between geometry and arithmetic rested upon the thought that the "point" is to spatial magnitude what the number i is to num ber. Numbers were thought of as collections of units, and vol umes, as, in like fashion, collections of points; that is, the point was conceived as a minimum volume. As the criticisms of Zeno showed, this conception was fatal to the specially Pythagorean science of geometry itself, since it makes it impossible to assert the continuity of spatial magnitude. (This, no doubt, is why Plato, as Aristotle tells us, rejected the notion of a point as a fiction.) There is also a difficulty about the notion of a number as a "collection of units," which must have been forced on Plato's at tention by the interest in "irrationals" which is shown by repeated allusions in the dialogues, as well as by the later anecdotes which represent him as busied with the problem of "doubling the cube" or finding "two mean proportionals." "'Irrational" square and cube roots cannot possibly be reached by any process of forming "collections of units," and yet it is a problem in mathematics to determine them, and their determination is required for physics (Epin. 99oc-991b).
This is sufficient to explain why it is necessary to regard the numbers which are the physicist's determinants as themselves de terminations of a continuum (a "great and small"), by a "limit" and why, at the same time "the One" can no longer be regarded as a "blend" of "unlimited" and "limit" but must be, itself, the factor of "limit." (If it were "the first result" of the blending, it would re-appear in all the further "blends"; all numbers would be "collections of one" and there would be no place for the "irrationals.") There is no doubt that Plato's thought proceeded on these general lines. Aristotle tells us that he said that num bers are not really "addible"(a) o-vutAnroin Elvac Tolje aptOgobs Typos CAMXovs, Met M 1083 a 34), that is that the integer-series is not really made by successive additions of and the Epinomis (loc. cit.) is emphatic on the point that contrary to the accepted opinion, "surds" are just as much numbers as integers. The un derlying thought is that numbers are to be thought of as gene rated in a way which will permit the inclusion of rationals and irrationals in the same series. In point of fact there are logical difficulties which make it impossible to solve the problem pre cisely on these lines. It is true that mathematics requires a sound logical theory of irrational numbers, and again, that an integer is not a "collection of units"; it is not true that rational integers and "real numbers" form a single series.
The Platonic number-theory was inspired by thoughts which have since borne fruit abundantly, but it was itself premature. We learn partly from Aristotle, partly from notices preserved by his commentators, that in the derivation of the integer-series, even numbers were supposed to be generated by the "dyad" which "doubles" whatever it "lays hold of," odd numbers in some way by "the One" which "limits" (6p4 or "equalises" but the interpretation of these statements is, at best, conjectural. In
the statement about the "dyad" there seems to be some confusion between the number 2 and the "indeterminate dyad," another name for the continuum also called the "great-and-small," and it is not clear whether this confusion was inherent in the theory itself, or has been caused by Aristotle's misapprehension.
Nor, again, is it at all certain exactly what is meant by the operation of "equalising" ascribed to the One.' It would be im proper here to propound conjectures which our space will not allow us to discuss.
schools of Athens and appropriated their emoluments. Plato's greatest scholar, Aristotle, finally went his own way and organised a school of his own in the Lyceum, claiming that he was preserv ing the essential spirit of Platonism, while rejecting the difficult doctrine of the Forms; the place of official head of the society was filled first by Speusippus, Plato's nephew B.c.), then by Xenocrates B.c.). Under Arcesilaus (276-241 B.c.) the Academy began its long-continued polemic against the sen sationalist dogmatism of the Stoics, which accounts both for the tradition of later antiquity which dates the rise of a "New" (some said "Middle") and purely sceptical "Academy" from Ar cesilaus, and for the eighteenth-century associations of the phrase "academic philosophy." In the first century B.C. the most interesting episode in the his tory of the school is the quarrel between its President, Philo of Larissa and his scholar Antiochus of Ascalon, of which Cicero's Academica is the literary record. Antiochus, who had embraced Stoic tenets, alleged that Plato had really held views indistin guishable from those of Zeno of Cittium, and that Arcesilaus had corrupted the doctrine of the Academy in a sceptical sense. Philo denied this. The gradual rapprochement between Stoicism and the Academy is illustrated from the other side by the work of Stoic scholars like Panaetius of Rhodes, and Poseidonius of Apamea, who commented on Platonic dialogues and modified the doctrines of their school in a Platonic sense.