90 
PYTHAGOREAN ARITHMETIC 
numbers in YIII. 11, 12, and for similar plane and solid num 
bers in YIII. 18, 19. Nicomachus quotes the substance of 
Plato’s remark as a ‘ Platonic theorem adding in explanation 
the equivalent of Eucl. VIII. 11, 12. 1 
(S) A theorem of Archytas. 
Another interesting theorem relative to geometric means 
evidently goes back to the Pythagoreans. If we have two 
numbers in the ratio known as e?nyoptos, or suiter particular is, 
i.e. the ratio of n + 1 to n, there can be no number which is 
a mean proportional between them. The theorem is Prop. 8 of 
Euclid’s Sectio Canonist and Boetius has preserved a proof 
of it by Archytas, which is substantially identical with that of 
Euclid. 3 The proof will be given later (pp. 215-16). So far as 
this chapter is concerned, the importance of the proposition lies 
in the fact that it implies the existence, at least as early 
as the date of Archytas (about 430-365 b.c.), of an Elements 
of Arithmetic in the form which we call Euclidean; and no 
doubt text-books of the sort existed even before Archytas, 
which probably Archytas himself and others after him im 
proved and developed in their turn. 
The ‘irrational’. 
We mentioned above the dictum of Proclus (if the reading 
dXoycor is right) that Pythagoras discovered the theory, or 
study, of irrationals. This subject was regarded by the 
Greeks as belonging to geometry rather than arithmetic. 
The irrationals in Euclid, Book X, are straight lines or areas, 
and Proclus mentions as special topics in geometry matters 
relating (1) to positions (for numbers have no position), (2) to 
contacts (for tangency is between continuous things), and (3) 
to irrational straight lines (for where there is division ad 
infinitum, there also is the irrational). 4 I shall therefore 
postpone to Chapter Y on the Pythagorean geometry the 
question of the date of the discovery of the theory of irra 
tionals. But it is certain that the incommensurability of the 
. 1 Nicom. ii. 24. 6, 7. 
2 Micsici Scriptores Graeci, ed. Jan, pp. 148-66; Euclid, vol. viii, ed. * 
Heiberg and Menge, p. 162. 
3 Boetius, De Inst. Musica, iii. 11 (pp. 285-6, ed. Friedlein); see Biblio 
theca Maihematica, vi s , 1905/6, p, 227. 
4 Proclus on Eucl. I, p. 60. 12—16.