the whole line was discovered before Plato’s time, if not as
early as the irrationality of V 2.
(0) The Geometrical Number.
This is not the place to discuss at length the famous passage
about the Geometrical Number in the Republic} Nor is its
mathematical content of importance; the whole thing
mystic rather than mathematical, and is expressed
rhapsodical language, veiling by fanciful phraseology a few
simple mathematical conceptions. The numbers mentioned
are supposed to be two. Hultsch and Adam arrive at the
same two numbers, though by different routes. The first
of these numbers is 216, which according to Adam is the sum
of three cubes 3 3 + 4 3 + 5 3 ;
Hultsch obtains it. 2
1 Republic, viii. 546 b-d. The number of interpretations of this passage
is legion. For an exhaustive discussion of the language as well as for
one of the best interpretations that has been put forward, see Dr. Adam’s
edition of the Republic, vol. ii, pp. 204-8, 264-312.
2 The Greek is ev co 7rpcorw av^rjaeis dvvdpevai re Ka\ SvvacTTevopevcu, rpels
aTrocrrdcretf, Terrapas 8e dpovs XajBovcrai opoLowrau re koi avopoiovvrcov nai
av^ovroov Kai (pdivovrav, ndvra npoarj-yopa Kai prjrd npos dXXrjXa aTre(pT]vav,
which Adam translates by ‘the first number in which root and
square increases, comprehending three distances and four limits, of
elements that make like and unlike and wax and wane, render all
things conversable and rational with one another ’. avgrjcreis are
clearly multiplications, bvvdpevai re ka\ duvaarevopevai are explained in
this way. A straight line is said diwaadai (‘to be capable of’) an area,
e. g. a rectangle, when the square on it is equal to the rectangle ; hence
bvvnpivi] should mean a side of a square, fivvaa-revopevr] represents a sort
of passive of 8vvapevrj, meaning that of which the bwnpevr) is ‘ capable ’;
hence Adam takes it here to be the square of which the Swaplvrj is the
side, and the whole expression to mean the product of a square and its
side, i. e. simply the cube of the side. The cubes 3 3 , 4 3 , 5 3 are supposed
to be meant because the words in the description of the second number
‘of which the ratio in its lowest terms 4:3 when joined to 5’ clearly
refer to the right-angled triangle 3, 4, 5, and because at least three
authors, Plutarch (De Is. et Os. 373 F), Proclus (on Each I, p. 428.1) and
Aristides Quintilianus {De mus., p. 152 Meibom. = p. 90 Jahn) say that
Plato used the Pythagorean or 1 cosmic ’ triangle in
his Number. The ‘three distances ’ are regarded
as ‘ dimensions ’, and the ‘ three distances and
four limits ’ are held to confirm the interpretation
‘ cube ’, because a solid (parallelepiped) was said to
have ‘three dimensions and four limits’ {Theol. Ar.,
p. 16 Ast, and Iambi, in Nicom., p. 93. 10), the limits
being bounding points as A, B, C, D in the accom
panying figure. ‘ Making like and unlike ’ is sup
posed to refer to the square and oblong forms. in which the second
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