Hoyrup argues very convincingly for a more geometrical reading of the Babylonian sources than the traditional arithmetical reading. Babylonian mathematical sources are extremely sparse in words, as is understandable since they are laboriously inscribed on clay. This extremely condensed, telegraphic style, allows for a considerable scope of interpretation. A typical phrase such as "30 a-na 7 ta-na-sima 210" (p. 5) can be interpreted alternately in purely arithmetical ("multiply 30 by 7; the result is 210") or geometrical terms ("form a rectangle with sides 30 and 7; it's area is 210"). The former option is the traditional one adopted by Neugebauer et al. The primary argument for this interpretation is the fact that the sources often add lengths and areas together, which is geometrically nonsensical. On the other hand the terms used for multiplication seem to point, linguistically speaking, to the geometrical interpretation ("to raise", "to hold", etc.), and indeed certain words for "multiplication" are only used for multiplying two lengths, never areas. Furthermore, certain words that can be read as "protrude", "break", etc., can be understood quite literally in the geometrical reading, whereas they are basically ignored in the arithmetical readings:
"We may say that the received interpretation made sense of the numbers occurring in the text. But it obliterated the distinction made in the texts which after all need not be synonymous unless the arithmetical interpretation is taken for granted; ... and it had to dismiss some phrases as irrelevant ... or to explain them by gratuitous ad-hoc hypotheses." (p. 13)
Consider an example: BM 13901 #1. Hoyrup's translation is as follows (p. 50).
"The surface and my confrontation I have accumulated: 45' it is."
It is to be understood that "the surface" means the area of a square, and the "confrontation" its side. So the problem is x^2+x=45'.
"1, the projection, you posit."
This step gives a concrete geometrical interpretation of the expression x^2+x. We draw a square and suppose its side to be x. Then we make a rectangle of base 1 protrude from one of its sides. This rectangle has the area 1*x, so the whole figure has the area x^2+x, which is the quantity known.
"The moiety of 1 you break, 30' and 30' you make hold."
We break the rectangle in half and attach the half we cut off to an adjacent side of the square. We have now turned our area of 45' into an L-shaped figure.
"15' to 45' you append: 1."
We fill in the hole in the L. This hole is a square of side 30', so its area is 15'. So when we fill in the hole the total area is 45'+15'=1.
"1 is equalside."
The side of the big square is 1.
"30' which you have made hold in the inside of 1 you tear out: 30' is the confrontation."
The side of the big square is x+30' by construction, and we have just seen that it is also 1. Therefore x must be 1-30'=30', and we have solved the problem.
The difference between the arithmetical and geometrical readings is important since only in Hoyrup's reading does it follow that "The procedure is ... algorithm and proof in one. ... [It] performs all steps in such a way that their correctness is obvious." (p. 98)
But the geometrical reading does not mean that the procedure is applicable to geometrical problems only. On the contrary, the procedure is "functionally abstract": there are examples where a segment represents a number, an area, a volume, or a commercial rate (p. 280). Virtually all texts "use the Sumerograms us and sag unerringly for the lengths and widths of the standard representation; 'real' linear dimensions (the length of a wall, the distance bricks are to be carried, the width of a canal), in contrast, may as well be written in syllabic Akkadian ... This suggests strongly that the Old Babylonian authors were explicitly aware of the functionally abstract character of their standard representation." (pp. 280-281) "In this sense, Neugebauer was right in considering the us and sag as equivalents of the symbols of modern algebra." (p. 10)