Archimedes’ Estimation

by James Buckland


Archimedes, famed Greek scientist/philosopher, once wrote a research paper, aptly titled The Sand Reckoner, in which he assertained the maximum number of grains of sand that could fit in the universe. This was, of course, totally ridiculous and inaccurate, but, as with most science of antiquity, it was the process, not the result, that produced the most fruitful work, such as hitherto unknown methods of counting extremely large numbers.

The motivation for this work lies in a contemporary philosophy, the assumption that sand was infinite, uncountable. In early systems of numerical measurement, counting systems often reached an upper bound at 1000 or so, anything greater being of little use in daily life and therefore being easily described as uncountable. This innate human instinct, incredibly, hasn’t died out since Ancient Greece. This is an easily acceptable proposition for the unscientific mind, but apparently Archimedes had other intentions.

The Sand Reckoner, in totality, calculates the number of grains of sand that could fit inside the universe, which was at the time defined roughly as a sphere whose center was the center of the Earth and whose radius was the distance between the centers of the Earth and Sun. Ultimately, he derives an approximation of 1063 grains of sand.

Let’s check his work. The number of grains of sand N would be equal to \dfrac{M}{m}, where M is the mass of a sand-filled universe, and m is the mass of a single grain of sand. Thus,

N = \dfrac{M}{m} = \dfrac{V\rho}{m} = \dfrac{\frac{4}{3}\pi r^3 \rho}{m}.

Given the mass of a grain of sand m = 6.48\times10^{-5} \text{ kg}, the radius of the given sphere r = 1\text{AU}\approx 1.5\times10^8\text{ km} , and the density of sand \rho = 1442\text{ kg/m}^3, we find N = 8.09 \times 10^{42} grains of sand, which is not so bad, for an ancient greek. Much of the disparity between our calculations has to do with our varying definitions of the size of the universe — his calculation also accounted for the sphere enclosing the paths of the distant stars, which he thought were not so much further from Earth than the Sun’s orbit.

There is another oddity — this calculation’s relevance to the Eddington number, a calculation for the number of protons in the universe, which is currently pegged at around 1080 protons; given the number of protons in a grain of sand, we find that Archimedes’ 1063 grains of sand in the universe times an approximate 1022 protons in a grain of sand gives us a number not so unlike the Eddington number, off by a factor of only 105. This is… a coincidence, but an extraordinary one.