May 23, 2017

Measuring the Circumference of the Earth: the Eratosthenes Method

† This is one I duplicated in High School that I first heard about on Carl Sagans’ Cosmos series.† Way back in the 3rd Century BC, the Greek philospher Eratosthenes of Cyrene devised a method of calculating the circumference of the Earth.

†††While studying at the Library of Alexandria, he came upon the a curious fact.† Apparently, on the date of the summer solstice, the sun passed directly overhead at the southern town of Syene, in†what is present day Aswan. It would†shine directly down wells, and nothing would cast a shadow due to the sun being at the zenith.

†† Eratosthenes†wondered if the same would happen at Alexandria.† To his suprise, objects there cast a 7 degree 12′ shadow at noon on the solstice.†He correctly deduced that a curved Earth would account for the discrepancy, the angle from Syene on the Tropic of Cancer at north latitude 23 degrees†26′ to Alexandria comprising about 1/50th of the Earths’ total circumference.† The only remaining factor is to measure the mileage from Alexandria to Syene. Eratosthenes†employed a slave to pace out the distance of 800 kilometers.† This computes to 50 x 800=40,000 kilometers, very close to the currently accepted distance of 41,425 kilometers. (The Earth isn’t perfectly round; its slightly oblate at the poles and bulges a tiny bit below the equator, but thats another discussion!)†Thats’ pretty good, by classical or modern standards!

†† This can be done by simply observing the angle of the sun when it transits on the day of the summer solstice.† One complicating factor;† with the advent of time zones and daylight savings, the sun may not transit (defined as passing through the meridian, or the imaginary north/south line in the sky) at local noon.††one way find the local transit time would be to find the†midday point between sunrise and sunset.† For example, today at St. Froid lake in northern Maine, the sun rises at 4:45 AM and sets at 8:35 PM, giving†a span of daylight 0950 minutes long. Divided by two yields†475 minutes, or 7 hours and 55 minutes. this added to the sunrise time yields a local transit time of 12:40 PM local. I know, I can hear legions of sundial fans saying; “what† about sun fast/slow”†calculations?”† Again thats’ another discussion.† Your goal here is to duplicate Eratosthenes, not set time on your atomic clock. Incidentally, this factor is known as the Equation of Time, and it equals zero†four times a year; April 16th, June 14th, September 2nd, and December 26th.†All of these are close to the two each†yearly equinoxes and solstices; this factor would only affect the transit of the sun by about a minute.† The path the sun circumscribes through out the year as a result of this is called an

†Analemma.†(See link)

†††http://antwrp.gsfc.nasa.gov/apod/ap070617.html

†† Another method could be to simply start measuring the angle of the sun at 11AM at five minute intervals and note when the sun is highest above the horizon. That would be the†local solar transit time. For equipment, I used a 60mm refractor with homemade setting circles; something as simple as a straw on a plastic protractor with a line bob would do. Always remember, if using a telescope, do not look directly at the sun; serious eye damage could result!

†† Now that you’ve got your one field observation,†its math time.† Lets say you live in Buffalo, New York, and the shadow observed by the transiting summer solstice sun was 19 degrees, 27′ from zenith. Don’t forget: their are 60 seconds in a arc minute; 60 arc minutes in a degree; and 360 degrees in a circle. This can be inferred to be the same angle that Buffalo sits above the Tropic of Cancer. This is about 1/18.5 of a circle (360 divided by 19.5; numbers have been rounded to protect†the innocent!)

†† All that remains is to find the distance from Buffalo to the Tropic of Cancer.† In 1982 when I originally did this experiment, I measured the distance on a map;† today, its possible to do it via computer, with no distance pacing slave required.† A good tool for this is Google Earth; with this program, you can simply use the Add tool to show the latitude/longitude lines and the Measure device to plot a line from Buffalo to†the Tropic in kilometers or miles.† I came out with approxiamately 2,160km. This , multiplied by 18.5, yields of circumference of 39,960 miles, within 40 miles of the Erarosthenes figure.

† A link below discribes the original experiement;†††††††††††††††††††††††††

http://en.wikipedia.org/wiki/History_of_geodesy

†† For taking the actual measurement of the sun’s local elevation†a device†costructed entirely of wood (for the mount),†cardboard (for the tube), a protractor for†measuring, and a small amount of metal (the axis and plum weight)†can be used.† It†can be †constructed in about two hours using common household items, and it yields an accuracy of about 1 degree of arc. A larger device could yield a higher accuracy.† A downloadable protractor can be found and printed at;

http://www.ossmann.com/protractor/

Be sure to precisely mark the central axis of the tube used to measure against the protractor; this can be done by measuring the diameter of the tube.† The shadow of the tube is used to point the instrument at the sun; photos of this are to come!

† Finally, you don’t have to wait for the summer solstice to do this experiment; it can also be done on the Vernal or Autumnal Equinox, when the sun is directly over the equator,†or the Winter Solstice, when the sun is directly overhead on the Tropic of Capricorn.† Just keep in mind to measure from†the correct latitudes on these dates.††

The Farmers Almanac is a good source of these dates;†††

http://www.almanac.com/astronomy/seasons.php

†† So try this one†next equinox/solstice;†build a solar altimeter, invite some flat earth advocates over, and measure the circumference of the Earth. A Tequila Sunrise might even be in order!

Trackbacks

  1. [...] or magnitudes of celestial objects could be measured. Similarly, Eratosthenes calculated the circumference of the Earth by observing the difference in the angle of the noon day Sun on the summer solstice at [...]

  2. [...] noon on the day of the solstice, the Sun shined straight down a local well, and cast no shadows. He went on to correctly deduce that the differing shadow angles between the two locales is due to the [...]

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