By Wilbur Richard Knorr
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Extra info for Ancient sources of the medieval tradition of mechanics: Greek, Arabic, and Latin studies of the balance (Monografia / Istituto e museo di storia della scienza)
J. , 1957, pp. 130-132 and my " Archimedes and the Elements ", Archive for History of Exact Sciences, 1978, 19, sect. II. 50 THE MEDIEVAL TRADITION OF MECHANICS of the balance 2. It is claimed that if the extended portion is replaced by the equal weight T suspended from its midpoint, the system remains in equilibrium 3. The proof is indirect: if it is not as claimed, we suppose at first that T is too light. We must add some weight L to its side to restore balance. We now subdivide the original extended segment into a number of equal parts, each being less than weight L.
At any rate, the Arabic line preserves not a hint of the important insight conveyed in the Latin. In the Arabic version the proof of this rule is clumsy and confused. It begins by introducing four auxiliary terms: E=(La-Lb) X (Wu+ Wb) ; F=E +(La+Lb); G=FX(La+Ib) and H=G+2Ib. One must show that H is the value of the counterweight which restores to equilibrium the beam divided into segments A, B. The writer next shows that E=Fx (La+Lb) and that (La-Lb): (La+Ib)=F: (Wu+ Wb). In this way he comes to see that F expresses the weight of the excess portion 2 Cf.
Nevertheless, as far as the content of the Excerptum is concerned, wherever the Arabic and Latin versions of Thabit's work diverge, it agrees with the Latin. For instance, Exc. (2) paraphrases 1. Kar. VII, a theorem missing from the Arabie; so also Exc. (4) agrees with V, even to the detail of considering specifically four equally distributed weights, while the comparable result does not have the status of a theorem in the Arabic. In Exc. (7) the notion of uirtus (" force ") is presented in the image of walkers traversing certain distances in given times; the same example appears in the Latin 1.