306 
PLATO 
The second number is described thus : 
‘ The ratio 4 : 3 in its lowest terms (‘ the base irvOp-pu, of 
the ratio knirpiTos) joined or wedded to 5 yields two harmonies 
when thrice increased (rph av^pQeis), the one equal an equal 
number of times, so many times 100, the other of equal length 
one way, but oblong, consisting on the one hand of 100 squares 
of rational diameters of 5 diminished by one each or, if of 
number is stated. 
Another view of the whole passage has recently appeared (A. G. Laird, 
Plato's Geometrical Number and the comment of Proclus, Madison, Wiscon 
sin, 1918). Like all other solutions, it is open to criticism in some 
details, but it is attractive in so far as it makes greater use of Proclus 
(in Platonis rem])., voi. ii, p. 36 seq. Kroll) and especially of the passage 
(p. 40) in which he illustrates the formation of the ‘ harmonies ’ by means 
of geometrical figures. According to Mr. Laird there are not tivo separ 
ate numbers, and the description from which Hultsch and Adam derive 
the number 216 is not a description of a number but a statement of a 
general method of formation of ‘ harmonies which is then applied to 
the triangle 3, 4, 5 as a particular case, in order to produce the one 
Geometrical Number. The basis of the whole thing is the use of figures 
like that of Eucl. YI. 8 (a right-angled triangle divided by a perpendicular 
from the right angle on the opposite side into two right-angled triangles 
similar to one another and to the original triangle). Let ABC be a 
right-angled triangle in which the sides CB, BA containing the right 
angle are rational numbers a, b respectively. 
Draw AF at right angles to ÌC meeting CB 
produced in F. Then the figure AFC is that of 
Eucl. VI. 8, and of course AB 2 =CB.BF. 
Complete the rectangle ABFL, and produce 
FL, CA to meet at K. Then, by similar tri 
angles, CB, BA, FB (—AL) and KL are four 
straight lines in continued proportion, and their 
lengths are a, b, b 2 /a, h 3 /a 2 respectively. Mul 
tiplying throughout by a 2 in order to get rid of 
fractions, we may take the lengths to be a 3 , 
a 2 b, ab 2 , b 3 respectively. Now, on Mr. Laird’s 
view, av£r](T€is òvvàpxvai are squares, as AB 2 , and 
nò^Tjcreii òvvacrrevófievai rectangles, as FB, BG, to 
u>hich the squares are equal. ‘ Making like and 
unlike ’ refers to the equal factors of a 3 , b 3 and the unequal factors of 
a 2 b, ab 2 ; the terms a 3 , a 2 b, ab 2 , b 3 are four terms (Spot) of a continued 
proportion with three intervals (an-oo-rao-ety), and of course are all ‘ con 
versable and rational with one another (Incidentally, out of such 
terms we can even obtain the number 216, for if we put a = 2, *b = 3, we 
have 8, 12, 18, 27, and the product of the extremes 8.27 = the product 
of the means 12.18 = 216). Applying the method to the triangle 3, 4, 5 
(as Proclus does) we have the terms 27, 36, 48, 64, and the first three 
numbers, multiplied respectively by 100, give the elements of the 
Geometrical Number 3600 2 = 2700.4800. On this interpretation rpis 
nv^rjdeis simply means raised to the third dimension or ‘ made solid ’ (as 
Aristotle says, Politics © (E). 12, 1316 a 8), the factors being of course 
3.3.3 = 27, 3.3.4 = 36, and 3.4.4 = 48 ; and ‘ the ratio 4 : 3 joined 
to 5 ’ does not mean either the product or the sum of 3, 4, 5, but simply 
the triangle 3, 4, 5. 
K