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If you are dropped at an arbitrary point on the globe and you don't know the date can you determine your latitude based only on observations of the sun? (Since I don't think it is possible to determine the hemisphere, let the question be latitude relative to the equator.) How quickly can you do so?

My first guess is that if you knew the earth's angle of inclination relative to the sun you could in theory get an answer as quickly as you can measure the rate of the sun's transit of the sky. (Which with increasingly precise measurements would approach an instantaneous answer.)

Without knowing the date I don't think you can quickly tell the earth's current inclination. But I'm also not sure I'm thinking about this question optimally.

(Let's ignore the boundary case where you find yourself in a polar region at a time of year when the sun doesn't appear at all.)

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    $\begingroup$ not an answer to the actual question, BUT you don't need the sun at all, a Foucault Pendulum en.wikipedia.org/wiki/Foucault_pendulum can tell you your latitude by measuring the rate of rotation of the plane of the pendulum's swing $\endgroup$ Jun 3 at 19:04
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You can do it in one or two days. You'll know compass directions roughly, which is good enough, by observing the Sun's east-west path. You measure the Sun's altitude at meridian passage on two successive days. The difference from one day to the next is maximum on the equinoxes, when the Sun's GP crosses the equator, and minimum half way between them on the first days of winter and summer. Just knowing the altitude difference is enough to determine four possible dates. Knowing whether the Sun's altitude is increasing or decreasing from one day to the next removes two of those possibilities. For example, if the sun is south of you and the altitude is increasing, then the day is after the first day of winter and before the first day of summer, but you wouldn't know whether it was some number of days before the first day of summer or roughly the same number of days after the first day of winter. There are two ways of resolving that ambiguity. If you're far enough from the equator, step outside: much of the time the temperature is a good enough indication. If you're somewhere like Hawaii, where the temperature is fairly constant year round, then you can observe altitude at meridian passage on a third day. From that you can calculate the rate of change of the altitude difference. If the rate of change in the example is negative, the date is between the first day of spring and the first day of summer.

It's not a good way to find the date. The Sun's altitude only changes by 23.7 minutes of arc on the biggest change days, and 23.6 on the day after. If you're a dab hand with a sextant you might be able to find altitude to within a tenth of an arc minute.

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  • $\begingroup$ Clever! How you can tell "which way you were facing" in order to determine a single date from the year? I suspect this is the same problem – and for the same reason – mentioned in the question of not being able to tell which hemisphere you are in without some additional reference data point. $\endgroup$
    – feetwet
    Jun 3 at 20:07
  • $\begingroup$ I think I got it: The sun's path determines (or, one might say, defines) a unique hemisphere. "When facing the equator, the sun appears to move from left to right in the Northern Hemisphere and from right to left in the Southern Hemisphere." (And handedness can be unambiguously defined for anyone not made of antimatter.) $\endgroup$
    – feetwet
    Jun 3 at 20:44
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    $\begingroup$ @feetwet the hemisphere you're in isn't the discriminator. It's the difference between its declination and your latitude. For example, if you're at 10 degrees N latitude and the Sun's at 23 degrees N declination you'll see it move right to left. $\endgroup$
    – stretch
    Jun 4 at 17:06

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