Following the completion of the first edition of Liber abbaci, Leonardo eventually became something of a celebrity, not just in his hometown of Pisa but throughout Italy. Much of his fame came from the depth of his scholarship. Other books described the Hindu-Arabic number system, a few dating from before Leonardo’s time, many more written later, and many of them found their way into classrooms for use as texts. But as the mathematician Laurence Sigler wrote in the introduction to his English-language translation of Leonardo’s masterpiece, none were as “comprehensive, theoretical, and excellent as the Liber abbaci of Leonardo Pisano.”1
Between the publication of the first edition of Liber abbaci in 1202 and the second edition in 1228, Leonardo published a number of other works that added to his reputation. De practica geometrie, (Practical geometry), completed in 1223, was written for professional people whose work involved surveying and land measurement.2 Though not as long as Liber abbaci, it was still a substantial work, and like Leonardo’s arithmetic text it included both practical instructions for performing various calculations written for artisans, as well as mathematical verifications of the methods he described for scholars. Leonardo dedicated the book to his friend Dominicus Hispanus, a mathematician in Frederick’s court. It contains a large collection of geometry problems, arranged into eight chapters, with theorems based on Euclid’s books Elements and On Divisions. Equations are expressed geometrically, in narrative form, so that 4x – x2 = 3 is described as “If from the sum of the four sides, the square surface is subtracted, then three rods remain.” Leonardo also included practical information for surveyors, giving the measurements used in Pisa, and a chapter on how to calculate the height of tall objects using similar triangles. In the final chapter he presented what he called “geometrical subtleties”, of which he wrote: “Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces … to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated.”
Overall, the book appears to have been heavily influenced by the work of Abraham bar Hiyya, who lived in Spain in the twelfth century. (Bar Hiyya wrote in Hebrew, but his work was one of the many translated into Latin during that period.)
In 1225, Leonardo published a book titled Flos (The flower), largely devoted to algebra, and containing his solutions to a series of problems posed to him in a contest organized for Frederick, to whom he sent a copy. Liber quadratorum (The book of squares), which he also published in 1225, is a book on advanced algebra and number theory, and is Leonardo’s most mathematically impressive work.3 It deals mainly with the solution of various kinds of equations involving squares, generally with more than one variable, where the solutions have to be whole numbers—the very kind of problem that led Pierre de Fermat to propose his famous “Last Theorem” in the seventeenth century, a conundrum that was eventually solved by Andrew Wiles in 1994.
These four books, together with a letter he wrote to Theodorus Physicus, the imperial philosopher (Epistola ad Magistrum Theodorum),4 comprise the only surviving works of Leonardo. The letter to Theodorus is undated and is preserved as a copy written in Milan in 1225. In it, Leonardo discussed three problems, one arithmetic, one geometric, and one algebraic. The first of the three is called the “Problem of the 100 Birds,” and Leonardo had already presented a solution in Liber abbaci. This time, however, he developed a general method for the solution of such indeterminate problems. The second problem is to inscribe a regular pentagon inside an equilateral triangle. He solved it in an algebraic fashion. The final problem is a linear equation with five unknowns, for which he gave a formulaic solution.
Other manuscripts Leonardo is known to have written, but which have been lost, include the Libro di minor guisa, a book about commercial arithmetic, and a discussion of book 10 of Euclid’s Elements, in which he provided a numerical discussion of irrational numbers that Euclid had dealt with geometrically.
So great did Leonardo’s reputation become in the years following the first publication of Liber abbaci that sometime in the mid-1220s, most likely around 1225, he had an audience with the emperor. Frederick heard about Liber abbaci and its author from the court scholars, among them the court astrologer Michael Scott, the court philosopher Theodorus Physicus, and the imperial astronomer Dominicus Hispanus, the man who suggested to Frederick that he meet Leonardo when the court was next in Pisa.
The occasion would have drawn a lot of attention. Frederick did everything on a grand scale. Popularly known as Stupor mundi (wonder of the world), the young king of Germany, Italy, Sicily, and Burgundy always traveled with a large entourage that included foot soldiers, knights, officials, pages, slaves, dancing girls, jugglers, musicians, and eunuchs, together with his exotic menagerie of lions, leopards, panthers, bears, and apes, all led on chains, hunting dogs, hawks, peacocks, parakeets, ostriches, and a giraffe. A caravan of camels transported his supplies, and Frederick himself would ride at the head of the procession, while at the rear an elephant carried a wooden platform on its back on which were perched trumpeters and archers with crossbows.5
Born in Sicily in 1194, Frederick had been crowned emperor by the pope in 1220.29 As a young man growing up in Sicily he had developed a passion for learning, particularly science and mathematics, interests that led him to found in 1224 the university in Naples that bears his name: University of Naples Frederick II (Universita degli Studi di Napoli Federico II).
In large part because of its location, Sicily had long been a meeting ground for the Christian and Muslim cultures of Europe and North Africa. The island had four official languages—Latin, Greek, Arabic, and French—as well as the Sicilian spoken by the ordinary people. The Sicilian intelligentsia had acquired from the Muslims an interest in science. In particular, Frederick’s grandfather, King Roger of Sicily, was in the habit of summoning to his court learned men from many lands, whereupon he would talk with them at length to discover all they knew, and have records made of all that was said. Roger’s successors, William I and William II, arranged for the translation into Latin of the ancient Greek texts on mathematics and astronomy, including the works of Euclid, Aristotle, Ptolemy, and Hero of Alexandria—usually from Arabic editions.
Frederick spoke six languages: Latin, Sicilian, German, French, Greek, and Arabic. He was an avid patron of the arts and acquired from his family and those around him a deep fascination with all matters scientific and mathematical, including astronomy, optics, geometry, algebra, natural science, and alchemy.
From his grandfather he also learned to adopt a skeptical approach, never accepting new knowledge without adequate evidence. For instance, he pursued his interest in natural history in an experimental fashion. He built incubating ovens to study the development of chick embryos, and he sealed the eyes of vultures to see whether they found their food by sight or by smell. After talking with experts in falconry, he wrote a book, De arte venandi cum avibus (On the art of hunting with birds), in which he discussed their classification, habits, migration patterns, and physiology.
The young king sent letters to Muslim rulers, phrasing some of his requests for information as puzzles. He contacted scholars in Egypt, Syria, Iraq, Asia Minor, and Yemen, seeking answers to scientific questions. He wrote to al-Ashraf, the sultan of Damascus, posing problems in mathematics and philosophy, to which the sultan replied giving solutions obtained by an accomplished Egyptian scholar. Scholars from many lands visited his court frequently.
Not surprisingly, then, when Frederick learned of Leonardo’s work, he summoned the author of Liber abbaci to his Pisan palazzo to discuss the book and give a demonstration of his mathematical ability. In the prologue to Liber quadratorum (The book of squares), which Leonardo wrote immediately after his meeting with the emperor, to whom he dedicated it, he wrote: “When I heard recently from a report from Pisa and another from the Imperial Court that your sublime majesty deigned to read the book I composed on numbers, and that it pleased you to listen to several subtleties touching on geometry and numbers …”6 In addition to his audience with the emperor, Leonardo was asked to give a public demonstration of mathematical ability, by responding to three challenge problems put to him in advance by Johannes of Palermo, one of the court mathematicians. Leonardo subsequently presented written accounts of his solutions, two in Flos, a copy of which he sent to Frederick, and one in Liber quadratorum.30
Johannes first asked Leonardo to find a rational number (i.e., a whole number or a fraction) such that, when 5 is added to its square, the result is the square of another rational number, and when 5 is subtracted from its square, the answer is also the square of a rational number. He had almost certainly found this question in an Arabic manuscript. The Arabic scholars seemed to like this kind of number puzzle, and many variations were known. This one was particularly tricky (using the techniques known at the time), and the solution Leonardo found (and subsequently published in Liber quadratorum) was both long and ingenious. The final answer was 3 + ¼ + 1⁄6, or 41⁄12. For its square increased by 5 gives the square of 4½, and decreased by 5 gives the square of 2 7⁄12.
Johannes’s second problem was another of a kind much loved by Arabic scholars, involving the solution of a cubic equation. In modern symbolic notation, the equation Leonardo was asked to solve is x3 + 2x2 + 10x = 20. (In fact, this very equation can be found in al-Khayyám’s Algebra, so Johannes was taking a risk that Leonardo had seen it earlier.) Though the ancient Greek mathematicians knew how to solve quadratic equations—using the same techniques taught to schoolchildren ever since—cubic equations are far more challenging.7 (It would be several centuries before general algebraic methods were developed to solve cubic equations.) Algebraic notation was still many centuries away, so Johannes posed the problem in words:
Find a number such that if it be raised to the third power, and the result added to twice the same number raised to the second power, and if that result be then increased by ten times the number, the answer is twenty.
Leonardo solved the equation by approximation. There is no record of the exact method he used, but there were a number of techniques available at the time. It is likely his argument went something like this: The first step is to observe that the unknown number must be between 1 and 2. For if it were 1, the answer to the computation would be less than 20, and if it were 2 the answer would exceed 20. So a reasonable initial guess is x = 1.5. The idea now is to iteratively refine this guess, gradually homing in on the correct answer. The key step is deciding how to improve each approximation.
If x solves the equation x3 + 2x2 + 10x = 20, then x(x2 + 2x + 10) = 20, and so
Hence, if xn is an approximation of the solution that is too high (respectively, too low), then
is an approximation that is too low (high). It follows that the average of these two approximations
is a better one. Thus, starting with an initial approximation of x0 = 1.5 (say), one calculates a sequence of approximations x0, x1, x2,…, xn,… which fairly quickly reaches an acceptably accurate approximation. Computing to fifteen decimal places of accuracy, this process yields the values:
and the final approximation in this list turns out to be correct to seven decimal places, after just twelve steps.
Leonardo performed the calculation using sexagesimal fractions (i.e., fractions expressed in base 60), following the practice used by astronomers since Ptolemy. In the sexagesimal notation he was using, the answer he obtained was 1022′ 7″42′″ 33IV4V40VI. In this notation, 22′ is 22⁄60, 7″ is 7⁄3600, 42′″ is 42⁄216,000, and so forth, each successive fraction being expressed as a higher power of 60. Expressed in words, as Leonardo presented it to the court, this reads: “One unit, 22 in the first fractional part, 7 in the second, 42 in the third, 33 in the fourth, 4 in the fifth, and 40 in the sixth.”
In modern decimal notation, Leonardo’s solution is 1.3688081075, which is correct to nine decimal places, a result that is far more accurate than the answer to the same problem that had been obtained (using the same method) by Arab mathematicians who had solved it previously (and more accurate than the answer presented above that was obtained using modern algebra and a computer spreadsheet).
The third problem Leonardo solved was the easiest of the three, being a computation where the unknown quantity is not raised to any power. (In modern parlance, it involves only linear equations.) Liber abbaci was full of such problems, though of course Johannes had chosen one that did not appear in Leonardo’s own book.
Three men owned a store of money, their shares being ½, 1⁄3, and 1⁄6. But each took some money at random until none was left. Then the first man returned ½ of what he had taken, the second 1⁄3, the third 1⁄6. When the money now in the pile was divided equally among the men, each possessed what he was entitled to. How much money was in the original store, and how much did each man take?
Leonardo solved the problem using the Direct Method. He began his solution, as subsequently recorded in Flos, by pointing out that
If you take away half of anything, you have an equal half left; similarly, if you take away a third, that third is half of the remaining two-thirds; likewise, if you take away a sixth, that sixth is a fifth of the remaining five-sixths.
Let us use the term res for the amount each man received when the pile of money was divided equally among them. Then it follows that after the three men had returned the given portions of their money, the first one had half of the money in the original store minus res. The second had a third of the original store, minus the same res. The third had a sixth of the original store, minus the same res.
Since the first man had already put back half of what he originally took, and kept one-half, the half that he kept was equal to one-half the original store minus res; in other words, the whole of the money he took from the store was equal to the store minus twice res.
Since the second man put back a third of what he had taken, and that third part was half of what he kept [half of the remaining two-thirds], which was a third part of the total store minus res, one-half plus one-sixth [or 2⁄3 reduced to unit fractions] of what he received equaled the third part of said store minus res. In other words, the amount the second man took was equal to one-half the total store of money minus one and a half res.
Since the third man put back a sixth part of what he took, and that sixth part was a fifth of what he had left [1⁄6 is a fifth of 5⁄6] the five-sixths he had left was equal to a sixth part of the total money minus res. In other words, the third man took a fifth of the total store minus one and one-fifth res.
Therefore if you add: the total store minus two res [the amount the first man took] and half the store minus one and a half res [the amount the second man took] and one-fifth the store minus one and one-fifth res [the amount the third man took], the total amount, the sum of the amounts the three men took, equals one and seven-tenths of the total money minus four and seven-tenths res. Therefore seven-tenths the total store equals four and seven-tenths res; therefore, multiplying seven-tenths of the store by ten, and four and seven-tenths res by ten, seven times the total store equals forty-seven res; therefore if you suppose res to equal seven [this amounts to selecting the smallest solution], the total money will be forty-seven [the smallest possible whole-number answer] … Therefore since the first man took the total money minus two res, or forty-seven minus two res, or fourteen, thirty-three will remain for what the first man received. Since the second man took one-half the total money minus one and one-half res, twenty-three and one-half, minus ten and one-half, or thirteen, will be what the second man received. Since the third took one-fifth the total money minus one and one-fifth res, nine and two-fifths, minus eight and two-fifths, or one, will be what the third man received. And thirty-three of the first man plus thirteen of the second plus one makes forty-seven, the total store.8
With that, Leonardo’s demonstration was complete. It is hard not to be impressed by Leonardo’s ability. Though a mathematician today would regard the three exhibition problems as basic and their solutions as routine, they presented a considerable mental challenge when not approached using modern symbolic notation. A present-day solution to the third problem would translate his words into algebraic symbols:
Let t be the total money in the store; let u be the amount each man received when the money left in the store was divided equally among them; and let x, y, z be the amounts each man took.
Note first that
u = 1⁄3 (½x + 1⁄3y + 1⁄6z)
At the end, the first man had what he was entitled to, namely, half the original amount, ½t. Therefore, before he received u, he had ½t – u. He had already put back half of what he originally took, or ½x, and kept ½x, so
½x = ½t – u
which can be rearranged as
x = t – 2u
Similarly, the second man ended with 1⁄3t. Before he was given u, he had 1⁄3t – u. He had already put back a third of what he originally took, or 1⁄3y, and kept 2⁄3y, so
2⁄3y = 1⁄3t – u
which can be rearranged as
y = ½t – 1½u
Finally, the third man finished with 1⁄6t. Before he received u, he had 1⁄6t – u. He had already returned a sixth of what he first took, or 1⁄6z, and kept 5⁄6z, so
5⁄6z = 1⁄6t – u
which can be rearranged as
z = 1⁄5t – 11⁄5u.
Adding these three equations:
t = |
x + y + z |
= |
t – 2u + ½t – 1½u + 1⁄5t – 11⁄5u |
= |
17⁄10t – 47⁄10u |
Rearranging gives
7⁄10t = 47⁄10u
so
7t = 47u
The smallest whole-number solution to this is where u = 7 and t = 47. In this case, the original amount is 47, the first man took x = t – 2u = 47 – 14 = 33, the second took y = ½t – 1½u = 13, and the third took z = 1⁄5t - 11⁄5u = 1. Note that 33 + 13 + 1 = 47.
Leonardo’s triumphant demonstration at the imperial palace is one of the last occasions where we have any reliable knowledge of his activities. The only later reference to the man succeeding generations would refer to as Fibonacci was a proclamation by the commune of Pisa in 1241, granting “the discreet and learned man, Master Leonardo” an annual honorarium of twenty Pisan pounds plus expenses for services to the city.9 Historians believe that this was in return for Leonardo’s service as auditor for the commune.
Frederick died in 1250. Speculation as to how Leonardo ended his days ranges from his being killed in the recurrent civil strife in Pisa to him living out his days peacefully as a revered and honored citizen. In any event, he did not witness Pisa’s glorious period come to an end. In 1284, just forty-three years after the last record of him, Pisa was defeated by its arch rival Genoa in the naval battle of Meloria. Fra Salimbene, a contemporary chronicler from Parma, described the battle with these chilling words: “They grappled their ships together after the fashion of sea fights, and there they fought with such slaughter on either side that even the heavens seemed to weep in compassion, and many on both sides were slain, and many ships sunk. But when the Pisans seemed to have the upper hand, more Genoese came and fell upon them, wearied as they were … At last the Pisans, finding themselves worsted, surrendered to the Genoese who slew the wounded and threw the rest into prison.” That the captives’ lives were spared was not due to any altruism on the part of the Genoese; rather it was to prevent their wives from remarrying back home, thereby reducing the Pisan birth rate.
The Melorian defeat was, however, merely a particularly brutal event that hastened what would in any case have been the end of Pisa’s reign as a glorious city-state and a center of maritime trade. Following the death of Frederick, the Hohenstaufen empire, long the source of much of Pisa’s security, went into decline, as one by one the emperor’s sons and grandsons died at the hands of their enemies. Wars with other cities together with the continuing strife inside the city itself led to a rapid decline in Pisa’s trade with the Tuscan interior. On top of that, malaria broke out in the swamps through which Leonardo had ridden to the Porto Pisano when he had embarked on his historic journey to North Africa, decimating the local population.
Meanwhile, Florence, the center of the growing wool industry, rose to become the leading city in Tuscany, while to the north Venice was taking over as the new trading capital of the world. Pisa slid rapidly to the status of the provincial town it has been ever since. But the real glory of Pisa is an intellectual one that lives on through the enormous legacies to humankind left by the city’s two most brilliant sons: Galileo Galilei in the sixteenth century, and before him Leonardo Pisano in the thirteenth.