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6. How Many Grains of Sand Does It Take To Fill the Universe
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51 HOW MANY GRAINS OF SAND DOES IT TAKE TO FILL THE UNIVERSE? VI In his short treatise entitled The Sand Reckoner, Archimedes proposes to count the grains of sand contained in a sphere that has the sun at its center and has the sky as its periphery, demarcated by fixed stars. To this end, Archimedes had to devise a numbering system capable of expressing very large numbers in some way equivalent to our binary calculus or, in fact, closer to what we call exponential notation [fig. 18]. He came to the following conclusion: It is clear therefore that the fullness of sand, having its volume equal to the sphere of fixed stars, as Aristarchus hypothesizes, is less than 1,000 thousands of the 8th order of numbers.1 And these, O King Gelon, I understand not to appear to be easily accepted by the rabble who have no familiarity with calculations, but rather by those who have considered the distances and sizes of the earth and the sun and the moon and the entire cosmos, that they may regard them as believable on account of 52 THE GREAT ARCHIMEDES VI Figure 18 Frontispiece of the first modern edition of Archimedes, published at Basel in 1544. It represents the Greek text of both works of Archimedes known up to that time, including the late antique commentary of Eutocius and a complete Latin translation. Thereafter the base of readers of Archimedes throughout Europe would expand and, within a few years, would form a daringly innovative scientific community. [3.239.214.173] Project MUSE (2024-03-19 04:17 GMT) HOW MANY GRAINS OF SAND DOES IT TAKE TO FILL THE UNIVERSE? 53 VI my demonstration. Therefore I thought that it would not be inappropriate for you to consider these things. Unlike his other works, this one is not dedicated to a scientist but to a ruler of Syracuse, with whom Archimedes clearly enjoyed a good deal of familiarity. This work turns out to be quite different in terms of style and content, and, in terms of its goal, it catches the reader off guard, virtually disorienting him with “a style of suspense and surprise.”2 To that end, if the problem ostensibly was to calculate the number of grains of sand necessary to fill the entire universe, Archimedes chiefly must have sought to calculate its size in terms of space. As Dijksterhuis has noted, “The work The Sand-Reckoner, though meant by the author as a contribution to Greek arithmetic, owes its historical interest not only to what it contains as such; it is no less valuable as a document of Archimedes’ astronomical activity. It was of course to be expected that he engaged in astronomy, though he has not left any work exclusively devoted to it: astronomy and mathematics in his day were scarcely distinguished as two different branches of science.”3 It is interesting how, at the opening of The Sand Reckoner , Archimedes gives the measurement, essentially correct , even if done with simple means, of the angle according to which the sun is visible in the sky (between 32 and 27 sixtieths of a degree): The diameter of the sun is approximately 30 times greater than the diameter of the moon and not more . . . in addition, the diameter of the sun is greater than the side of a thousand-sided polygon inscribed on the biggest circle that the cosmos could possibly contain. This calculation is rendered by means of optics, a science which forms a bridge between geometry and astronomy . In his demonstration, Archimedes explicitly refers 54 THE GREAT ARCHIMEDES VI to Aristarchus, the astronomer who, at the beginning of the third century BC, first formulated the hypothesis for the heliocentric solar system4 and had found that the sun occupies approximately the seven hundred and twentieth part of the circle of the zodiac. The Sand Reckoner also reveals the autobiographical detail that Archimedes’ father had worked specifically on astronomy. In addition, from The Sand Reckoner we learn about the practical tools used for these calculations, i.e., the straight edge and a sliding cylinder: When the sun is almost on the horizon and is able to be observed, a straight edge is turned toward the sun, and one places one’s eye at the flat of the straight edge. Then a cylinder, placed between the sun and one’s eye, obscures the sun. The cylinder is then moved little by little away from one’s eye, ending where a...