MATH 110-2 – Algebra Through History Viète’s Take on Diophantos II, 8 November 20, 2019

Background

To make sure we understand what Viète 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!)

Questions

Suppose we want to start with F = 39 and write F^{2} = 1521 as a sum of two other squares.

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).

1. What is the ratio \frac{F}{Z} here?

2. What are X = \frac{BF}{Z} and Y = \frac{DF}{Z}? Check that X^{2} + Y^{2} = F^{2}.

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?)

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 Viète is aware of what you said? Did it seem as though Diophantos was?