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Fibonacci and Square Numbers - The Court of Frederick II

Patrick Headley

The Book of Squares is addressed to Frederick II, the Holy Roman Emperor, ruler of Germany and southern Italy. The outline of Leonardo's life is given elsewhere on this web site, but it is worthwhile to consider the context of this particular work in more detail. Frederick II had a tremendous hunger for knowledge. In spite of his German roots, he was most at home in Sicily, an intersection point for Latin, Greek, and Arabic culture. Scholars from all three were brought to his court. Frederick was called stupor mundi, "the wonder of the world," because his behavior was so astonishing by medieval standards. As an example, we have a rare instance of mathematics being used as a tool of diplomacy. Frederick felt obligated to participate in the Crusades, but instead of attacking Jerusalem with his superior force he set up camp outside and patiently worked out a peaceful takeover. Frederick and Sultan al-Kamil arranged what we might call cultural exchanges to ease tensions as the negotiations proceeded, and at one point a list of mathematical problems was sent to the Arabic scholars of Jerusalem as a friendly challenge [p. 217]. Click here for more information on Frederick's crusade. Click here for more biographical information on Frederick.


Leonardo was clearly delighted by the spirit present at Frederick's court. His book Liber abbaci was dedicated to Michael Scott, Frederick's closest advisor on scientific matters [pp. 306-7]. Two other scholars associated with Frederick, John of Palermo and Master Theodore, appear in The Book of Squares. In fact, the book begins with Leonardo's explanation of how a question from John prompted his research, and an answer to a question from Theodore brings the book to a close. The initial challenge from John of Palermo was this:

 Find a square number from which, when five is added or subtracted, always arises a square number [p. 3].

In modern notation this means finding \(x,y,\) and \(z\) so that \(x^2 + 5 = y^2\) and  \(x^2 - 5 = z^2.\)

This is easy enough if you allow irrational solutions, for instance taking \(x=\sqrt{17},\) \(y=\sqrt{22},\) and \(z=\sqrt{12}.\) But further reading makes it clear that Leonardo is looking for a rational solution (it is not hard to see that a solution in integers is impossible). His work then leads him to consider many more questions about sums and differences of squares.

Patrick Headley, "Fibonacci and Square Numbers - The Court of Frederick II," Convergence (August 2011)