After these examples of firstdegree methods we shall now go on with the principal part of Old Babylonian algebra—postponing once more the precise determination of what “algebra” will mean in a Babylonian context. In the present chapter we shall examine some simple problems, which will allow us to discover the fundamental techniques
“Surface” and “confrontation” are heaped. This addition is the one that must be used when dissimilar magnitudes are involved, here an area (two dimensions) and a side (one dimension). The text tells the sum of the two magnitudes—that is, of their measuring numbers: 45′. If c stands for the side of the square and (c) for its area, the problem can thus be expressed in symbols in this way:
Figure 3.1 shows the steps of the procedure leading to the solution as they are explained in the text:
A: 1, the projection, you posit. That means that a rectangle
(c,1) is drawn alongside the square
(c). Thereby the sum of a length and an area, absurd in itself, is made geometrically meaningful, namely as a rectangular area
. This geometric interpretation explains the appearance of the “projection,”
C: 30′ and 30′ you make hold. The outer half of the projection (shaded in grey) is moved around in such a way that its two parts (each of length 30′) “hold” the square with dotted border below to the left. This cutandpaste
D: 15′ to 45′ you join: 1. 15′ is the area of the square held by the two halves (30′ and 30′), and 45′ that of the gnomon. As we remember from page
by 1, 1 is equal. In general, the phrase “by Q, s is equal” means (see page
30′ which you have made hold from the inside of 1 you tear out.
The new translation calls for some observation. We take note that no explicit argument is given that the cutandpaste
The essential stratagem of the Old Babylonian method is the completion of the gnomon as shown in Figure 3.2. This stratagem is called a “quadratic completion”;
It is obvious that a negative solution would make no sense in this concrete interpretation. Old Babylonian algebra was based on tangible quantities even in cases where its problems
Certain general expositions of the history of mathematics claim that the Babylonians did know of negative numbers. This is a legend based on sloppy reading. As mentioned, some texts state for reasons of style not that a magnitude A exceeds another one by the amount d but that B falls short of A by d; we shall encounter an example in BM 13901 #10
To “tear out”
1, the projection, you posit. In Figure 3.3, B, the rectangle ) is composed of a (white) square and a (shaded) “excess” rectangle whose width is the projection 1.
The moiety of 1 you break. The excess rectangle, presented by its width 1, is divided into two “moieties”; the one which is detached is shaded in Figure 3.3, C.
Cutting and pasting this rectangle as seen in Figure 3.3, D we once again get a gnomon with the same area as the rectangle , that is, equal to 14‵30.
30′ and 30′ you make hold, 15′. The gnomon is completed with
the small square (black in Figure 3.3, E) which is “held” by the two
moieties. The area of this completing square
We notice that this time the “confrontation”
Seconddegree problems
The above exercise belongs to the latter type—if we neglect the fact that it does not deal with a rectangle at all but with a pair of numbers belonging together in the table
One might expect the product of igûm and igibûm
It is important to notice that here the “fundamental representation”
The next steps are remarkable. The “moiety” that was detached and moved around (the “madehold,” that is, that which was “made hold” the complementary square) in the formation of the gnomon is put back into place. Since it is the same piece which is concerned it must in principle be available before it can be “joined.” That has two consequences. Firstly, the “equal”
In BM 13901 #1
We now return to the tablet containing a collection of problems
Formulated differently, the ratio between the two sides is as 7 to 6. This is the basis for a solution based on a “false position”
Even this problem deals with two squares (lines Obv. II.44–45).^{6} The somewhat obscure formulation in line 45 means that the second “confrontation” equals twothirds of the first, with additional 5′ nindan. If and stands for the two “confrontations,” line 44 informs us that the sum of the areas is , while line 45 states that .
This problem cannot be solved by means of a simple false position in which a hypothetical number is provisionally assumed as the value of the unknown—that only works for homogeneous problems.^{7} The numbers 1 and 40′ in line 46 show us the way that is actually chosen: and are expressed in terms of a new magnitude, which we may call :
That corresponds to Figure 3.6. It shows how the problem is reduced to a simpler one dealing with a single square . It is clear that the area of the first of the two original squares ( ) equals , but that calculation has to wait until line Rev. I.1. The text begins by considering , which is more complicated and gives rise to several contributions. First, the square in the lower right corner: 5′ and 5′ you make hold, . This contribution is eliminated from the sum of the two areas: inside you tear out: 25′ you inscribe. The 25′ that remains must now be explained in terms of the area and the side of the new square .
, as already said, is times the area : 1 and 1 you make hold, 1.^{8} After elimination of the corner remains of , on one hand, a square , on the other, two “wings” to which we shall return imminently. The area of the square is : 40′ and 40′ you make hold, 26′40″. In total we thus have times the square area : 26′40″ to 1 you join: 1°26′40″.
This equation confronts us with a problem which the Old Babylonian author has already foreseen in line Rev. I.2, and which has caused him to postpone until later the calculation of the wings. In modern terms, the equation is not “normalized,” that is, the coefficient of the seconddegree term differs from 1. The Old Babylonian calculator might correspondingly have explained it by stating in the terminology of TMS XVI
The Babylonians got around the difficulty by means of a device shown in the righthand side of figure 3.7: the scale
an equation of the type we have encountered in BM 13901 #1
is thus the area of the completed
As 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
Line 2 speaks of the “surface” as 10′. This shows that the student is once more supposed to know that the discussion deals with the rectangle (30′,20′). 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.
#2 teaches how to confront a more complex situation; now the sum of the area and both sides is given (see Figure 3.10). Both length and width are prolonged by 1; that produces two rectangles ( ,1) and ( ,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 (= 1°30′), width (= 1°20′) and area ; 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).”
[1]
Léon Rodet, Journal asiatique, septième série 18, p. 205.
[2]
The inverse of the “heaping” operation, on the other hand, is no subtraction at all but a separation into constitutive elements. See note 3, page
[3]
The verb in question (nadûm) has a broad spectrum of meanings. Among these are “to draw” or “to write” (on a tablet) (by the way, the word lapātum, translated “to inscribe,” has the same two meanings). Since what is “laid down” is a numerical value, the latter interpretation could seem to be preferable—but since geometrical entities were regularly identified by means of their numerical measure, this conclusion is not compulsory.
[4]
Here we see one of the stylistic reasons that would lead to a formulation in terms of fallingshort instead of excess. It might as well have been said that one side exceeds the other by one sixth, but in the “multiplicativepartitive” domain the Babylonians gave special status to the numbers 4, 7, 11, 13, 14 and 17. In the next problem on the tablet, one “confrontation” is stated to exceed the other by one seventh, while it would be just as possible to say that the second falls short of the first by one eighth.
[5]
One might believe the underlying idea to be slightly different, and suppose that the original squares are subdivided into 7
7 respectively 6
6 smaller squares, of which the total number would be 1‵25, each thus having an area equal to
and a side of 30′. However, this interpretation is ruled out by the use of the operation “to make hold”: Indeed, the initial squares are already there, and there is thus no need to construct them (in TMS VIII #1 we shall encounter a subdivision into smaller squares, and there their number is indeed found by “raising”—see page
[6]
This part of the tablet is heavily damaged. However, #24 of the same tablet, dealing with three squares but otherwise strictly parallel, allows an unquestionable reconstruction.
[7]
In a simple false position, indeed, the provisionally assumed number has to be reduced by a factor corresponding to the error that is found; but if we reduce values assumed for and with a certain factor—say, —then the additional 5′ would be reduced by the same factor, that is, to 1′. After reduction we would therefore have .
[8]
This meticulous calculation shows that the author thinks of a new square, and does not express in terms of and .
[9]
This device was used constantly in the solution of nonnormalized problems, and there is no reason to suppose that the Babylonians needed a specific representation similar to Figure 3.7. They might imagine that the measuring scale was changed in one direction—we know from other texts that their diagrams could be very rough, mere structure diagrams—nothing more than was required in order to guide thought. All they needed was thus to multiply the sum by , and that they could (and like here, would) do before calculating .
[10]
The quotient is called ba.an.da. This Sumerian term could mean “that which is put at the side,” which would correspond to way multiplications were performed on a tablet for rough work, cf. note 11, page
[11]
That the value of c_{1} is calculated as 1⋅c and not directly identified with c confirms that we have been working with a new side c.
[12]
The tablet is rather damaged; as we remember, passages in ^{¿}…^{?} are reconstructions that render the meaning (which can be derived from the context) but not necessarily the exact words of the original.
[13]
The word ki.gub.gub is a composite Sumerian term that is not known from elsewhere and which could be an ad hoc construction. It appears to designate something stably placed on the ground.
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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