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Frank,

Thanks for the comprehensive reply.  I did not mean to imply that  
Bowditch's equations (as presented by Joel) would not work at the  
time of the equinoxes. Its just that I could think of one case where  
if both altitudes were zero (sunrise and sunset) it could not work.   
This is the well known emergency procedure of finding latitude by  
length of day, which gets less accurate as the equinoxes are  
approached.  I was wondering if the equations fell apart because  
declination was zero, or because the two altitudes were zero and/or  
equal.  Its mainly a math question pertaining to Joel's presentation.

Ken

Ken
On Jun 22, 2008, at 6:38 PM, frankreed@HistoricalAtlas.net wrote:

>
> 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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