106 bernard vitrac
arithmetical books, the Ishâq–Th âbit version (itself inspired by Heron) is the
best evidence of this ‘betterment’. Th e consideration of changes in order con-
fi rms the complexity of the phenomenon. In Book vi , Propositions vi .9–13
(according to the numbers of the Heiberg edition), resolve the fi ve problems
listed in the table above.
In the indirect tradition, the order of presentation runs 13–11–12–9–10.
Th e solutions of the problems are independent of each other. Th us the
inversion has no infl uence on the deductive structure, but vi .13 uses (part
of ) vi .8 Por.:
From this it is clear that, if in a right-angled triangle a perpendicular be drawn
from the right angle to the base, the straight line so drawn is a mean proportional
between segments of the base. 95
Th e Proposition has thus been moved in order to place it in contact with
the used result. Since there are clearly two groups – one concerning pro-
portionality, the other about sections – the coherence of the two themes has
been maintained by also moving vi .11–12 (or, in the case of Adelard’s trans-
lation, only vi .11 because it lacks vi .12 as a result of a ‘Hajjajian’ lacuna). 96
Th is order of the indirect tradition appears to be an improvement over the
Greek.Greek order Medieval order
9: From a given straight line to cut off
a prescribed part.
10: To cut a given uncut straight line
similarly to a given cut straight line.13: To two given straight lines to fi nd
a mean proportional.
11: To two given straight lines to fi nd
a third proportional.
11: To two given straight lines to fi nd
a third proportional.12: To three given straight lines to fi nd
a fourth proportional.
12: To three given straight lines to fi nd
a fourth proportional.9: From a given straight line to cut off
a prescribed part.
13: To two given straight lines to fi nd
a mean proportional.10: To cut a given uncut straight line
similarly to a given cut straight line.95 In the majority of Greek manuscripts, a second assertion declares that each side of a right angle
is also the mean proportional between the entire base and one of the segments of it (which has
a common extremity with the aforementioned side). It is absent in V , for example. Heiberg
considered it inauthentic and bracketed it (see EHS ii : 57.1–3). Both parts exist in the Ishâq–
Th âbit version and Adelard of Bath and Gerard of Cremona, but the complete Porism does not
fi gure in the Leiden Codex (the an-Nayrîzî version). Moreover a scholium, attributed to Th âbit,
explains that the Porism had not been found among the Greek manuscripts. Without a doubt,
this is in error. In (at least) two mss of the Ishâq–Th âbit version, a gloss indicates that Th âbit
had not found what corresponds to only the second part of the Porism (excised by Heiberg).
See Engroff 1980 : 28–9.
96 Th is we know thanks to the recension of pseudo-Tûsî. See Lévy 1997 : 222–3.