The Sand Reckoner

Lore & Background

In The Sand Reckoner, Archimedes first invented a system for naming large numbers, as the existing number system could only express numbers up to a myriad (10,000). He called numbers up to 10^8 'first order' and 10^8 itself the 'unit of the second order,' continuing this pattern up to a myriad-myriad times the unit of the 108th order, which became the unit of the second period. He eventually arrived at the largest number named: 10^(8·10^16). Archimedes also discovered and proved the law of exponents, b^m b^n = b^(m+n), necessary for manipulating powers of 10.

Reader's Guide

The Sand Reckoner is significant as one of the few surviving references to the heliocentric model of Aristarchus of Samos, which posited that the Sun remains unmoved while the Earth orbits it. Archimedes used this model to estimate an upper bound for the size of the universe, assuming the universe was spherical and that the ratio of its diameter to the Earth's orbit equaled the ratio of the Earth's orbit to the Earth's diameter. He concluded the universe's diameter was no more than 10^14 stadia (about 2 light years) and that it would require no more than 10^63 grains of sand to fill it. The work demonstrates Archimedes' ability to handle extremely large numbers and his method of rounding up to ensure an upper bound, emphasizing his meta-goal of showing how to calculate with previously impossibly large numbers rather than achieving precise accuracy.

Did You Know?

• The Sand Reckoner is considered the most accessible work of Archimedes.
• Archimedes named numbers up to a myriad-myriad times the unit of the 108th order, which he called the unit of the second period.
• Archimedes estimated that the diameter of the universe was no more than 10^10 stadia, about 2 light years.