**Archimedes,The Sand-Reckoner (Arenarius), ch. 2 (sects. 1-4)**©- translated by Henry Mendell (Cal. State U., L.A.)

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[1] Given these as supposed, the following will also be proved, namely the diameter of the world is less than ten-thousand-times the diameter of the earth, and furthermore the diameter of the world is less than 100 ten-thousand-times ten-thousand stadia [100,0000,0000].

For since it is supposed that the diameter of the sun is not larger than thirty-times the diameter of the moon, but that the diameter of the earth is larger than the diameter of the moon, it is clear that the diameter of the sun is less than thirty-times the diameter of the earth. Again, since the diameter of the sun was proved to be larger than the side of a chiliagon inscribed in the greatest circle of those on the world, it is obvious that the perimeter of the mentioned chiliagon is smaller than one-thousand-times the diameter of the sun. But the diameter of the sun is smaller than thirty-times the diameter of the earth. Thus the perimeter of the chiliagon is smaller than three-ten-thousand-times [3,0000] the diameter of the earth.

[2] And so since the perimeter of the chiliagon is smaller than thirty-thousand-times the diameter of the earth, it is larger than three-times the diameter of the world.

For it has, in fact, been proved that because the diameter of every circle is smaller than a third part of the perimeter of every polygon which is equilateral and having more angles than the hexagon inscribed in the circle. The diameter of the world would then be smaller than ten-thousand-times the diameter of the earth. And so the diameter of the world has been proved smaller than ten-thousand-times the diameter of the earth, [3]while it is clear from this that the diameter of the world is smaller than 100 ten-thousand-times ten-thousand stadia. For since it is supposed that the perimeter of the earth is not larger than three-hundred ten-thousand (300,0000), but the perimeter of the earth is larger than three-times the diameter due to the fact that the circular-arc of every circle is larger than three times the diameter of every circle, it is clear that the diameter of the earth is smaller than 100 ten-thousand stadia (100,0000). And so since the diameter of the world is smaller than ten-thousand-times the diameter of the earth, it is clear that the diameter of the world is smaller than 100 ten-thousand-times ten-thousand stadia.

Summary

D(x) = the diamter of x, P(x) = the perimeter of x. In the text and addenda,
commas are placed every fourth digit, in mimicry of Archimedes usage. Here they
are placed every third, in accordance with our usage.

1 | D(sun) ≤ 30 D(moon) | Supposition |

2 | D(earth) > D(moon) | Supposition |

3 | D(sun) < 30 D(earth) | from 1, 2 |

4 | D(sun) > S(1000-gon in world) | Supposition-proved from observation |

5 | P(1000-gon in world) < 1000 D(sun) | from 4 |

6 | D(sun) < 30 D(earth) | see 3 |

7 | P(1000-gon in world) < 30,000 D(earth) | from 5, 6 |

8 | note: D(circle) = 1/3 6 Radius(circle) = 1/3 P(hexagon) < 1/3 P(n-gon, for n > 6) | |

9 | P(1000-gon in world) > 3 D(world) | from 8 |

10 | D(world) < 10,000 D(earth) | from 7, 9 |

11 | P(earth) ≤ 3,000,000 stadia | Supposition |

12 | P(earth) > 3 D(earth) | Weaker supposition than the Dimension
of the Circle, but standard in measurements and enough for present purposes |

13 | D(earth) < 1,000,000 stadia | lines 11, 12 |

14 | D(world) < 10,000 D(earth) | see 10 |

15 | D(world) < 10,000,000,000 stadia | lines 13, 14 |

[4] And so I suppose these concerning the magnitudes and distances, while concerning the sand these. If there is a magnitude composed from the sand not larger than a poppy-seed, the number of it is not larger than ten-thousand, while the diameter of the poppy-seed is not larger than a fortieth-part of an inch [lit. finger].

I suppose this after examining it in this way. Poppy-seeds were placed on a smooth ruler in a straight line, lying one by one and touching one another. 25 poppy-seeds took up more place than an inch length. And so, having put the diameter of the poppy-seed as smaller, I suppose it to be about a fortieth-part of an inch and no smaller, since I wish also through these to prove indisputably what's proposed.

Summary

10000 grains of sand = 1 poppyseed

40 poppy-seeds = 1 inch in length

Hence, 1 inch x 1 inch = 10000 x 40 x 40 = 1600,0000 grains of sand