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Page 1: Islamic Contributionsmathed.byu.edu/~williams/Classes/300W2012/PDFs/PPTs/... · 2012. 2. 14. · A Jug of Wine, a Loaf of Bread‐‐and Thou Beside me singing in the Wilderness‐‐

Islamic Contributions

Page 2: Islamic Contributionsmathed.byu.edu/~williams/Classes/300W2012/PDFs/PPTs/... · 2012. 2. 14. · A Jug of Wine, a Loaf of Bread‐‐and Thou Beside me singing in the Wilderness‐‐

Islamic Empire

• About 570 AD – Birth of Muhammad• About 610 AD – Muhammad’s first vision in a cave near Mecca; he begins to preach.

• 622 AD – Muhammad and followers flee to Medina 

• 630 AD – Mecca captured by Muslims• 632 AD – Death of Muhammad

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Islamic Empire – Spain to India

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The Golden Age

• The birth of Islam and its subsequent spread through the Arabian Peninsula, Northern Africa (as well as Spain) and central Asia, gave rise to what some have called the “Golden Age of Islam.”

• Umayyad Caliphate, centered at Damascus and later at Córdoba.  

• Abbasid Caliphate, centered at Baghdad.

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Microscopic Timeline

2500 BCE 2000 BCE 1500 BCE 1000 BCE 500 BCE 0 CE 500 CE 1000 CE 1500 CEal-Khwariz mi al-TusiKhayyam

al-KashiMongolsCrusades

ISLAMGolden Age

SuiQin Qin JiushaoLiu HuiGnomon, Nine Chapters

INDIA

CHINA Yuan / MingSongTangWarring StatesWarringStates

ZhouShang Han

500 CE0 CE500 BCE2500 BCE 2000 BCE 1500 BCE 1000 BCE 1000 CE 1500 CE

Muslim / Kerala SchoolImperial Guptas / Classical

1500 CE1000 CE

Jainism / BuddhismIndo-European / HinduIndus Valley Civilization

1000 BCE1500 BCE2000 BCE2500 BCE 500 BCE 0 CE 500 CE

MycenaeanMinoan GREECEChristianRomanHellenisticClassicalArchaicDark

500 CE0 CE500 BCE2500 BCE 2000 BCE 1500 BCE 1000 BCE

MESOPOTAM IA

EGYPTInt

Int

1000 BCE1500 BCE2000 BCE2500 BCE

New KingdomMiddle KingdomIntOld Kingdom

AssyriaOld BabylonAkkadiaaria

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More Readable Timeline

1500 BCE 1000 BCE 500 BCE 0 CE 500 CE 1000 CE 1500 CEal-Khwariz mi al-TusiKhayyam

al-KashiMongolsCrusades

ISLAM Golden Age

SuiQin Qin JiushaoLiu HuiGnomon, Nine Chapters

INDIA

CHINA Yuan / MingSongTangWarring StatesWarringStates

ZhouShang Han

500 CE0 CE500 BCE1500 BCE 1000 BCE 1000 CE 1500 CE

Muslim / Kerala SchoolImperial Guptas / Classical

1500 CE1000 CE

Jainism / BuddhismIndo-European / Hindu

1000 BCE1500 BCE 500 BCE 0 CE 500 CE

MycenaeanGREECE ChristianRomanHellenisticClassicalArchaicDark

500 CE0 CE500 BCE1500 BCE 1000 BCE

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House of Wisdom

• For us, an important aspect of this was the establishment of the “House of Wisdom” in Baghdad beginning in 762 by Caliph al‐Mamun.  He and his successors invited scholars and artists from many areas to this new “university.”  For the next half century, scholarship flourished here, particularly by comparison to the relatively impoverished Dark Ages in Europe.  

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House of Wisdom

• Written works from all over were collected and translated here, including those from the Greek and Roman empires and India.  Much of what we know about ancient Greek culture was preserved in this way.   

• We should note that Córdoba was also an important cultural and learning center of the Muslim world in medieval times.

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Abu Ja’far Muhammad Ibn Musa al‐Khwarizmi

• It was in this environment that a man known simply as Abu Ja’far Muhammad Ibn Musa al‐Khwarizmi (meaning, “Muhammad, father of Ja’far and son of Musa, whose family is from Khwarizm”) lived and worked.  

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Abu Ja’far Muhammad Ibn Musa al‐Khwarizmi

• We know little of his life, except that he was born in about 780 and died in roughly 850 AD.  He was appointed First Astronomer and Head Librarian at the House of Wisdom.  He wrote several books:– four books on astronomy, including works on sundials and astrolabes.

– a book on trigonometry, Zīj al‐Sindhind– two very influential books on algebra and arithmetic  

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al‐Kitāb al‐mukhtaṣar fī ḥisāb al‐jabrw’al‐muqābala

Perhaps the most important book was al‐Kitāb al‐mukhtaṣar fīḥisāb al‐jabr w’al‐muqābala or “The Compendious (Condensed) Book on Calculation by Completion* and Balancing*.”

* Or something.

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al‐Kitāb al‐mukhtaṣar fī ḥisāb al‐jabrw’al‐muqābala

• The term al‐jabr from which our modern word algebra is derived, may refer to the act of removing a subtracted term from an equation by adding the it to the other side.  For example, 

becomes • And  al‐muqābala may refer to “comparing” or bringing quantities of the same type to the same side of the equation (e.g. becomes  ).  

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al‐Kitāb al‐mukhtaṣar fī ḥisāb al‐jabrw’al‐muqābala

• The book dealt with the solution of linear and quadratic equations by reducing them to one of six standard forms using these processes.   

• Contains algebraic traditions, seemingly, from Mesopotamia, India, and Greece – which is a common theme of your textbook chapter.

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The Second Book

• The second book was a book explaining the Hindu positional number system, including the use of “0,” as well as arithmetic calculations in that system.  The original Arabic version of the book has not survived, so we have only a Latin translation, probably from the twelfth century.  

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The Second Book

• The original Arabian title of the book was Kitāb al‐jam‘wal tafrīq bi ḥisāb al‐Hind, or Book on Addition and Subtraction after the Method of the Indians.  

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The Second Book

• The Latin version was not titled, so it is known variously as Dixit Algorizmi (“So said Al‐Khwarismi”) or Liber Algorizmi (“The book of Al‐Khwarismi”).  From these names we get the word algorithm to describe any systematic process of doing arithmetic.  It introduced the decimal position system of numeration developed in India.

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Abū Kāmil Shujāʿ ibn Aslam

• Lived from about 850 – 930 AD• Wrote Kitāb fi al‐jabr wa al‐muqābala, or The Book of Algebra, in which he accepted irrational numbers, in the form of a square root or a fourth root, as both solutions and as coefficients in quadratic equations.  

• Also did work on indeterminate equations, and wrote a lost work on double false position, among many others.

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Omar Khayyám

• Another Persian mathematician from this era was Ghiyās od‐DīnAbul‐Fatah Omār ibnIbrāhīm KhayyāmNishābūrī, or Omar Khayyam for short.  He lived from about 1040 to 1120 AD, and is known as a poet, mathematician, and astronomer.  

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Omar Khayyám

• He wrote Treatise on Demonstration of Problems of Algebra in 1070.  In it he generalizes and systematizes the geometric methods used in Greece for solving cubic equations by intersecting conics.  

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Omar Khayyám

• First Persian mathematician to call the unknown shiy (meaning thing or something in Arabic) which was transliterated into  Spanish as xay and later shortened to x.  

• In the Islamic world, Pascal’s triangle is known as “Khayyam’s Triangle.”  

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Omar Khayyám

• Wrote Explanations of the Difficulties in the Postulates in Euclid's Elements in 1077.  

• He did not try to prove Euclid’s fifth postulate, but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."

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Omar Khayyám

• He also studied what we now call Saccheriquadrilaterals (but which should probably be called Khayyám‐Saccheri quadrilaterals), and succeeded in deriving Euclid’s fifth postulate from his postulate. On the way, he proved many theorems of what is now hyperbolic geometry.  

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Aside:  The RubaiyatTranslated by Edward FitzGerald (with a fair degree of editorial license).  

• The Moving Finger writes; and, having writ,Moves on: nor all thy Piety nor WitShall lure it back to cancel half a Line,Nor all thy Tears wash out a Word of it.

• A Book of Verses underneath the Bough,A Jug of Wine, a Loaf of Bread‐‐and ThouBeside me singing in the Wilderness‐‐Oh, Wilderness were Paradise enow!

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Aside2:  The Problem with Translations

• Literal Translation:• Signs of destiny have always beenThose hands inscribed both good and meanWhat was written, came from the unseenThough we tried without and worried within.

• FitzGerald:• The Moving Finger writes; and, having writ,Moves on: nor all your Piety nor WitShall lure it back to cancel half a Line,Nor all your Tears wash out a Word of it.

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Aside2:  The Problem with Translations

• Edward Whinfield:• Give me a skin of wine, a crust of bread, A pittance bare, a book of verse to read; With thee, O love, to share my lowly roof, I would not take the Sultan's realm instead! 

• Arthur Talbot:• Give me a scroll of verse, a little wine, With half a loaf to fill thy needs and mine, And with the desert sand our resting place, For ne'er a Sultan's kingdom would we pine. 

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Aside2:  The Problem with Translations

• FitzGerald:• A Book of Verses underneath the Bough,A Jug of Wine, a Loaf of Bread‐‐and ThouBeside me singing in the Wilderness‐‐Oh, Wilderness were Paradise enow!

• Wendy Cope:• Here with a Bag of Crisps beneath the Bough, A Can of Beer, a Radio ‐ and Thou Beside me half asleep in Brockwell Park; And Brockwell Park is Paradise enow. 

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OK, enough silliness.

• From the preface to Khayyam’s Algebra:

“In the name of God, gracious and merciful! Praise be to God, lord of all Worlds, a happy end to those who are pious, and ill‐will to none but the merciless.  May blessings repose upon the prophets, especially upon Mohammed and all his holy descendants.”

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Nașīr al‐Dīn al‐Țūsī

• 1201 – 1274• In 1258, Mongols who had taken over Russia took over Baghdad.  

• al‐Țūsī continued to work under the new ruler Hulegu, grandson of Genghis Khan. 

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Nașīr al‐Dīn al‐Țūsī

• First to treat trigonometry as a mathematical discipline separate from astronomy; developed plane and spherical geometry into a more sophisticated subject.

• Wrote Al‐risala al‐shafiya'an al‐shakk fi'l‐khutut al‐mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250)

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Nașīr al‐Dīn al‐Țūsī

• Wrote detailed critiques of the parallel postulate and of Omar Khayyám's attempted proof a century earlier. Nașīr al‐Dīn attempted to derive a proof of the parallel postulate by contradiction. He worked with the same (Saccheri) quadrilaterals and attempted to reach a contradiction from the hypotheses of the acute and obtuse angles.  

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Șadr al‐Dīn al‐Țūsī

• Nașīr al‐Dīn al‐Țūsī had a son Șadr al‐Dīn who for reasons I have been unable to ascertain was sometimes referred to as Pseudo‐Țūsī.  

• In 1298 he wrote a work based on Nașīr al‐Dīn’s later thought, containing yet another attempt to prove the EPP, based on another hypothesis equivalent to EPP:

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Pseudo‐Țūsī’s Hypothesis

• Given two lines                   with A*G*B and C*H*D, with                 and        not perpendicular to       , then the perpendiculars dropped from      to       have length greater than GH on the side on which       makes an obtuse angle with       and less than GH on the other side.   

and AB CD

GH CD GHAB

AB

CD

GHAB

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Less than GH                Greater than GH

G

A

B

C D

H

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Pseudo‐Țūsī and Later Authors

• Șadr al‐Dīn’s work was published in Italy in 1594 in Arabic, with a Latin title page.  It was known to both John Wallis and GirolamoSaccheri, and could well have been part of the foundation for their work on the EPP.

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Ulugh Beg

• 1393 – 1449• Ruler and grandson of Tamerlane, whose ancestor married Ghengis Khan’s granddaughter.  

• He was also an astronomer and mathematician.  Very accurate catalogues of stars, trig tables.

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Jamshid al‐Kashi

• ~1380 – 1429 AD • Astronomy, Trigonometry• Became a member of Ulugh Beg’s “House of Learning” or university.  His best work was likely done under Ulugh Beg’s support.

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Jamshid al‐Kashi

• Wrote The Treatise on the Chord and Sine, in which he: – Provided the solution of a cubic equation to find the 

sine of one degree accurate to about 12 decimal places. 

– Showed a method for extracting n‐th root of a number (similar to Horner’s method)

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Jamshid al‐Kashi

• Wrote Miftah al‐hisab, or The Key of Arithmetic, containing:– The first systematic treatment of decimal fractions– Approximation of π correct to 9 sexagecimaldigits(16 decimal places), by circumscribing a circle by a polygon having sides. Best approximation for next 200 years.

– Properties of binomial coefficients

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Islamic Accomplishments

• Fully developed decimal place value system, including decimal fractions

• Systematized the study of algebra, and began to study the relations between algebra and geometry

• Advanced the major Greek works on geometry (Euclid, Apollonius, Archimedes)

• Improvements in plane and spherical trigonometry