A traveller starts out from the Earth's equator, heading exactly northeast. Undeterred by mountains, oceans and political boundaries, he continues on a northeasterly heading until he can go no further.

Where does he end up?

How far did he go?

How many times did he circumnavigate the earth? (For these purposes, this means travel through 360 degrees of longitude.)

(In reply to re: re Solution by TomM)

[TomM] I agree that because a sphere can't be flattened like a cylinder can, the "straight" trig answer does not apply.

You're right that straight trig does not rigorously provide the answer. For mathematical rigor in this situation, you need calculus. You observe that at every latitude, except exactly at the north pole, a infintesimal change in latitude requires exactly √2 times that must distance travelled. You then integrate. (Yes, I know that the calculus based on infintesimals is itself not very rigorous and has been replaced by a calculus based on limits, but it's much harder to come up with words to express these concepts in the limits-based calculus.)

[Cheradenine] I cannot get beyond the intuitive fact it is not possible to reach a certain point if never moving straight at it.

There are lots of examples to refute this. Consider, for example, a unit square. You're trying to get from the lower left corner to the upper right corner. You go 1/2 unit up, then 1/2 unit right, 1/4 unit up, 1/4 unit right, 1/8 unit up, 1/8 unit right, etc. After travelling exactly 2 units, you'll be at the upper right corner. But you were never travelling directly toward it.

Posted by Jim Lyon on 2002-10-02 08:01:31