THE GEOMETRICAL NUMBER 
307 
irrational diameters, by two, and on the other hand of 100 
cubes of 3.’ 
The ratio 4 : 3 must be taken in the sense of ‘ the numbers 
4 and 3 ’, and Adam takes ‘joined with 5 ’ to mean that 4, 3 
and 5 are multiplied together, making 60 ; 60 ‘ thrice increased ’ 
he interprets as ‘60 thrice multiplied by 60 ’, that is to say, 
60x 60 x 60 x 60 or 3600 2 ; ‘so many times 100’ must then 
be the ‘equal’ side of this, or 36 times 100; this 3600 2 , or 
12960000, is one of the ‘harmonies’. The other is the same 
number expressed as the product of two unequal factors, an 
‘oblong’ number; the first factor is 100 times a number 
which can be described either as 1 less than the square of the 
‘ rational diameter of 5 ’, or as 2 less than the square of 
the ‘ irrational diameter ’ of 5, where the irrational diameter 
of 5 is the diameter of a square of side 5, i. e. V50, and the 
rational diameter is the nearest whole number to this, namely 
7, so that the number which is multiplied by 100 is 49 — 1, or 
50 — 2, i. e. 48, and the first factor is therefore 4800; the 
second factor is 100 cubes of 3, or'2700; and of course 
4800 x 2700 = 3600 2 or 12960000. Hultsch obtains the side, 
3600, of the first ‘ harmony ’ in another way; he takes 4 and 3 
joined to 5 to be the sum of 4, 3 and 5, i. e. 12, and rply av^rjOeii, 
‘ thrice increased ’, to mean that the 12 is ‘ multiplied by three’ 1 
making 36 ; ‘so many times 100 ’ is then 36 times 100, or 3600. 
But the main interest of the passage from the historical 
1 Adam maintains that rp'is av^deis cannot mean ‘ multiplied by 3 ’. He 
observes (p. 278, note) that the Greek for ‘ multiplied by 3 ’, if we 
use av^avcn, would be rpidSi av^rjSeis, this being the construction used by 
Nicomachus (ii. 15. 2 Iva o 6 rp'is y a>v TrdXiv rpiddi fir’ dXXo dinar/] pa 
iiv^rjdrj ku'l yeiiijTtn d k£) and in Theol. At. (p. 39, Ast e£ddi avt-rjdeis). Never 
theless I think that rp'is av^dtis would not be an unnatural expression for 
a mathematician to use for ‘ multiplied by 3 ’, let alone Plato in a passage 
like this. It is to be noted that TroXXarrXnaidCu) and TroXXanXdaios are 
likewise commonly used with the dative of the multiplier; yet Io-ukis 
7ioXXmrXda-ios is the regular expression for ‘ equimultiple ’. And av^dvco is 
actually found with roaavrdKis: see Pappus ii, p. 28. 15, 22, where roanv- 
thkis av^croptv means ‘ we have to multiply by such a power ’ of 10000 or 
of 10 (although it is true that the chapter in which the expression occurs 
may be a late addition to Pappus’s original text). On the whole, I prefer 
Hultsch’s interpretation to Adam’s, rpls nv^Oeis can hardly mean that 
60 is raised to thq fourth power, 60 4 ; and if it did, ‘ so many times 100 
immediately following the expression for 3600 2 , would be pointless and 
awkward. On the other hand, ‘so many times 100’ following the ex 
pression for 36 would naturally indicate 3600. 
x 2