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428.29–430.1): ‘it is of course easier, when we have previously acquired, by
the method, some knowledge of the questions, to supply the proof, than it
is to fi nd it without any previous knowledge’. We should note that what is at
stake is not just the question of admissible methods, but that of what counts
as a proper demonstration.
(3.3) For my third example I turn to Hero of Alexandria. 9 A l t h o u g h h e
frequently refers to Archimedes as if he provided a model for demonstra-
tion, his own procedures sharply diverge, on occasion, from his. In the
Metrica , for instance, he sometimes gives an arithmetized demonstration
of geometrical propositions, that is he includes concrete numbers in his
exposition. Moreover in the Pneumatica especially he allows exhibiting a
result to count as a proof. Th us at 1 16.16–26 and at 26.25–28 he gives what
we would call an empirical demonstration of propositions in pneumatics,
expressing his own clear preference for such by contrast with the merely
plausible reasoning used by the more theoretically inclined investigators.
In both respects his procedures breach the rules laid down by Aristotle in
the Posterior Analytics , both in that he permits ‘perceptible’ proofs and does
not base his arguments on indemonstrable starting points and in that he
moves from one genus of mathematics to another. If we think of precedents
for his procedures, then they have more in common with the suggestion
that Socrates makes to the slave-boy in Plato’s Meno (84a), namely that if
he cannot give an account of the solution to the problem of doubling the
square, he can point to the relevant line.
(3.4) Fourthly there is Ptolemy’s redeployment of the old dichotomy
between demonstration and conjecture in two contexts in the opening
books of the Syntaxis and of the Te t r a b i b l o s. In the former ( Syntaxis 1 1,
1 6.11–7.4) he discusses the diff erence between mathēmatikē , ‘physics’ and
‘theology’. Th e last two studies are conjectural, ‘physics’ because of the insta-
bility of what it deals with, ‘theology’ because of the obscurity of its subject.
Mathēmatikē , by contrast, which here certainly includes the mathemati-
cal astronomy that he is about to expound in the Syntaxis , alone of these
three is demonstrative, since it is based on the incontrovertible methods
of geometry and arithmetic. Whatever we may think about the diffi culties
that Ptolemy himself registers, in practice, in living up to this ideal when
it comes, for instance, to his account of the movements of the planets
in latitude, it is clear what his ideal is. Moreover when in the Te t r a b i b l o s
(1 1, 3.5–25, 1 2, 8.1–20) he speaks of the other branch of the study of the
heavens, that which engages not in the prediction of the movements of the9 Cf. Tybjerg 2000 : ch. 3.