4.4: TMS XIII
- Page ID
- 47612
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As TMS VII #2, this problem is rather difficult. It offers an astonishing example of application of the geometrical technique to a non-geometrical question.
1 2 gur 2 pi 5 bán of oil I have bought. From the buying of 1 shekel of silver,
2 4 silà, each (shekel), of oil I have cut away.
3 \(\frac{2}{3}\) mina of silver as profit I have seen. Corresponding to what
4 have I bought and corresponding to what have I sold?
5 You, 4 silà of oil posit and 40, (of the order of the) mina, the profit posit.
6 igi 40 detach, \(1^{\prime} 30^{\prime \prime}\) you see, \(1^{\prime} 30^{\prime \prime}\) to 4 raise, 6′ you see.
7 \(6^{\prime}\) to \(12‵50\), the oil, raise, \(1‵17\) you see.
8 \(\frac{1}{2}\) of 4 break, 2 you see, 2 make hold, 4 you see.
9 4 to \(1‵17\) join, \(1‵21\) you see. What is equal? 9 is equal.
10 9 the counterpart posit. \(\frac{1}{2}\) of 4 which you have cut away break, 2 you see.
11 2 to the 1st 9 join, 11 you see; from the 2nd tear out,
12 7 you see. 11 silà each (shekel) you have bought, 7 silà you have sold.
13 Silver corresponding to what? What to 11 ¿silà? may I posit
14 which \(12‵50\) of oil gives me? \(1‵10\) posit, 1 mina 10 shekel of silver.
15 By 7 silà each (shekel) which you sell of oil,
16 that of 40 of silver corresponding to what? 40 to 7 raise,
17 \(4‵40\) you see, \(4‵40\) of oil.
This is another problem which, at superficial reading, seems to reflect a situation of real practical (here, commercial) life. At closer inspection, however, it turns out to be just as artificial as the preceding broken-reed question: a merchant has bought \(M=2\mathrm{gur} 2\mathrm{pi} 5\mathrm{bán}\) (= \(12‵50 \mathrm{sìla}\)) of fine oil (probably sesame oil). We are not told how much he paid, but the text informs us that from the quantity of oil which he has bought for one shekel (a) he has cut away 4 sìla, selling what was left ( \(v=a-4\)) for 1 shekel; \(a\) and \(v\) are thus the reciprocals of the two prices—we may speak of them as “rates” of purchase and sale. Moreover, the total profit \(w\) amounts to \(\frac{2}{3}\) mina = 40 shekel of silver. For us, familiar with algebraic letter symbolism, it is easy to see that the total purchase price (the investment) must be \(M \div a\), the total sales price \(M \div v\), and the profit in consequence \(w=(M \div v)-(M \div a)\). Multiplying by \(a \cdot v\) we thus get the equation
\(M \cdot(a-v)=w \cdot a v\),
and since \(v=a-4\), the system
\(a-v=4 \quad, \quad a \cdot v=(4 M) \div w\).
This system—of the same type as the one proposed in YBC 6967, the igûm-igibûm problem (page 46)—is indeed the one that is solved from line 8 onward. Yet it has certainly not been reached in the way just described: on one hand because the Babylonians did not have our letter symbolism, on the other because they would then have found the magnitude \((4 M) \div w\) and not, as they actually do, \((4 \div w) \cdot M\).
The cue to their method turns up towards the end of the text. Here the text first finds the total investment and next the profit in oil (\(4`40\) sìla). These calculations do not constitute a proof since these magnitudes are not among the data of the problem. Nor are they asked for, however. They must be of interest because they have played a role in the finding of the solution.
Figure 4.8 shows a possible and in its principles plausible interpretation. The total quantity of oil is represented by a rectangle, whose height corresponds to the total sales price in shekel, and whose breadth is the “sales rate” \(v\) (sìla per shekel). The total sales price can be divided into profit (40 shekel) and investment (purchase price), and the quantity of oil similarly into the oil profit and the quantity whose sale returns the investment.
The ratio between the latter two quantities must coincide with that into which the quantity bought for one shekel was divided—that is, the ratio between 4 sìla and that which is sold for 1 shekel (thus \(v\)).
Modifying the vertical scale by a factor which reduces 40 to 4, that is, by a factor \(4 \div w=4 \div 40=6^{\prime}\), the investment will be reduced to \(v\), and the area to \((4 \div w) \cdot M=1` 17\). In this way we obtain the rectangle to the right, for which we know the area ( \(a \cdot v=1` 17\)) and the difference between the sides ( \(a-v=4\)), exactly as we should. Moreover, we follow the text in the order of operations, and the oil profit as well as the investment play a role.
On the whole, the final part of the procedure follows the model of YBC 6967 (and of other problems of the same type). The only difference occurs in line 10: instead of using the "moiety" of \(a-v\) which we have "made hold" in line 8, \(a-v\) is "broken" a second time. That allows us to "join" first (that which is joined is already at disposal) and to "tear out" afterwards.
In YBC 6967, the igûm-igibûm problem (page 46), the geometric quantities served to represent magnitudes of a different nature, namely abstract numbers. Here, the representation is more subtle: one segment represents a quantity of silver, the other the quantity of oil corresponding to a shekel of silver.