Appendix A: Problems for the Reader

 

The problems presented in Chapters 25 were so different one from the others that it was necessary to accompany each of them with a copious commentary. In order to allow the reader who may like to explore some Old Babylonian texts without being held firmly by the hand, this appendix contains problems in translation only, or at most accompanied by the most necessary clarifications. Some are counterparts of problems that were presented above and come from the same tablets. 

TMS XVI #2

 

13The 4th of the width to that by which the length goes beyond the width, to join, 

1415′. You, 15′ to 4 raise, 1 you see, what is it? 

154 and 1 posit. 

1615′ scatter. 10′, the going-beyond, and 5′, the joined, posit. 20′, the width, 

17to 10′, the going-beyond, join, 30′ the length, and 20′, to tear out, posit. 5′ to 4 raise, 

1820′ you see. 20′, the width, to 4 raise, 1°20′ you see. 

1930′, the length, to 4 raise, 2 you see. 20′, the width, 

20from 1°20′ tear out, 1 you see. 1 

21from 2, the lengths, tear out, 1 you see, what is it? 

22From 4, of the fourth, 1 tear out, 3 you see. igi 4 detach, 15′ you see. 

2315′ to 3 raise, 45′ you see, as as much as (there is) of widths posit. Posit to tear out. 

241 as as much as (there is) of lengths posit. [...] 1 take, to 1 length 

25raise, 30′ you see. 20′ the width, 20′ to 45′, (as much as (there is) of) widths, raise, 

2615′ you see, 15′ to 15′ join, 30′ you see, 30 the length. 

Commentary: see #1 of the same tablet, page 27. 

TMS VII #1

 

1The 4th of the width to the length I have joined, its 7 th to 10 I have gone, 

2as much as the heap of length and width . You, 4 posit; 7 posit; 

310 posit; 5′ to 7 raise, 35′ you see. 

430′ and 5′ single out. 5′, the step, to 10 raise, 

550′ you see. 30′ and 20′, posit. 5′, the step, to 4, of the fourth of the width, 

6raise: 20′ you see, 20′, the width. 30′ to 4, of the fourth, 

7raise, 2 you see. 2 posit, lengths. 20′ from 20′ tear out, 

8and from 2, 30′ tear out, 1°30′ you see. 

9From 4, of the fourth, 1 tear out, 3 {…} you see. 

10igi 3 detach, 20′ you see. 20′ to 1°30′ raise: 

1130′ you see, 30′ the length. 30′ from 50′ tear out, 20′ you see, 20′ the width. 

12Turn back. 7 to 4, of the fourth, raise, 28 you see. 

1310 from 28 tear out, 18 you see. igi 3 detach, 

1420′ you see. 20′ to 18 raise, 6 you see, 6 (for) the length. 

156 from 10 tear out, 4 (for) the width. 5′ to 6 raise, 

1630′ the length. 5′ to 4 raise, 20′ you see, 20′ the width . 

Commentary: see #2 of the same tablet, page 34. 

VAT 8389 #1

 

Obv. I 

1From 1 bùr 4 gur of grain I have collected, 

2from 1 second bùr 3 gur of grain I have collected. 

3grain over grain, 8‵20 it went beyond 

4My plots I have accumulated: 30‵. 

5My plots what? 

630‵, the bùr, posit. 20‵, the grain which he has collected, posit. 

730‵, the second bùr, posit. 

815‵, the grain which he has collected, 

98‵20 which the grain over the grain went beyond, 

10and 30‵ the accumulation of the surfaces of the plots posit: 

1130‵ the accumulation of the surfaces of the plots 

12to two break: 15‵. 

1315‵ and 15‵ until twice posit: 

14igi 30‵, of the bùr, detach: . 

15 to 20‵, the grain which he has collected, 

16raise, 40′ the false grain; to 15‵ which until twice 

17you have posited, 

18raise, 10‵ may your head hold! 

19igi 30, of the second bùr, detach, . 

20 to 15‵, the grain which he has collected, 

21raise, 30′ the false grain; to 15 which until twice 

22you have posited, raise, 7‵30. 

2310‵ which your head holds 

24over 7‵30 what goes beyond? 2‵30 it goes beyond. 

252‵30 which it goes beyond, from 8‵20 

26which the grain over the grain goes beyond, 

Obv. II 

1tear out: 5‵50 you leave. 

25‵50 which you have left 

3may your head hold! 

440′, the change, and 30′, the change, 

5accumulate: 1°10′. The igi I do not know. 

6What to 1°10′ may I posit 

7which 5‵50 which your head holds gives me? 

85‵ posit. 5‵ to 1°10 raise. 

95‵50 it gives to you. 

105‵ which you have posited, from 15‵ which until twice 

11you have posited, from one tear out, 

12to one join: 

13The first is 20‵, the second is 10‵. 

1420‵ (is) the surface of the first plot, 10‵ (is) the surface of the second plot. 

15If 20‵ (is) the surface of the first plot, 

1610‵ the surface of the second plot, their grains what? 

17igi 30‵, of the bùr, detach: . 

18 to 20‵, the grain which he has collected, 

19raise, 40′. To 20‵, the surface of the first plot, 

20raise, 13‵20 the grain of 20‵, the surface of the plot. 

21igi 30‵, of the second bùr, detach: . 

22 to 15‵, the grain which he has collected, raise, 30′. 

2330′ to 10‵, the surface of the second plot 

24raise, 5 the grain of the surface of the second plot. 

2513‵30 the grain of ¿the surface? of the first plot 

26over 5 the grain of ¿the surface? of the second plot 

27what goes beyond? 8‵20 it goes beyond. 

This problem belongs to one of two twin tablets, containing a total of ten problems about the rent paid for two parcels of a field. On one parcel the rent is 4 gur of grain per bùr, on the other it is 3 gur per bùr. The present problem informs us also that the total area is 30‵ (sar = 1 bùr), and that the difference between the total rents of the two parcels is 8‵20 (sìla). The other problems give, for instance, the two areas, or the difference between the areas together with the total rent. 

As explained on page 17, the bùr and the gur are units belonging to practical life. In order to work in the place-value system we need to convert them into the standard units sar and sìla (1 bùr = 30‵ sar, 1 gur = 5‵ sìla); as we see, the difference between the two rents is already given in sìla, and the total area in sar. 

A modern reader may find it strange that the two rents per bùr, which in lines I.1–2 are given in gur (per bùr), are translated into sìla in lines I.6–7 without multiplication; in general, as we see, the text skips no intermediate step. The explanation is that the conversion is made by means of a “metrological table” (probably a table learned by heart). Precisely because such conversions had to be made so often, scribes had tables which not only stated the converted values of the practical units but also of their multiples. However, they had no tables for combined conversions, and therefore the final conversion into sìla per sar asks for calculation. 

The modern reader may also wonder that the text does not indicate once for all the value of the bùr in sìla and its igi. Once more the reason is that the text describes the Old Babylonian calculational technique: the calculator writes on a small tablet for rough work the three numbers 20 (20‵ sìla per bùr), 30 ( sar per bùr) and 2 ( , igi 30‵)—and afterwards, by means of the multiplication table, the product 40 (20‵ sìla per sar). 

A small explanation may be necessary in order to facilitate understanding of the procedure: first the text determines what the difference between the two rents would be if the two parcels had been equal in area, that is, 15‵ sar each. This difference is not large enough—it is 2‵30 sìla, 5‵50 sìla too small—and therefore the first parcel must be enlarged. Each time a sar is transferred from the second to the first parcel, the difference grows by 40′+30′ sìla (the two “changes” of II.41); the number of sar that must be transferred is then found by division. 

At the end we find a numerical verification. Such verifications are not rare in the Old Babylonian texts even though their presence is not a general norm. 

VAT 8390 #1

Obv. I 

1Length and width I have made hold: 10‵ the surface. 

2The length to itself I have made hold: 

3a surface I have built. 

4So much as the length over the width went beyond 

5I have made hold, to 9 I have repeated: 

6as much as that surface which the length by itself 

7was made hold. 

8The length and the width what? 

910‵ the surface posit, 

10and 9 (to) which he has repeated posit: 

11The equalside of 9 (to) which he has repeated what? 3. 

123 to the length posit 

133 to the width posit. 

14Since “so much as the length over the width went beyond 

15I have made hold,” he has said 

161 from 3 which to the width you have posited 

17tear out: 2 you leave. 

182 which you have left to the width posit. 

193 which to the length you have posited 

20to 2 which to the width you have posited raise, 6. 

21igi 6 detach: 10′. 

2210′ to 10‵ the surface raise, 1‵40. 

23The equalside of 1‵40 what? 10. 

Obv. II 

110 to 3 which to the length you have posited 

2raise, 30 the length. 

310 to 2 which to the width you have posited 

4raise, 20 the width. 

5If 30 the length, 20 the width, 

6the surface what? 

730 the length to 20 the width raise, 10‵ the surface. 

830 the length together with 30 make hold: 15‵. 

930 the length over 20 the width what goes beyond? 10 it goes beyond. 

1010 together with 10 make hold: 1‵40. 

111‵40 to 9 repeat: 15‵ the surface. 

1215‵ the surface, as much as 15‵ the surface which the length 

13by itself was made hold. 

As support for the interpretation, a diagram may serve (Figure 1). Then the text almost explains itself, in particular if one keeps in mind BM 13901 #10 (page 46) and BM 15285 #24 (page 93). 

1 The geometry of VAT 8390 #1. 

One should take note of the use of the multiplicative operations “make hold,” “raise” and “repeat.” That “making hold” really implies a construction is underlined in I.3, as we have also seen in AO 8862 #2 (page 60). The “raising” in I. 20 and II.7 is of special interest: it finds the area of rectangles, but as these are already in place, there is no need to construct them. Therefore the area is merely calculated. 

VAT 8520 #1

 

Obv. 

1The 13th from the heap of the igûm and the igibûm 

2to 6 I have repeated, from the inside of the igûm 

3I have torn out: 30′ I have left. 1 the surface. The igûm and the igibûm what? 

4Since “the thirteenth of the heap of the igûm and the igibûm 

5to 6 I have repeated, from the inside of the igûm 

6I have torn out: 30′ I have left,” he has said, 

713, of the thirteenth, posit; 6 to which he has repeated posit; 

81, the surface, posit; and 30′ which he has left posit. 

9From 13, of the thirteenth, 6 to which he has repeated 

10tear out. 7 you leave. 

117 which you leave and 6 to which you have repeated, 

12may your head hold! 

137 to 6 raise, 42 to 1, the surface, raise, 42. 

1442, may your head hold! 

1513, of the thirteenth, to 30′ which he has left 

16raise, 6°30′ to two break: 3°15′. 

173°15′ together with 3°15′ make hold: . 

18To , 42 which your head holds 

19join, . 

20The equal of what? 7°15′. 

217°15′ and 7°15′, its counterpart, lay down: 

223°15′, the made-hold, from one tear out, to the other join: 

23The first is 10°30, the other is 4. 

24What to 7, which your head holds, should I posit 

25which 10°30′ gives me? 1°30′ posit. 1°30′ to 7 raise, 

2610°30′ it gives you. 1°30′ which you have posited is the igûm. 

27igi 6, which your head holds, detach, 10′. 

2810′ to 4 raise, 40′ is the igibûm. 

29Since 1°30′ is the igûm, 40′ is the igibûm, the surface is what? 

301°30′, the igûm, to 40′, the igibûm, raise, 1 is the surface. 

311°30, the igûm, and 40′, the igibûm, heap: 2°10′. 

Rev. 

1The thirteenth of 2°10′ what? 10′. 

210′ to 6 repeat: 1, from 1°30, 

3the igûm, tear out: 30′ you leave. 

Like YBC 6967 (page 46), this problem deals with a number pair from the table of reciprocals. Both texts speak of their product as “the surface,” in agreement with the geometric representation. But there is a difference: this time the product is 1, not 1‵ as in YBC 6967. 

As regards the mathematical structure and the procedure, one may compare with TMS IX #3 (page 57). 

Str 368

 

Obv. 

1I have taken a reed, its measure I do not know. 

21 kùš I have cut off. 1 sixty (steps along) the length I have gone. 

3(With) what I have cut off I have enlarged it 

4with 30 (steps) of that (along) the width I have gone. 

56‵15 is the surface. The head (initial length) of the reed what? 

6You, by your proceeding, 

71‵ and 30 posit. (For) the reed which you do not know 

81 posit, to 1 sixty which you have gone 

9you raise: 1‵ is the false length. 

1030 to this 1 raise, 30 is the false width. 

1130, the false width to 1‵, the false length, 

12raise, 30‵ the false surface. 

1330‵ to 6‵15, the true surface, 

Rev. 

1raise: it gives you. 

25′ which you have cut off to the false length raise, 

35 it gives you. 5 to the false width raise, 

42‵30 it gives you. of 2‵30 break, 1‵15 

51‵15 make encounter,  

6to join, . 

7What is equal? 13‵45 is equal. 

81‵15 which you have made encounter to the inside join, 

915‵ it gives you. igi 30‵, the false surface, detach, . 

10 to 15‵ raise, 30′ is the head of the reed. 

This is the rectangle version of the “broken reed” (see page 70), similar to VAT 7532. In this variant, the field is rectangular, and the reed breaks a single time only. 

YBC 6504 #1

 

Obv. 

1So much as length over width goes beyond, I have made confront itself, from the inside of the surface 

2I have torn it out: . Length over width 10′ goes beyond. 

3By your proceeding, 10′ you make hold: 

4 to you join: 10′ you posit. 

5Half of 10′ you break: 5′ you posit. 

65′ you make hold: you posit. 

7, the surface, to 10′ you join: you posit. 

8By , 25′ is equal. 5′ to 25′ you join: 

930′, the length, you posit. 5′ from 25′ your tear out: 

1020′, the width, you posit. 

This problem deals with the same mutilated rectangle as #4 of the same tablet (see page 79): Together, indeed, the four problems of the tablet represent an interesting variant of the closed group where the “surface” of a rectangle is given together with the length; with the width; with the sum of the sides; or with their difference (see note 3, page 108). In the present tablet, the “surface” is replaced everywhere by the same mutilated rectangle. 

In this first problem, we know the side of the square that has been “torn out.” It is therefore easily reduced to the type we know from YBC 6967 (page 46). In following the operations one should keep in mind that the number 10′ occurs in two different roles. 

Exceptionally in this type, the “joining” of 5′ precedes the “tearing out.” The tablet seems to belong to the same early phase and text group as AO 8862 , and it shares this particularity with three texts from Eshnunna (thus belonging to an even earlier phase). It seems indeed that the school is responsible for the request that operations should always be concretely meaningful, just as it was responsible for outlawing broad lines—this request is not evidence of “a primitive intellect not yet ready for abstraction,” as has been supposed, but of a critical mind reflecting upon how to justify what is done. 

YBC 6504 #3

 

Rev. 

1So much as length over width goes beyond, made encounter, from inside the surface I have torn out, 

2. 30′ the length, its width what? 

330′ made encounter: 15′ you posit. 

4 from inside 15′ you tear out, you posit. 

5Half of 30′ you break: 

615′ made encounter: you posit. 

7 to you join: you posit. 

8By , 25′ is equal. 15′ from 25′ you tear out: 

910′ you posit. 10′ from 30′ you tear out: 

1020′, the width, you posit. 

This is the third problem from the same tablet. It makes use of a ruse which is both elegant and far from every routine (see Figure 2): elimination of the mutilated rectangle from the square on the length leaves a remainder that can be decomposed as a square and a rectangle . These can be reconfigured as a gnomon, as shown in the diagram. We may look at the process as a “change of variable”—the problem now concerns a square and 30 of its sides, and its solution follows the book for such problems. 

2 The geometry behind YBC 6504 #3, in slightly distorted proportions. 

BM 85200+VAT 6599 #23

 

Rev. I 

19An excavation. So much as I have made confront itself, and 1 kùš, going beyond, that is the depth. 1°45′ of dirt I have torn out. 

20You, 5′, going beyond, to 1, the conversion, raise, 5′ you see; to 12 raise, 1 you see. 

215′ make confront itself, you see. to 1 raise, you see. igi 25 detach, 

222‵24 you see. 2‵24 to 1°45′ raise, 4‵12 you see. 

23from “equal, 1 joined,” 6 ¿1? is/are equal(s). 6 to 5′ raise, 30′ you see, confronts itself. 6 (error for 7) the depth. 

24The procedure. 

This problem comes from the same tablet as the “excavation problem” BM 85200+VAT 6599 #6 that was dealt with above (page 89), and its solution follows the same principles. Now the “ground” is square, and the depth exceeds the side by 1 kùš. As “reference body” a cube of side 1 kùš is chosen, which allows the use of a table of , called “equal, 1 joined.” Such tables have been found. 

Db2–146

 

Obv. 

1If, about a (rectangle with) diagonal, (somebody) asks you 

2thus, 1°15 the diagonal, 45′ the surface; 

3length and width corresponding to what? You, by your proceeding, 

41°15′, your diagonal, its counterpart lay down: 

5make them hold: comes up, 

6 ¿may your? hand ¿hold? 

745′ your surface to two bring: 1°30′ comes up. 

8From cut off: {…} the remainder. 

9The equal of take: 15′ comes up. Its half-part, 

10 comes up, to raise: comes up 

11 your hand. 45′ your surface over your hand, 

12 comes up. The equal of take: 

13 comes up, its counterpart lay down, 

14 which you have made hold to one 

15join: from one 

16cut off. 1 your length, 45 the width. If 1 the length, 

1745 the width, the surface and the diagonal corresponding to what? 

18You, by your making, the length make hold: 

191 comes up … may your head hold. 

Rev. 

20… : 45′, the width, make hold: 

21 comes up. To your length join: 

22 comes up. The equal of take: 

231°15′ comes up. 1°15′ your diagonal. Your length 

24to the width raise, 45′ your surface. 

25Thus the procedure. 

26

 

This is one of the texts from the Eshnunna region, and thus belongs to the earliest phase (and as we see, it uses the phrase “to one join, from one cut off,” not respecting the “norm of concreteness”). With fair precision it can be dated to c. 1775 bce. The problem is one of the riddles which the Old Babylonian school borrowed from the Akkadian surveyors (see pages 106 and 107); it turns up, solved in precisely the same way, in a Hebrew manual from 1116 ce, that is, 1900 years later. In the text we see several reminiscences of this origin—for instance the introductory passage “If, about a (rectangle with) diagonal, (somebody) asks you thus” and the reference to the square on the length in line 21 simply as “your length”; both features reverberate in BM 13901 #23. 

3 The geometry of Db2-146. 

Lines 1–9 find the difference between the length and the width of the rectangle; the method is shown in the upper part of Figure 3. Afterwards, the sides are found from this difference and the area by the procedure which we already know perfectly well, for instance from YBC 6967 (see page 46), and which corresponds to the lower diagram in the figure. (However, the use of “raising” in Obv. 10 shows that the procedure is supposed to be supported by the already existing upper diagram.) 

The “hand” of lines 6 and 11 is a reference to the reckoning board on which the calculator performed his additions and subtractions. The “half-part” of line 9 (muttatum) is a synonym for “moiety.” 

In the end we have a proof with an unmistakeable trace of the “Pythagorean rule” in abstract formulation (the length make hold, without the usual identification of its numerical value). 

Footnotes

[1]

The tablet is damaged at this point, but the traces of signs that remain could well come from the word takkirtum, which means “change” or “modification” but does not occur in other mathematical texts. In any case, this philological doubt does not touch the interpretation of the mathematical procedure.