Lines 19 and 20 present a system of two equations about a rectangle, one of the first and one of the second degree. The former is of the same type as the one explained in TMS XVI #1
Lines 28–30 repeat the trick used in section #2 of the text (see Figure 3.10): the length and the width are prolonged by 1, and the square that is produced when that which the two “joined”^{1} “hold” is “joined” to the “heap” ; out of this comes a “surface 2,” the meaning of which is again explained in lines 30–33.
The lines 34–37 are very damaged, too damaged to be safely reconstructed as far as their words are concerned. However, the numbers suffice to see how the calculations proceed. Let us introduce the magnitudes and . The text refers to them as the length and width “of the surface 2”—in other words, . Further,
Once again, a cutandpaste
One part of the new square
(16°15′) is constituted by the gnomon, whose area results from recombination of the original rectangle
; this area is hence 2‵6. We also know the area of the outer square,
(lines 40 and 41). When the gnomon is “torn out” (lines 41 and 42),
remains for the square contained by the gnomon. Its side (that which “is equal”) is 11°45′, which must now be “joined” to one of the pieces 16°15′ (which gives us
) and “torn out” from the other, its “counterpart” (which gives us
). This time, however, it is not the same piece that is “joined”
The first two words of the first line (I.30) tell us that we are dealing with a figure that is fully characterized by its length and its width, that is, with a rectangle (cf. page
Before studying the procedure, we may concentrate on certain aspects of the formulation of the text. In line I.31 we see that the operation “to make hold”
The text, almost certainly from Larsa, seems to be from c. 1750 bce and thus to belong to the early phase of the adoption of algebra by the southern scribe school
The upper part of Figure 4.2 illustrates this situation, with 2 and 3 “inscribed as inscription” of respectively of the “projections”^{3} 1 of the length and the width (lines II.2–5); the heavily drawn configuration thus has an area equal to 15.
However, the present text does not proceed like that—Old Babylonian
algebra was a flexible instrument
It would be easy to subtract
from
, but that may not have been deemed sufficiently informative.^{4} In any case, the text introduces a detour by the phrase “Do not go beyond!” (the same verb as in the “subtraction by comparison”). A rectangle
(3,2) is constructed (perhaps one should imagine it in the corner where 2 and 3 are “inscribed” in Figure 4.2; in any case Figure 4.3 shows the situation). Without further argument it is seen that the half (three small squares) exceeds the third (two small squares) by one of six small squares, that is, by a sixth—another case of reasoning by “false position.”
Once more we therefore have a rectangle of which we know the area and the sum of length and width. The procedure is the same as in the final part of TMS IX #3
This problem
Everything else, however—that is, that the area of the field is known before it is measured, and also the ways to indicate the measures of the pieces that break off from the reed—shows which
ruses the Old Babylonian school masters had to make use of in order to produce seconddegree problems
For once, Figure 4.5 reproduces a diagram that is traced on the tablet itself. In general, as also here, diagrams are only drawn on the tablets
The piece that broke off last is put back into place, and the “(lower) width” (evidently to the right) is paced out (line Obv. 7) as 36 r. Finally we learn that the area of the field is 1 bùr = 30‵ sar (1 sar = 1 nindan^{2}), see page
Another false position
Yet the text does not calculate this area: The surface to 2 repeat. Instead it doubles the trapezium so as to form a rectangle (see the left part of Figure 4.6), and the lines Obv. 14–16 calculate the area of this rectangle (the “false surface”
If the reed had not lost an ulterior piece of
kùš, we might now have found the solution by means of a final false position similar to that of BM 13901 #10
From here onward, the procedure coincides with that of BM 13901 #2
The area
corresponds to the rectangle of (height)
and breadth
. Half of the excess of the height over the breadth is “broken” and repositioned as seen in the diagram: lightly shaded in the original positions, heavily shaded where it is moved to. The construction of the completing square is described with one of the synonyms
After the usual operations we find that , and in line Rev. 5 that . We observe, however, that the “moiety” that was moved around is not put back into its original position, which would have reconstituted in the vertical direction. Instead, the other “moiety,” originally left in place, is also moved, which allows a horizontal reconstitution : 6‵ which you have left to 2‶30‵ join, 2‶36‵ it gives you.^{7}
One might believe this problem type to be one of the absolute favorites of the Old Babylonian teachers of sophisticated mathematics. We know four variants of it differing in the choice of numerical parameters. However, they all belong on only two tablets sharing a number of terminological particularities—for instance, the use of the logogram for the “moiety,” and the habit that results are “given,” not (for example) “seen” or “coming up.” Both tablets are certainly products of the same locality and local tradition (according to the orthography based in Uruk), and probably come from the same school or even the same hand. A simpler variant with a rectangular field, however, is found in an earlier text of northern origin, and also in a text belonging together with the trapezium variants; if not the favorite, the broken reed was probably a favorite.
This is another problem
This system—of the same type as the one proposed in YBC 6967
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” (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.
Modifying the vertical scale
On the whole, the final part of the procedure follows the model of YBC 6967
In YBC 6967, the igûmigibûm problem (page
This problem comes from the collection of problems
The problem could have been solved by means of the diagram shown in Figure 4.10, apparently already used to solve problem
However, the author chooses a different method, showing thus the flexibility of the algebraic technique. He takes the two areas and as sides of a rectangle, whose area can be found by making 10′ and 10′ “hold” (see Figure 4.10):
We now have a rectangle for which we know the area and the sum of the two sides, as in the problems TMS IX #3
What is to be taken note of in this problem is hence that it represents areas by line segments and the square of an area by an area
This is no real contradiction. The present problem
Figure 4.12 makes clear the procedure: 4c is represented by 4 rectangles
; the total
thus corresponds to the crossshaped configuration where a “projection”
Lines 12–13 prescribe to cut out of the cross (demarcated by a dotted line) and the “joining” of a quadratic complement to the gnomon that results. There is no need to “make hold,” the sides of the complement are already there in the right position. But it is worthwhile to notice that it is the “projection” itself that is “joined”: it is hence no mere number but a quadratic configuration identified by its side.
The completion
The archaizing aspect, it should be added, does not dominate completely. Line 12, asking first for the “inscription” of 4 and stating afterwards its igi
However , once again the calculator shows that he has several
strings on his bow, and that he can choose between them as he finds
convenient
Line 6 finds , and from here onward everything follows the routine, as can be seen on Figure 4.14: 28 will be equal to 2°20′, and hence to 5′.^{12} Therefore, the length will be , and the width .
The mistake seems difficult to explain, but inspection of the geometry of the argument reveals its origin (see Figure 4.15). On top the procedure is presented in distorted proportions; we see that the “joining” of
presupposes that the mutilated rectangle be cut along the dotted line and opened up as a pseudognomon. It is clear that what results from the completion
This mistake illustrates an important aspect of the “naive”
[1]
As the “tobejoined” of page
[2]
We should observe that the half that appears here is treated as any other fraction, on an equal footing with the subsequent third. It is not a “moiety,” and the text finds it through multiplication by 30′, not by “breaking.”
Let us also take note that the half of the length and the third of the width are “joined” to the “surface,” not “heaped” together with it. A few other early texts share this characteristic. It seems that the surveyors thought in terms of “broad lines,” strips possessing a tacitly understood breadth of 1 length unit;
this practice is known from many preModern surveying traditions, and agrees well with the Babylonian understanding of areas as “thick,” provided with an implicit height of 1 kùš (as inherent in the metrology of volumes, which coincides with that for areas—see page
[3]
The absence of this notion from the text should not prevent it from using it as a technical term of general validity.
[4]
Alternatively, the trick used by the text could be a leftover from the ways of surveyors not too familiar with the placevalue system; or (a third possibility) the floatingpoint character of this system might make it preferable to avoid it in contexts where normal procedures for keeping track of orders of magnitude (whatever these normal procedures were) were not at hand.
[5]
It is not quite to be excluded that the text does not directly describe the construction but refers to the inscription twice of 3°25′ on a tablet for rough work, followed by the numerical product—cf. above, note 11, page
[6]
The position of the “upper” width to the left is a consequence of the new orientation of the cuneiform script (a counterclockwise rotation of 90°) mentioned in the box “Cuneiform writing.” On tablets, this rotation took place well before the Old Babylonian epoch, as a consequence of which one then wrote from left to right. But Old Babylonian scribes knew perfectly well that the true direction was vertically downwards—solemn inscriptions on stone (for example Hammurabi’s law) were still written in that way. For reading, scribes may well have turned their tablets 90° clockwise.
[7]
This distinction between two halves of which one is “left” is worth noticing as another proof of the geometric interpretation—it makes absolutely no sense unless understood spatially.
[8]
By error, line 30 of the text has instead of ; a partial product 25 has been inserted an extra time, which shows that the computation was made on a separate device where partial products would disappear from view once they had been inserted. This excludes writing on a clay surface and suggests instead some kind of reckoning board.
The error is carried over in the following steps, but when the square root is taken it disappears. The root was thus known in advance.
[9]
In the original, the word is “surface” marked by a phonetic complement indicating the accusative. An accusative in this position is without parallel, and seems to allow no interpretation but the one given here.
[10]
For once, the determinate article corresponds to the Akkadian, namely to an expression which is only used to speak about an inseparable plurality (such as “the four quarters of the world” or “the seven mortal sins”).
[11]
The use of a “raising” multiplication shows that the calculator does not construct a new rectangle but bases his procedure on a subdivision of what is already at hand—see the discussion and dismissal of a possible alternative interpretation of the procedure of BM 13901 #10 in note 5, page
[12]
Line 10 speaks of this as 5′ the length—namely the side of the small square. Some other texts from Susa also speak of the side of a square as its “length.”
Table of Contents
1 Introduction: The Issue – and Some Necessary Tools
2 Techniques for the First Degree
3 The Fundamental Techniques for the Second Degree
4 Complex Seconddegree Problems
5 Application of QuasialgebraicTechniques to Geometry
9 A Moral
Appendix A: Problems for the Reader
Appendix B: Transliterated Texts
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Series 
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