    _8.92623____                       9.80627_______
    
                             inv log    ____.08438______ inv log    
    ___.64010_______
                                +
                                                               
    ....................>>> __.08438_______
    
     
    Hc 46-26__________<<<<<<<........................inv sin  
    __.72448_______           
    
                       
    ......................................................>>>>>>>>>>>>>>>   
    log sec Hc + 10.16166
    
    Z 
    __57.2___________<<<<<<<<<..................................................... 
    .inv log sin __9.92459
    
    Note that the azimuth is solved the same way using either method. This 
    azimuth formula is the same one that George complains gives ambiguous 
    answers when near 90º but this is the standard formula given in the 1928 
    edition of Bowditch for the Haversine- cosine and for the Sine-Cosine 
    methods.
    
    
    
    This method required referring to only eight pages of trig tables and 
    log tables, the addition of
    three six digit numbers twice, and the addition of two six digit numbers 
    twice.
    
    The Haversine 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.
    
    
    They can also be solved using the following key stroke sequence on a 
    calculator with three
    memories:
                        
    
    Assumed Lat
    2nd DMS-D.D (changes to decimal degree format)
    STO 1  (stored A. LAT in 1)
    Declination
    2nd DMS-D.D
    STO 2   (stored DEC in 2)
    GHA
    2nd DMS-D.D
    -
    Assumed Longitude
    2nd DMS - D.D
    = (computed LHA)
    STO 3   (LHA stored in 3)
    COS
    X
    RECALL 2  (recalled declination)
    COS
    X
    RECALL 1  (recalled Assumed latitude)
    COS
    +
    RECALL 1 (recalled A. LAT)
    SIN
    X
    RECALL 2  (recalled DEC)
    SIN
    =
    2nd SIN  (ARCSIN, computed Hc)
    
    2nd D.D-DMS (changes decimal degree Hc to degree-minute-second so it
    can be written down)
    
    2nd DMS - D.D  (changes it back)
    
    
    1/x   (converts COS Hc to SEC Hc)
    RECALL 3  (recalled LHA)
    SIN
    X
    RECALL 2  (recalled DEC)
    COS
    =
    2nd SIN  (ARCSIN, computed Z)
    
    *
    gl
    
    
    
    Gary LaPook wrote:
    > 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 ______________<<<
    >
    >
    >
    >
    >
    > 
    ---------------------------------------------------------------------------------------------------------
    >
    > Corrected form:
    >
    >                             Hc                                      Az
    >
    > LHA ___________        log hav LHA  ___________        log sin LHA  ______________          
    > Lat ___________        log cos Lat   ____________ 
    > Dec  __________        log cos Dec + ___________       log cos Dec  ______________                   
    >                                     ______________               
    >
    >                                hav  _______________  
    <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<  corrected line (could use inv log instead)
    > (L ~ D) ___________    hav (L~D)  + ____________ 
    >
    >         89-60
    >
    > ZD -  ___________<<<<  inv hav  _____________>>> >>>>>>log csec ZD  
    +___________    <<<<<<< this line is correct  
    >                                                
    >  
    > Hc ____________           
    >                    
    > Ho-_____________
    > A______________                     Z ______________<<<
    >
    >
    >
    > gl
    >
    >
    >
    > Gary LaPook wrote:
    >   
    >> 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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