The Sand Reckoner

From Academic Kids

The Sand Reckoner is probably the most accessible work of Archimedes, in some sense, it is the first research-expository paper. In this work, Archimedes sets himself the challenge of debunking the then commonly held belief that the number of grains of sand is too large to count. In order to do this, he first has to invent a system of naming large numbers in order to give an upper bound, and he does this by starting with the largest number around at the time, the myriad myriad or one hundred million (a myriad is 10,000). Archimedes' system goes up to

<math>10^{8* 10^{16}}<math>

which is a myriad myriad to the myriad myriadth power, all taken to the myriad myriadth power. This gives a good indication of the notational difficulties encountered by Archimedes, and one can propose that he stopped at this number because he did not devise any new ordinal numbers (larger than 'myriad myriadth') to match his new cardinal numbers. Archimedes also discovered and proved the law of exponents

<math> 10^a 10^b = 10^{a+b}<math>

necessary to manipulate powers of 10. Archimedes then sets about estimating an upper bound for the number of grains of sand. Not wanting to be outdone, he counts not only the grains of sand on a beach, but on the entire earth, the earth filled with sand, and then in a universe filled with sand. He then estimates this for the largest model of the universe yet proposed, the heliocentric model of Aristarchus of Samos (in fact, this now lost work is known due to this reference). The reason for this is that a heliocentric model must be much larger if stellar parallax is not clearly measurable. Archimedes proceeds by giving upper bounds for the diameter of the earth, the distance from the earth to the sun, and the diameter of the universe. In order to do this last step, he assumes that the ratio of the diameter of the universe to the diameter of the orbit of the earth around the sun equals the ratio of the diameter of the orbit of the earth around the sun to the diameter of the sun. This simply says that stellar parallax equals solar parallax, and one can interpret this as Archimedes' reason for using this assumption, which is not clearly explained in the text. The resulting estimate is that the radius of the universe is about one light year which is consistent with current estimates for the radius of the solar system. Archimedes' final estimate gives an upper bound of <math> 10^{64} <math> for the number of grains of sand in a filled universe.

Archimedes makes some interesting experiments and computations along the way. One experiment estimates the angular size of the sun, as seen from the earth. Archimedes' method is especially interesting as it may be the first known example of experimentation in psychophysics, the branch of psychology dealing with the mechanics of human perception, and whose development is generally attributed to Hermann von Helmholtz (this work of Archimedes is not well known in psychology). In particular, Archimedes takes into account the size and shape of the eye in his experiment measuring the angular diameter of the sun. Another interesting computation accounts for solar parallax, in particular, the differences in distance from the sun, whether taken from the center of the earth or from the surface of the earth at sunrise. Once again, this may be the first known computation dealing with solar parallax.

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