Robson launches into her main argument with the following remarks, pgs 183-184:
The terms igum and igibum have been a problem for all translators, from Neugebauer to the present. Neugebauer avoids translation of the words in YBC 6967, as do many others since. He merely transliterates. At first glance they look like "the short side" and "the long side" but this may fail to capture the nuance intended by the OB scribe. In his Vocabulary, pg 164, Neugebauer shows them both as "reciprocal" and this is the source of Robson's translation. (For further discussion see below.) Hoyrup, pgs 44 and 55, also refrains from translating them. Sometimes he uses them in the simple sense of "one side" and the "other side," pgs 57. I further examined Hoyrup's remarks. Rectangles can be defined in the Sumerian as us and sag, "long side" and "narrow side," pg 34, or "length" and "width", Neugebauer Vocabulary, pgs 170 and 174. But on pg 35 Hoyrup admits that (igum and igibum) may actually be the length and the width of an excavation. As he states:
igum-igibum thus represents a standard problem which may be varied in more or less intricate ways; but it refers back to the conventional standard representation of measurable line segments.
Before considering Robson's main thesis I shall use Hoyrup's and Neugebauer's translations of YBC 6967, in order to gain some insight into the OB methods. One can follow the line designators by the slashes "/" in Robson's translation above. This is just a short diversion into OB mathematics because I want to show that Hoyrup's (and Robson's) idea of primitive mathematics does those old people a bad disservice.
For Robson's linguistic comparison see above.
Line No. |
Hoyrup | Neugebauer | Translation Notes |
---|---|---|---|
The Obverse. | |||
1 | The igibum over the igum. 7 it goes beyond. | The igibum exceeds the igum by 7. | eli denotes relative position, "over or above," and should be expressed. "Igibum over the igum exceeds . . ." iter and its inflected form iteru means "exceeds." Hoyrup sees this as "goes beyond," giving it a more primitive expression. |
2 | igum and igibum what? | What are the igum and igibum? |
minum means "what?" |
3 |
You, 7 which the igibum |
As for you, halve 7, by which the igibum |
atta is a clearly a Semitic personal pronoun, "you." The word sa means "which" or "of." Neugebauer "halve" is lifted from two lines down. |
4 |
over the igum goes beyond |
exceeded the igum, |
The Sumerian anagram UGU means "over" or "above." |
5 |
to two break: 3;30 |
and the result is 3;30 |
ana is a term that denotes a mathematical operation. Here it means "multiply." sina means "two." hepema is translated as "result" by Neugebauer. He does a halving of the 7 two lines up; Hoyrup here sees it as "break," persistently forcing a more primitive meaning on the ancient scribes. |
6 |
3;30 together with 3;30 |
Multiply together 3;30 with 3;30 |
itti is translated "with." |
7 |
make hold: 12;15 |
and (the result is) 12;15 |
sutakilma is translated "result," "come up," or "make hold." Again we see Hoyrup's desire to make the meaning more primitive. |
8 |
To 12;15 which comes up for you |
To 12;15 which resulted for you |
ana should mean "add." Again and again Hoyrup alters the meaning of the phrases, ilikum "comes up" for "resulted." |
9 |
1 the surface append: 1 12;15 |
add 1, the product, and (the result is) 1 12;15 |
Hoyrup visualizes a geometric figure. Neugebauer understands a mathematical operation. A partially restored void in the clay gives "surface" to Hoyrup, "product" to Neugebauer. |
10 |
The equal side of 1 12;15 what 8;30 |
What is the square root of 1 12;15? Answer: 8:30 |
ibsi is viewed by Hoyrup as "equal side," by Robson as "square side," and Neugebauer as "square root." Neugebauer translates terms in their mathematical meaning; Hoyrup and Robson are unwilling to do this, and confine themselves to geometric interpretation. |
11 |
8;30 and 8;30, its counterpart, lay down |
Lay down 8;30 and 8;30, its equal and then subtract |
Robson sees "equivalent," which probably captures the sense better. |
Reverse: | |||
1 |
3;30 make hold |
3;30, the takiltum, |
This word presents difficulties. Hoyrup reverts to "make hold." |
2 |
from one tear out, |
from the one, |
Hoyrup repeatedly imposes meaning on the scribes, forcing them into conceptualization they did not intend, here "tear out." |
3 |
to one append. |
add it to the other, |
ana, a mathematical operation, is translated "append" by Hoyrup. |
4 |
The first is 12, the second is 5 |
One is 12, the other 5 |
isten, sanum = first or one, second or other. |
5 | 12 is the igibum, 5 is the igum | 12 is the igibum, 5 the igum |
Hoyrup persistently and repeatedly imposes his primitive mathematical conceptualization upon the ancient scribes. He sees them as exclusively geometric in their thinking; Neugebauer sees them as mathematical. Robson borrows Hoyrup's attitude and believes those ancient people could not think "mathematically." For us to translate the old documents in mathematical terms, she believes, forces them into a modern mental framework that does not respect their primitive methods, and takes the documents out of their historical setting.
I do not know the Akkadian language. I must resort to dictionaries. Here I have used the limited dictionaries provided by Hoyrup and Neugebauer, who do not include all the words they use. To resolve the differences between the "new" translations and the old would require linguists with good working knowledge of the ancient Semitic languages, and the borrowing of Sumerian anagrams. To properly assess the Akkadian terms we would need to compare with Hebrew, Arabic, and other Near East languages that would show us in deeper context the intent of the words. I do not trust Hoyrup and Robson to be intellectually honest. They have an agenda to demonstrate their views of human evolution. A basic dishonesty exists in modern scholarship when it forces on those ancient people attitudes about that evolution. Neugebauer came from a generation that was willing to accept at face value the words he encountered. Hoyrup and Robson have now transformed the meanings into a different evolutionary framework. We see it in numerous examples above: goes beyond for exceed, break for halving (denoted by an explicit two), surface for product, equal side or square side for square root, tear out for no apparent reason. I ask: do we two break or do we break in half? If I encounter an arithmetic square, and then determine its mathematical properties, do I square side it or do I take the square root? Hoyrup and Robson have imposed on those ancient people what they think the terms should mean.
This is not the imposition of modern mathematical methods on the ancients; it is plain common sense.
Examination of this example of geometric (or mathematical) operation should show how the ancient scribe thought.
He has a rectangle in which the one side exceeds the other in length by 7 parts. The entire area of the rectangle is 60, (sexagesimal 1).
How did he determine the lengths of the two sides?
The level of the problem may be found in modern High Schools, as a written problem in algebra, or presented as a drawing in geometry class.
The scribe first breaks the 7 parts into two halves of 3.5 parts each. He then moves one of the broken parts down under the remaining area to produce the figure shown by Robson. (I am assuming that the figure produced by both Hoyrup and Robson accurately represents the figure on the clay tablet, since I do not have a copy available to me.) This provides him with an incomplete square. He notes that the area of this missing part is 3.5 X 3.5, or 12.25. He then adds the original area to that number to form a complete square, 1 12;15. Thus he has created a square of 72.25 area. He asks what are the lengths of the sides of this square. They are 8.5. He then subtracts from this new number 3.5 to obtain 5. He then adds to this new number the same 3.5 to obtain 12. Thus the original lengths were 5 and 12.
How did the scribe know this would give him a correct solution? It certainly is not apparent to me. Can we justify it? I shall now show Neugebauer's understanding of the problem.
Here we obtain a better appreciation of the terms igum and igibum. This is important to Robson's analysis of Plimpton 322 and her treatment of reciprocals, which we shall consider in the next paper. Hoyrup was unwilling to take the meaning to this level, and saw igum and igibum applied mathematically more generally than what Neugebauer states.
Neugebauer notes that we have two mathematical statements: xy = 60 and x - y = 7, where x and y represent the two sides of the rectangle. This produces a quadratic whose solution then gives 12 and 5 as the answers. He says that the statement of the problem by the OB scribe produces a formula which is followed exactly by the text.
Neugebauer's shows the result of the quadratic, as we would do it in modern formulation.
What did Neugebauer mean that this is followed exactly by the text?
We can follow it. We must assume the igibum and igum multiplied against one another provide an area of sexagesimal 1, or 60.
On the Obverse:
Line 1: One side = 7. This is given as a statement of the problem.
Line 3 to Line 5: Break 7 into two parts for 3;30. This is seen in the quadratic 7/2.
Line 6: Multiply 3;30 by itself.
Line 7: This yields 12;15. We see this in the quadratic (7/2)^{2}.
Lines 8 - 9: Add 1. We see this in the quadratic beneath the radical.
Line 10: Take the square root to give 8;30. Neugebauer shows this as part of the solution to the quadratic.
Line 11: You can see that the scribe specifies that there are two parts to this 8;30, which is technically accurate.
On the Reverse, all of which follow the modern quadratic expression:
Line 1: Take hold of the 3;30 you obtained above.
Line 2: From the one (or first) 8;30 subtract .
Line 3: To the other (or second) 8;30 add.
Line 4: The one is 12 and the other 5.
Line 5: The igibum is 12 and igum is 5.
Indeed, the modern quadratic solution is followed exactly by the text, operation by operation.
When I work through the statements by the old Semitic Akkadian scribe I see that he is doing exactly what we in the modern world are doing, anticipating every step. There is one major difference: he represents his solution with a geometric figure while we represents ours in algebraic notation. But the same steps are there.
In the modern quadratic solution we "set up" the problem by formulating the mathematical conditions before we begin solving.
xy = 60 and x - y = 7. These are statements of the problem. Then:
y = 60/x
x - 60/x = 7
x^{2} - 60 = 7x
x^{2} - 7x - 60 = 0
We then obtain the expression shown by Neugebauer.
These steps are missing in the method used by the OB scribe. He begins by breaking up the geometry into two 3;30 pieces. He then moves them to create an additional artificial area. But how did he know his two 3;30 pieces, and the new area, would result in the proper solution?
If the igibum and igum denote an area of 1 (60), and we limit ourselves to integers, how many different ways can we express that area?
1 X 60
2 X 30
3 X 20
4 X 15
5 X 12
6 X 10
and then repeat. So we have only six possible igibum and igum with integer definitions.
Solutions exist for all of these, with the difference between igibum and igum (x - y) at 59, 28, 17, 11, 7, and 4 respectively.
Now that we have generalized the problem we are more familiar with the possible solutions.
I ask again: How could the OB scribe have known that his solution was correct? How did he know that his procedure would produce the proper answer? How did he know to set up the problem? Robson makes the remark: This large square must have length of 8;30, so the lengths of the original rectangle must be 8;30 + 3;30 = 12 and 8;30 - 3;30 = 5 respectively. She jumps to this statement without showing us the logic of how she got there. I certainly would not have been able to predict that this was a correct solution. In order to accept that his solution was correct I had to retranslate back into my algebraic notation. Is that because I am restrained by my indoctrination into modern algebra ? Is it that I cannot get into the mind set of those minds?
Or is it possible that the OB scribe did not originally solve the problem by this geometric method, but saw geometry as a way to represent the solution? Did he work it out on a separate piece of clay in a form similar to our quadratic expression, and then transform that representation into the geometric figure? Was this his way of teaching students who did not know how to set the problem?
Hoyrup offers other examples, BM 13901, #1 and #2, pg 50, as similar in solution to this, in geometric rearrangement of pieces. Hence it was a general procedure, not limited to problems with igibum and igum.
The statement of methods offered by Hoyrup does not help me understand. His presentations are done at the level of reporting on a phenomenon, and not on logic. If those ancient minds employed geometric solutions to quadratic problems, they were more sophisticated than modern minds, not less, and made leaps of logic that we do not follow. Hoyrup goes on to discuss Scholastization, pg 380. He admits that they used "completion tricks" and "isolation tricks." Some of the examples he illustrates are "truly astonishing" and "beautiful illustrations." As he says about those ancient people:
"concreteness of thought" which it presupposes is no indication of "primitivity" or failing capability to think in abstract terms; it is the consequence of metamathematical choice, and thus comparable to the eviction of fractions from Greek theoretical arithmetic and to Viete's insistence on homogeneity.
The many illustrations offered by Neugebauer on elements of mathematics, up to and including logarithms and exponentials, shows a sophistication that we have not even begun to fathom.