Logic doesn't apply to the real world.
D. R. Hofstadter and D. C. Dennett (eds.), The Mind's I, 1981.
Fibonacci and Square Numbers
Fibonacci is one of the best-known names in mathematics, and yet Leonardo of Pisa (the name by which he actually referred to himself) is in a way underappreciated as a mathematician. When hearing the name we are most likely to think of the Fibonacci sequence, and perhaps Leonardo's problem about rabbits that began the sequence's rich history. Leonardo's role in bringing the ten-digit Hindu-Arabic number system to the Christian nations of Europe might also come to mind. While these two contributions are undoubtedly enough to guarantee him a lasting place in the story of mathematics, they do not show the extent of Leonardo's enthusiasm and genius for solving the challenging problems of his time, and his impressive ability to work with patterns of numbers without modern algebraic notation. In this article, we will try to shed light on this side of Leonardo's work by discussing some problems from Liber quadratorum, written in 1225, using the English translation The Book of Squares made by L. E. Sigler in 1987. All page references in what follows are to that book . Questions for student investigation are at the end of this article.
The Court of Frederick II
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 a picture of some of Frederick's remaining castles. 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:
¼ 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
This is easy enough if you allow irrational solutions, for instance taking x = Ö17, y = Ö22, and z = Ö12. 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.
How can we find two squares that sum to a square? Leonardo answers this question in several ways, with the first method following from a simple observation that provides inspiration throughout the book:
I thought about the origin of all square numbers and discovered that they arise out of the increasing sequence of odd numbers; for the unity is a square and from it is made the first square, namely 1; to this unity is added 3, making the second square, namely 4, with root 2; if to the sum is added the third odd number, namely 5, the third square is created, namely 9, with root 3; and thus sums of consecutive odd numbers and a sequence of squares arise together in order [p. 4].
Thus, the sums 1 + 3 + 5 + 7 = 16 and 1 + 3 + 5 + 7 + 9 = 25 are both squares. Since we add the square 9 to the first sum in order to get the second, we have16 + 9 = 25 as a sum of two squares adding to a third square. Leonardo explains that we could use any odd square in place of 9 to do the same thing. For instance, using 49, we have 1 + 3 +¼+ 47 = 576 and 1 + 3 +¼+ 49 = 625, so 576 + 49 = 625 is another sum of the same form. Leonardo goes on to note that the final two or more terms of these odd-number sums can also sum to a square. For instance, an even square can be partitioned into consecutive odd numbers, an example being 17 + 19 = 36. Since 1 + 3 +¼+ 15 = 64 and 1 + 3 +¼+ 19 = 100, we get 64 + 36 = 100.
Taking things further, Leonardo poses the following:
I wish to find three squares so that the sum of the first and the second as well as all three numbers are square numbers [p. 105]
He explains his solution this way:
I shall find first two square numbers which have sum a square number and which are relatively prime. Let there be given 9 and 16, which have sum 25, a square number. I shall take the square which is the sum of all odd numbers which are less than 25, namely the square 144, for which the root is the mean between the extremes of the same odd numbers, namely 1 and 23. From the sum of 144 and 25 results, in fact, 169, which is a square number. And thus is found three square numbers for which the sums of the first two and all three together are square numbers [p. 105].
In fact, Leonardo points out that this method can be extended to any number of squares, since (1 + 3 + ¼ + 167) + 169 = 7056 + 169 = 842 + 132 = 852 = 7225, and (1 + 3 + ¼ + 7223) + 7225 = 36122 + 852 = 36132. Thus, we get the following sequence of squares:
and so on.
The solution to John of Palermo's problem requires several steps and some of the most intricate work in The Book of Squares. The first step seems at first to be unrelated:
If two numbers are relatively prime and have an even sum, and if the triple product of the two numbers and their sum is multiplied by the number by which the greater number exceeds the smaller number, there results a number which will be a multiple of twenty-four [p. 48].
Leonardo often represents numbers as lengths. Here he considers points a, b, and g on a line, with the two numbers being the lengths from a to b and from b to g. Thus, in our notation, the expression under discussion is
The purpose of the congruous numbers is shown a little later while addressing the following problem:
Find a number which added to a square number and subtracted from a square number yields always a square number [p. 53].
This is John of Palermo's problem except it does not require the number to be 5. Let us continue using ab and bg as before. We throw in the additional condition that bg·(bg-ab) < ab·(ab+bg), which will keep us in the realm of positive integers in what follows. Leonardo's method involves the bg·(bg-ab) consecutive odd numbers centered at ab·(ab + bg) and also the ab·(bg - ab) consecutive odd numbers centered at bg·(ab + bg). In the example given in the book, ab = 3 and bg = 5, so we get 10 consecutive odd numbers centered at 24:
and also 6 consecutive odd numbers centered at 40:
These two sequences have an amazing relationship. They are next to each other in the list of all odd numbers, and they have the same sum, 240, which is the congruous number ab·bg·(ab + bg)·(bg - ab) = 3·5 ·8 ·2. Thus,
This solves the problem. Now, how can Leonardo use this idea with 5 instead of 240?
Leonardo realizes that what he needs is a congruous number that is 5 times a square, because then he can divide through by that square and leave just the 5 [p. 76]. The values ab = 4 and bg = 5 almost work, since 4·5 ·(4+5) ·(5-4) = 180 = 5 ·62. But 4+5 is not even as is required for a congruous number, the problem being that there are not 5 consecutive odd numbers centered at 36, or 4 centered at 45. Leonardo gets around this by doubling all the values. There are 10 consecutive odd numbers centered at 72, namely those from 63 to 81. There are also 8 consecutive odd numbers centered at 90, those from 83 to 97. Both sequences sum to 720=5·122, and we get
Finally, Leonardo has his answer. He writes
There is for the first square 6 97/144, with root 2 7/12, which results from dividing 31 by the root of 144, which is 12, and there is for the second, which is the sought square, 11 97/144, with root 3 5/12, which results from dividing 41 by 12, and there is for the last square 16 97/144 with root 4 1/12 [p. 78].
In modern notation, we could write
 Fibonacci, Leonardo Pisano. The Book of Squares. Translated by L. E. Sigler. Boston: Academic Press, 1987.
 Grant, Edward. A Source Book in Medieval Science. Cambridge, MA: Harvard University Press, 1974.
 Monteferrante, Sandra. "Leonardo of Pisa: Bunny Rabbits to Bull Markets ." Convergence. 31 July 2006.
 Vogel, Kurt. "Fibonacci, Leonardo or Leonardo of Pisa." Dictionary of Scientific Biography. Vol. 4. New York: Chas. Scribner's Sons (1970), pp. 604-613.
Questions for Investigation
1. Find all the ways to express 225 as a sum of consecutive odd integers. Use your results to find the squares that can be added to 225 to produce another square. What determines the number of ways in which a given number can be expressed as a sum of consecutive odd numbers?
2. Show that 336 is a congruous number. Use your results to find a rational number x such that x2 – 21 and x2 + 21 are both squares of rational numbers. Can you find examples with numbers other than 5 (shown in the text) and 21?
3. There is a correspondence between ordered triples (a,b,c) with a2 + b2 = c2 and ordered triples (p,q,r) with p2, q2, r2 forming an arithmetic progression. The triple (a,b,c) = (3,4,5) corresponds to (p,q,r) = (1,5,7), the triple (a,b,c) = (5,12,13) corresponds to (p,q,r) = (7,13,17), and the triple (a,b,c) = (8,15,17) corresponds to (p, q, r) = (7,17,23).
Discover the rule for this correspondence and explain why it works.
4. Triangular numbers can be found by the taking the sum of all integers from 1 to n , so we get 1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4= 10, and so on. Adapt as many of Leonardo’s results as you can to the case of triangular numbers.