Eleanor Robson and Plimpton 322

Part IV - Mathematical Concepts - Compasses

Robson admits examples of the use of compasses.

Her Fig 3 gives an illustration of the use of compasses. Hoyrup also illustrates the use of compasses, pg 265.

Here I use Hoyrup's Figure 72. I quote from him:

The circle on the obverse of the tablet is drawn with a compass, and the sides and height of the inscribed triangle traced after a ruler (and the angle between the height and the base is as right as can be controlled on the photo). What the text does is to find the radius r of the circle which is circumscribed about the double 304050-triangle which, as pointed out by Bruins in the commentary, provides us with incontestable evidence that the Susian calculators of the Old Babylonian age were able to conceptualize both the circumscribed circle and the height of an isosceles triangle.

This is a remarkable illustration of analytical OB mathematics. We can see that it refutes Robson's entire line of reasoning about circles. A radius is defined. Whether Robson sees this as a "rotatable line" becomes an academic argument. What is meant by that term? When I rotate a compass I do not see a line dangling from my instrument; I merely rotate it to create the circle. If I did not use a compass, but put a peg in the ground, stretched out a string to the radius of my circle, and then rotated that, would that help Robson understand what a "rotatable line" is? When I use a compass I do the same thing except that I have lifted my string off the ground, pinned it to some convenient handle that preserves proportions, and then rotated the handle.

Is it possible to rotate a string, or a compass, and not arrive at the concept of angular measure? Were the ancient OB scribes that primitive in their thinking? I can go part way around the circle, 1/8 or 1/3 or 3/5 or any measurable amount and I inherently invoke the idea of angular measure. I can measure that as an angle, or portions of a radius, radian measure, or portions of a circumference, inferred radian measure, but I do measure it. The paucity of circle problems in OB mathematics does not rule out knowledge of circular measure.

Note that this problem is not set up as a slope problem. It is not proposed in terms of a vertical distance and horizontal distance, but has a specified radius. Arguments have raged around Egyptian understanding of angular measure. Did they know only slopes? We can see that this one problem, in 1800 BC, shows an understanding of angular measure. Surely the Egyptians were able to borrow intellectual ideas from the Mesopotamians.

The Susa tablet sets out a problem with two 3-4-5 triangles to create an isosceles triangle with sides 50, 50 and 60. (Note the sizing to 30, 40, 50 and 60 susi, or 1, 4/3, 5/3 and 2 cubits.) The problem is to find the radius of the circle that circumscribes the figure. We can see that the problem is abstracted into a mathematical statement that does not require pegs in the ground, strings, or other real paraphernalia. Most of OB mathematical illustrations are abstracted in like manner. The dimensions merely provide means to quantify the problem in the mind. That is the reason OB problems are often illustrated in convenient and repetitive measurable dimensions. Consider the perfect square in the preceding paper with 30 susi (1 cubit) on a side, and this illustration with 30-40-50 susi.

Here is where Robson becomes so adamant about transferring our modern algebraic notions back to Old Babylon. By habit, it is easy for us to use algebraic notation to represent their problems and their thinking. Neugebauer did just that. Hoyrup also employs algebraic notations because it makes the representation so easy of  what and how the OB scribes thought. Refer to his discussion about this isosceles problem on pgs 265-268.

This is also where we meet the evidence of the lack of general procedures. There was no activity in OB mathematics that would generalize the problems into algebraic notation as a statement. Every mathematical illustration is real life, hard reality. Nevertheless, the representation of the problem is abstracted in illustration. This current problem is not stated in terms of a real circle inscribed in the earth, but as a general case, with dimensions of 50-50-60. The act of abstracting to convenient susi shows how they approached these problems.

This raises an important question. Where is the line that divides the illustration of hard reality from the abstract? The OB scribes could abstract a practical mathematical question, but why did they not take the next step and abstract to pure algebraic notation? Especially after repeated representation of problems.

The easiest way to solve this particular problem is through the Pythagorean relationship. If we try any other method we quickly become involved in complex solutions. Hoyrup recognized this fact, pg 266. He admits to solutions on arithmetical grounds using a rule that Euclid stated a thousand years later, Elements, II.7. . . . it is virtually certain that it is already known no later than 2200 BC . . . See his discussion on pg 267. But this invokes the use of squares and square roots.

We proceed thusly:

r2  = 302  + (40-r)2 (stated by Hoyrup)

But (40-r) = 8 45 in sexagesimal, (or 8.75 in decimal), is given in the problem. Then, calculating decimally:

r2 = 900 + 76.5625 = 976.5625

(I could just as well state this expression in sexagesimal numbers.)

r = 31.25 or in sexagesimal, 31 15.

The question that now arises is if the circle was first created, and then the isosceles triangle inserted within the circle. Or was the isosceles created and the circle circumscribed around it? The clay figure is heavily damaged, with the Reverse containing numbers that appear to be intermediate results related to the figure, but so badly destroyed one cannot make deductions from them. Since the radius is sought we naturally assume that the isosceles triangle was first set out, and then the circle circumscribed around it.

The concept of a radius is central to the statement of this problem. Furthermore, the problem illustrates a conceptualization that requires recognition of something worked from the inside, not from the outside. Again Robson reveals her lack of understanding of OB mathematics and her desire to reduce those ancient Semitic scribes to primitive people.

Note that three points define a circle; we do not know if the OB scribes knew this. Since they give a distance from the baseline of the triangle to the center of the circle, they are pointing us to that center. Once this is known, the compasses can easily find the radius, and then draw the circle. The problem was to determine the distance the compasses should be set to obtain the result.

If the OB scribes understood isosceles triangles did they also understand equilateral triangles? What else was in their circular and triangular repertoire? This example points to a wider mathematical knowledge. I get the impression that the extant evidence is only skimming the surface of OB knowledge.