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Brad and Lu,

I also think that these simple methods are not as simple if you try to
implement them in real life.

a) Taking the altitude of Polaris.
For this you need to divide a circle, say to 1 degree accuracy.
Can you do this without instruments? I cannot. I need at least a divider,
and even then this is a highly non-trivial task.
Alternatively, you can make an instrument of straight sticks (kind of
cross staff). Then the same problem arises: to measure the length
of these sticks. Dividing a ruler is no simpler task than dividing
a circle.
All this is BEFORE you "use trigonometry". But how exactly you use
trigonometry without a calculator/tables? You compute a table of sines
and or tangents? With Ptolemy method, or using a Taylor series?

It is true, you will need the tangent of only one angle when measuring
Polaris altitude with a cross staff, but the problem of making a ruler
remains.

I already mentioned on this list a Russian captain Golovnin who spent 10
years in captivity in Japan in the early XIX century. He taught the Japanese
some modern navigation, and
his mate computed tables of trig functions for the Japanese.
I wonder how many modern captains or mates are capable of doing this:-)

b) Using Sun. Certainly the method is not practical on a ship.
To find the noon altitude of the Sun with a gnomon (vertical tick),
you need a flat horizontal ground, a plumb line to install your stick,
then you probably have to observe for several days to find the noon
altitude on a certain day.
Declination does not seem to be such a big problem in principle:
you can compute it approximately from the dates of the solstices
and the angle between the equator end ecliptic.
(Assuming that Sun rotates on a circle not an ellipse, as the ancients
did). If you can make trigonometric tables, then you probably can
do this as well.

With any of these methods, determination of latitude, say to 1-2 degrees
will take many days and only possible from land.

By the way, all this is described with some details in Jules Verne novel
Mysterious Island. According to Jules Verne, Cyrus Smith used his height
(which he knew precisely) as a unit of length. But the details how he
divided this unit into smaller units are omitted:-)
Jules Verne apparently did not understand that to measure an angle
you can use arbitrary unit of length, and its relation to feet or
inches is irrelevant. The main difficulty is in dividing this unit
with sufficient accuracy.

Alex.


> Brad:
>
> I must respectfully disagree.   If I plant a stick in the ground and are
> willing to measure its shadow for a full year (or at least from one
> solstice to the other), I can indeed deduce my latitude.   If I want to
> do it in less time (say in just a day or in a week) I must have a
> declination table  and an idea of what date it is.   While the latter
> was not necessarily forbidden by the statement of the problem, the former
> certainly is.
>
> I had originally wondered if I could have applied the Rule of Twelfths to
> approximating the sun's declination for any particular date, but it turns
> out that the curve of the sun's declination is NOT a sine wave (nice
> explanation in Wikipedi

   

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