Free Thinking in Numbers: How Maths Illuminates Our Lives by Daniel Tammet Page B

escapes numbering’. He was expressing the same idea that would lead his fellow Greeks to coin the term ‘sand hundred’ for an inconceivably great quantity.
    Pindar’s claim remained unassailable for some two centuries, which of course is not bad at all as far as a line of poetry goes. The eventual refutation, composed in the middle of the third century BCE , can be fairly listed among the finest achievements of the mathematician Archimedes.
    Introducing his academic paper– the first in recorded history – to the king of his day, Archimedes made a spectacularly audacious argument.
     
Some people believe, King Gelon, that the grains of sand are infinite in number. I mean not only the sand in Syracuse and the rest of Sicily, but also the sand in the whole inhabited land as well as the uninhabited. There are some who do not suppose that they are infinite, but that there is no number that has been named which is so large as to exceed its multitude . . . I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us . . . exceed . . . the number of grains of sand having a magnitude equal to the earth filled up.
     
    Archimedes’s estimations for the measurements of the Earth, moon, sun, and the other stars were generous: for example, making the Earth’s perimeter ten times larger than the calculations of earlier astronomers. Similarly, Archimedes went to great lengths to provide a capacious margin for error concerning the estimated size of a grain of sand. He compared ten thousand grains to the scale of a poppy-seed, and then patiently lined the seeds end to end on a smooth ruler. In this way he measured the number of poppy-seeds required to reach an inch as being twenty-five. This figure he adjusted still further, changing it to forty seeds per inch-length, so as to ‘prove indisputably what is proposed’. Thus he calculated as sixteen million (10,000 x 40 x 40) the maximum number of grains of sand that could fill one square inch.
    Archimedes assumed that the universe was spherical. He estimated a value for the diameter of the universe using calculations for the diameter of the Earth’s orbit around the sun. The universe, according to his reckoning, had a diameter no greater than 100,000,000,000,000 stadia (about two light years). 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grains of sand would more than saturate the whole of space.
    Next, Archimedes showed that the Greek term ‘myriad’ (ten thousand or a hundred hundreds) more than sufficed for the purpose of counting even the largest worldly quantities. The phrase ‘myriad myriads’, he pointed out, allowed the counter to reach the equivalent of one hundred million – the largest named number in his time. But, he continued, if it is possible to count in myriads, it should be equally possible to count in ‘myriad myriads’ so that multiplying the latter by itself the counter could attain ‘myriad myriad myriad myriads’ or 10,000,000,000,000,000. And considering this new figure likewise as a unit, as respectable as a ‘myriad’ or ‘myriad myriads’, the counter could multiply ‘myriad myriad myriad myriads’ by itself and proceed to: ‘myriad myriad myriad myriad myriad myriad myriad myriads’ or 100,000,000,000,000,000,000,000,000,000,000.
    Up till now we have multiplied a myriad by itself a total of eight times. Archimedes’s next step possessed all the elegance of simple logic: multiply a myriad myriads by itself myriad myriads times over. The ‘1’ that starts the resulting number is tailed by eight hundred million zeroes.
    Doggedly pursuing his logic, Archimedes proposed multiplying this new number by itself up to as many as a myriad myriads times over: a number requiring the insertion of eighty quadrillion (80,000,000,000,000,000) zeroes after the one.
    Archimedes concluded his paper in confident, if understated tones.
     
King Gelon,