Thursday Eratosthenes Blogging: Measuring Latitude and Longitude with a Sundial

As I keep saying in various posts, I'm teaching a class on timekeeping this term, which has included discussion of really primitive timekeeping devices like sundials, as well as a discussion of the importance of timekeeping for navigation. To give students an idea of how this works, I arranged an experimental demonstration, coordinated with Rhett at Dot Physics. We've been trying to do this literally for months, but the weather wouldn't cooperate. Until this past weekend, when we finally managed to make measurements that allow us to do some cutting-edge science. For 200 BC, anyway...

So, what did we do? Well, we each made a sundial, and shot time-lapse video of it using a webcam. Here's mine-- note the Lego gnomon, graciously donated to science by SteelyKid (whose attempts to help with "Daddy's 'spermint" weren't enough to earn a co-author credit, but do rate this acknowledgement):

The too-bright first few frames are because I forgot to adjust the exposure initially, and the greying out at the end is because some thick clouds rolled in. This was shot in our back yard in Niskayuna, and simultaneously (in some frame of reference, anyway), Rhett was taking video of his own sundial, in Hammond, LA. I took both videos, and ran them through Tracker video to measure the position of the end of the shadow for each frame, and produced the following results:


You can see that both datasets show the expected behavior: the total length of the shadow: the shadow starts out long, and gets shorter, reaching its minimum length at noon. After solar noon, the shadow gets longer again.

I took these datasets and fit a parabola to each (not for any really sound theoretical reason, but because it was easy to do a parabolic fit in Excel, my demo version of SigmaPlot having expired). Using the fit values to find the minimum gave the time of solar noon as 12:10 for me, and 12:19 for Rhett.

I found the minimum length by averaging nine points near the minimum time from the fit (the exact point predicted as the minimum, and four frame to either side), and compared to the length of the gnomon to find the latitude. My Lego stack was 20.5 cm high, with a minimum shadow length of 37.6 cm; Rhett's gnomon was a nail with a height of 6.2 cm and a minimum shadow length of 7.2 cm.

The ratio of shadow length to gnomon length gives the tangent of the latitude, which come out to 61.4 and 49.3 degrees for my data and Rhett's, respectively. This is the latitude relative to the point on the Earth where the Sun is directly overhead at noon, though, and needs to be corrected to get the real latitude. The online nautical almanac gives the declination for the Sun on that date as approximately 18 degrees south of the equator. Using this correction, the latitudes determined from the measurements come out to 43.4 and 31.3 degrees, in reasonably good agreement with the known values (from Google Maps) of 42.8 and 30.5 degrees.

Finding longitude is, famously, more difficult, and relies on knowing the time of noon at your location compared to the time of noon at a reference location, conventionally chosen to be Greenwich, England. Looking up the time of noon at Greenwich on Jan. 29th, 2012 gives a time of 12:13 GMT. My measurement of 5:10 GMT (12:10 EST) is thus four hours and fifty-seven minutes off, giving a longitude of 74.3 degrees west. Rhett's one time zone west of me, so his data give a noon time of 6:19 GMT (12:19 CST), corresponding to a longitude of 91.5 degrees west. Again, these values are in reasonably good agreement with the accepted values of 73.9 and 90.5 degrees west, respectively.

The resulting positions aren't too bad. Using SunCalc, because it was easy to see how to put in latitude and longitude points, you can see that the data put my position and Rhett's somewhat north and west of our actual locations. The position difference is around 45 miles in my case, and 85 in Rhett's. So, it's not exactly GPS levels of precision, but then, I didn't work all that hard to get these measurements. A little more care would undoubtedly get closer to the mark.

So, we've accomplished a couple of things, here. First, with some household items and cheap webcams, we have conclusively demonstrated that the Earth is round, after the method of Eratosthenes. And we've also shown that with some simple tools and appropriate references, you can locate yourself on the surface of the Earth with pretty good accuracy.

There are some obvious ways to do a better job of this, but this was plenty good enough as a proof-of-principle for my class. And that's as much as I need for a blog post, too...

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Great experiment, I see a science fair project coming (once Minimu gets trig). But your determination of longitude is still dependent on having an accurate time for two points on the globe. How did they determine longitude before there were accurate clocks to take with you on a trip?

There is (at least) one other thing you could do with your data. You can use spherical trig to convert your latitude/longitude measurements to the arc angle of a great circle between Niskayuna and Hammond. Then use that arc angle and a known distance between the two to estimate the circumference of the earth. For a class of physics/math majors, you'd probably want to make them derive the relevent formulas themselves, but you could just give the formulas to this class.

You state, " And we've also shown that with some simple tools and appropriate references, you can locate yourself on the surface of the Earth with pretty good accuracy"

I would not call using webcams, the internet, the nautical almanac (online version at that) etc. simple! There's a couple of thousand years of the evolution of science and a hell of a lot of blood, sweat and tears invested in producing your simple things.

An interesting related calculation was done in ancient China: while Eratosthenes used the assumption that the Earth is round and the sun infinitely far away to compute the Earth radius from the gnomon distances and angles, ancient Chineses used the assumption that the Earth is flat to compute the sun distance from the same data.
It is for example discussed in
this academic peper (pdf), p 27 / 160.

By Frédéric Grosshans (not verified) on 02 Feb 2012 #permalink

How does one measure longitude with no accurate timepiece at your location (but one at Greenwich, a good almanac, and a good sextant/astrolabe)?

Jovian satellite transitions? Moon phase at horizon/zenith? Is this another exercise for your class?

Those results are excellent! Were you both careful to have the gnomon perfectly vertical? That would contribute to your uncertainty, as does the fact that the minimum is pretty flat even in winter.

1) Do you know about the book "Seize the Daylight"? Although supposedly about daylight savings time, it address the whole issue of standard time (including the distinction between the "mean time" of noon and the local time of noon that you noted was 13 minutes off at Greenwich on that date) along with the political decisions about choosing time zones. You are both surprising close to the center of your respective time zones!

2) Regarding @1: With great difficulty. A singular astronomical event (appearance of a moon of Jupiter or occultation of a star by the dark part of the Moon's disk) can be used to synchronize clocks at distant locations, so you can compare the time from that event to local noon.

3) Regarding @3: You only need the web cam if you lack the "time" to do it more carefully by marking points on a smooth level surface every minute. Comparison of the data sets can be done by mail. The biggest challenge is synchronizing your clocks (I LOL'd at "in some reference frame" aside) without using any of the methods available since the invention of the telegraph and radio. See #2.

By CCPhysicist (not verified) on 02 Feb 2012 #permalink

As I recall, Dava Sobel mentions, but does not explain in much depth, a method of determining longitude which was called "lunars." This involved compiling and carrying extensive almanacs of the moon's predicted position relative to the stars at various longitudes, and was considered a useful and viable alternative to the method requiring a chronometer.

By Patrick Dennis (not verified) on 03 Feb 2012 #permalink

Thanks, that explains how they got longitudes in the 16th century (before the Jovian moons were discovered). Also explains why Columbus didn't realize he was still half a world from Asia.

Wonderful project! I absolutely love it. One clarification though - Eratosthenes is crediting for using such a measurement to estimate the circumference of the earth, ASSUMING it was round (not proving that it was round), also requiring knowledge of the distance between the two sites (well, the distance of the north-south projection). You can actually get the same effect from a flat Earth and a nearby sun, where the different shadow heights (angles) give you the distance to the sun relative to the distance between points.

But even without all that (finding Earth's circumference), the project lets you get lat/long, as well as illustrating time-zone / UTC, and seasonal progression. You could repeat over the course of the year to measure tilt of Earth's axis.

What a great project!