Leonardo of Pisa (Fibonacci) · Mathematics
Fibonacci’s masterpiece, the <em>Liber Quadratorum</em> (Book of Squares, 1225), on congruent numbers and sums of squares; the mathematical contest before Emperor Frederick II that provoked it; and the case that Fibonacci was a genuine, original number theorist - not merely the rabbit man - centuries ahead of his time.
If you know Fibonacci only for the sequence, you know him for his least sophisticated work. His own contemporaries, and many later mathematicians, regarded a different book as his masterpiece: the Liber Quadratorum, the ‘Book of Squares,’ completed in 1225. It is a work of pure number theory - the study of the deep properties of whole numbers - and it is strikingly original, tackling questions about square numbers with a rigor and inventiveness that would not be matched in Europe for centuries. Where the Liber Abaci taught calculation for the marketplace, the Book of Squares pursues mathematics for its own sake, chasing subtle patterns among the integers. It is the clearest proof that Fibonacci was not merely a transmitter of others’ methods and a poser of puzzles, but a creative mathematician of the first rank, capable of results that were genuinely new.
The central problem of the Book of Squares is genuinely hard, and it came from a challenge (more on that shortly). Find a square number such that, when you add a certain fixed number to it, you again get a square, and when you subtract that same fixed number, you get a square yet again. The fixed number that makes this work is called a congruent number (Fibonacci’s term was congruum). The specific challenge was: find a square that remains a square when 5 is added and when 5 is subtracted. Fibonacci solved it. Take the square to be (41/12) squared = 1681/144. Add 5 (that is, 720/144) and you get 2401/144 = (49/12) squared; subtract 5 and you get 961/144 = (31/12) squared. Both results are perfect squares. Finding such a number is far from obvious, and Fibonacci gave general methods for generating congruent numbers, showing a command of the problem that goes well beyond a lucky guess.
The scene is worth picturing: the most powerful ruler in Europe, a court famed for its scholars, and a challenger firing hard problems at the mathematician from Pisa. Fibonacci answered them all. Beyond the congruent-number problem, the challenge that most impresses modern mathematicians is the cubic equation x cubed + 2x squared + 10x = 20. Fibonacci first proved, by careful reasoning, that no solution could be a whole number, a simple fraction, or any of the square-root quantities found in Euclid - a remarkable piece of theory in itself, showing the answer to be a new kind of number. Then, unable to write it exactly, he computed an approximation in the sexagesimal (base-60) notation of astronomers, accurate to about nine decimal places - an astonishing feat of hand calculation for the thirteenth century, and one nobody in Europe would clearly surpass for centuries. This is not the work of a puzzle-monger; it is the work of a formidable and original mathematician operating at the frontier of what was known.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Fibonacci’s Liber Quadratorum (1225) tackled congruent numbers and sums of squares, that it grew out of a mathematical contest at Frederick II’s court (the challenge to find a square staying square when 5 is added or subtracted, solved by 41/12), and that Fibonacci stated the identity - already known…
Leads to Brahmagupta.
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