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  ______________                   
                                    ______________               

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

        89-60

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____________