Pythagoras

An ancient Greek natural scientist, mathematician, and religious leader. Born on the island of Samos in the Aegean Sea. There are various theories about his origins, but it is said that he visited Egypt in his youth. After returning to Greece, he had a bad relationship with Polykrates, the tyrant of Samos, and moved to the Greek colony of Croton in southern Italy. After being active there, he moved to Metapontum, where he died.

In Croton, he organized a cult that followed the trend of Orphism at the time. Its doctrine was based on the immortality of the soul, transmigration, and retribution after death, and placed emphasis on purification and salvation of the soul. The members were closely united with Pythagoras at the apex, and within the cult they lived ascetic and strict lives under various precepts, and were extremely exclusive. In addition, the principle was that property should be shared, and this was also applied to the results of academic research within the cult, so by the time of Aristotle it had become difficult to distinguish between Pythagoras' achievements and those of his disciples.

Pythagoras and his school studied music, mathematics, astronomy, and medicine, and many of their accomplishments have left their mark on the history of science. However, their research was primarily auxiliary to the pursuit of doctrine. Because of this, even their highly regarded research in mathematics sometimes mixed rationality with mysticism. For example, odd numbers were considered male and even numbers female, and the sum of the male number 3 and the female number 2, 5, was considered the number that symbolized marriage.

Pythagoras believed that the origin of all things, which Greek natural scientists of the time were exploring, was "number." The background to this is thought to be his discoveries in music, such as the fact that in the case of chords on the harp, the length of the strings has a simple numerical proportion, and that shapes are created by combining several points (i.e., the positive number 1). In fact, he and his school discovered through research into music theory that the three numbers a, b, and c are,
If ab=bc, then a, b, and c are an arithmetic progression.
If a:b=b:c, then a, b, and c are a geometric progression.
If (ab):(bc)=a:c, then a, b, and c are a harmonic sequence.
I knew that.

He also invented triangular numbers (the sum of a sequence of natural numbers, 1+2+3+……+n=n(n+1)/2), rectangular numbers (the sum of a sequence of even numbers starting with 2, 2+4+6+……+2n=n(n+1)), pentagonal numbers (the series 4+7+10+…… with a common difference of 3), and hexagonal numbers (the series 5+9+13+…… with a common difference of 4). He also discovered a pair of perfect numbers (a number whose sum of all its factors, including 1, is equal to itself) and amicable numbers (two numbers each of which is the sum of all the factors of the other), 284 (=1+2+4+5+10+11+20+22+44+55+110) and 220 (=1+2+4+71+142). He also learned that the regular polygons that can completely fill the area around a single point are the equilateral triangle, square, and regular hexagon, and it is said that he knew all three of the regular polyhedrons - the regular tetrahedron, regular cube, and regular dodecahedron - as well as the regular octahedron and regular icosahedron, making a total of five.

The famous "Pythagoras' Theorem" (the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides) was discovered by Pythagoras himself or one of his disciples, but its rigorous proof was later provided by Euclid. However, the discovery of the Pythagoras' Theorem brought a difficult problem to this school. The relationship between one side of a square and its diagonal was found to be 1:, which was difficult to accept in this school, which only considered positive numbers to be numbers. Furthermore, such numbers also appeared in the mean and extremity ratios (golden section) used to draw regular pentagons. Therefore, they called these "unspeakable numbers" irrational numbers alogos, and tried not to reveal this secret outside the school.

The Pythagorean view of the universe can be seen in the writings of his disciple Philolaus. They adopted a spherical theory rather than the traditional flat Earth theory, and advocated an irregular heliocentric theory rather than a geocentric theory. They considered 10 to be a perfect number (1+2+3+4), the number of chord ratios, and a sacred number, but they also introduced the number of celestial bodies, the fixed star sphere, the five planetary spheres (Saturn, Jupiter, Mars, Mercury, Venus), the Sun, the Moon, the Earth, and the Earth-Planet, making it 10. These 10 bodies revolve around the "Central Fire," which is at the center of the universe, controls the activities of the universe, gives life to the Earth, and has a kind of creative power. The reason that the Central Fire is invisible from the Earth is because humans live only on the hemisphere of the Earth, and that hemisphere always rotates so that it does not face the Central Fire (the Earth does not rotate on its axis). The Sun is glassy, ​​and reflects the Central Fire to transmit light and heat to the Earth, and the Moon shines by receiving the light of the Sun. He also said that the movements of the heavenly bodies produce huge chords, but that humans cannot hear these chords because they have been listening to them since birth. Although this view of the universe had many weaknesses, it had a considerable influence on later generations, for example, it stated that the Earth moves around the center of the universe like the planets, that all celestial bodies, including the Earth, are spherical, and it distinguished between planets and stars.

[Hirata Hiroshi]

"The History of Greek Mathematics" by T. Heath, translated by Hiroshi Hirata (1959/Reprint edition, 1998, Kyoritsu Shuppan)" ▽ "The Origin of Science" by Hiroshi Hirata (1974, Iwanami Shoten)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

古代ギリシアの自然学者、数学者、宗教家。エーゲ海のサモス島の生まれ。出身については諸説あるが、青年期、エジプトを訪れたといわれる。帰国後サモス島の僭主(せんしゅ)ポリュクラテスPolykratesと折り合いが悪く、南イタリアのギリシアの植民市クロトンに移住、この地で活躍してのち、メタポンツムに移り、没した。

　クロトンにおいて、当時流行したオルフェウス教の流れをくむ一つの教団を組織した。その教義は、魂の不滅、輪廻(りんね)、死後の応報にあり、魂の浄化、救済を重視し、団員はピタゴラスを頂点に緊密に団結し、内部にあってはさまざまな戒律の下に禁欲的、厳格な生活を送り、きわめて排他的であった。また財産の共有を原則とし、それを教団内の学問研究の結果にも適用したため、ピタゴラスの業績と門弟の業績とを区別することは、すでにアリストテレスのころには困難となった。

　ピタゴラスおよびその学派は、音楽、数学、天文学、医学を研究し、そのなかには科学史に残る業績も少なくないが、彼らにとっての研究の本来は教義を追究するための補助的なものであった。そうした彼らの研究であったがために、とくに評価の高い数学の研究にさえも、合理性のなかにときとして神秘性が混在している。たとえば、奇数は男性、偶数は女性とみなし、男性数3と女性数2の和である5は結婚を象徴する数としたたぐいである。

　ピタゴラスは、当時のギリシアの自然学者が探究した万物の根源を「数」だとした。その背景には、たとえば音楽において、和音が一絃琴(いちげんきん)の場合、絃の長さが簡単な数比例をなすこと、またものの形は点（すなわち1の正数）をいくつか組み合わせるとできあがること、などの発見があったと考えられる。事実、彼または彼の学派は、音楽理論の研究から三つの数a、b、cが、
　　a-b=b-cを満足すれば、a、b、cは等差数列である、
　　a:b=b:cを満足すれば、a、b、cは等比数列である、
　　(a-b):(b-c)=a:cを満足すれば、a、b、cは調和数列である、
ということを知っていた。

　また、点の配置から、三角形数（自然数の数列の和、1+2+3+……+n=n(n+1)/2になる）、長方形数（2から始まる偶数の数列の和、2+4+6+……+2n=n(n+1)になる）や、さらに、五角形数（公差が3の4+7+10+……の級数）、六角形数（公差が4の5+9+13+……の級数）などを考え出した。完全数（その数の1を含むすべての因数の和が、その数に等しいもの）や友愛数（2数のそれぞれが、他の数のすべての因数の和になるもの）として284(=1+2+4+5+10+11+20+22+44+55+110)と220(=1+2+4+71+142)の一対を発見している。また1点の周りをびっしりと埋め尽くす正多角形は、正三角形、正方形、正六角形であることを知り、正多面体については正四面体、正六面体、正十二面体の三つとも、さらに正八面体と正二十面体とを加えて五つを知っていた、ともいわれる。

　有名な「ピタゴラスの定理」（直角三角形の斜辺の平方は他の2辺のそれぞれの平方の和に等しい）は、ピタゴラス自身か、その門人かの発見であるが、その厳密な証明は、後のユークリッドがしたものである。ところがピタゴラスの定理の発見は、この学派に難問をもたらした。それは正方形の1辺とその対角線との関係が 1: という、正数だけを数とみなすこの学派では認めがたいものをみつけたことで、さらにこういった数は、正五角形の作図の際に使う中外比（黄金分割）の場合にも現れた。そこで彼らは、こうした「口にできない数」を無理数alogosとよび、この秘密を学派外に口外しないようにしたという。

　ピタゴラス（学派）の宇宙像は、門人フィロラオスの著作にうかがえる。彼らは従来の大地の平板説をとらず球状説を採用し、天動説ではなく変則的な地動説を唱えた。彼らは10が完全数（1+2+3+4）であり、和音の比の数でもあり、神聖な数とみなしたが、天体の数についても、恒星球、五つの惑星球（土星、木星、火星、水星、金星）、太陽、月、地球と、対地球という天体を導入して10個とした。この10個は宇宙の中心にあって宇宙の活動を管理し、地球に生命を与え、一種の創造力をもつ存在の「中心火」の周りを回っている。中心火が地球から見えないのは、地球の半球面だけに人間が住み、その半球面はつねに中心火には向かないように回転している（地球の自転はない）からである。太陽はガラス状で、中心火を反射して地球に光と熱を伝え、月は太陽の光を受けて輝く。また天体の動きは巨大な和音を生じているが、人間は生まれて以来聞き続けているので、その和音は聞こえない、とした。この宇宙像は多くの弱点をはらんでいたが、地球が宇宙の中心の周りを惑星と同様に運行すること、地球を含むすべての天体が球形だとしたこと、惑星と恒星を区別したことなど、後世に少なからず影響を与えた。

［平田　寛］

『T・ヒース著、平田寛訳『ギリシア数学史』（1959／復刻版・1998・共立出版）』▽『平田寛著『科学の起原』（1974・岩波書店）』

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