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Ken, you wrote:
"Regarding Joel Silverberg's talk on latitude by double altitudes, I asked 
the question of how this works on the two days of equinoxes.  On these days 
there will be 12 hours between the zero altitudes of sunrise and sunset at 
all latitudes.  So how could a unique latitude be determined?  Perhaps Joel 
will give us an answer to this." 

I don't know if Joel is following the list regularly right now, so I hope 
you don't mind if I address this. 

Latitude by double altitudes (as originally understood c.1750-1850) is an 
observation composed of two different altitudes of the same body and the 
time interval between the sights as recorded by a common watch. Watches 
sufficient for this purpose were available and commonly carried by officers 
at sea from the beginning of the 18th century. 

As an example, on March 25, 2008 I see the Sun low in the east (let's assume 
it's rather close to true azimuth 90) a little after sunrise. I measure its 
altitude and get 5� 00'. Exactly thirty minutes later, I measure its 
altitude again and get 10� 00'. Both altitudes have already been corrected 
for dip, refraction, etc. What's my latitude? Well, even without calculation 
you can take a good guess. Obviously if I'm on the equator near the equinox, 
the Sun would be rising at nearly 15 degrees per hour. If I'm near the pole, 
the Sun would hardly change its altitude in an hour. Instead, in this case, 
we're seeing a rate of 5� in half an hour or 10� per hour. Since it's rising 
more slowly than the equatorial rate, it must be climbing at an angle 
relative to the horizon, probably close to 45 degrees. And in fact, the 
instantaneous rate of change of altitude in the general case is just 
(15�/hour)*cos(Latitude)*sin(Azimuth). This implies that our latitude must 
be around 48� since the azimuth is near 90�. Notice that it doesn't depend 
on the length of the day. But there IS a major problem with this method. 
When the observer is near the equator, the rate of change of altitude is 
going to be very close to 15� per hour over a wide range of latitudes (near 
the equinoxes). So "latitude by double altitudes" is not very useful near 
the equator. Note that I am simplifying here. The actual process for 
clearing this sight and getting a latitude is more elaborate (see the method 
in Bowditch which Joel outlined in his talk), primarily because we're 
looking at the change in altitude over a significant time interval instead 
of the instantaneous rate, but the principle is basically the same. Also 
note that we can get a complete position fix from these sights, not just 
latitude. A 19th century navigator could use either of those altitudes as a 
"time sight" to get longitude, too, assuming he has access to GMT. 

Another approach when dealing with these old 19th century methods is to turn 
them into 20th century equivalents, which are almost always more general. 
How would I use those two altitudes above with modern sight reduction 
techniques? Each sight generates an ordinary LOP. If the Sun is near the 
prime vertical, the LOPs will be running more or less north-south so either 
one yields a good longitude (that explains the time sight aspect of the 19th 
century approach). If there is sufficient change in azimuth between the two 
sights, the LOPs will cross at an angle and we can also get a latitude out 
of the pair. If we're near the equator, especially when near an equinox, and 
the Sun is rising on a nearly constant azimuth, we will get only a very 
small angle between the LOPs even if we wait an hour or more between sights 
so the ability to fix latitude would be much reduced. And that's really all 
there is to it. Unfortunately for the progress of navigation, 19th century 
navigators and mathematicians had a very hard time seeing the advantage of 
using LOPs as in Sumner's method. They saw it as a "trick" or a cheap 
substitute for the spherical trigonometry of the double altitude method. 

 -FER 



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