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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