3.6: TMS IX #1 and #2
- Page ID
- 47606
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1 The surface and 1 length I have heaped, \(40^{\prime}\). ¿30, the length,? \(20^{\prime}\) the width.
2 As 1 length to \(10^{\prime}\) the surface, has been joined,
3 or 1 (as) base to \(20^{\prime}\), the width, has been joined,
4 or \(1^{\circ} 20^{\prime}\)′ ¿is posited? to the width which \(40^{\prime}\) together with the length ¿holds?
5 or \(1^{\circ} 20^{\prime}\) toge 
ther 
with \(30^{\prime}\) the length holds, \(40^{\prime}\) (is) its name.
6 Since so, to \(20^{\prime}\) the width, which is said to you,
7 1 is joined: \(1^{\circ} 20^{\prime}\) you see. Out from here
8 you ask. \(40^{\prime}\) the surface, \(1^{\circ} 20^{\prime}\) the width, the length what?
9 \(30^{\prime}\) the length. Thus the procedure.
#2
10 Surface, length, and width I have heaped, 1. By the Akkadian (method).
11 1 to the length join. 1 to the width join. Since 1 to the length is joined,
12 1 to the width is joined, 1 and 1 make hold, 1 you see.
13 1 to the heap of length, width and surface join, 2 you see.
14 To \(20^{\prime}\) the width, 1 join, \(1^{\circ} 20^{\prime}\). To \(30^{\prime}\) the length, 1 join, \(1^{\circ} 30^{\prime}\).
15 ¿Since? a surface, that of \(1^{\circ} 20^{\prime}\) the width, that of \(1^{\circ} 30^{\prime}\) the length,
16 ¿the length together with? the width, are made hold, what is its name?
17 2 the surface.
18 Thus the Akkadian (method).
As TMS XVI #1, sections #1 and #2 of the present text solve no problem.12 Instead they offer a pedagogical explanation of the meaning to ascribe to the addition of areas and lines, and of the operations used to treat second-degree problems. Sections #1 and #2 set out two different situations. In #1, we are told the sum of the area and the length of a rectangle; in #2, the sum of area, length and width is given. #3 (which will be dealt with in the next chapter) is then a genuine problem that is stated and solved in agreement with the methods taught in #1 and #2 and in TMS XVI #1.
Figure (3.9) is drawn in agreement with the text of #1, in which the sum of a rectangular area and the corresponding length is known. In parallel with our symbolic transformation
\(\ell \cdot w+\ell=\ell \cdot w+\ell \cdot 1=\ell \cdot(w+1)\),
the width is extended by a "base."13 That leads to a whole sequence of explanations, mutually dependent and linked by “or … or … or,” curiously similar to how we speak about the transformations of an equation, for example
\(" 2 a^{2}-4=4, \quad \text { or } \quad 2 a^{2}=4+4, \quad \text { or } \quad a^{2}=4, \quad \text { or } \quad a=\pm \sqrt{4}=\pm 2^{\prime \prime}\).
Line 2 speaks of the "surface" as \(10^{\prime}\). This shows that the student is once more supposed to know that the discussion deals with the rectangle 
(\(30^{\prime}\),\(20^{\prime}\)). The tablet is broken, for which reason we cannot know whether the length was stated explicitly, but the quotation in line 6 shows that the width was.
In the end, lines 7-9 shows how to find the length once the width is known together with the sum of area and length (by means of a division that remains implicit)..
#2 teaches how to confront a more complex situation; now the sum of the area and both sides is given (Figure 3.10). Both length and width are prolonged by 1; that produces two rectangles 
(\(\ell\),1) and (\(w\),1), whose areas, respectively, are the length and the width. But it also produces an empty square corner (1,1). When it is filled we have a larger rectangle of length \(\ell+1\left(=1^{\circ} 30^{\prime}\right)\), width \(w+1\left(=1^{\circ} 20^{\prime}\right)\) and area \(1+1=2\); a check confirms that the rectangle "held" by these two sides is effectively of area 2.
This method has a name, which is very rare in Old Babylonian mathematics (or at least in its written traces). It is called "the Akkadian (method)." "Akkadian" is the common designation of the language whose main dialects are Babylonian and Assyrian (see the box "Rudiments of general history"), and also of the major non-Sumerian component of the population during the third millennium; there is evidence (part of which is constituted by the present text) that the Old Babylonian scribe school took inspiration for its "algebra" from the practice of an Akkadian profession of surveyors (we shall discuss this topic on page 108). The "Akkadian" method is indeed nothing but a quadratic completion albeit a slightly untypical variant, that is, the basic tool for the solution of all mixed second-degree problems (be they geometric or, as with us, expressed in number algebra); and it is precisely this basic tool that is characterized as the "Akkadian (method).