4.8: YBC 6504 #4
- Page ID
- 47616
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11 So much as length over width goes beyond, made encounter, from inside the surface I have torn out:
12 \(8^{\prime} 20^{\prime \prime}\). \(20^{\prime}\) the width, its length what?
13 \(20^{\prime}\) made encounter: \(6^{\prime} 40^{\prime \prime}\) you posit.
14 \(6^{\prime} 40^{\prime \prime}\) to \(8^{\prime} 20^{\prime \prime}\) you join: \(15^{\prime}\) you posit.
15 By \(15^{\prime}\), \(30^{\prime}\) is equal. \(30^{\prime}\), the length, you posit.
So far, everything we have looked at was mathematically correct, apart from a few calculational and copying errors. But everybody who practises mathematics sometimes also commits errors in the argument; no wonder then that the Babylonians sometimes did so.
The present text offers an example. Translated into symbols, the problem is the following:
\((\ell, w)-\square(\ell-w)=8^{\prime} 20^{\prime \prime} \quad, \quad w=20^{\prime}\).
Astonishingly, the length is found as that which “is equal by”
\((\ell, w)-\square(\ell-w)+\square(w)\)—that is, after a transformation and expressed in symbols, as \(\sqrt{(3 w-\ell) \cdot \ell}\).
The mistake seems difficult to explain, but inspection of the geometry of the argument reveals its origin (Figure 4.15). On top the procedure is presented in distorted proportions; we see that the “joining” of \(\square(w)\) presupposes that the mutilated rectangle be cut along the dotted line and opened up as a pseudo-gnomon. It is clear that what results from the completion of this configuration is not \(\square(\ell)\) but instead—if one counts well—\((3 w-\ell, \ell)\). Below we see the same thing, but now in the proportions of the actual problem, and now the mistake is no longer glaring. Here, \(\ell = 30^{\prime}\) and \(2 = 20^{\prime}\), and therefore \(\ell-w=w-(\ell-w)\). In consequence the mutilated rectangle opens up as a true gnomon, and the completed figure corresponds to \square(t)—but only because \(\ell=\frac{3}{2} w\).
This mistake illustrates an important aspect of the "naive" geometry: as is generally the case for geometric demonstrations, scrupulous attention must be paid so that one is not induced into error by what is "immediately" seen. the rarity of such errors is evidence of the high competence of the Old Babylonian calculators and shows that they were almost always able to distinguish the given magnitudes of a problem from what more they knew about it.