Our main topic will be the Old Babylonian treatment of seconddegree
equations.^{1} However, the solution of seconddegree equations or equation systems often asks for firstdegree
The equation thus concerns a length and a width. That tells us that
the object is a rectangle—from the Old Babylonian point of view, the
rectangle is the simplest figure determined by a length and a width
alone
Something, however, is lost in this translation. Indeed, the length and the width is a condensed expression for a “heaping,”
Once the length and the width have been “heaped,” it is possible to “tear out”
Line 1 shows the nature of a Babylonian
Next, lines 1 and 2 ask the student to multiply the 45′ (on the
righthand side of the version in symbols) by 4: You, 45′ to 4 raise, 3 you see. To “raise,”
The answer to this question is found in lines 2–5. 4 and 1 posit: First, the student should “posit”
50′ and 5′, to tear out, posit: the numbers 50′ and 5′ are placed on level «1» of the diagram. This should surprise us: it shows that the student is supposed to know already that the width is 20′ and the length is 30′. If he did not, he would not understand that and that (that which is to be torn out) is 5′. For the sake of clarity not only the numbers 50′ and 5′ but also 30′ and 20′ are indicated at level «1» in our diagram even though the text does not speak about them.
Lines 3–5 prove even more convincingly that the student is supposed to know already the solution to the problem (which is thus only a quasiproblem). The aim of the text is thus not to find a solution. As already stated, it is to explain the concepts and procedures that serve to understand and reduce the equation.
This calculation can be followed in Figure 2.4, where the numbers on level «1» are multiplied by 4, giving thereby rise to those of level «4»:
Finally, the individual constituents of the sum are identified, as shown in Figure 2.5 2, the lengths, and 1, 3 widths, heap, 3 you see: 2, that is, 4 lengths, and 1, that is, widths, are added. This gives the number 3. We have now found the answer to the question of line 2, 3 you see. 3, what is that?
Figure 2.6 shows that this corresponds to a return to level «1»:
15′ to 1 raise, 15′ the contribution of the width. (line 7): 1, that is, 3 widths, is multiplied by , which gives 15′, the contribution of the width to the sum 45′. The quantity of widths to which this contribution corresponds is determined in line 8 and 9. In the meantime, the contributions of the length and the width are memorized: 30′ and 15′ hold—a shorter expression for may you head hold, the formulation used in other texts. We notice the contrast to the material taking note of the numbers 1, 4, 50′ and 5′ by “positing” in the beginning.
The explanation could be the following: a true field might measure
30 [nindan] by 20 [nindan] (c. 180 m by 120 m, that is,
bùr
Before leaving the text, we may linger on the actors that appear, and which recur in most of those texts that state a problem together with the procedure leading to its solution.^{5} Firstly, a “voice” speaking in the first person singular describes the situation which he has established, and formulates the question. Next a different voice addresses the student, giving orders in the imperative or in the second person singular, present tense; this voice cannot be identical with the one that stated the problem, since it often quotes it in the third person, “since he has said.”
In a school context, one may imagine that the voice that states the problem is that of the school master, and that the one which addresses the student is an assistant or instructor—“edubba texts,”^{6} literary texts about the school and about school
This is an “indeterminate”
We take note that is “joined” to the length; that we take of the outcome; and that afterwards we “go” this segment 11 times. What results “goes beyond” the “heap” of length and width by 5′. The “heap” is thus no part of what results from the repetition of the step—if it were it could have been “torn out.”
The solution begins with a pedagogical explanation
However, the Babylonians did not operate with such equations; they are likely to have inscribed the numbers along the lines of a diagram (see Figure 2.8); that is the reason that the “coefficient” does not need to appear before line 29.
As in the first problem of the text, a solution to the homogeneous equation is found by identification of the factors “to the left” with those “to the right” (which is the reason that the factors have been inverted on the lefthand side of the last equation): (now called “the length” and therefore designated in Figure 2.8 thus corresponds to 5°40′, while (referred to as “the heap” of the new length and a new width , that is, ) equals 11; must therefore be . Next the text determines the “tobejoined” (wāṣbum) of the length, that is, that which must be joined to the length in order to produce the original length : it equals , since . Further it finds “the tobetornout” (nāsum) of the width, that is, that which must be “torn out” from in order to produce . Since , must equal ; the “tobetornout” is thus .
But “joining” to and “tearing out” from only gives a possible solution, not the one which is intended. In order to have the values for and that are aimed at, the step 5′ is “raised” (as in the first problem) to 5°40′ and 5°20. This gives, respectively, and ; by “joining” to the former its “tobejoined” and by “tearing out” from the latter its “tobetornout” we finally get .
We must take note of the mastery with which the author avoids to make use in the procedure of his knowledge of the solution (except in the end, where he needs to know the “step” in order to pick the solution that is aimed at among all the possible solutions). The numerical values
[1]
As in the case of “algebra” we shall pretend for the moment to know what an “equation” is. Analysis of the present text will soon allow us to understand in which sense the Old Babylonian problems can be understood as equations.
[2]
“you see ” translates ta mar . The scribe thus does not omit a word, he uses the first syllable (which happens to carry the information about the grammatical person) as a logogram for the whole word. This is very common in the texts from Susa, and illustrates that the use of logograms is linked to the textual genre: only in mathematical texts can we be reasonably sure that no other verbs beginning with the syllable ta will be present in this position.
[3]
A right triangle is certainly also determined by a length and a width (the legs of the right angle), and these two magnitudes suffice to determine it (the third side, if it appears, may be “the long length”). But a triangle is always introduced as such. If it is not practically right, the text will give a sketch.
The word “practically” should be taken note of. The Babylonians had no concept of the angle as a measurable quantity—thus, nothing corresponding to our “angle of 78°.” But they distinguished clearly “good” from “bad” angles—we may use the pun that the opposite of a right angle was a wrong angle. A right angle is one whose legs determine an area—be it the legs of the right angle in a right triangle, the sides of a rectangle, or the height and the average base of a right trapezium.
[4]
“Simple” because there is also a “double false position” that may serve to solve more complex firstdegree problems. It consists in making two hypotheses for the solution, which are then “mixed” (as in alloying problems) in such a way that the two errors cancel each other (in modern terms, this is a particular way to make a linear interpolation). Since the Babylonians never made use of this technique, a “false position” always refers to the “simple false position” in what follows.
[5]
The present document employs many logograms without phonetic or grammatical complements. Enough is written in syllabic Akkadian, however, to allow us to discern the usual scheme which, in consequence, is imposed upon the translation.
[6]
The Sumerian word é.dub.ba means “tablet house,” that is, “school.”
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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