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George Huxtable wrote in NavList 1314 (with a correction in 1323): 

"But what solutions do NavList members arrive at, I wonder, if they pick
up that same challenge, using any method they think appropriate?"

The problem was defined as:

"1. An observer, at position P1, measures the altitude of a star S1,
at(Dec1 = 0, GHA1 = 0), to be 30 degrees.
2. Then he travels due North by 60 nautical miles (= 1 degree), to
P2.
3. From there, he observes another star S2 (then at Dec2 = N 1 
degree, GHA2 = W 45 degrees) to be at an altitude of 45 degrees.
Where on Earth is he then?"

---
Assume position P1 to be located at Lat, Long. Then P2 will be at
Lat+1d, Long. 

We use the altitude formula sinAlt=sinLat*sinDec+cosLat*cosDec*cosLHA,
where LHA=GHA+Long, GHA is counted westwards and longitude eastwards.

Now, at P1
sin30d=sinLat*sin0d+cosLat*cos0d*cos(0d+Long)
which can be simplified to
1/2=cosLat*cosLong

At P2 we have
sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d*cos(45d+Long)

Now let's make a guess: Long=-45d, i.e. 45 degrees westerly longitude.
With this guess we find that 
sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d=(cosLat-cos(Lat+2d)+cos(Lat+2
d)+cosLat))/2=cosLat

The last equation can be simplified to
1/sqrt(2)=cosLat, and then Lat=+/-45d. This result satisfies the
equation at P1 as well, and thus we are quite sure it is correct, but in
order to be fully convinced we can verify the solution(s) by calculating
the altitudes:

sinAlt1=sin45d*sin0d+cos45d*cos0d*cos(0d-45d)=(cos45d)^2=1/2 => Alt1=30d
sinAlt1=sin(-45d)*sin0d+cos(-45d)*cos0d*cos(0d-45d)=(cos45d)^2=1/2 =>
Alt1=30d
sinAlt2=sin46d*sin1d+cos46d*cos1d*cos(45d-45d)=cos45d=1/sqrt(2) =>
Alt2=45d
sinAlt2=sin(-44d)*sin1d+cos(-44d)*cos1d*cos(45d-45d)=cos45d=1/sqrt(2) =>
Alt2=45d

The altitudes are correct, thus the guess was right and the two
solutions are:

P1 45N, 45W
P2 46N, 45W
or
P1 45S, 45W
P2 44S, 45W
---

Lars 
59N 18E
  






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