The Mayans

Mactutor

The Mayan system was base twenty with a zero symbol. The zero symbol represents a shell.

Positional values: 1, 20, 18∙20, 18∙20², 18∙20³, ...

Tzolkin: 13 months of 20 days. The months named after gods.

Haab: 12 months of 30 days with 5 day Wayeb. The Wayeb was unlucky.

52 civil years (or 73 ritual years) cycle. Venus 104 years.

Long count of days since 12 August 3113 BC. You can think of a year as 360 days.

Astronomical observations. Year length: 365.242 days (the modern value is 365.242198 ).

Lunar month: 81 lunar months lasted 2392 days. This gives 29.5308 days as the length of the lunar month (the modern value is 29.53059 days).

Palenque -- temple of the inscriptions.

Chaac, the god of Rain and thunder.

Tikal

A Selection of Medieval Mathematicians

1. Gerbert

He was involved in politics, and was the first Frenchman to become pope (Sylvester II). He was also active in educational reform. He wrote elementary books on arithmetic and geometry, which represented the state of knowledge of mathematics in western Europe at the time (lagging behind other parts of the world).

BRITANNICA.
born: c. 945, near Aurillac, Auvergne, France
died: May 12, 1003, Rome
original name Gerbert Of Aurillac
Summary: French head of the Roman Catholic church (999–1003), renowned for his scholarly achievements, his advances in education, and his shrewd political judgment.

WIKI. Silvester II (c. 950 – May 12, 1003), born Gerbert d'Aurillac.
Summary: Prolific scholar of the 10th century. Introduced Arab knowledge of arithmetic and astrology to Europe. The first French Pope (999 to his death).

After he died, legends about his great learning surfaced. Some attributed his learning to magical arts learned in Spain, some to the devil's coaching, some to an artificial head that answered his questions.

Legend of Gerbert and the Devil.

2. Omar Khayyām (عمر الخيام) (Persian عمر خیام)

Biography in Encyclopaedia Britannica. [Free Web Version]

4. Fibonacci

As his real name suggests, Fibonacci was from Pisa in Italy. His father Guglielmo (William) was a merchant from the Bonacci family. When Leonardo was a boy, his father was in charge of the community of Pisan merchants in the North African port of Bugia (now in northeastern Algeria). There, young Leonardo studied arithmetic with an Arab master. As he grew up he traveled to several Arabic and non-Arabic land (he mentions "Egypt, Syria, Greece, Sicily and Provence"). During his travels he became convinced that the Hindu-Arabic numerals were the best for calculation, and learned how to use them in their various forms. He called them the Indian numerals.

He is most famous for his Liber abaci. "book of Calculation" (1202) which concerned arithmetic and basic algebra. It is famous for popularizing the Hindu-Arabic numerals. Before then, only a few people knew about them from translations of al-Khwarizmi. It explains the notation and the concept of the place values of numerals, and explains how to calculate with them. Then problems are solved involving commerce, for example, conversion of weights, measure, and currency between units that were in use in the area. Also gave techniques (for example, the rule of three) for solving simple algebraic problems, including quadratic equations and systems of linear equations. Extraction of roots both geometrically and via approximations (in base 60). Perfect numbers. Summing arithmetic and geometric sequences. This book generated much interest. I was widely copied, and in those days copies were all made by hand. In addition, other authors imitated his work.

One problem was the following: "A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?" The solution to this problem involves the famous Fibonacci sequence. Later (noticed by Robert Simson at the University of Glasgow in 1753) it was realized that the ratio of successive terms approach the golden ratio. Scientists began to discover such sequences in nature: in pine cones, in the spirals of seeds on sunflowers, in snail shells, in the branching of trees, in animal horns, in considering the ancestors of a male bee. (This sequence arose earlier in India, but apparently from a different motivation: as a combinatorics problem counting the possible quantitative meters in poetry).

The Holy Roman emperor, Frederick II, was impressed with Fibonacci's book, and invited him to meet with him (around 1225). The emperor's scientific advisor challenged Fibonacci with a series of problems. Fibonacci included three of them in his later books. Two were Diophantine equations. The third was a cubic equation. In modern notation: x³ + 2x² + 10x = 20. At the time, no one knew how to solve, algebraically, the general cubic equations, but Fibonacci was able to come up with a method to approximate very closely the solution. His solution was expressed in base 60 as 1; 22, 7, 42, 33, 4, 40. This equation actually was old in Fibonacci's time. It was solved geometrically by Omar Khayyam by intersecting a circle and a hyperbola.

His interaction with the emperor and the emperor's scholars was not limited to this one meeting, but was continued by correspondence. His book, Liber quadratorum ( “Book of Square Numbers”) of 1225 was dedicated to the emperor Frederick.

Liber quadratorum ( “Book of Square Numbers”, 1225) Concerns square integers and rational numbers. For example, it studies Pythagorean triples. But it goes well beyond that. In fact, it is Fibonacci's masterpiece: many of the theorems are original to Fibonacci himself. On type of problem involves finding a number that when you add and subtract it to a square, both results are also squares. He showed that x² + y² and x² - y² cannot both be squares. He showed that the difference between two fourth powers cannot be a square. This book is in the tradition of Diophantus, and would today be classified as number theory.

Other works: Practica geometriae (Practice of Geometry) (1220), a short work based on Euclid. It gives precise proofs, but also practical techniques for surveyors and others. For example, on how to calculate heights using similar triangles. Flos (1225) mentions his solution to the cubic mentioned above. He gives proofs that the solution cannot be an integer or a rational number, so approximating the solution is the best one can hope for.

Biography in Dictionary of Scientific Biography
Encyclopaedia Britannica (also: Academic Edition, requires campus password)

MACTUTOR. Leonardo Pisano Fibonacci
Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)

BRITANNICA. Leonardo Pisano
Born: c. 1170, Pisa?
Died: after 1240
English Leonardo of Pisa , original name Leonardo Fibonacci

WIKI. Leonardo of Pisa (c. 1170 or 1180 – 1250)
also known as Leonardo Pisano, or Leonardo Bonacci Leonardo Fibonacci, or simply Fibonacci
Summary: Considered by some "the most talented mathematician of the Middle Ages." Today best known for for:
(1) Popularizing the Hindu-Arabic numeral system in Europe, primarily through the publication of the Liber Abaci.
(2) The Fibonacci sequence.

5. Yang Hui

Born: about 1238 in Qiantang (now Hangzhou), Chekiang province, China
Died: about 1298 in China

6. Oresme

Oresme was of Norman origin, and he seems to have studied at the In 1356 he became grand master of the College of Navarre at the University of Paris. In 1378 he was consecrated bishop of Lisieux. He became a friend to Charles, later King Charles V of France (1364). Charles was both intellectual and religious. When Charles became king, Oresme became the kings chaplain and councillor. Later he was appointed Bishop of Lisieux (1377).

Charles V appointed him to translate several of Aristotle's works into French. In his commentaries on Aristotle, and in other works, he presented his economic ideas. He is generally considered to be the greatest medieval economist. Charles also asked him to write many works in French to popularize the sciences and learning in France. Because of this, and his translations into French, he introduced several new French words that became the French equivalents for various Latin technical terms.

He was also an important philosopher. At this time, scholars generally believed that Aristotle was essentially correct. However, Oresme disagreed with Aristotle on many matters. For example, he argued that the earth could be rotating, but still came down on the side of a stationary earth. (This was in Livre du ciel et du monde(1377)) He identified the infinite in space and time with God. He opposed Astrology for scientific and religious reasons. He also sought natural explanations of seemingly marvelous phenomena

He showed that the harmonic series diverges.

De proportionibus proportionum (“On Ratios of Ratios”) Came up with the idea of fractional powers.

Tractatus de configurationibus qualitatum et motuum (“Treatise on the Configurations of Qualities and Motions”). In this work he came up with the idea of the graph of a function. He called the coordinates latitudo and longitudo. He related the graph of the function to uniform or nonuniform motion. In many ways he anticipated, and may have influenced, the discovery of analytic geometry by René Descartes (1596–1650). He used the idea of a graph to prove that the distance traveled under uniform acceleration is the same as if the body moved at a uniform speed equal to its speed at the midpoint of the period. Oresme might have been a big influence on Galileo (1564–1642).

Encyclopaedia Britannica (also: Academic Edition, requires campus password)
Crater Oresme

MACTUTOR. Nicole d' Oresme
Born: 1323 in Allemagne (west of Riez), France
Died: 11 July 1382 in Lisieux, France
Summary: Nicole Oresme was a French mathematician. Invented coordinate geometry long before Descartes. The first to use a fractional exponent and also worked on infinite series.

BRITANNICA. Oresme, Nicholas
Born: c. 1320, Normandy
Died: July 11, 1382, Lisieux, France
French Nicole Oresme
Summary: French Roman Catholic bishop, scholastic philosopher, economist, and mathematician. Work provided some basis for the development of modern mathematics and science and of French prose, particularly its scientific vocabulary.

WIKI. Nicole Oresme or Nicolas d'Oresme (c. 1323 - July 11, 1382)
Summary: One of the most famous and influential philosophers of the later Middle Ages. An economist, mathematician, physicist, astronomer, philosopher, psychologist, and musicologist, a passionate theologian and Bishop of Lisieux. A competent translator, counselor of King Charles V of France, one of the principal founders and popularizers of modern sciences, and probably one of the most original thinkers of the 14th century.

7. al-Kāshī (الكاشي) and Ulugh Beg (Persian الغ‌بیگ)

[Victor Katz]
Ghiyāth al-Dīn Jamshīd al-Kāshī (d 1429).
Lived in Samarkand. Although al-Samawʼal understood the decimal place value system, the full development was not complete until al-Kāshī. He used a convenient notation, the vertical line (while al-Samawʼal indicated each power of ten in words), and had total command of the idea. (The Chinese originated decimal fractions, then al-Uqlīdīsī used them in simple situations. al-Samawʼal seemed to have a better facility. al-Kāshī had full mastery).

MACTUTOR. Ghiyath al-Din Jamshid Mas'ud al-Kashi (1390 - 1450). Iran.

Ulugh Beg (Persian الغ‌بیگ)

Crater Ulugh Beigh

MACTUTOR. Ulugh Beg (1393 - 1449).
Born: Soltaniyeh, Timurid, Persia (now Iran).
Died: 27 Oct 1449 in Samarkand, Timurid empire.
Summary: His grandfather, Timur, came from a Mongol tribe living in Transoxania, today Uzbekistan. Led a union of Turko-Mongol tribes to conquest: Iran, Iraq, Turkey, even India (Delhi), Syria. His grandfather died in 1405 trying to conquer China. The capital was Samarkand. Ulugh Beg often travelled with his grandfather on his military campaigns. Ulugh Beg's father moved the capital to Herat (in western Afghanistan), but gave Samarkand to Ulugh Beg when Ulugh Beg was 16. In 1420 he finished building a madrasah (center for higher education) and appointed the best scientists he could find. For example, al-Kashi and Qadi Zadi and 60 others. Ulugh Beg led meetings to discuss astronomy. Often only al-Kashi could solve the problem. He started his observatory in 1428. It was a circle over 50 m in diameter and 35 m high. It had 3 levels. It had a huge quadrant, and other impressive instruments. al-Kashi and others worked there. Cubic equations an the binomial theorem, Ulugh Beg's accurate sine and tangent tables (8 decimal places) in one degree intervals, formulas in spherical geometry. A new star catalogue, the first since Ptolemy (published 1437, 992 stars). He found the sine of 1 degree using a cubic equation. Year length 365 days 5 days 49 min 15 sec. Unable to retain power. His own son instigated his death. Tomb: discovered in 1941.

Encyclopaedia Britannica: Ulugh Beg (1394- Oct. 27, 1449)
Born in Soltaniyeh, Timurid Iran
Died in Samarkand, Timurid empire [now in Uzbekistan])
Grandson of Timur (Tamerlane). Interested in arts and intellectual matters. The Timurid dynasty of Iran reached its cultural peak under his rule. His father captured Samarkand, and gave it to Ulugh Beg. Ulugh Beg made it into a ceter of Muslim culture. Wrote poetry, history. Studied the Qur'an. Interested in astronomy. Built observatory (begun in 1428) at Samarkand. Found errors in Ptolemy. He was not successful as a ruler. He was put to death by his son.

WIKI His name (Persian الغ‌بیگ) means "Great Ruler" or "Patriarch Ruler" . Real name Mirza Mohammad Taragai bin Shahrukh (Mīrzā Mohammad Taragai bin Shāhrukh). Founded a madrasah (مدرسة). The observatory was called the Gurkhani Zij. The catalogue of stars was edited by Thomas Hyde (1665). Tabulae longitudinis et latitudinis stellarum fixarum ex observatione Ulugbeighi. The year was measured in 1437, 365d 6h 10m 8s (an error +58s). In 1525, only 88 years later, Nicolaus Copernicus improved it by by 28s. Grandfather: Timur bin Taraghay Barlas (Tīmūr) (Persian: تیمور لنگ) Timur the Lame.

Samarkand (Russian: Самарканд, Persian and Arabic: سمرقند‎ Samarqand) is the third-largest city in Uzbekistan. About 90 percent of the inhabitants are Tajik (related to the Persians). It used to be more populous in the middle ages, and is very old. It was a Persian city before Alexander the Great conquered it. It was one of the most important cities of the later (Sassanid) Persian Empire. The first paper mill was set up after 751. Destroyed by the Mongol conqueror Genghis Khan (1220). Timur (Tamerlane) made it his capital in 1370. It became the most important economic and cultural centre in Central Asia. The city was abandoned in 18th century, but became reoccupied, and was part of the Russian Empire. The old city contains some of the finest monuments of Central Asian architecture from the 14th to the 20th century. It was on the silk road: junction of trade routes from China and India.

Here is a statue of Ulugh Beg in Riga, Latvia.

Observatory:

Ulugh Beg Madrasa around 1910 (in Samarkand)

(Developed 2006 by Professor Wayne Aitken for Math 330).