Petronian Numbers Huxley, George Greek, Roman and Byzantine Studies; Spring 1968; 9, 1; ProQuest pg. 55

Petronian Numbers 

George Huxley 

PLUTARCH remarks that a certain Petron of Himera declared the 
universe to consist of 183 kosmoi arranged so as to form a tri­
angle with sixty worlds along each side and the remaining 

three at the corners (De defectu oraculorum 22.422B). This remarkable 
theory was known to Plutarch indirectly from the writings of Ph ani as, 
the Aristotelian scholar of Eresos in Lesbos, who quoted Hippys of 
Rhegion on Petron's cosmology.l The date of Hippys is not known­
there is no special reason to date him as early as the first half of the 
fifth century B.c.2-but we need not doubt that Phanias reported a 
genuine piece of western Greek (or possibly Sikel)3 philosophical doc­
trine. Petron's theory deserves closer examination. 

Plutarch was puzzled by the statement that the kosmoi were 
a1T'TOI-'EVOV~ &,u~,\wv Ka'Tct a'TOLXELOV (422n), but he earlier states (422B) 
that they were in contact with each other and rotated cc as in a dance," 
" e "" , '-1.. c~ '\\ 1\ ' 1 I" , , a1TTEa aL OE TOV~ E'f'Et;'YJ~ a/\I\'YJI\WV aTpEl-'a 1TEpLLOV'Ta~ wa1TEp EV XOPEUf. 

The meaning therefore is that each world rotated on a fixed axis but 
touched its neighbour, which therefore rotated at the same speed but 
in the opposite sense, if all the kosmoi were of the same size. Plutarch's 
words suggest that the three kosmoi, one at each of the corners, also 
rotated, but here a problem would have demanded Petron's attention 
if he thought his system out thoroughly. Three spheres in contact, 
having parallel axes and equal diameters, cannot drive one another. 
The solution is to place spheres at the corners, each having a greater 
diameter than the two other spheres in contact with it. At each 
corner there would thus be two, not three, points of contact. 

Petron's difficulties would not have ended there, however, for a 
continuous series of mutually driving, contra-rotating spheres must 

1 De defor. 23.422D. 
2 Cf Jacoby on FGrHist 554 T 1 and F 5. The Sud a s.v. -J.rr (rr) v, 'P7]yivo, (Hippys T 1) state: 

taToptK6" Y0'0vtiJ, brt TWV IIEputKwv. Ka, 7rPWTO, ~YProPE T<XS' L'tKEAtK<X, 7rp&.gn" fl, VaTEPOV 
M07], J1TET'P.ETO. 

8 E. A. Freeman, The History of Sicily from the Earliest Times II (Oxford 1891) 159-160. 

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56 PETRONIAN NUMBERS 

be of an even number if one adjacent pair is not to rotate in the same 
sense and so create friction at the point of contact. Petron's system 
had 183 spheres if Phanias reported his theory correctly, and so if all 
the spheres were in contact, each with its neighbour, there must have 
been friction at one point. It is possible that Phanias or Hippys mis­
understood Petron, who perhaps gave 61, not 62, spheres to each side 
of his triangular universe. The sum of spheres in it would then have 
been (59X 3)+3=180, an even number, permitting a free running 
system. 

Petron's system is of little astronomical interest, although it is 
remarkable that Leukippos seems also to have contemplated a tri­
angular universe.4 The Petronian triangle does, however, have a place 
in early Greek number theory, because it is different from normal 
Pythagorean triangular numbers. The typical Pythagorean number is 
a sum of successive terms in the series of natural numbers. Thus 
1+2+3+ ... +n=-!n(n+l) is a Pythagorean triangular number of 
side n. The Pythagorean series begins 1,3,6, 10, 15 ... Petronian 
triangular numbers are of the form 1, 3, 6, 9, 12, 15 ... Thus 

are Petronian triangles, but 

are Pythagorean. Erwin Schroedinger failed to understand this dis­
tinction when he wrote: HOne of the early Pythagoreans, Petron, 
contended that there were altogether 183 worlds, arranged in a 
triangle, though, by the way, this is not a triangular number."5 The 

4 Diels, Vorsokr. ll 67 A 24 line 33. 
5 Nature and the Greeks (Cambridge 1954) 36. W. Nestle, RE 37 (1937) 1191 S.V. PETRON, 

states that there were 163 Petronian KOU!-'Ot, but that seems to be a mistake. 



GEORGE HUXLEY 

corresponding Petronian square numbers would be of the form: 

So this series would begin 1, 4, 8, 12, 16 ... 

THE QUEEN'S UNIVERSITY OF BELFAST 

September, 1967 

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