ATTACKING THE FIFTH POSTULATE
Omar Khayyam and other Arab mathematicians had a near obsession with Euclid’s fifth postulate, which states that if two lines that, intersecting a third line, make two interior angles whose sum is less than the sum of two right angles—that is, 180 degrees—then the lines, if extended far enough, will intersect on the side of the third line on which the sum of the two angles is smaller than the sum of two right angles. To these mathematicians, the fifth postulate was not an axiom, but rather something that, like a theorem, was provable based on the previous four postulates Euclid had put forward. But no one could prove the fifth postulate.
The Arab mathematician Alhazen (ca. 965–1039), who had written the highly influential Book of Optics, attacked what is now known as the “parallel postulate.” After constructing a tri-rectangular quadrilateral, he thought he could prove that the fourth angle must be a right angle. What he did not realize was that he was operating on an assumption that, itself, was equivalent to Euclid’s fifth postulate—the result he was trying to prove.
Omar Khayyam criticized Alhazen’s attempted proof, saying that Aristotle had ruled out the viability of motion in geometry. In his own attempt at a solution to this problem, he constructed a quadrilateral whose two sides were perpendicular to the base, and concluded that the two interior angles formed by the fourth side must be equal to each other. Referencing the Aristotelian principle that two converging lines must intersect, he ruled out the possibility that the two angles could be either acute or obtuse and concluded that they were right angles. However, the Aristotelian principle he invoked is, in fact, equivalent to Euclid’s fifth postulate, so both mathematicians failed in their respective “proofs.”
A century later an Arab mathematician named Nasir al-Din al-Tusi (1201–74) mounted the most consequential attack on the problem of the fifth postulate. Tusi was the astronomer of the Mongol ruler Hulagu Khan, grandson of Genghis Khan and brother of Kublai Khan. Tusi proceeded from the same geometrical starting point as his predecessors, the quadrilateral, and his “proof” also depended on Euclid’s assumption, but his analysis would be useful in later centuries in the development of non-Euclidean geometries.
AL-KASHI
Major Arab and Muslim contributions to mathematics lasted into the fifteenth century. The last important Muslim mathematician of historical times was Jamshid al-Kashi (ca. 1380–1429). Al-Kashi was born in Kashan, central Persia. Around the time of his birth, the Mongol empire was expanding and conquering parts of Persia, and life was difficult in the realm. Al-Kashi, who early in life showed great promise in mathematics, became a wandering scholar eking out a living by teaching mathematics in the villages and towns in the region. We know that on June 2, 1406, he observed and recorded an eclipse of the moon, and his note has been used to date other events with accuracy. He became a respected astronomer after making many observations of the sky and publishing a book in 1407, the abbreviated title of which was The Stairway of Heaven. It dealt with measuring angular distances between heavenly bodies and performing calculations in astronomy.
His renown brought al-Kashi to the attention of Ulugh Beg, a grandson of the Mongol emperor Tamerlane. Ulugh Beg was the ruler of the ancient city of Samarkand, in present-day Uzbekistan, and he was very interested in science and mathematics. He also had great plans for the city, including founding a university and building an observatory, so he invited al-Kashi to join him there, along with other mathematicians and thinkers. Al-Kashi accepted the invitation and became the most prominent mathematician and astronomer in Samarkand, the “new Baghdad.”
Ulugh Beg (1394–1449) is pictured alongside the observatory he built in Samarkand, the modern-day capital of the Samarqand Province of Uzbekistan. This Russian commemorative stamp also features the date of his initial calculation of the sidereal year: 1437.
Al-Kashi specialized in the solution of quadratic equations. He seemed to relish doing long calculations, and he is credited with bringing decimal notations and computations, ideas that were later further pursued in Europe, to the fore. He computed roots of equations to an accuracy never before reached by any of his predecessors. He not only established the use of decimal fractions but also computed π to fourteen decimal places: 3.14159265358979.
Two centuries before the French mathematician and physicist Blaise Pascal published his treatise on the triangle of binomial coefficients, now called Pascal’s triangle, al-Kashi discussed its properties in his Key to Arithmetic. In fact, the triangle had been discovered in China a century earlier. Al-Kashi died in Samarkand in 1429 and was sorely missed by Ulugh Beg, who considered him the greatest scientist of the time. His death signaled the final decline of Arab and Islamic mathematics and the ascendance of the West.
The Beijing Ancient Observatory, completed in 1442, contains many pretelescopic astronomical instruments, such as this bronze armillary sphere, used to calculate the coordinates of celestial bodies.
FIVE
MEDIEVAL CHINA
Believed to be as old as the Babylonian and Egyptian cultures, the Chinese civilization is far more ancient than that of Greece. Although dates areuncertain, we know that mathematics has been studied in China for thousands of years. An early Chinese mathematical text of an uncertain date contains work on fractions and descriptions of the properties of right triangles, as seen in the mathematics of ancient Greece. It also recounts a debate between a ruler and one of his ministers in which the minister tells the ruler that the art of numbers originates in two figures: the circle and the square. The square represents the earth, and the circle represents heaven. This example of mathematical mysticism is reminiscent of the Pythagorean order’s conviction that “God is number.”
THE MAGIC SQUARE
Part of the magic of numbers manifested itself in the very ancient Chinese practice of constructing magic squares, which is believed to have originated before the first century CE—some say much earlier. The Chinese seem to have been the first to invent the magic square and to construct many such squares for their own entertainment and for mystical uses—much as they used compass needles in the ancient discipline of feng shui, the art of orienting buildings and furniture for maximum spiritual harmony. Magic squares were believed to have mystical qualities, and in ancient times some people wore them around their necks or kept them displayed on walls or doorways. Below is one of the first magic squares discovered in China:
This magic square, or Lo Shu square, is of Tibetan origin. In the center of the tablet are the numerals of the magic square, written in Tibetan characters and emblazoned on the back of a turtle.
Note that the sum of all rows, columns, and diagonals is 15. This particular square also has mythological origins. According to Chinese tradition, a turtle from the River Lo brought this magic square to humankind in the days of Emperor Yii, about whom there were many legends. This magic square was given the name Lo Shu, which means “river map,” although its derivation is obscure.
In early antiquity—certainly before the first century—the Chinese began to use rods of bamboo, iron, or ivory to perform calculations. Officials often carried such rods with them to calculate taxes and other financial data. These rods were manipulated with immense dexterity and were described by onlookers as “flying so quickly that the eye couldn’t follow their movement.”1 The abacus seems to have been derived from this ancient Chinese system of calculation, though it is now believed that, because the rods were so efficient, they were used for much longer than originally thought, and the abacus emerged only relatively late in Chinese history.
Chinese mathematicians in the Middle Ages used counting rods to perform various calculations. This illustration of Yang Hui’s triangle (a triangular arrangement of the binomial coefficients, later called Pascal’s triangle) uses counting-rod numerals within its structure.
NINE CHAPTERS
The most important Chinese mathematical text—and one of the oldest in existence—is the “Nine Chapters on Mathematical Art,” a tr
eatise dating from about 250 BCE, although some believe it is much older and may have been initiated by a series of mathematicians as early as the tenth century BCE. In contrast with the abstract mathematics of the Greeks, the Nine Chapters centers on practical issues of a mathematical nature, such as problems of taxation, agriculture, and engineering. The book provides rules for the areas of triangles, circles, and trapezoids. There is also a discussion of how many equations are needed to solve a problem with a given number of unknowns. For example, one problem presents four equations with five unknowns (which we now know is impossible, since the number of equations must equal the number of unknowns if we want unique solutions).
A commentator on the Nine Chapters who lived in the third century CE estimated π to be 3.14 by using a regular polygon of 96 sides, and later obtained an even better approximation, 3.14159, by using a polygon of 3,072 sides. The polygons allow us to calculate areas, and when these areas are added, an estimate of the area of the circumscribing circle is obtained, leading to an estimate of π. Estimating π seems to have been a major occupation of Chinese mathematicians, who came up with better and better estimates as time passed. The Nine Chapters also gave us the famous “Chinese remainder theorem,” which is used in number theory, relating equations and their solutions with properties of numbers.
The opening page of chapter 1 of “Nine Chapters on Mathematical Art,” one of the oldest mathematical treatises in existence.
LI ZHI
Li Zhi (sometimes spelled Li Chih) was an important mathematician of the thirteenth century. He later changed his name to Li Ye to avoid confusion with the third emperor of the Tang dynasty. His father was an official of the Jurchen empire, which encompassed Manchuria and northern China. Li Zhi was born in Beijing, the Jurchen capital, in 1192. When he was a teenager, the Mongols under Genghis Khan attacked northern China and took Beijing, so his family sent him to attend school in Hopeh Province.
The Mongol emperor Kublai Khan—who, allegedly, would ride into battle in an armored fortification atop four elephants, as depicted in this 1874 illustration—was a great champion of mathematics who twice offered Li Zhi a position in his government.
When he finished his education, Li Zhi took the examinations for civil service in 1230. He was about to take a position as registrar of the district of Kaoling, but persistent Mongol attacks prevented him from assuming the promised job. In a strange twist of fate, he was then offered a better position in the civil service: that of governor of the Jun prefecture in Henan Province. The job didn’t last long—after continued Mongol attacks, he barely escaped certain death when a colleague managed to convince a Mongol soldier to let him live.
For more than a decade, Li Zhi lived in abject poverty in yet another part of China to which he had to escape: Shanxi Province. Still, he was so gifted that even under such dire circumstances, he managed to complete a major work in mathematics, the Ce Yuan Hai Jing (Sea Mirror of Circle Measurements), in 1248. Teaching mathematics and distributing his book eventually improved his economic situation, so he moved back to Hopeh Province, where he had hoped to live in peace. In 1257 Kublai, grandson of legendary Mongol conqueror Genghis Khan, sent emissaries to Li Zhi—by then a famous mathematician—asking him for advice on restructuring the civil-service examinations. Apparently, he liked Li Zhi’s proposal, because in 1260, after he became the Mongol leader Kublai Khan, he offered Li Zhi a top administrative job controlling his vast empire.
Already in his sixty-ninth year, Li politely declined the offer, saying he was too old to be a top administrator. Four years later Kublai Kahn appointed Li Zhi to his newly established scientific academy, but the aging mathematician apparently felt unwell and resigned shortly after joining the Mongol academy. He returned to his old home in a peaceful mountain region of China, where he lived out his last years and died in 1279, at the age of eighty-seven.
Li Zhi’s book included 170 problems that had never before been solved. He was interested in the relationships between the sides of triangles and the radii of the inscribing or inscribed circles. Such analysis, usually involving the Pythagorean theorem—which is now believed to have been discovered independently in China in early antiquity—led to equations of varying degrees. Li Zhi therefore studied equations as high as the sixth order. Such an equation is obtained when one relationship about triangles involving an unknown quantity is inserted into another. Repetitive application of such a procedure quickly raises the order of the unknown in the equation. While Li Zhi showed how to set up equations in order to solve a problem, he did not describe his methods of solution. This focus on equations of a higher and higher order would become very important as Europe emerged from the Dark Ages and built upon the mathematics of antiquity and the Near East.
PART III
RENAISSANCE MATHEMATICS
Leonardo da Vinci’s ca. 1487 drawing Vitruvian Man has come to symbolize the creative, inventive spirit of the Renaissance. Da Vinci’s affinity for mathematics informed his friendship with Luca Pacioli, another great Italian mathematician of the period.
SIX
ITALIAN SHENANIGANS
According to historian of mathematics Carl Boyer, the main problem with the transmission of mathematical ideas in historical times was language. The Arabs, according to Boyer, made their big breakthrough when they were finally able to translate into Arabic the important foundational work of the Greeks. The Europeans would not be able to penetrate the Arab language barrier for another three hundred years. When this finally happened, it changed the world.
FIBONACCI
As interest in astronomy grew in the twelfth and thirteenth centuries, European scholars began to study Arabic so that they could understand the mathematical works of the Arabs. It was thus that al-Khwarizmi’s work on algebra was brought to the attention of European mathematicians.
A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares whose dimensions correspond to the numbers in the Fibonacci sequence.
Prime among these new European mathematicians was Leonardo of Pisa (ca. 1170–1250), better known as Fibonacci (son of Bonacci). Fibonacci was the first European mathematician to use the Hindu-Arabic numerals we use today. He found these numerals in the works of al-Khwarizmi and other Arab scientists, and thanks to Fibonacci’s writings, these numerals—or, rather, the version of them that evolved in Europe—replaced the very cumbersome and inefficient use of letters as numbers. Using the new numerals allowed Fibonacci to formulate his famous sequence in simpler notation (i.e., 1, 1, 2, 3, 5, 8, 13, 21 … rather than I, I, II, III, V, VIII, XIII, XXI …). In Fibonacci’s sequence—first derived as a series of numbers representing an idealized scenario in which a pair of rabbits mates and produces a new pair of rabbits every month—each number represents the sum of the two numbers immediately preceding it. As it turns out, the ratio between each term and its predecessor approaches 1.618 …, the golden ratio, which appears commonly in nature and in art.
How did he do it? Relative to most mathematicians, Fibonacci had an advantage: his father, Guglielmo Fibonacci, was a successful merchant and trader who traveled to North Africa and often took his son with him. To aid his father, Leonardo learned Arabic and thus had a much better handle on the language than other European mathematicians of the time. He also gained access to many Arabic texts and assimilated their ideas into his work. The rediscovery and blending of ideas through the Arabic texts and translations of ancient Greek mathematics allowed Fibonacci to single-handedly jump-start European mathematics.
As the son of a wealthy family, Fibonacci was able to travel easily—and his father’s trade relations in the Mediterranean helped him there as well. As a young adult, Fibonacci embarked on a multiyear adventure, visiting many ports in the Mediterranean and studying with Arab mathematicians. He also became proficient in the use of the Hindu-Arabic numerals. In the year 1200, when he was around thirty years old, he returned to his native Italy. Two years later he published his masterpiece, the Liber
Abaci (Book of Calculation), which promoted the use of the Hindu-Arabic numerals, discussed his explorations in number theory and geometry, and introduced his famous sequence. The book described the “nine Indian figures”—the digits 1 through 9—and the symbol that designates the concept of zephirum (zero): 0. Fibonacci did not use decimal fractions, however; they would appear later in European mathematics.
The Liber Abaci also described the extraction of roots in the Arab tradition of al-Khwarizmi, and covered problems in finance and trade, including foreign-exchange calculations. An example of the kind of problem that appears in the Liber Abaci is the following:
Seven old women went to Rome; each woman had seven mules; each mule carried seven sacks; each sack had seven loaves; and with each loaf were seven knives; each knife had seven sheaves.
The question was to find the sum of all of these: women, mules, sacks, loaves, knives, and sheaves. A very similar problem appears in the Egyptian Ahmes papyrus, dating from about 1600 BCE. The next example, from the Liber Abaci, gave rise to Fibonacci’s sequence:
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