The sand reckoner
How many grains of sand would you need to fill the universe? Archimedes shows off his number prowess in the 3rd century BC. Let us hear his argument and consider also other such fantastical questions.
How many grains of sand would you need to fill the universe? How much coffee is made each day in New York City? Would it fill the Statue of Liberty? If you stacked quarters to the Moon, how much would it be worth? Would they be more massive than the Earth? Would the coins fit in Central Park? How much does the Empire State Building weigh? How many babies are born each day in New York? If you gathered all the hot air you have breathed in your life into a hot-air balloon, would it lift you and all your possessions?
A universe of sand
How many grains of sand would you need to fill the universe? This whimsical question is the principal concern of The Sand Reckoner, a remarkable treatise of Archimedes written in the 3rd century BC.
Ostensibly focussed on the calculation of a definite numerical answer, the treatise also advances what is clearly a further underlying goal for Archimedes, namely, to showcase his number representation system and its capacity for handling extremely large numbers. This was powerful new number technology, transcending the comparatively primitive Greek number system of his time—it amounts presciently to a place-value system in base myriad, or 10,000. My reading of the text is that Archimedes was eager to show off his agility with large numbers, and the grains-of-sand problem was an entertaining choice to exhibit his powers of calculation with numbers far larger than others had previously considered.
Archimedes sets out his project, dedicated to Gelon of the royal house of his native Syracuse, like this:
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