Euclid and Pythagoras

Gould, Euclid and Pythagoras , Classical Weekly, 36 (1942:Oct.-1943:June) CLASSICAL WEEKLY any prime since, whenever we tried to divide it by a prime, we would get a remainder of one. This important theorem has always been held in honor by mathematicians. Let us for our part try to see its connections within and buffeted from without by concepts, it lost its determinate hold on the religious beliefs of the Roman populace. Yer, when it finally succumbed to the spintual teachings of Christianity, it asserted its true nature by imposing on these a wealth of traditionally Roman cult with the religious doctrines of the Pythagoreans. One of the most noticeable features of these doctrines is the emphasis laid upon the fact thar every number is pera even or 1.5 (see, eg. Aristotle, Metaph. 17: roù Sè dpelpod da ré Te dpriov ral rò wepirréy). Pythagoras re | moreover, that the odd numbers are superior to the even ones (e.g. Ps. ceremony. Plut. v. Hom. 145: rav épcéie dpr rôv pèv dprioy FRANEUN B. Krauss évbcá xai dre, róv Sì reprody wÄrpy re Kai réÂuoy) and he explained this assertion by saying that odd PENNSYLVANIA STATE COLLEGE numbers do not allow themselves to be aided into two ual (ibid: Scafpeow obx émBéxeras). It is clear ds lie tk in the anne ny ak DR Douce mame Euclid and Pythagoras There is divinity in odd numbers.—Sir John Falstaff “There is luck in odd numbers." This saying, considered as superstition by some and as religious dogma by others, has had a long history in many different have its roots deep in human nature, for it has been held by different races over the face of the globe. And we shall see that he has given it the right explanation. IE odd numbers owe their superior qualities to the fact that they cannot be divided into two equal parts nations. It was current sosti the: Genes Vergil quotes then the prime numbers must enjoy a very great superit in his Eclogues (8.76): numero deus im Goutd,S.H. Pa 2 02 Yan\ Our own | offers many examples, like gaging quatrain from the Irish novelist, Samuel Lover: this en- “Now, Rory, leave off, sit; you'll hug me no morc; That's eight times to-day that you've kissed me before.” “Then here goes another,” says he, “to make sure, For there's luck in odd numbers,” says Rory O'More. Light is thrown.on the explanation of this belief by a study of the relation between Euclid and Pythagoras. The contents of Euclid's Elements were*determined to a great extent by the mathematical discoveries of Pythagoras and his school. Proclus so far as to say thar Euclid's whole thirteen books were devoted to explaining the five “divine solids of Plato,” first investigated by Pythagoras. But this is a one-sided view of the relation between the two men. The Elements contain a great dea} beside solid geometry. Books VII, VIII and deal with the theory of numbers considered as falling iority, since they do not allow division of any kind. Pythagoras used to represent a number with a row of dots and then iment with the ways in which these dots could be rearranged to form a rectangle. He was much impressed by the distinctive character of any row which represented a prime number; it could not be formed into a at all. In a usually misunderstood passage of the Metaphysics, Aristotle mentions the importance which Pythagoras attached to the fact that every number except the primes could be formed as the product of two numbers. The odd numbers are better than the even numbers because are not divisible by two;. the prime numbers are best or “oddest” of all, because they are not divisible by any number. Beyond comparison, the most “sacred” of all the prime numbers is seven. To what does this number owe into two classes, the prime numbers, 2, 3, 5, 7,11... which are not the product of other numbers, and the its gene med Ay dl do » eh eS jar position? What religious experience led the Hebrew people to the feeling that it is añ e Pr ad to rest one Pda ivisible by one or more primes. The question arises whether the list of prime numbers ever comes to an end. Pico io given by Philohus, a ma Acad Place Are there infinitely many primes or is one of them His answer is of interest to us since our week still has seven days; there can be no doubt that it gives an accurbigger than all the others? Theorem 20, Book IX, proves that no matter how far we go along the list of primes, there still remain others. There is no hy if there were, the absurd result would saa For that we could find a composite number not divisible by any prime! To find such a number we would need only: to multiply together all the primes up to and including our bi prime and edd to this result the number one. large number so obtained would be composite, since larger than the biggest prime, and not divisible |by Copyright (c) 2004 ProQuest Information and Learning Company Copyright (c) Classical Association of the Atlantic States, Inc. anacion for the ate and complete ex the most sacred number worship of number tion. He says that seven is because it is the largest prime number Jess than ten (see, e.g., Joannes Lydus, De Mens. ii.13, p. 72). It is natural to take ten as a limit since, having ten fingers, we count up to ten and then begin over again (cf. iySexa, Sößexa etc.), while the importance of having a prime can be seen in the following way. Suppose chat our week had not seven days but six. Then the impressiveness to our,minds of the first

CLASSICAL WEEKLY without lanauon, perhaps because he ed Pandora with Anesidora the carth goddess) or holy day would be lessened by its having a rival, so to speak in the fourth day, seeing that each of them Ilaria, would BapynAlouny Eyxurov m stand at the beginning of a half week of three days. The same trait of the mind can be observed in music. In three-quarter time the first of the three notes in a bar is the only one to receive an accent; but in six-eight time the fourth note also receives some accent because of an inevitable though perhaps subconscious realization on the of both and listener that it stands at the beginning of the second half-bar. In a week of six days the unavoidable singling out of the fourth day would give it a psychological im- Flom che cocommanding positon tion of the fstfirst or or holy dey‘ The number of minutes in an hour is xy, a number that a hae des a quarter-hour, etc. al contain an exact number of minutes. But if the beginning of a new hour had the same religious significance as the beginning of a new week, its importance on the face of the clock would be detracted by the of certain other periods which naturally stand out above the rest, namely the half-hour and the quarterIt was because Pytha was interested in questions theorems about the divisibility of numbers which make up Books VII, VIII and IX of Euclid’s Elements. In the theorem quoted above, about the infinitude of the number of primes, mathematicians have always recognized a cornerstone of the theory of numbers, a stone en which voted to the have erected a magnificent edifice deof distribution of primes. But from our point of view, as students of Greek religion, we may also honor this theorem as the one in which Euclid proves that God has created an inexhaustible supply of the best, the most divine, the “oddest” of all numbers, the primes. S. H. Gouin UNIVERSITY OF TORONTO considered obscure and difficult. Athenacus (9379) quotes our passage as well as the fg. Ananatos as evi question is The of. wee Pandan sacras icky "a doc or soft cake: the word seems to be used only as an adjective; it could here be an adverbial accusative or it may be used as a noun. Thus the dative 5 would be. instrumental. But in that case it is hard’ to understand why the ecaping culprit (?) or lover (?) should seck refuge with or implore the plant itself, It seems better, then, to take 4 as the indirect object of Wicoxe (note the iterative), so that Pandora (a hetaira? The name is not listed in RE 8s.v. Hetairai.)* offers a libation to the plant itself. Have we here a survival of che tree and plant cult? That Ananaios swears by the cabbage might support this view, were it not for the fact that we know of the cuphemistic subsitution of animate and inanimate objects for the name of a god. Ernst Rress SCARSDALE, NEW YORK The Anabasis and Katabasis in Eleusis The Eleusinian festivals, celebrated in the days of Athenian greatness in honor of Demeter and Persene, were held in Eleusis, a town of Attica not far rom Athens. These rites, according to different authors in classical times and in the Christian era, had great influence on the religion of the ancient world. poa Raglio pr say Rc mdrr peter of this cult, called Eleusinian mysteries a mystical drama. Some hostile writers have suggested chat the mysteries were a dramatic representation of the experiences of Perse and Demeter, showing their passage through Hades and the coming of a new birth and a new order. The Homeric H to Demeter is the ancient evidence of this belief and this gives only a part of the story, and that part elaborated and adorned with poetic ornament. We must supplement the Hymn from other sources and then separate the essential of the myth, which are likely to be just those in we can detect the symbolic } The Sanctity of the Cabbage The fragment of Hi D==40B is 3D—4B for the holiness of the plant, of which no doubt seems 113 le. In the same place Achewhi but Athens was the starting-point, Every year there were two celebrations, the Greater Eleusinia, the Katabasis, celebrated in the autumn, and the Lesser Eleusinia, the Anabasis, celebrated in the spring at Agrae on the banks of the Ilissus. nacus states that cabbage a (the expressi wapeoxeväalero means ‘was cooked’) to protect lying-in women against witchcraft. In Mia were considered only as a the sanctity a leaves of the to be reenforced by the seven t, for which, however, I can find no points of similarity, even the Lesser Mysteries to the Greater (Scholiast on Aristophanes, Plut. 849: for: rà Pape Gore botanical evidence (cf. W. Roscher, SAWA 1906, 41). Mirri were nd uni a pe pal A scale, but The difficulty of the fragment lies in lines 2 and 3: D Weoxe HavBúpy (Schwenn RGVV XV 36.5 reads initiation in the Lesser was generally required before che candidate could present himself for initiation into Copyright (c) 2004 ProQuest Information and Learning Company Copyright (c) Classical Association of the Atlantic States, Inc. mpoxda e xai pod s Tay peyd. Lesser