Popular Symbolic / Mystical Numbers in Antiquity

LAROCHE, Roland A. — Popular symbolic / mystical numbers in antiquuv LARSCHE R A: Latomus 1995 54 (3) : 568-576. + A brief overview of the ten most important myS\ AAT tical / symbolic numbers favoured by Greek and Roman authors (3.4, 5. 7,9, 10 12, 17, 30, 40). Most of the numbers examined reveal ultimate Pythagore an influence, but the number 40 stems from a different tradition which is NearEastern in origin. 050 Populir Symbolie/Mystical Numbers in Antiquity Having long wondered why ancient Greek and Roman authors. especialiv writers OÏ history and epics, advance the numerical touuls which they do in hieir works have carefuils analvseéd these works und found out ihui diese author naurallv favor certain svmbolic: mystical totals both implictly and expliciuiy (0, I this paper Î will present à brief overview of the 10 most important mystical; symbolic numbers favored in antiquitv. A symbole, mysticai number is one Which suggests something In addition to the number per se by reason of association, convention, etc, Such 4 munber makes no specific reference to the actual quantits involved. Rarher it is the visible sign of something invisible. such as an idea or a quality. and, in fact. it is a symbol. I is necessary to distinguish between the arithmetical and the symbolic number; the former defines an object by its quantity or order but says nothing about its nature, whereas the latter expresses an inner link with the object it defines by virtue of a mystic relationship between what is enumeraied and the number Useil, Religious numerical values Varv with longitude and latitude and it is possible 10 identify manv different numbers which were, and in many cases still are, invested with occult sienificance in different places and at different times. From very ancient times numbers represented symbolic ideas and the interpretation vf numbers is one of the vldest svmbolic sciences, Also “ie peculiar fascination which numbers us individuals have exertcé où the mind of man from time immemoriel was the main obstacie in the war of developing 4 collecüve theory of numbers, ie. an arithmetic"{?}. For example, the Pytha- | POPOIRR NY MBORIC NES RON NE NIRERS ES ONS TIRE LE voreans and thé Grecks were verv much preuccupied wii the symbolise of numbers und Plulo regurded unmMbers ds dit Essetite oi drinoie. doid Bu ent as the basis of 1he cosmos and of man. Pythagoreanism was the influenitai doctrine of the followers of Pvythagoras of Samos, a Greek philosopher and mathematician of the 6 century B.C. None of Prthagoras own work has survived but the ideas attributed to him by his foilowers are among the most portent in modern bstor,, Pure knuwlcdgc, tie Pithagorcuns argued, was the purification {oarharsisr of the soul. This meant rising above the data ot the human senses. The pure essential reality. they said, was found on, in the realrm of numbers. The simple, wonderful proportions of numbers wouid explain the harmonies of music which were the beauty of the eur, For thai reason thev introduced the musical terminologs of the octave, the tifth. the fourth. expressed as 2H. El. and 44 For astronomy the Prthagorean adoration of numbers carried an overwhelming message. “They sas that the inings themselves are numbers, was Arisiulle's succinet sunimart in his Metaplhusies, ‘and do not place the obiects of mathematics between Forms and sensibie things. Since, again, thev saw that the modifications and the ratios ot the musical seules were expressible in numbers-since, then, ali other chings seemed in their whole nature to be modelléd on aumbers. and numbers secm to be the first things in the whole of nature, thev supposed the elements of numbers 10 be the elements of all things. and the whole heaven 10 be a musical scale and a number … and the whoic arrangement of the heavens they collected and litted into choir scheme : and if therc was a gap anywhere, they readily made additions so as to make their whole theory coherent7 (. Plato. and his mystic followers. the Neopiatenists, héllevef that ail the data of the senses were mere unsubstantial Shadows and thai the real world was a worid of ideal forms. The Neoplatonists buiit their whole view of the world on an ideal mathematics. Numbers offered the best human vision Of God and the wortisont AÏ mathematical snecies … have à primars subsisience in the soul”. observed Procius tA. D. 4464 7 4KS), the last and vreutest of the Greek exponents of Neoplatonisn, ‘so that. before sensible numbers. there 41) R. À Larüoue has treutvd this topic in articles on Grecx and Roman literature and chronologs : Falerius Antias and [lis Numerical Totals : A Reappraisalin Historia XXVE EH 1977, p. 358-308 ; The han King-List in Dionvsius 1 70-71: À Numerical Analysis in Hisioria NXXT TL. 982 pp 112-120, Æurky Korean Chronclogr : lis Schemaric Nature in €. Deroux (ed), Sradies in Latin Lierature and Roman Hisors HE. 1983, p. 5-28: Evidence of a Ali Year Generation Value in Books EX of Tivy in €, Dérocx td), Srudies in Latin Literature and Roman History UV. 1886. p. 3145 : Meonbor Sosienis, Mmber Mrsicisns and Numerical Pracrices in Lit Books NX) and Related Gresk and Ronian Writers (PAD. Dissertation: Tufts Universiy 1972: on microliim at Ann Arbor. Michigan). 1) CET. Daxrzac. Number: The Lanvuage 01 Soence, New York. 18544. p. 36 and in general Chamer HI Number-lore" Fhe reader should sonsuil any standard work on symbols for additional information and lor illustrative exampies of statements made in this paper about symbolic numbers. In the present investigation have are Lo be found in her inmost recesses, self-moving numbers. ideal proportions speciale relied un tie totlowing sources EF Hoepee. Medieval Nionher Simhofisin. New York. IJRR 2 R. ArsenDs, Le summhodisme des nombres. Paris, 194K 5 R. GU'ENOX. Lésoterisme de Dante, Paris. 1049: PB Thet, Je srmtholivne dans da mvhotogie grecque. Paris. 1952: ML Ersape, /mages er Sunbues, Paris. 1887: M. Guyka Philo saphie er mrstique du nombre, Pans. 1952: 1. Pantin La cmhoüque des nombres dans l'inconscient, Paris, PSE: CG. JUNG. Man ani His Sinibrois, Paris. 1954 : anc sspecait LE Ciaror 4 Dierrers 0 Srnhols, Nos Vork. 1962 und Dicrionmiairre des srmholes, èd J. Canvas and À. Girärur ns, Parts, 1973 4 Volumes with enicusive bibliograpayi. 63 CT Daniel J. Boowsus. he Laxcoverers. New York 1853. p. 298-209.

POPLLAR SYMBOLIC: MYSTICAL NUMBERS IN ANTIQUITY of harmony previous Lo concordant sounds: and invisible orbs, prior to the bodies which revolve in a circle … we must follow the docirine of Timaeus, who derives the origin. and consummates the fabric of the soul, from mathematical forms. and reposes in her nature the causes of evervthing which exists"? (#), Plato was particularly interested in the mysticism of numbers and in the should also be an arithmetician. d elements : fire, air, water. and earth. The 2 essential prototypes of symbolic quantity are not à and 2 which are insignificant for symbolic/ mystical considerations and which were not considered numbers by the Pvthagoreans. but rather 3 and 4 which signifv dynamism and stability respectively and it can be argued that their sum and multiplication yield the 2 numbers that are next in symbolic importance: 7 (4 + 3) and 12 (4x3) (*). Often 4 is symbolic of dearh and closely connected with death rituals. The number 5 : both an odd and a prime number it was significant for the Pythagoreans but not considered as mysterious as either 3 or 7. Às the fifth it represented one of the 3 basic harmonies of ancient Greek music and Pythagoras selected the 5-pointed star. the pentagram, as the symbol for his brotherhood since it is proportioned in such a way that comparisons between its different parts almost always vicld the Golden Nuraber, ie. 618. The number 5 is symbolic of man and it was called the nuptial number by the Pythagoreans since it represents the union of heaven (3) and the Wugna Aarer (2). The number 5 corresponds to penlagonal symimetry, à common characteristic of organic nature, and to the 5 senses which represent the 5 forms of matter. The number 7 : both an odd and a prime number. 7 was the most prevalent mystical number in antiquity after the number 3 and it was favorcd by the The following is a list of the 10 most important mystical; symbolic numbers favored in antiquity. In most cases Greek and Roman authors favored Pythagoreun mystical numbers. The number 3 : both an odd and a prime number and like all prime numbers an irreducible datum. It was the most important Pythagorean. Greek, and Roman mystical number. In fact. the dynamic and svmbolic richness of 3 he was born on the 74 day of the month and it is a key biblical sacre number (cf. God rested on the 7% day). Undoubledlv. a very mystérious importance was attributed to 7 because of such vatural phenumena as the 7 planets, the 7 intervals on the musical scale. the 7 basic colors, etc. The number 7 svmbolizes the entire universe in motion since à is the sum 0f3 570 Republic (VHI, 546) he speaks in an obscure fashion of a certain mystical number but does not reveal what this number is. The solutions proposed to date for this mystical Platonic number are unsatisfactory. According Lo Cicero (Tiusculan Disputarions 1. 17) Plato travelled to [taly to become acquainted with the Pvythagoreans and he met Archvias and Timaeus and learned from them all the tenets of the Pythagoreans. Some commentators () believe that the mystical Platouic number played an important role in the mysticism of the Hindus and the Babylonians and that perhaps Pythagoras found it on the banks of the Euphrates, if he really studied there, and that he took it with him to Crotona, passing it along 10 his disciples who finally told il to Plato and his followers. Additionally, Plato states in the Republic (VE, 525f.) that the study of numbers has a great and elevating effect since it compels the mind to reason about abstract numbers and that a philosopher is so exceptional that it cannot be overemphasized. The number 3 represents the synthesis of ! and 2 and thus solves the conflict posed by dualism. The number 3 plays au important role in Greek and Roman rituel, legend, and early history. L The number 4 : as the first square (2?) it was considered a magical, mysterious number by the Pythagoreans. As the fourth it represented one of the 3 basic harmonies of Greek music (6), The sum of the first 4 numbers formed the sacred tesrakrvs and this holy fourfoldness supposedly represented thé (4) CT Boonstix [n. 3], p. 299. {5} Cf D. E. Srora. History of Mathematies. New York, 1958, Vol. 1. p. &9. (6) According 10 ancient Greek, and Pythagorcan, musical theory 4, 5, and 8 represent the 3 harmonies (cuupeviur) ef ancient Greck music: the fourth, fifih, and octave. For further information on this topic, cf. Paul Kraus, Jübir ibn Hayyän : Hebrews, Greeks, Pythagoreans. and Romans. li was sacred to Apollo since (heaven) and 4 (the Earth}, Hippocrates (fl. €. 460 B.C.) states chat “the number 7. because of is occult virtues, tends to bring ail things into being: it is the dispenser of life and the source of all change- for the moon itself changes üts phase every 7 days. This number influences all divine beings” (5). Si Augustine (De Civitate Dei XVII, 4) has an extensive discussion on the special significance of 7 (?). 9 : as the triplicatiou of the triple, 3 x 3, it was a sacred number The number for the Gresks and the Romans but il was feared by he Pythagoreans because of its many surprising qualities, a superstition extending from ca. 500 B.C. to the present. The number 9 is a key svmbolie number in Greek myths and an important biblical number (ef. Christ died at the 9h hour), often symbolizing the end of a cycle, the completion of a task- undoubtediy revealing influence from the decimal system of counting since # new series starts after 9, 19. conmtriburion à l'histoire des idées scientifiques dans l'{slam (Cairo 1943), Vol. 11, p. 207- 309. These 3 numbers were variously combined as 4 = 4 and $ + 8 = 13 which yields pe anis Sr 15=17; and abo as 8=8 and d +35 =9 which vields the units +9= 17 47 CE. Cut [n. 21 sv. “numbers”. {8) CE Cirsor En. 2], p. 271. (9) C£ aiso his Serrnon 249 for a lengthy discussion of the importance of 7 and 17.

would receive one of the 17 letters of the ancient Greek alphabet (the 17 only 17 souls would be resurrected at the end of the world and each soul of the first 10 numbers which means that 17 = 7. According 10 the Pythagoreans goreans and thev believed that the numbers over 10 were simply a repetition aspeats, in part, because it is the sum of 2 very important mystical numbers :\ 3 and 10. Also it is the 7 prime number according to the following classification which 1 believe existed al some time in antiquity although there is no documentation of this fact: 1.3,5.7.11,13. and 17: later Aristote and Euclid replaced 1 by 2 as the first. and by necessity only. even prime number. The décimal system of counting had a mysterious importance for the Pytha- The number 17: both an odd and a prime number it had many mystical The number 12 : as the product of 3 x 4 it is closely linked with the wheel and the circle and it is especially important for spatiotemporal divisions. It designates the 12 signes of the Zodiac- one of the most widespread symboisand symbolizes cosmic order. In fact. systems based on the cirele tend to have 12 as the end-limit as in 12 hours. 12 months, F2 Olvmpians, etc. ft is also à popular biblical sacred number (cf. 12 apostles, 12 tribes of Israel, 12 doors in Jerusalem, etc). 1.000. 103. special to it, are necessary 10 its perfection”. Many muhiples of ID are symbolically significant : especially 30. 3 x 10: 40. 4x 10 ; 100, 19 x 10: and perfect ought 1o have. while there are manv characteristics which, while not triangle and our oath”. Pythagoras further states thai “10 is a perfect number. and it is both right and according 1o Nature that we Greeks and all men arrive at this number in ail kinds of wavs when we count. though we make no effort to de so: for it has many special qualities which a number thus to count to à, exclaimed “Do you see ? What you think is 4 is 10, à perfect by the Pythagoreans and it symbolized universal creation and the human soul. This number was worshipped and invoked in their oaths under the formula : “The Yésrakivs in which is found the source and the roots of eternal Nature”. The Pythagoreans believed that all was derived from 10 and that all went back to 10. The decimal system of counting had a mysterious importance for them and they considered the numbers over 10 to be simply a repetition of ïhe first 10 numbers. Ten signified perfect harmony and it was the source of ail since it represented 1 + 2+3+ 4. It was invoked as a deiiy and called the “Sacred Ten” which contains the key 10 cverything, Pythagoras (apud Lucian. Auction of Souls 1V), after ordering à new disciple appears graphically as +. , the number 10 was called the most perfect number The number 10 : as the etraktys, Le. the sum of the first 4 numbers, which [ÈS] 29, etc, It symbolizes truth since when it is multipliéd by any number it reproduces itself by mystic adéition and it is associated with the measure of gestations in humans. the 9-month interval from conception to birth. LA POLLLAR SYMBOLIC. MYSTICAL NUMBERS IN ANTIQUITY number 17 had an ambivalent the number of fish in the distinguished literary tradition s dans la itiérarure épico-religieuse (10) Cf. lrène Marisorr. Nombres symbolique 1962. p. 442. des Tures d'Anatolie in Journal Asiatique. s.v. “dix-sept”. LL shoulé be noted that (tt) CE. Cusvaren [n. 2], Vol. I, p. 203-204, would be resurrected at the end of the according to the Islamic tradition only F7 souls 17 letters of the alphabet which spelled world and each one would receive one of the ing ultimate Pythagorean intluence. out the perfect name of God — obviousty reveal rnassus (1. 45, 3) Romulus îs the (123 However, according 10 Dionysius of Halica were so chen it wouki be a question (Tr king in descent from Aentas and if this 7? Roman kings. À situation which would of 17 pre-Alban and Alban kings followed by influence from 1e mystical numbers ? mirror the Greek alphabet and reveal definite there are lé pre-Alban and Alban kings and 17. According to traditional chronologv King-List in Dionvsius 1 70-71: 4 followed by 7 Roman Kings — dt. The Alban topic. Nuncrical Analysis (n. 1] for a discussion ot this since it appeared for the 3 time after miraculous catch which was made when Jesus tine (Sermon 249) comments his resurrection to 7 of his disciples. St. Augus and since he had à touch of at length about the special significance of 153 bv the fact that the addition of the Pythagorean about him he was thrilled miraculous catch. Lto 17 = 153 which was the number of fish in the cal number in antiquitv : In general, the number 17 was an important mysti Aeneid: in Athens there were eg. Vergil cites 17 different oumbers in the s. 1he last King: in tal. there 17 kings from Cecrops, the !* kine, to Codru as to Numitor, and so Romulus were 16 kings before Romulus. ie. from Aene was the 171 King overall (P) : etc. is cited in the Bible (/ofm XXI, 3-14) as The number 153 in its own right has a long. ers: 1+2+3.+17= 153. Furthermore, the sum of the first 17 numb vice versa. it was feared because XVII could tation in antiquity since in Roman times also be interpreted as VIXI = mortutes Sir. of the Anatolian Turks and j should be noted that in the religious epics an almost magical quality and in the Istamic tradition the number 17 has worshipped the aumber 17 (10), the mystical Shiites from à very early time of everything in the world was 17. They believed that the shape or form (sa) the Shiïtes stems from ancient Some believe that this veneration among s of the ancient Greek alphabet Pythagorean speculations based on the letter s in this alphabet (11). However. since there are 17 consonants and 7 vowel 7 vowels in the ancient Gretk E believe that there are 17 consonants and cal numbers 7 and 17 and not alpirabet because of influence from the mysti cosmological questions. However, the t name of God. The Pythagoreans cossonants ?) which spelled out the perfec +8. it is the sum of 4 + 5 (=9) attributed a special importance 10 17 since and in their theory about the harthe 3 key numbers in their music theory constantiy recurs in ancient Pythamony of the spheres: also the ratio 9:8 ns on grammar, musical, and gorean and Greek arithmetical speculatio conno-

The number 30: as the product of 3%X10, 2 very important mystical numbers, il was a sucred number in amiquity- especially in Greek sources to specify the duration of a generation (#3). The number 30 was also a special number for the Latins (and subsequently for the Romans) and it is prominent in early Roman legend. For example, there is the famous prodigy of the white sow with 30 piglets which was Said to have appeared to Aeneas and the subsequent building of Alba Longa 30 years, or one generation, after the founding of Lavinium (Dionysius of Halicarnassus I, 66, 1 = Cato frag. 13 (15). Varro (R.R. 2, 4: L.£. 5. 144) explains that the white sow represents Alba Longa and the 30 piglets the 30 years that would elapse between the building of Lavinium and Alba Longa. This prodigy is mentioned by Fabius Pictor (contemporary of the Hannibalic War : frag. 4) who evidently was the source began as an aetiological explanation of the league of 30 Latin cities (cf. Pliny, for Dionysius’ statement which reeurs in the Aereid (VIII. 44). This legend N.H. 111, 69} and obviously Rome, anxious to reduce the prestige and the standing of the 30 Latin cities, succeeded in proposing a new interpretation FOPLULAR SYMBOLIC; MYSTICAL NUMBERS IN ANTIQUETY un Jesus was taken to the temple 40 days after ins birth, he preached for 40 months, he rose up after 40 hours in the tomb ; ete. it was a key mystical number in antiquity and it stems from a tradition that is Near-Eastern in origin. The Hebrews considered 40 to be a special, magical number (1) and consequently they utilized a 4û-vear generation value (1é). (45) For example, the great déluge lnsted for 40 days and 40 niglts; Moses, at the age of 40, conferred with Jehovah for 40 days and 40 nights on Mount Sinai: récorum Romanorum Reliquice, Laipzig, 19142, Voi, 1. (13) In chronology 2 generation refers tu the interval of time between the birel of a father and the birth of" son. The use of a 30-veur generation value in antiquity is well atiested to: cf. eg. Liomeër (iae 1. 250 Sch.. in Dixvorr L. p.38): Hesiob (D. 695-498): Snrox (frag. 27, 1.9): Dioxvsius oF Maricarxassus (1 31, LE, 74, 4-6: [1 27. 42 IL 67, 2: ete.); Davy (1 3, 3-4: IX, 3, 2: IX, 37. 12:}. For secondary sources: cf. eg. K. J. Dover, Thucydides and His History, Oxford, 1965, p. dff.: À. R. Buen, Dates in Early Greek Historr in JHS LV, 1935, p. 141; D. W. PRaRkEn, Srudies in Greek Gencalogicat Chronology. Lancaster, 1943, p. 20 and 102 : W. Des Bon. Laconian Srudies, Amsterdam, 1954, p. 10: A. E. SAMCEL. Greek and Roman Chronology. Munich, 1972. especially p. 241ff. “Generational Reckoning” and his statement on p. 244 that “Although genealogical calculation must be assume to have served the purposes of almost all historical chronology before the fourth century, and must also underlie most of the precise chronology of the pre-fifth century Greece in all subsequent sources. it is extremely difficult to apply the control of generation reckoning Lo an understanding of the means à specific author used to arrive at a date”, CE also The Alban King-List in Dionvsius L 70-71: A Numerical Anulvsis [n. 1] which examines the pervasive influence of a 30-vear generation value in the formulation of the Alban king-list and in early Roman (and pre-Roman) events. {14) Roman fragments in this paper are from Hermannus Purer, Ferernwn Histouse of a 40-vear generation value travelled a route similar to that followed by the 12-month calendar. ie. these practices originated in the Near-Eust and that the Etruscan preference for 40 as a special number and the closely related The number 40 is conspicuous in early Roman legend and history and its presence in the narrative usually indicates ultimate Etruscan influence. 1 believe indicates Near-Fastern influence since the custom of inspecting the liver of sacrificial animals for divination was popular in Asia Minor and in Babylonia. generation value in Eratosthenes back to Hecataeus of Miletus (fl 500 B.C. : wrote Genealogies), ie, to Asia Minor. According to Herodotus (1, 94) there was a belief in antiquity that the Etruscans originally came from Asia Minor. speciticaily from Lydia. It is possible, but not as likely, that a 40-vear generation value was adopted independentiy by various peoples, Cumulative evidence indicates that 40 was a sacred number for the Etruscans and treaties assigned to them are generally expressed in terms of 40-vear intervals, ie. 40-year generations. Further evidence ofinfluence from the sacred number 40 is furnished by Etruscan augury : e.g. the Piacenza Liver is divided into 40 compartments, ie. 24 inner and 16 outer divisions. This practice clearly adopted this practice. In fact, some auihorities (1/} trace the use of à 40-vear Subsequenuy. other peoples. e.g. the Greeks and the Etruscans, seem Lo have Dé] (16) For cxampk. the children of lsracl wandered in the wilderness for 40 vears — onc generation; Saul, David. and Solomon each reigned tor 4f) years — one generation : D. W. Praxken {n. 13] p. 3, states hu “the wrers of the Old Testament employed a generation of 40 years”. E. P. Braur. Abingdon Bible Handbook, New York. 1975, p. 103, states that “ancient peoples. such as the Phoenicians and the Carthaginians often reckoned a generation by the round number 4f years”. He bclieves that undoubæædly the children of Israel wandered in the wilderness for a generation but that ihe length of à generation was actually about 20-25 vears. Thus, in his opinion. the chronoloey of “Numbers” is traditional and not strictly historical. (7) Cf. E. Maven. Forschungen zur Alien Geschichte. Halle, 1892-1899, Vol. 1. p. 151. for the use of a 40-vear generation value by Hecataeus: Burx En. 13], p. 130146, accepts Mevers theory and further proposes that subsequent}y Eratosthenes adopted à 4Ü-vear generation value rather than (he more common interval of 30 years. PRARKEN [n. 13], p. 68. and 102f.. proposes that many ofthe calculations of Herodotus are based on a 40-vear generation value and that they stem ultimately from Hecataeus, who made the first consistent use of genealogical chronology. and that the first such chronological system originated in Anatoïia ; he also finds traces of a 40-vear generation value in Thucydides and he believes that perhaps the Persians followed a 40-year generation value. W. Dex Boër [n. 13] p. 5-93. accepts Meyer’s theory and then finds traces of a 40-yeur generation value in Herodotus and Thucydides ; he further believes that che use of 4 40-vear generation value was a very ancient practice in Sparta (cf. especially p. 88-95 “Fonty-Years Generations"}, For a recent. comprehensive, and detailed ‘account of these and related copies, cf. Kurt Von Frrrz. Die griechisehe Geschichuschreibung, Berlin, 1967. The number 40: as the product of 4 x 10, 2 important mystical numbers, by which the 30 piglets represented the 30-vear interval between the founding of Lavinium and Alba Longa.

were subsequentiy transmitted by the Eiruscans 10 the Romans (8). In several instances Roman regnal chronclogy reveals influence from a 40-vear generation value and periods of years in Livv. especially for treaties, are often schematic and expressed according to 40-year intervals. The evidence indicates that ancient Greek and Roman authors often cited certain numbers because of their symbolic'mystical dimension and 1hat the influence from these numbers was felt both implicitly and explicitiy. Most of the symbolic/mystical numbers examined in this paper reveal ultimate Pythagorean influence but the number 40 stems from a different tradition which is Near-Eastern in origin. Albany Academy, Roland À. LAROCHE. Albany, NY. U.S.A. (18) Cf. R. M. Ocnvie, À Commentary on Liv: 1-5, Oxford, 1970, p. 95-56, who speculates that “the change (Le, in Rome during Numa’s veign) to a 12-month calendar was inspired not merely by a desire to correlate the lunar and the solar vear but by a more exact computation of both undertaken principally by the Babylonians and mediated through the Etruscans”.