The Sand Reckoner - Estimation of The Size of The Universe

Estimation of The Size of The Universe

Archimedes then estimated an upper bound for the number of grains of sand required to fill the Universe. To do this, he used the heliocentric model of Aristarchus of Samos. The original work by Aristarchus has been lost. This work by Archimedes however is one of the few surviving references to his theory, whereby the Sun remain unmoved while the Earth revolves about the Sun. In Archimedes' own words:

His hypotheses are that the fixed stars and the Sun remains unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.

The reason for the large size of this model is that the Greeks were unable to observe stellar parallax with available techniques, which implies that any parallax is extremely subtle and so the stars must be placed at great distances from the Earth (assuming heliocentrism to be true).

According to Archimedes, Aristarchus did not state how far the stars were from the Earth. Archimedes therefore had to make an assumption; he assumed that the Universe was spherical and that the ratio of the diameter of the Universe to the diameter of the orbit of the Earth around the Sun equalled the ratio of the diameter of the orbit of the Earth around the Sun to the diameter of the Earth. This assumption can also be expressed by saying that the stellar parallax caused by the motion of the Earth around its orbit equals the solar parallax caused by motion around the Earth.

In order to obtain an upper bound, Archimedes used overestimates of his data by assuming:

• that the perimeter of the Earth was no bigger than 300 myriad stadia (~5·105 km).
• that the Moon was no larger than the Earth, and that the Sun was no more than thirty times larger than the Moon.
• that the angular diameter of the Sun, as seen from the Earth, was greater than 1/200th of a right angle.

Archimedes then computed that the diameter of the Universe was no more than 1014 stadia (in modern units, about 2 light years), and that it would require no more than 1063 grains of sand to fill it.

Archimedes made some interesting experiments and computations along the way. One experiment was to estimate the angular size of the Sun, as seen from the Earth. Archimedes' method is especially interesting as it takes into account the finite size of the eye's pupil, and therefore may be the first known example of experimentation in psychophysics, the branch of psychology dealing with the mechanics of human perception, whose development is generally attributed to Hermann von Helmholtz. Another interesting computation accounts for solar parallax and the different distances between the viewer and the Sun, whether viewed from the center of the Earth or from the surface of the Earth at sunrise. This may be the first known computation dealing with solar parallax.