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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Discussion 3 -- ``Plimpton 322''}
\centerline{October 8, 2010}
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{\it Background}
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The picture below is a modern drawing (made by a historian named Eleanor Robson)
of the Old Babylonian mathematical tablet known as ``Plimpton 322.''  There is
a similar drawing in Rudman, {\it The Babylonian Theorem}, but this one
seems a bit better to me.  (By the way, the name ``Plimpton 322'' 
comes from the fact that this is catalog number 322 in the G.A.Plimpton 
collection at Columbia University in New York.)
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There is a photograph of the original tablet on our class homepage.  
As you will see, the reflection of light off the surface
of the clay and the depth of the stylus marks make that somewhat 
difficult to read.  Unfortunately, the tablet also has some damaged 
regions where chips of the original surface have been lost (indicated 
by the hatched areas here).  So I think you will find that this version 
is much easier to interpret and work with.  

The top two rows of cuneiform on the tablet contain a somewhat cryptic 
text whose exact meaning is not completely understood, and that we will 
not consider for now.  The rest of the tablet consists of 5 columns 
cuneiform, mostly number symbols.  Let's call those columns A,B,C,D,E, starting 
{\it from the right}.    
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{\it Questions}
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\item{A)} Look at the column on the right hand edge (column A).  What
are those numbers and why do you suppose that column was included?
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Column B contains the same cuneiform symbol repeated in each row.  
This is a word related to the text at the top.  We will ignore this.
The next three columns are entirely composed of numbers.  They will
be our focus in today's work (especially columns C and D).  For instance
you should see that the numbers in columns C, D in row 1 are
$$(1:59)_{60} = (119)_{10} \qquad {\rm and} \qquad (2:49)_{60} = (169)_{10}$$
Note that 
$$169^2 - 119^2 = 14400 = 120^2$$
is another perfect square(!)   
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\item{B)}  Identify and translate the rest of the numbers in columns C and D.  
Write them in our convention for base 60 numbers, and then as base 10 numbers.
If there is some ambiguity or the surface of the tablet is damaged in 
one of those entries, you can either try to guess to fill that entry 
in or just indicate that there was a problem with that entry.  
\item{C)}  What happens if you take the number on a row in column C, square it,
and subtract the square of the number on the same row in column D.  What is 
true about the numbers you get?  (There should be some ``exceptions'' to 
a general pattern you notice in most rows.  In fact, it appears that there
are some {\it errors} in the table!)  
\item{D)} (Harder!)  What about column E (the left most column).  Unfortunately note
that there is a big "chunk" missing out of the first several rows on the 
left.  However, the lower portion of that column is better.  Look at
row 11 carefully, for instance.  From the rows above, it appears that there
should always be a  1 at the start, so this number is 
$$(1 : 33: 45)_{60}.$$
How does that relate to the entries $c,d$ on that row in columns C and D and 
the number $c^2 - d^2$?  Try a few of the other rows that followed the 
pattern from part C and see if it always works this way. 
\bye
