Number of sand: the number of grains of sand it would take to fill the universe, as calculated by Archimedes. The number of sand was often used by the Greeks as a trope for infinity. Archimedes introduces his treatise on the topic with these words:
"Some people believe, King Gelon, that the number of sand is infinite in multitude [...] There are some who do not suppose that it is infinite, and yet that there is no number that has been named which is so large as to exceed its multitude."*
To calculate this number, Archimedes had to tackle two problems. First, he had to figure out the volume of the ►universe. Second, he had to develop a way of expressing the number of grains of sand in the first place. This second task was even more of a challenge than the first, for the Greeks were not yet familiar with decimal notation. The largest number that Greek mathematicians could use was one myriad myriads or, in contemporary notation, one hundred million (100,000,000 or 108).
Archimedes found a strategy that would be revived more than two thousand years later by ►Georg Cantor in the mathematics of transfinite ►ordinal numbers. He called this largest known number, that is, 108, a "number of the first order" and assigned it its own numerical symbol, namely, ω (Omega). He then used ω in turn as a basic unit for composing numbers of the second order and continued counting with the help of this number. This enabled him to count to 1016, which he then treated as a new unit, ω2. With this unit he went on to count numbers of the third order ω3 , and so on.
The idea of turning a number, that is, a multitude, into a unit, then continuing to count with that unit just as one had with the number 1, was revolutionary at the time. With the introduction of that idea, the decimal system was but a small step away (see ►number systems); all that was missing was the number zero. However, although zero had been invented by Indian mathematicians long before, it would be another thousand years for it to be introduced to Europe by Islamic scholars as a basis for number systems.
To determine the radius of the universe, Archimedes used the distance between the Earth and the sun, though he wound up overestimating that difference. In this way he arrived at 1063 as the number of grains of sand that would fill the universe. Incidentally, this quantity of sand comes relatively close to the matter known today to fill up the ►observable part of the universe, which contains 1078 atoms.
* Archimedes, The Sandreckoner, trans. by Henry Mendell, http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/SandReckoner.html