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\centerline{MATH 110-2 -- Algebra Through History}
\centerline{Vi\`ete's Take on Diophantos II, 8}
\centerline{November 20, 2019}
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\noindent
{\it Background}
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To make sure we understand what Vi\`ete is saying here, 
let's work out a couple of examples.  (Note:  This is also 
related to one of the problems on this week's problem set!)
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\noindent
{\it Questions}
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Suppose we want to start with $F = 39$ and write $F^2 = 1521$ as a sum of two other
squares. 
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\item{A.} Probably the most convenient right triangle to use is the $(5,12,13)$-right 
triangle, or $B = 5$, $D = 12$, $Z = 13$  (note that $B^2 + D^2 = 25 + 144 = 169 = 13^2$, 
so this is a valid ``Pythagorean triple'' to use).  
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\itemitem{1.}  What is the ratio ${F\over Z}$ here?  
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\itemitem{2.}  What are $X = {BF\over Z}$ and $Y = {DF\over Z}$?  Check that $X^2 + Y^2 = F^2$.  
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\item{B.}  Repeat the calculations in part A if you started from the $(3,4,5)$-right triangle
instead, or $B = 3$, $D = 4$, $Z = 5$.  (Why did I say that the choice in A was the most convenient?)
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\item{C.}  Thinking about A and B together and generalizing, how many different ways are there to 
write $F^2$ as a sum of two squares (of rational numbers)?  Does it seem clear that Vi\`ete is
aware of what you said?  Did it seem as though Diophantos was? 
\bye
