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We shall now search for signs that Thabit took advantage of that work in composing his Kitab al-Qarastun. The last proposition of K. Qar. invites comparison with L. Can. III, in that both express the solution in the form of a computational rule . In K. Qar. VIII, the counterweight W is worked out according to a series of multiplications and divisions ; these may be compiled as a formula W = (La-Lb) X (Wa + Wb) + (La + Lb) X (La + Lb)+2Lb, where the terms denote the lengths and weights of the segments A, B of the beam, as before.

I and L. Kar. VII); 3b - converse of 3a (d. 1. Can. II); 3c - alternative form of rule for the counterweight (d. 1. Can. III and 1. Kar. VIII) SOLUTION OF THE PROBLEM OF THE WEIGHTED BEAM 39 The contents of Kitab al-Qarastun, as extant in the Beirut manuscript, are keyed in column QB, parallel to the contents of Liber Karastonls (LK). E = enunciation; P = proof; X = numerical example. The propositions are numbered in accordance with Liber Karastonis (ed. Clagett). Correspondence between the enunciations of theorems is generally literal; proofs may agree closely, or with only minor discrepancies (as in II, III, IV) ; but just as often the Latin reveals signs of major re-editing (as in VI, VIII); and in some cases, the Latin holds propositions missing from the Arabic (V, VII).

368-371). 6 On the Archimedean techniques embodied in the Arabic K. Qar. VI, see jaouiche, op. , pp. 94-101, U8£' Jaouiche assumes that Thabit devised this proof as an original effort. But he seems unaware of the problems entailed by this assumption. 54 THE MEDIEVAL TRADITION OF MECHANICS This is precisely how the step is managed in K. Qar. VI. Given two magnitudes, a multiple of the smaller is taken which exceeds the larger; then the corresponding submultiple of the larger is taken. This submultiple is a part of the larger which is less than the smaller given magnitude, as the proof requires.

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