104 bernard vitrac
enrichment was not confi ned to the fi nal, more complicated portion of the
text in question.
It is even probable that the entire treatise has been subjected to such treat-
ment. For example, the arithmetic books of the Ishâq–Th âbit and Gerard of
Cremona versions possess four supplementary Propositions with respect to
the Greek. Ishâq–Th âbit ix.30–31 are added to improve (Heib.) ix .30–31,
and Ishâq–Th âbit viii .24–25 are the converses of (Heib.) viii .26–27.
In fact, the proof of (Ishâq–Th âbit) viii .24 (plane numbers) is nothing more
than the second part of (Heib.) ix .2! Hence the idea, again suggested by
Heron, to remove this portion in order to introduce it as a Proposition in
its own right and to do the same for the converse of viii .27 (solid numbers)
to simplify the proof of ix .2. 89
Insofar as the Euclidean approach is deductive, the work just described
represents a real improvement of the text as much from a logical perspec-
tive as from a mathematical point of view. A number of implicit presump-
tions which might be described as harmless but real deductive lacunae have
been identifi ed and eliminated. However, the logical concerns have been
sometimes pushed beyond what is reasonable. For example, in the desire
to make the contrapositives appear in the text, Propositions viii .24–27
in the Ishâq–Th âbit version expect the reader to know that two numbers
are similar plane numbers if and only if they have the ratio that a square
number has to a square number to one another. Th e Lemma x.9/10 – an
addition probably connected to Ishâq–Th âbit viii .24–25 – thence deduces
that non-similar plane numbers do not have the ratio that a square number
has to a square number to one another.
Likewise, the (important) Propositions x .5–6 establish that the ‘com-
mensurable magnitudes have to one another the ratio which a number has
to a number’ (5) and the inverse (6). In the Greek manuscripts, but not in
the primary indirect tradition, two other Propositions (Heib.) x .7–8 have
been inserted: ‘Incommensurable magnitudes have not to one another
the ratio which a number has to a number’ (7, contrapositive of 6) and its
inverse (8, contrapositive of 5)!
Propositions viii .14–15 show that ‘if a square (resp. cube) [number]
measures a square (resp. cube) [number], the side will also measure
the side; and, if the side measures the side, the square (resp. cube) will
also measure the square (resp. cube)’. In the Greek manuscripts these
Propositions are followed by their contrapositives (Heib. viii .16–17, for
example): ‘If a square number does not measure a square number, neither89 See Vitrac 2004 : 25.