Friday, June 20, 2014

Squaring the Circle, part 2 -- the Mesopotamians

Plimpton 322
Image via Wikipedia

Reading about mathematics in Mesopotamia has been pretty interesting.  They seem to have done a lot of math just for the fun of it.  They were fond of tables comparing numbers and their relationships to one another – multiplication tables, tables of reciprocals, squares, cubes, and square and cube roots.  Algebra was their specialty. 

They had a base 60 system, apparently because sixty can be divided so nicely in so many different ways – into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths (did I miss any?).  We still use base-60 for telling time and for measuring angles, which is interesting because it doesn’t seem that the Babylonians had the concept of measuring an angle.

For numbers above 59 they used a place value system which worked the way ours does.  The first place to the right is “ones,” the numbers from 1 to 59 in their case.  The place to the left of that is for the base units – 10 in our case, 60 in theirs.  To the left of that is the base squared – 100 for us, 3600 for them.  (This looks like x(60)2 + x(60) + x in Math-ese, but I prefer English.)  They eventually developed a zero which they used as a place holder in the middle of long numbers, but they didn’t use it in the far right position if it was needed there, so 1, 60, and 3600 look like the same thing and have to be guessed from context.

This place value system was also used for fractions and is the reason for the unfamiliar notation I mentioned last week.  In the example quoted, the first number is the whole number.  The semi-colon serves the way a decimal point does for us – to separate the whole number from the “less than one” part to the right.  The commas are needed to distinguish the places since each place might have anything from a 1 to a 59 in it.

Those are some of the basics of math in Mesopotamia, but this is supposed to be about finding the area of a circle, right?

Remember all those tables and lists I mentioned in the first paragraph?  The clay tablet pictured at the top of this post is a table that describes the relationship of the squares drawn on the sides of a triangle, sort of like this:

This is such a handy way of understanding shapes that the Mesopotamian mathematicians noted this kind of relationship with all the regular (equilateral) polygons with three to seven sides, comparing the area of each shape to the square that could be made along its side.  One tablet lists all these relationships and gives very accurate ratios describing the relationships.

The ratio of the area of the pentagon, for example, to the square on the side of the pentagon, is given as 1;40, a value that is correct to two significant figures. For the hexagon and the heptagon, the ratios are expressed as 2;37,30 and 3;41, respectively.  In the same table, the scribe gives 0;57,36 as the ratio of the perimeter of the regular hexagon to the circumference of the circumscribed circle, and from this, we can readily conclude that the Babylonian scribe had adopted 3;7,30, or 3 1/8, as an approximation for π.
[Merzbach and Boyer, p. 35]


Me, trying to learn Math.
And yes, that does mean a hexagon with a circle drawn around it.
I checked.  :-)

So it turns out that there’s a good reason for comparing various shapes to squares, and circles to various regular polygons – it helps us understand the relationships between different shapes, which helps us understand the shapes themselves.  And noting these relationships gave the Babylonians, like the Egyptians before them, a fairly accurate way of figuring out the area of any circle.

But one thing that confused me in A History of Mathematics’ chapters on the Ancient Greeks and Mesopotamians was the phrase “an approximation for π,” which the authors use several times.  They make it sound like π is a thing that everyone knows about and only needs to be clearly defined or standardized, the way a “foot” was eventually standardized as a unit of measure.  It was pretty exciting when I finally realized why they keep saying this, but that will be in the next post.

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Click here for Part 1, which is about the Egyptians


  1. This is really interesting. We are studying ancient history this year, and I actually have History of Mathematics on the shelves. I should pull it off and look at it again during the weekend. I *think* I know what approximation of pi refers to but I am not going to venture a guess or look online -- rather, I'll wait for the post about it : ).

    1. Oh, Willa, I've been reading John Mark Reynolds' book When Jerusalem Met Athens in preparation for the CiRCE conference, so I might not be able to make the next post until after I get back in town, which will be the end of the month, what with working in visits to the relatives along with the conference trip. Thanks for the encouragement!


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