『砂粒を数えるもの』は、アルキメデスの著作の一つで、「砂の計算者」とも呼ばれます。この著作は、シラクサの王であるゲロン（ヒエロン2世の息子）に宛てたもので、宇宙を埋め尽くすのに必要な砂粒の個数を概算したものです。
大数の体系化: アルキメデスは、宇宙の大きさと砂粒の大きさを仮定し、それに基づいて宇宙を埋め尽くすのに必要な砂粒の個数を計算しました。彼の数の体系は、万を基本としており、その数は非常に大きなものでした。アルキメデスは、数を段階的に「第1期の数」「第2期の数」などに分類し、これを用いて砂粒の個数を議論しました。彼の数の体系は、万進法を基本としており、ギリシア語、中国語、日本語の万進法と共通点があるとされています。
指数法則: アルキメデスは、後の計算のために指数法則についても言及しています。
宇宙の大きさの仮定と概算: アルキメデスは宇宙の大きさについて以下の仮定を立てました。
- 地球の周囲は約300万スタディアで、それ以上ではない。
- 地球の直径は月の直径よりも大きく、太陽の直径は地球の直径よりも大きい。
- 太陽の直径は月の直径の約30倍で、それ以上ではない。
- 太陽の直径は、宇宙球の大円に内接する正1000角形の一辺よりも大きい。
これらの仮定のうち、1, 2, 4は正しく（ただし1は大きく見積もりすぎ）、3は誤りであると現代的な知識から知られています。しかし、これらの仮定を元にして宇宙の大きさに関する概算を行い、誤った仮定から得られた結果として正しい命題もありました。
砂粒の大きさの仮定と概算: アルキメデスは砂粒の大きさに関しても仮定を立てました。
- 1個のケシ粒の体積は、1万個の砂粒の体積よりも大きくはない。
- 1個のケシ粒の直径は、1⁄40ディジットよりも小さくはない。
これらの仮定から、直径1ディジットの球を満たす砂粒の個数は、10億より少ないと導かれました。また、アリスタルコスの考えを取り入れて、諸恒星までの距離を考慮した恒星球の砂粒の個数も概算されました。
注目すべき記述: アルキメデスは宇宙の大きさを見積もるために複数の天文学者の説を紹介し、それが当時の貴重な資料となっています。また、自身の父を「私の父フェイディアス」としているが、その父が有名な彫刻家であることから、アルキメデスの祖父も芸術家である可能性が指摘されています。
総じて、この著作はアルキメデスの数学的な洞察と観察力を示すものであり、当時の宇宙観や数の概念についての理解を深めるうえで重要な文献とされています。
"The Sand Reckoner" (or "On the Sand Reckoner") is one of Archimedes' works, also known as "The Sand Counter" or "The Sand Calculator." Among Archimedes' writings, it is the most straightforward in content. The treatise assumes the knowledge about the universe available at the time and attempts to estimate the number of sand grains required to fill the entire universe. It is presented in a formal format addressed to Hieron II, the king of Syracuse.
Systematization of Large Numbers: Archimedes established a system for handling large numbers. Assuming a larger size for the universe and a smaller size for sand grains based on the astronomical knowledge of the time, he discussed the approximate number of sand grains needed to fill the universe. Before Archimedes, there were specific names for numbers up to a myriad, and by counting myriads, numbers up to 10^16 were referred to as the "second class of numbers." This progression continued, reaching the "second myriad class of numbers." The final number in this system is denoted as P and is given by 10^8 raised to the power of 10^8.
Furthermore, Archimedes introduced terms such as "first-period numbers" and "second-period numbers" to classify numbers systematically. He used this system to estimate the quantity of numbers up to P^2, a colossal number resembling tetration, reflecting the ancient understanding of extremely large numbers.
Archimedes employed a numbering system based on myriad as the basic unit, which shares commonalities with the myriad-based system in Greek, Chinese, and Japanese, as opposed to the modern English decimal system.
Notable Descriptions: To estimate the size of the universe, Archimedes introduced various descriptions from different astronomers, providing valuable insights into the astronomical knowledge of the time. For instance, he mentioned Aristarchus of Samos, who proposed a heliocentric model with the Sun at the center of the universe.
While little information is available about Archimedes' ancestry, he referred to his father as "my father Phidias" in this treatise. Scholars, such as Rivka Nets, suggest that Phidias might have been a renowned sculptor. This assumption arises from the fact that during that period, naming a son after a famous artist was likely limited to those in the field of arts.
Assumptions and Approximations: Archimedes made assumptions regarding the size of the universe, assuming it to be a sphere with the Earth at its center. Notable assumptions include:
- The Earth's circumference is approximately 3 million stadia and not more.
- The Earth's diameter is larger than the Moon's diameter, and the Sun's diameter is larger than the Earth's diameter.
- The Sun's diameter is about 30 times that of the Moon's diameter, and not more.
- The Sun's diameter is greater than the side of a regular 1000-sided polygon inscribed in the celestial sphere.
While assumptions 1, 2, and 4 align with contemporary knowledge (with assumption 1 being an overestimate), assumption 3 is incorrect. Archimedes used these assumptions to deduce that the diameter of the universe is smaller than 10 billion (10^10) stadia.
Regarding the size of sand grains, Archimedes assumed:
- The volume of a poppy seed is not greater than that of 10,000 sand grains.
- The diameter of a poppy seed is not smaller than 1/40 of a digit.
From these assumptions, he concluded that the number of sand grains filling a sphere with a diameter of 1 digit is less than a billion (10^9). Additionally, incorporating Aristarchus' belief that the distances to stars are vast, Archimedes estimated the number of sand grains needed to fill a sphere with the radius extending to the stars.
Conclusion: Archimedes' work demonstrates his mathematical insight and observational skills. While some assumptions were incorrect, his conclusions from these assumptions led to accurate propositions, showcasing his ability to reason through complex problems. "The Sand Reckoner" is considered an important document providing insights into the understanding of the universe and numerical concepts during that era.