Agus Subandi,Drs.MBA
'Abd al-Hamīd ibn Turk
Abd al-Hamīd ibn Turk (fl. 830), known also as Abd al-Hamīd ibn Wase ibn Turk Jili was a ninth century Turkic Muslim mathematician. Not much is known about his biography. The two records of him, one by Ibn Nadim and the other by al-Qifti are not identical. However al-Qifi mentions his name as ʿAbd al-Hamīd ibn Wase ibn Turk Jili. Jili means from Gilan.[1]
He wrote a work on algebra of which only a chapter called "Logical Necessities in Mixed Equations", on the solution of quadratic equations, has survived.
He authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr.[2] The manuscript gives exactly the same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.[2] The similarity between these two works has led some historians to conclude that algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.[2]
[edit] References
1. ^ Ibn Turk in Dāʾirat al-Maʿārif-i Buzurg-i Islāmī, Vol. 3, no. 1001, Tehran. To be translated in Encyclopædia islamica.
2. ^ a b c Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. p. 234. ISBN 0-471-54397-7. "The Algebra of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on Al-jabr wa'l muqabalah which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's Algebra and in one case the same illustrative example x2 + 21 = 10x. In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century."
• Høyrup, J. (1986). "Al-Khwarizmi, Ibn Turk and the Liber Mensurationum: On the Origins of Islamic Algebra". Erdem 5: 445–484.
• Sayili, Aydin (1962). Abdülhamit İbn Türk'ün Katışık Denklemlerde Mantıki Zaruretler Adlı Yazısı ve Zamanın Cebri. (Logical necessities in mixed equations by ʿAbd al Hamīd ibn Turk and the algebra of his time.). Ankara: Türk Tarih Kurumu Basımevı. Rev. by Jean Itard in Revue Hist. Sci. Applic., 1965, I8:123-124.
[hide]v • d • eMathematics in medieval Islam
Mathematicians 'Abd al-Hamīd ibn Turk • Abd al-Rahman al-Sufi • Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī • Abū al-Wafā' al-Būzjānī • Abū Ishāq Ibrāhīm al-Zarqālī • Abū Ja'far al-Khāzin • Abū Kāmil Shujā ibn Aslam • Sind ibn Ali • Abu'l-Hasan al-Uqlidisi • Abu-Mahmud al-Khujandi • Abu Nasr Mansur • Abū Rayḥān al-Bīrūnī • Abū Sahl al-Qūhī • Ahmad ibn Yusuf • Al-Abbās ibn Said al-Jawharī • Al-Birjandi • Al-Ḥajjāj ibn Yūsuf ibn Maṭar • Alhazen • Ibn Muʿādh al-Jayyānī • Al-Karaji • Al-Khazini • Al-Kindi • Al-Mahani • Al-Nayrizi • Al-Saghani • Al-Sijzi • Yaʿīsh ibn Ibrāhīm al-Umawī • Alī ibn Ahmad al-Nasawī • Ali Qushji • Avicenna • Banū Mūsā • Brethren of Purity • Hunayn ibn Ishaq • Ibn al-Banna al-Marrakushi • Ibn al-Shatir • Ibn Sahl • Ibn Tahir al-Baghdadi • Ibn Yahyā al-Maghribī al-Samaw'al • Ibn Yunus • Ibrahim ibn Sinan • Jamshīd al-Kāshī • Kamāl al-Dīn al-Fārisī • Kushyar ibn Labban • Muhammad Baqir Yazdi • Muhammad ibn Jābir al-Harrānī al-Battānī • Muḥammad ibn Mūsā al-Khwārizmī • Muhyi al-Dīn al-Maghribī • Nasīr al-Dīn al-Tūsī • Omar Khayyám • Qāḍī Zāda al-Rūmī • Al-Khalili • Shams al-Dīn al-Samarqandī • Sharaf al-Dīn al-Tūsī • Sinan ibn Thabit • Taqi al-Din • Thābit ibn Qurra • Ulugh Beg • Yusuf al-Mu'taman ibn Hud • Na'im ibn Musa • Qotb al-Din Shirazi • Ibn al‐Haim al‐Ishbili
Treatises Almanac • Book of Fixed Stars • Book of Optics • De Gradibus • Encyclopedia of the Brethren of Purity • Tables of Toledo • Tabula Rogeriana • The Compendious Book on Calculation by Completion and Balancing • The Book of Healing • Zij • Zij-i Ilkhani • Zij-i-Sultani
Centers Al-Azhar University • Al-Mustansiriya University • House of Knowledge • House of Wisdom • Istanbul observatory of Taqi al-Din • Madrasah • Maktab • Maragheh observatory • University of Al-Karaouine
Influences Babylonian mathematics • Greek mathematics • Indian mathematics
Influenced Byzantine mathematics • European mathematics • Indian mathematics
Abd al-Rahman al-Sufi
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'Abd al-Rahman al-Sufi (Persian: عبدالرحمن صوفی) (December 7, 903 – May 25, 986) was a Persian astronomer also known as 'Abd ar-Rahman as-Sufi, or 'Abd al-Rahman Abu al-Husayn, 'Abdul Rahman Sufi, 'Abdurrahman Sufi and known in the west as Azophi; the lunar crater Azophi and the minor planet 12621 Alsufi are named after him. Al-Sufi published his famous Book of Fixed Stars in 964, describing much of his work, both in textual descriptions and pictures.
Contents
[hide]
• 1 Biography
• 2 Astronomy
• 3 Sufi Observing Competition
• 4 See also
• 5 Notes
• 6 References
• 7 External links
[edit] Biography
His name implies that he was a Sufi Muslim. He lived at the court of Emir Adud ad-Daula in Isfahan, Persia, and worked on translating and expanding Greek astronomical works, especially the Almagest of Ptolemy. He contributed several corrections to Ptolemy's star list and did his own brightness and magnitude estimates which frequently deviated from those in Ptolemy's work.
He was a major translator into Arabic of the Hellenistic astronomy that had been centred in Alexandria, the first to attempt to relate the Greek with the traditional Arabic star names and constellations, which were completely unrelated and overlapped in complicated ways.
[edit] Astronomy
Further information: Book of Fixed Stars
The constellation Sagittarius from The Depiction of Celestial Constellations.
He identified the Large Magellanic Cloud, which is visible from Yemen, though not from Isfahan; it was not seen by Europeans until Magellan's voyage in the 16th century.[1][2] He also made the earliest recorded observation of the Andromeda Galaxy in 964 AD; describing it as a "small cloud".[3] These were the first galaxies other than the Milky Way to be observed from Earth.
He observed that the ecliptic plane is inclined with respect to the celestial equator and more accurately calculated the length of the tropical year. He observed and described the stars, their positions, their magnitudes and their colour, setting out his results constellation by constellation. For each constellation, he provided two drawings, one from the outside of a celestial globe, and the other from the inside (as seen from the earth).
Al-Sufi also wrote about the astrolabe, finding numerous additional uses for it : he described over 1000 different uses, in areas as diverse as astronomy, astrology, horoscopes, navigation, surveying, timekeeping, Qibla, Salah prayer, etc.[4]
[edit] Sufi Observing Competition
Main article: Sufi Observing Competition
Since 2006, Astronomy Society of Iran – Amateur Committee (ASIAC) hold an international Sufi Observing Competition in the memory of Sufi. The first competition was held in 2006 in the north of Semnan Province[5] and the 2nd moe observing competition was held in summer of 2008 in Ladiz near the Zahedan. More than 100 observers from Iran and Iraq participated in this event.[6]
[edit] See also
• List of Muslim scientists
• List of Iranian scientists
• Astronomy in Islam
• Sufi Observing Competition
[edit] Notes
1. ^ "Observatoire de Paris (Abd-al-Rahman Al Sufi)". http://messier.obspm.fr/xtra/Bios/alsufi.html. Retrieved 2007-04-19.
2. ^ "Observatoire de Paris (LMC)". http://messier.obspm.fr/xtra/ngc/lmc.html. Retrieved 2007-04-19.
3. ^ Kepple, George Robert; Glen W. Sanner (1998). The Night Sky Observer's Guide, Volume 1. Willmann-Bell, Inc.. pp. 18. ISBN 0-943396-58-1.
4. ^ Dr. Emily Winterburn (National Maritime Museum) (2005). "Using an Astrolabe". Foundation for Science Technology and Civilisation. http://www.muslimheritage.com/topics/default.cfm?ArticleID=529. Retrieved 2008-01-22.
5. ^ http://www.asiac.ir/en/news/?NewsID=-333647997
6. ^ http://www.jamejamonline.ir/newstext.aspx?newsnum=100948106338
[edit] References
• Mitton, Jacqueline (2007). Cambridge Illustrated Dictionary of Astronomy. Cambridge University Press. ISBN 978-0-521-82364-7
[edit] External links
• Kunitzsch, Paul (2008) [1970-80]. "Al-Ṣūfī, Abu’I-Ḥusayn ‘Abd Al-Raḥmān Ibn ‘Umar Al-Rāzī". Complete Dictionary of Scientific Biography. Encyclopedia.com.
• biography at SEDS.ORG
[hide]v • d • eAstronomy in medieval Islam
Astronomers
Persian
Avicenna • Omar Khayyám • Al Beruni • Alhazen • Kushyar Gilani • Abd al-Rahman al-Sufi • Abū al-Wafā' al-Būzjānī • Abū Ja'far al-Khāzin • Abu-Mahmud al-Khujandi • Abu Nasr Mansur • Abū Sahl al-Qūhī • Al-Birjandi • Al-Ghazali • Al-Khazini • Al-Mahani • Al-Nayrizi • Al-Saghani • Al-Sijzi • Ali Qushji • Banū Mūsā • Habash al-Hasib al-Marwazi • Jamshīd al-Kāshī • Kamāl al-Dīn al-Fārisī • Muhammad ibn Mūsā al-Khwārizmī • Nasīr al-Dīn al-Tūsī • Shams al-Dīn al-Samarqandī • Sharaf al-Dīn al-Tūsī • Yaʿqūb ibn Ṭāriq • Zakariya al-Qazwini • Abu Said Gorgani • Ahmad Nahavandi • Khalid Ben Abdulmelik • Al-Kharaqī • Anvari • Ibn Kathīr al-Farghānī • Mashallah ibn Athari • Abu Ma'shar al-Balkhi • Jamal ad-Din • Ibrahim al-Fazari • Muhammad al-Fazari • Iranshahri • Naubakht • Qotb al-Din Shirazi
Arab
Ahmed ibn Yusuf • Ibn Matar • Al-Kindi • Ibn al-Shatir • Ibn Yunus • Ibrahim ibn Sinan • Albatenius • Muhyi al-Dīn al-Maghribī • al-Khalīlī
Others al-Zarqālī (Visigoth) • Ibn al-Banna al-Marrakushi (Berber) • Ibn Yahyā al-Maghribī al-Samaw'al (Jewish) • Piri Reis (Turkish) • Qāḍī Zāda al-Rūmī (Turkish) • Taqi al-Din (Turkish/Arab) • Sinan ibn Thabit (Mandean) • Thābit ibn Qurra (Mandean) • Ulugh Beg (Barlas/Persian) • Said Al-Andalusi • Ibn al‐Haim al‐Ishbili
Works
ʿAjā'ib al-makhlūqāt wa gharā'ib al-mawjūdāt • Arabic star names • Book of Optics • Encyclopedia of the Brethren of Purity • Islamic calendar • Star chart • Tabula Rogeriana • The Book of Healing
Zij: Alfonsine tables • Almanac • Astronomical catalog • Book of Fixed Stars • Star catalogue • Toledan Tables • Trigonometry table • Zij-i Ilkhani • Zij-i-Sultani
Instruments
Alidade • Analog computer • Aperture • Armillary sphere • Astrolabe • Astronomical clock • Celestial globe • Compass • Compass rose • Dioptra • Equatorial ring • Equatorium • Globe • Graph paper • Magnifying glass • Mural instrument • Navigational astrolabe • Nebula • Planisphere • Quadrant • Sextant • Shadow square • Spherical astrolabe • Sundial • Telescope • Triquetrum
Concepts Almucantar • Apogee • Astrophysics • Axial tilt • Azimuth • Celestial mechanics • Celestial spheres • Circular orbit • Deferent and epicycle • Earth's rotation • Eccentricity • Ecliptic • Elliptic orbit • Equant • Galaxy • Geocentrism • Gravitational potential energy • Gravity • Heliocentrism • Inertia • Islamic cosmology • Moonlight • Multiverse • Muslim views on astrology • Obliquity • Parallax • Precession • Qibla • Salat times • Specific gravity • Spherical Earth • Starlight • Sublunary sphere • Sunlight • Supernova • Temporal finitism • Trepidation • Triangulation • Tusi-couple • Universe
Centers Al-Azhar University • House of Knowledge • House of Wisdom • Islamic observatories • Istanbul observatory of Taqi al-Din • Madrasah • Maragheh observatory • Observatory • Research institute • Samarkand observatory • Umayyad Mosque • University of Al-Karaouine
Influences Babylonian astronomy • Egyptian astronomy • Hellenistic astronomy • Indian astronomy
Influenced Byzantine astronomy • Chinese astronomy • European astronomy • Indian astronomy
Retrieved from "http://en.wikipedia.org/wiki/Abd_al-Rahman_al-Sufi"
Categories: 903 births | 986 deaths | Astronomers of medieval Islam | Persian astronomers | Clockmakers | Greek–Arabic translators
Hidden categories: Articles containing Persian language text
Abū al-Ḥasan ibn Alī al-Qalaṣādī
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Abū al-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qalaṣādī (1412 in Baza, Spain – 1486 in Béja, Tunisia) was a Muslim mathematician from Al-Andalus specializing in Islamic inheritance jurisprudence. Al-Qalaṣādī is known for being one of the most influential voices in algebraic notation since antiquity and for taking "the first steps toward the introduction of algebraic symbolism." He wrote numerous books on arithmetic and algebra, including al-Tabsira fi'lm al-hisab (Arabic: التبصير في علم الحساب "Clarification of the science of arithmetic").[1]
Contents
[hide]
• 1 Early life
• 2 Symbolic algebra
• 3 See also
• 4 Notes
• 5 References
• 6 External links
[edit] Early life
Al-Qalaṣādī was born in Baza, an outpost of the Emirate of Granada. He received education in Granada, but continued to support his family in Baza. He published many works and eventually retired to his native Baza. He spent seven years living in Tlemcen, where he studied under the local Berber scholars, the most important of which was a man named Ibn Zaghu.
His works dealt with Algebra and contained the precise mathematical answers to problems in everyday life, such as the composition of medicaments, the calculation of the drop of irrigation canals and the explanation of frauds linked to instruments of measurement. The second part belongs to the already ancient tradition of judicial and cultural mathematics and joins a collection of little arithmetical problems presented in the form of poetical riddles
In 1480 the Christian forces of Ferdinand and Isabella, Los Reyes Católicos ("The Catholic Monarchs"), raided and often pillaged the city, al-Qalasādī himself served in the mountain citadels which were erected in the vicinity of Baza. al-Qalasādī eventually left his homeland and took refuge with his family in Béja, Tunisia. Baza was eventually besieged by the forces of Ferdinand and Isabella and its inhabitants sacked.
[edit] Symbolic algebra
Like his predecessors al-Qalaṣādī made attempts at creating an algebraic notation. Certainly symbols were not the invention of al-Qalaṣādī. Ibn al-Banna two centuries earlier made such an attempt, just like Diophantus and Brahmagupta in ancient times.[1] Al-Qalaṣādī represented mathematical symbols using characters from the Arabic alphabet, where:[1]
• ﻭ (wa) means "and" for addition (+)
• ﻻ (illa) means "less" for subtraction (-)
• ف (fi) means "times" for multiplication (*)
• ة (ala) means "over" for division (/)
• ﺝ (j) represents jadah meaning "root"
• ﺵ (sh) represents shay meaning "thing" for a variable (x)
• ﻡ (m) represents mal for a square (x2)
• ﻙ (k) represents kab for a cube (x3)
• ﻝ (l) represents yadilu for equality (=)
As an example, the equation 2x3 + 3x2 − 4x + 5 = 0 would have been written using his notation as:
2ﻙ ﻭ 3ﻡ ﻻ 4ﺵ ﻭ 5 ﻝ 0
[edit] See also
• Islamic mathematics
[edit] Notes
1. ^ a b c O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html.
[edit] References
• Rebstock, Ulrich (1990). "Arabic Mathematical Manuscripts in Mauretania". Bulletin of the School of Oriental and African Studies (University of London) 52 (3): 429–441. JSTOR 618117.
• Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7.
[edit] External links
• Saidan, A. S. (2008) [1970-80]. "Al-Qalaṣādī (or Al-Qalaṣādī), Abu ’L-Ḥasan ‘Alī Ibn Muḥammad Ibn ‘Alī". Complete Dictionary of Scientific Biography. Encyclopedia.com.
[hide]v • d • eMathematics in medieval Islam
Mathematicians 'Abd al-Hamīd ibn Turk • Abd al-Rahman al-Sufi • Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī • Abū al-Wafā' al-Būzjānī • Abū Ishāq Ibrāhīm al-Zarqālī • Abū Ja'far al-Khāzin • Abū Kāmil Shujā ibn Aslam • Sind ibn Ali • Abu'l-Hasan al-Uqlidisi • Abu-Mahmud al-Khujandi • Abu Nasr Mansur • Abū Rayḥān al-Bīrūnī • Abū Sahl al-Qūhī • Ahmad ibn Yusuf • Al-Abbās ibn Said al-Jawharī • Al-Birjandi • Al-Ḥajjāj ibn Yūsuf ibn Maṭar • Alhazen • Ibn Muʿādh al-Jayyānī • Al-Karaji • Al-Khazini • Al-Kindi • Al-Mahani • Al-Nayrizi • Al-Saghani • Al-Sijzi • Yaʿīsh ibn Ibrāhīm al-Umawī • Alī ibn Ahmad al-Nasawī • Ali Qushji • Avicenna • Banū Mūsā • Brethren of Purity • Hunayn ibn Ishaq • Ibn al-Banna al-Marrakushi • Ibn al-Shatir • Ibn Sahl • Ibn Tahir al-Baghdadi • Ibn Yahyā al-Maghribī al-Samaw'al • Ibn Yunus • Ibrahim ibn Sinan • Jamshīd al-Kāshī • Kamāl al-Dīn al-Fārisī • Kushyar ibn Labban • Muhammad Baqir Yazdi • Muhammad ibn Jābir al-Harrānī al-Battānī • Muḥammad ibn Mūsā al-Khwārizmī • Muhyi al-Dīn al-Maghribī • Nasīr al-Dīn al-Tūsī • Omar Khayyám • Qāḍī Zāda al-Rūmī • Al-Khalili • Shams al-Dīn al-Samarqandī • Sharaf al-Dīn al-Tūsī • Sinan ibn Thabit • Taqi al-Din • Thābit ibn Qurra • Ulugh Beg • Yusuf al-Mu'taman ibn Hud • Na'im ibn Musa • Qotb al-Din Shirazi • Ibn al‐Haim al‐Ishbili
Treatises Almanac • Book of Fixed Stars • Book of Optics • De Gradibus • Encyclopedia of the Brethren of Purity • Tables of Toledo • Tabula Rogeriana • The Compendious Book on Calculation by Completion and Balancing • The Book of Healing • Zij • Zij-i Ilkhani • Zij-i-Sultani
Centers Al-Azhar University • Al-Mustansiriya University • House of Knowledge • House of Wisdom • Istanbul observatory of Taqi al-Din • Madrasah • Maktab • Maragheh observatory • University of Al-Karaouine
Influences Babylonian mathematics • Greek mathematics • Indian mathematics
Influenced Byzantine mathematics • European mathematics • Indian mathematics
Abū al-Wafā' Būzjānī
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Abu al-Wafa' al-Buzjani
Born June 10, 940
Buzhgan
Died 997 or 998 CE
Baghdad
Era Islamic Golden Age
Region Islamic civilization
Main interests Mathematics and Astronomy
Notable ideas • Tangent function
• Law of sines
• Several trigonometric identities
Major works Almagest of Abū al-Wafā'
Influenced by[show]
•
Abu al-Wafa' Buzjani (Persian: ابوالوفا بوزجانی, full name: Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī, Persian: ابوالوفا محمد بن محمد بن یحیی بن اسماعیل بن العباس البوزجانی; June 10, 940 – July 1, 998[1] ) was a Persian Muslim mathematician and astronomer.
He studied mathematics and worked principally in the field of trigonometry. He wrote a number of books, most of which no longer exist. He also studied the movements of the moon.
Contents
[hide]
• 1 Biography
• 2 Astronomy
• 3 Mathematics
o 3.1 Almagest
• 4 Legacy
• 5 Notes
• 6 References
• 7 Further reading
• 8 External links
[edit] Biography
He was born in Buzhgan, (now Torbat-e Jam) in Iran. In 959 AD, he moved to Iraq. He was a contemporary of the distinguished scientists Al-Quhi and Al-Sijzi who were in Baghdad at the time and others like Abu Nasr ibn Iraq, Abu-Mahmud Khojandi, Kushyar ibn Labban and Al-Biruni.[2]
He died either in 997 or 998 CE in Baghdad.[2]
[edit] Astronomy
Abu Al-Wafa' was the first to build a wall quadrant to observe the sky,[2] for the accurate measurement of the declination of stars.[citation needed] It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kitāb az-Zīj.[2]
He also introduced the tangent function,[note 1] which helped to solve problems involving right-angled spherical triangles, and developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.[3]
[edit] Mathematics
He established several trigonometric identities such as sin(a ± b) in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords.[4]
He also discovered the law of sines for spherical triangles:
where A, B, C are the sides and a, b, c are the opposing angles.[4]
[edit] Almagest
Only the first seven treaties of the Almagest of Abu al-Wafa' are extant.[5] The work covers numerous topics in the fields of plane and spherical trigonometry and solutions to determine the direction of Qibla.[2]
[edit] Legacy
The crater Abul Wáfa on the Moon is named after him.
[edit] Notes
1. ^ Other sources, including the book chapter by Jacques Sesiano cited elsewhere in this article, give the credit for this innovation to Habash al-Hasib al-Marwazi.
[edit] References
1. ^ "بوزجانی". Encyclopaediaislamica.com. http://www.encyclopaediaislamica.com/madkhal2.php?sid=2053. Retrieved 2009-08-30.
2. ^ a b c d e Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy (Cambridge University Press) 21 (1). doi:10.1017/S095742391000007X.
3. ^ Hashemipour, Behnaz (2007). "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 188–189. ISBN 978-0-387-31022-0.
4. ^ a b Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
5. ^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
[edit] Further reading
• Youschkevitch, A.P. (1970). "Abū’l-Wafāʾ Al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn Al-ʿAbbās". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 39–43. ISBN 0684101149. http://www.encyclopedia.com/doc/1G2-2830900031.html.
[edit] External links
• O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Abu'l-Wafa.html.
Trigonometry
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Jump to: navigation, search
"Trig" redirects here. For other uses, see Trig (disambiguation).
The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.
Trigonometry
History
Usage
Functions
Generalized
Inverse functions
Further reading
Reference
Identities
Exact constants
Trigonometric tables
Laws and theorems
Law of sines
Law of cosines
Law of tangents
Pythagorean theorem
Calculus
Trigonometric substitution
Integrals of functions
Derivatives of functions
Integrals of inverse functions
v • d • e
Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1] or from Sanskrit trikon "triangle" + miti "measurement" = trikonmiti[2]) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[3]
Trigonometry is usually taught in middle and secondary schools either as a separate course or as part of a precalculus curriculum. It has applications in both pure mathematics and applied mathematics, where it is essential in many branches of science and technology. A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.
Contents
[hide]
• 1 History
• 2 Overview
o 2.1 Extending the definitions
o 2.2 Mnemonics
o 2.3 Calculating trigonometric functions
• 3 Applications of trigonometry
• 4 Standard Identities
• 5 Angle Transformation Formulas
• 6 Common formulas
o 6.1 Law of sines
o 6.2 Law of cosines
o 6.3 Law of tangents
o 6.4 Euler's formula
• 7 See also
• 8 References
o 8.1 Bibliography
• 9 External links
[edit] History
Main article: History of trigonometry
The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry."[4]
Ancient Egyptian and Babylonian mathematicians lacked the concept of an angle measure, but they studied the ratios of the sides of similar triangles and discovered some properties of these ratios. The ancient Nubians used a similar methodology.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
Ancient Greek mathematicians such as Euclid and Archimedes studied the properties of the chord of an angle and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics.[10] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[11] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.[13]
[edit] Overview
In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
• Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
• Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
• Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".
With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.
[edit] Extending the definitions
Fig. 1a - Sine and cosine of an angle θ defined using the unit circle.
The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
ex + iy = ex(cosy + isiny).
See Euler's and De Moivre's formulas.
•
Graphing process of y = sin(x) using a unit circle.
•
Graphing process of y = tan(x) using a unit circle.
•
Graphing process of y = csc(x) using a unit circle.
[edit] Mnemonics
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e. "SOH-CAH-TO-A").[14] Another method is to expand the letters into a phrase, such as "Some Old Horses Chew Apples Happily Throughout Old Age".[15]
[edit] Calculating trigonometric functions
Main article: Generating trigonometric tables
Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.
[edit] Applications of trigonometry
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
Main article: Uses of trigonometry
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
[edit] Standard Identities
Identities are those equations that hold true for any value.
Triangle with sides a,b,c and respectively opposite angles A,B,C
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.
[edit] Law of sines
The law of sines (also known as the "sine rule") for an arbitrary triangle states:
where R is the radius of the circumscribed circle of the triangle:
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
[edit] Law of cosines
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
or equivalently:
[edit] Law of tangents
The law of tangents:
[edit] Euler's formula
Euler's formula, which states that eix = cosx + isinx, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
[edit] See also
• Generalized trigonometry
• List of triangle topics
• Trigonometric functions
• Aryabhata's sine table
• List of trigonometric identities
• Trigonometry in Galois fields
• Unit circle
• Uses of trigonometry
• Small-angle approximation
• Skinny triangle
[edit] References
1. ^ "trigonometry". Online Etymology Dictionary. http://www.etymonline.com/index.php?search=trigonometry.
2. ^ "trikonmiti". Sanskrit Documents. http://sanskritdocuments.org/articles/AncientHinduCivilizationMathematicsDas.html.
3. ^ R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
4. ^ Boyer (1991). "Greek Trigonometry and Mensuration". p. 162.
5. ^ A history of ancient mathematical astronomy: in three parts. By Otto Neugebauer. pg. 744
6. ^ "The Beginnings of Trigonometry". Rutgers, The State University of New Jersey.
7. ^ Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004)."Sherlock Holmes in Babylon: and other tales of mathematical history". MAA. p.36. ISBN 0883855461
8. ^ Boyer p215
9. ^ Boyer p237, p274
10. ^ Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 0-393-32030-8.
11. ^ Groundbreaking Scientific Experiments, Inventions, and Discoveries
12. ^ William Bragg Ewald (2008)."From Kant to Hilbert: a source book in the foundations of mathematics". Oxford University Press US. p.93. ISBN 0198505353
13. ^ Kelly Dempski (2002)."Focus on Curves and Surfaces". p.29. ISBN 159200007X
14. ^ Weisstein, Eric W., "SOHCAHTOA" from MathWorld.
15. ^ Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0192806750.
[edit] Bibliography
• Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. ISBN 0471543977.
• Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.
• Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld. Weiner.
[edit] External links
Find more about Trigonometry on Wikipedia's sister projects:
Definitions from Wiktionary
Images and media from Commons
Learning resources from Wikiversity
News stories from Wikinews
Quotations from Wikiquote
Source texts from Wikisource
Textbooks from Wikibooks
• Trigonometric Delights, by Eli Maor, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
• Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
• Benjamin Banneker's Trigonometry Puzzle at Convergence
• Dave's Short Course in Trigonometry by David Joyce of Clark University
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Abū Ishāq Ibrāhīm al-Zarqālī
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"Arzachel" redirects here. For other uses, see Arzachel (disambiguation).
Abū Isḥāq Ibrāhīm ibn Yaḥyā al-Naqqāsh al-Zarqālī (1029–1087), Latinized as Arzachel, was an instrument maker and one of the leading theoretical and practical astronomers of his time. Although his name is conventionally given as al-Zarqālī, it is probable that the correct form was al-Zarqālluh.[1] He lived in Toledo in Castile, Al-Andalus (now Spain), moving to Córdoba later in his life. His works inspired a generation of Islamic astronomers in Andalusia.
The crater Arzachel on the Moon is named after him.
Contents
[hide]
• 1 Early life
• 2 Astronomy
o 2.1 Instruments
o 2.2 Theory
• 3 See also
• 4 Notes
• 5 Further reading
• 6 External links
[edit] Early life
Al-Zarqālī was born to a family of Visigoth converts to Islam in a village near the outskirts of Toledo, then a famous capitol of the Taifa of Toledo, known for its co-existence between Muslims and Christians.
Muladi art from Toledo in Al-Andalus depicting the Alcázar in the year 976.AD
He was trained as a metalsmith and due to his skills he was nicknamed Al-Nekkach (in Andalusian Arabic "the engraver of metals"). According to the historians of Al-Andalus he was a mechanic and metal-craftsman very crafty with his hands. As an instrument maker he first entered the service of the Cadi Said Al-Andalusi, in the year 1060 he wrote on how to create delicate instruments used in Astronomy, and very soon built instruments for a "prestigious group of scholars", but when they realized his great youthful intellect, they became interested in him. After two years of education in the various Maktabs in the city patronized by Al-Mamun of Toledo he eventually became a member of that prestigious group. In the following years he began to invent his own instruments and correct mathematical calculations made by his predecessors and would eventually compile the Tables of Toledo.[2]
He was particularly talented in Geometry and Astronomy. He is known to have taught and visited Córdoba on various occasions his extensive experience and knowledge eventually made him the foremost foremost astronomer of his time. Al-Zarqālī was not only just a Theoretical scientist but an inventor as well. His inventions and works put Toledo at the intellectual center of Al-Andalus.
In the year 1085 Toledo was sacked by Alfonso VI of Castile Al-Zarqālī like his colleagues such as al‐Waqqashi (1017–1095) of Toledo had to flee for his life. It is unknown whether the aged Al-Zarqālī fled to Cordoba or died in a Moorish refuge camp.
His works profoundly influenced the works of: Ibn Bajjah (Avempace), Ibn Tufail (Abubacer), Ibn Rushd (Averroës), Ibn al-Kammad, Ibn al‐Haim al‐Ishbili and Nur ad-Din al-Betrugi (Alpetragius).
[edit] Astronomy
[edit] Instruments
There is a record of an al-Zarqāl who built a water clock, capable of determining the hours of the day and night and indicating the days of the lunar months,[3] though some scholars think that this is a different person.[1]
Al-Zarqālī also wrote two works on the construction of an instrument (an equatorium) for computing the position of the planets using diagrams of the Ptolemaic model. These works were translated into Spanish in the 13th century by order of King Alfonso X in a section of the Libros del Saber de Astronomia entitled the "Libros de las laminas de los vii planetas".
He also invented a perfected kind of astrolabe known as "the tablet of the al-Zarqālī" (al-ṣafīḥā al-zarqāliya), which was famous in Europe under the name Saphaea.[4][5]
[edit] Theory
Al-Zarqālī corrected Ptolemy's geographical data, specifically the length of the Mediterranean Sea.[citation needed] In his treatise on the solar year, which survives only in a Hebrew translation, he was the first to demonstrate the motion of the solar apogee relative to the fixed background of the stars. He measured its rate of motion as 12.9 seconds per year, which is remarkably close to the modern calculation of 11.6 seconds.[6] Al-Zarqālī's model for the motion of the Sun, in which the center of the Sun's deferent moved on a small, slowly-rotating circle to reproduce the observed motion of the solar apogee, was discussed in the thirteenth century by Bernard of Verdun[7] and in the fifteenth century by Regiomontanus and Peurbach. In the sixteenth century Copernicus employed this model, modified to heliocentric form, in his De Revolutionibus Orbium Coelestium.[8]
Al-Zarqālī also contributed to the famous Tables of Toledo, an adaptation of earlier astronomical data to the location of Toledo along with the addition of some new material.[1] Al-Zarqālī was famous as well for his own Book of Tables. Many "books of tables" had been compiled, but his almanac contained tables which allowed one to find the days on which the Coptic, Roman, lunar, and Persian months begin, other tables which give the position of planets at any given time, and still others facilitating the prediction of solar and lunar eclipses.
He also compiled an almanac that directly provided "the positions of the celestial bodies and need no further computation". The work provided the true daily positions of the sun for four Julian years from 1088 to 1092, the true positions of the five planets every 5 or 10 days over a period of 8 years for Venus, 79 years for Mars, and so forth, as well as other related tables.[9][10]
His work[which?] was translated into Latin by Gerard of Cremona in the 12th century, and contributed to the rebirth of a mathematically-based astronomy in Christian Europe. It[which?] was later incorporated into the Tables of Toledo in the 12th century and the Alfonsine tables in the 13th century.[9]
In designing an instrument to deal with Ptolemy's complex model for the planet Mercury, in which the center of the deferent moves on a secondary epicycle, al-Zarqālī noted that the path of the center of the primary epicycle is not a circle, as it is for the other planets. Instead it is approximately oval and similar to the shape of a pignon.[11] Some writers have misinterpreted al-Zarqālī's description of an earth-centered oval path for the center of the planet's epicycle as an anticipation of Johannes Kepler's sun-centered elliptical paths for the planets.[12] Although this may be the first suggestion that a conic section could play a role in astronomy, al-Zarqālī did not apply the ellipse to astronomical theory and neither he nor his Iberian or Maghrebi contemporaries used an elliptical deferent in their astronomical calculations.[13]
[edit] See also
• Islamic astronomy
• Islamic scholars
• List of Arab scientists and scholars
[edit] Notes
1. ^ a b c s.v. "al-Zarqālī", Julio Samsó, Encyclopaedia of Islam, New edition, vol. 11, 2002.
2. ^ http://www.muslimheritage.com/topics/default.cfm?articleID=729
3. ^ John David North, Cosmos: an illustrated history of astronomy and cosmology, University of Chicago Press, 2008, p. 218 "He was a trained artisan who entered the service of Qadi Said as a maker of instruments and water-clocks."
4. ^ M. T. Houtsma and E. van Donzel (1993), "ASṬURLĀB", E. J. Brill's First Encyclopaedia of Islam, Brill Publishers, ISBN 9004082654
5. ^ Hartner, W. (1960). "ASṬURLĀB". Encyclopaedia of Islam. 1 (2nd ed.). Brill Academic Publishers. pp. 726. ISBN 90-04-08114-3. "It is, therefore, really al-Zarḳālī who must be credited with the invention of this new type of an astrolabe. Through the Libros del Saber (Vol. 3, Madrid 1864, 135-237: Libro de le acafeha) the instrument became known and famous under the name Saphaea. It is practically identical with Gemma Frisius's Astrolabum ...".
6. ^ Toomer, G. J. (1969), "The Solar Theory of az-Zarqāl: A History of Errors", Centaurus 14 (1): 306–36, doi:10.1111/j.1600-0498.1969.tb00146.x, at pp. 314–17.
7. ^ Toomer, G. J. (1987), "The Solar Theory of az-Zarqāl: An Epilogue", Annals of the New York Academy of Sciences 500: 513–519, doi:10.1111/j.1749-6632.1987.tb37222.x.
8. ^ Toomer, G. J. (1969), "The Solar Theory of az-Zarqāl: A History of Errors", Centaurus 14 (1): 306–336, doi:10.1111/j.1600-0498.1969.tb00146.x, at pp. 308–10.
9. ^ a b Glick, Thomas F.; Livesey, Steven John; Wallis, Faith (2005), Medieval Science, Technology, and Medicine: An Encyclopedia, Routledge, p. 30, ISBN 0415969301
10. ^ Toomer, G. J. (1969), "The Solar Theory of az-Zarqāl: A History of Errors", Centaurus 14 (1): 306–336, doi:10.1111/j.1600-0498.1969.tb00146.x, at p. 314.
11. ^ Willy Hartner, "The Mercury Horoscope of Marcantonio Michiel of Venice", Vistas in Astronomy, 1 (1955): 84–138, at pp. 118–122.
12. ^ Asghar Qadir (1989). Relativity: An Introduction to the Special Theory, pp. 5–10. World Scientific. ISBN 9971506122.
13. ^ Samsó, Julio; Mielgo, Honorino (1994), "Ibn al-Zarqalluh on Mercury", Journal for the History of Astronomy 25: 292, http://adsabs.harvard.edu/abs/1994JHA....25..289S
[edit] Further reading
• E. S. Kennedy. A Survey of Islamic Astronomical Tables, (Transactions of the American Philosophical Society, New Series, 46, 2.) Philadelphia, 1956.
• "Zarqālī, Abū Isḥāq Ibrāhīm Ibn Yaḥyā al-Naqqāsh al-". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. 1970–80. ISBN 0684101149.
[edit] External links
• Vernet, J. (2008) [1970–80]. "Al-Zarqālī (or Azarquiel), Abū Isḥāqibrāhīm Ibn Yaḥyā Al-Naqqāsh". Complete Dictionary of Scientific Biography. Encyclopedia.com.
• Muslim Scientists Before the Renaissance: Abū Ishāq Ibrāhīm al-Zarqālī (Arzachel)
• 'Transmission of Muslim astronomy to Europe'
• 'An Extensive biography'
[show]v • d • eAstronomy in medieval Islam
Persian
Avicenna • Omar Khayyám • Al Beruni • Alhazen • Kushyar Gilani • Abd al-Rahman al-Sufi • Abū al-Wafā' al-Būzjānī • Abū Ja'far al-Khāzin • Abu-Mahmud al-Khujandi • Abu Nasr Mansur • Abū Sahl al-Qūhī • Al-Birjandi • Al-Ghazali • Al-Khazini • Al-Mahani • Al-Nayrizi • Al-Saghani • Al-Sijzi • Ali Qushji • Banū Mūsā • Habash al-Hasib al-Marwazi • Jamshīd al-Kāshī • Kamāl al-Dīn al-Fārisī • Muhammad ibn Mūsā al-Khwārizmī • Nasīr al-Dīn al-Tūsī • Shams al-Dīn al-Samarqandī • Sharaf al-Dīn al-Tūsī • Yaʿqūb ibn Ṭāriq • Zakariya al-Qazwini • Abu Said Gorgani • Ahmad Nahavandi • Khalid Ben Abdulmelik • Al-Kharaqī • Anvari • Ibn Kathīr al-Farghānī • Mashallah ibn Athari • Abu Ma'shar al-Balkhi • Jamal ad-Din • Ibrahim al-Fazari • Muhammad al-Fazari • Iranshahri • Naubakht • Qotb al-Din Shirazi
Arab
Ahmed ibn Yusuf • Ibn Matar • Al-Kindi • Ibn al-Shatir • Ibn Yunus • Ibrahim ibn Sinan • Albatenius • Muhyi al-Dīn al-Maghribī • al-Khalīlī
Others al-Zarqālī (Visigoth) • Ibn al-Banna al-Marrakushi (Berber) • Ibn Yahyā al-Maghribī al-Samaw'al (Jewish) • Piri Reis (Turkish) • Qāḍī Zāda al-Rūmī (Turkish) • Taqi al-Din (Turkish/Arab) • Sinan ibn Thabit (Mandean) • Thābit ibn Qurra (Mandean) • Ulugh Beg (Barlas/Persian) • Said Al-Andalusi • Ibn al‐Haim al‐Ishbili
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Astronomy in medieval Islam
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Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age (8th–15th centuries),[1] and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and later in the Far East and India. It closely parallels the genesis of other Islamic sciences in its assimilation of foreign material and the amalgamation of the disparate elements of that material to create a science with Islamic characteristics. These included Sassanid, Hellenistic and Indian works in particular, which were translated and built upon.[2] In turn, Islamic astronomy later had a significant influence on Indian,[3] Byzantine[4] and European[5] astronomy (see Latin translations of the 12th century) as well as Chinese astronomy[6] and Malian astronomy.[7][8]
A significant number of stars in the sky, such as Aldebaran and Altair, and astronomical terms such as alhidade, azimuth, and almucantar, are still referred to by their Arabic names.[9] A large corpus of literature from Islamic astronomy remains today, numbering approximately 10,000 manuscripts scattered throughout the world, many of which have not been read or catalogued. Even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed.[10]
Contents
[hide]
• 1 History
o 1.1 700-825
o 1.2 825-1025
o 1.3 1025-1450
o 1.4 1450-1900
• 2 Observatories
• 3 Instruments
o 3.1 Celestial globes and armillary spheres
o 3.2 Astrolabes
o 3.3 Sundials
o 3.4 Quadrants
o 3.5 Equatorium
• 4 Muslim astronomers
• 5 Famous Muslim astronomy books
• 6 Notes
• 7 References
• 8 See also
• 9 External links
[edit] History
Pre-Islamic Arabs had no scientific astronomy. Their knowledge of stars was only empirical, limited to what they observed reagrding the rising and setting of stars. The rise of Islam provoked increased Arab thought in this field.[11]
Science historian Donald. R. Hill has divided Islamic Astronomy into the four following distinct time periods in its history.
[edit] 700-825
The period of assimilation and syncretisation of earlier Hellenistic, Indian, and Sassanid astronomy.
During this period many Indian and Persian texts were translated into Arabic. The most notable of the texts was Zij al-Sindhind,[12] translated by Muhammad ibn Ibrahim al-Fazari and Yaqub ibn Tariq in 777. Sources indicate that the text was translated after, in 770, an Indian astronomer visited the court of Caliph Al-Mansur. Another text translated was the Zij al-Shah, a collection of astronomical tables compiled in Persia over two centuries.
Fragments of text during this period indicate that Arabs adopted the sine function (inherited from India) in place of the chords of arc used in Greek trignometry.[13]
[edit] 825-1025
This period of vigorous investigation, in which the superiority of the Ptolemaic system of astronomy was accepted and significant contributions made to it. Astronomical research was greatly supported by the Abbasid caliph al-Mamun. Baghdad and Damascus became the centers of such activity. The caliphs not only supported this work financially, but endowed the work with formal prestige.
The first major Muslim work of astronomy was Zij al-Sindh by al-Khwarizimi in 830. The work contains tables for the movements of the sun, the moon and the five planets known at the time. The work is significant as it introduced Ptolemaic concepts into Islamic sciences. This work also marks the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge. Al-Khwarizmi's work marked the beginning of nontraditional methods of study and calculations.[14]
In 850, al-Farghani wrote Kitab fi Jawani (meaning "A compendium of the science of stars"). The book primarily gave a summary of Ptolemic cosmography. However, it also corrected Ptolemy based on findings of earlier Arab astronomers. Al-Farghani gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun and the moon, and the circumference of the earth. The books was widely circulated through the Muslim world, and even translated into Latin.[15]
[edit] 1025-1450
The period when a distinctive Islamic system of astronomy flourished. The period began as the Muslim astronomers began questioning the framework of the Ptolemaic system of astronomy. These criticisms, however, remained within the geocentric framework and followed Ptolemy's astronomical paradigm; one historian described their work as "a reformist project intended to consolidate Ptolemaic astronomy by bringing it into line with its own principles."[16]
In 1070, Abu Ubayd al-Juzjani published the Tarik al-Aflak. In his work, he indicated the so-called "equant" problem of the Ptolemic model. Al-Juzjani even proposed a solution for the problem. In al-Andalus, the anonymous work al-Istidrak ala Batlamyus (meaning "Recapitulation regarding Ptolemy"), included a list of objections to the Ptolemic astronomy.
The most important work, however, was Al-Shuku ala Batlamyus (meaning "Doubts on Ptolemy"). In this, the author summed up the inconsistencies of the Ptolemic models. Many astronomers took up the challenge posed in this work, namely to develop alternate models that evaded such errors. The most important of these astronomers include: Muayyad al-Din Urdi (circa 1266), Nasir al-Din al-Tusi (1201-74), Qutb al-Din al Shirazi (circa 1311), Sadr al-Sharia al-Bukhari (circa 1347), Ibn al-Shatir (circa 1375), and Ala al-Qushji (circa 1474).[17]
[edit] 1450-1900
The period of stagnation, when the traditional system of astronomy continued to be practised with enthusiasm, but with rapidly decreasing innovation of any major significance.
A large corpus of literature from Islamic astronomy remains today, numbering around some 10,000 manuscript volumes scattered throughout the world. Much of which has not even been catalogued. Even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed.
[edit] Observatories
Medieval manuscript by Qotbeddin Shirazi depicting an epicyclic planetary model.
The first systematic observations in Islam are reported to have taken place under the patronage of al-Mamun. Here, and in many other private observatories from Damascus to Baghdad, meridian degrees were measured, solar parameters were established, and detailed observations of the Sun, Moon, and planets were undertaken.
In the 10th century, the Buwayhid dynasty encouraged the undertaking of extensive works in Astronomy, such as the construction of a large scale instrument with which observations were made in the year 950. We know of this by recordings made in the zij of astronomers such as Ibn al-Alam. The great astronomer Abd Al-Rahman Al Sufi was patronised by prince Adud o-dowleh, who systematically revised Ptolemy's catalogue of stars. Sharaf al-Daula also established a similar observatory in Baghdad. And reports by Ibn Yunus and al-Zarqall in Toledo and Cordoba indicate the use of sophisticated instruments for their time.
It was Malik Shah I who established the first large observatory, probably in Isfahan. It was here where Omar Khayyám with many other collaborators constructed a zij and formulated the Persian Solar Calendar a.k.a. the jalali calendar. A modern version of this calendar is still in official use in Iran today.
The most influential observatory was however founded by Hulegu Khan during the 13th century. Here, Nasir al-Din al-Tusi supervised its technical construction at Maragha. The facility contained resting quarters for Hulagu Khan, as well as a library and mosque. Some of the top astronomers of the day gathered there, and from their collaboration resulted important modifications to the Ptolemaic system over a period of 50 years.
File:Ulugh.jpg
Ulugh Beg, founder of a large Islamic observatory, honoured on this Soviet stamp.
In 1420, prince Ulugh Beg, himself an astronomer and mathematician, founded another large observatory in Samarkand, the remains of which were excavated in 1908 by Russian teams.
And finally, Taqi al-din bin Ma'ruf founded a large observatory in Istanbul in 1575, which was on the same scale as those in Maragha and Samarkand.
In modern times, Turkey [1][2]has many well equipped observatories, while Jordan [3], Palestine [4][dead link], Lebanon [5][dead link], UAE [6], Tunisia [7], and other Arab states are also active as well. Iran has modern facilities at Shiraz University and Tabriz University. In Dec 2005, Physics Today reported of Iranian plans to construct a "world class" facility with a 2.0 m telescope observatory in the near future.[8]
[edit] Instruments
Our knowledge of the instruments used by Muslim astronomers primarily comes from two sources. First the remaining instruments in private and museum collections today, and second the treatises and manuscripts preserved from the middle ages.
Muslims made many improvements to instruments already in use before their time, such as adding new scales or details. Their contributions to astronomical instrumentation are abundant.
[edit] Celestial globes and armillary spheres
Celestial globes were used primarily for solving problems in celestial astronomy. Today, 126 such instruments remain worldwide, the oldest from the 11th century. The altitude of the sun, or the Right Ascension and Declination of stars could be calculated with these by inputting the location of the observer on the meridian ring of the globe.
An armillary sphere had similar applications. No early Islamic armillary spheres survive, but several treatises on “the instrument with the rings” were written. In this context there is also an Islamic development, the spherical astrolabe, of which only one complete instrument, from the 14th century, has survived.
[edit] Astrolabes
An 18th century Persian Astrolabe, kept at The Whipple Museum of the History of Science in Cambridge, England.
Brass astrolabes were developed in much of the Islamic world, chiefly as an aid to finding the qibla. The earliest known example is dated 315 (in the Islamic calendar, corresponding to 927-8). The first person credited for building the Astrolabe in the Islamic world is reportedly Fazari (Richard Nelson Frye: Golden Age of Persia. p163). He only improved it though, the Greeks had already invented astrolabes to chart the stars. The Arabs then took it during the Abbasid Dynasty and perfected it to be used to find the beginning of Ramadan, the hours of prayer, and the direction of Mecca.
The instruments were used to read the rise of the time of rise of the Sun and fixed stars. al-Zarqall of Andalusia constructed one such instrument in which, unlike its predecessors, did not depend on the latitude of the observer, and could be used anywhere. This instrument became known in Europe as the Saphaea.
[edit] Sundials
Muslims made several important improvements to the theory and construction of sundials, which they inherited from their Indian and Greek predecessors. Khwarizmi made tables for these instruments which considerably shortened the time needed to make specific calculations.
Sundials were frequently placed on mosques to determine the time of prayer. One of the most striking examples was built in the 14th century by the muwaqqit (timekeeper) of the Umayyid Mosque in Damascus, ibn al-Shatir.[18]
[edit] Quadrants
Several forms of quadrants were invented by Muslims. Among them was the sine quadrant used for astronomical calculations and various forms of the horary quadrant, used to determine time (especially the times of prayer) by observations of the Sun or stars. A center of the development of quadrants was ninth-century Baghdad.[19]
[edit] Equatorium
The Equatorium is an Islamic invention from Andalusia. The earliest known was probably made around 1015. It is a mechanical device for finding the positions of the Moon, Sun, and planets, without calculation using a geometrical model to represent the celestial body's mean and anomalistic position.
[edit] Muslim astronomers
Main article: Muslim astronomers
[edit] Famous Muslim astronomy books
• al-Khwarizmi (c. 830), Zij al-Sindhind
• al-Farghani (d. c. 850), Kitab fi Jawami Ilm al-Nujum
[edit] Notes
1. ^ (Saliba 1994b, pp. 245, 250, 256–257)
2. ^ (Gingerich 1986)
3. ^ Sharma, Virendra Nath (1995), Sawai Jai Singh and His Astronomy, Motilal Banarsidass Publ., pp. 8–10, ISBN 8120812565
4. ^ Joseph Leichter (June 27, 2009). "The Zij as-Sanjari of Gregory Chioniades". Internet Archive. http://www.archive.org/details/TheZijAs-sanjariOfGregoryChioniades. Retrieved 2009-10-02.
5. ^ Saliba (1999).
6. ^ van Dalen, Benno (2002), "Islamic Astronomical Tables in China: The Sources for Huihui li", in Ansari, S. M. Razaullah, History of Oriental Astronomy, Springer Science+Business Media, pp. 19–32, ISBN 1402006578
7. ^ African Cultural Astronomy By Jarita C. Holbrook, R. Thebe Medupe, Johnson O. Urama
8. ^ Medupe, Rodney Thebe; Warner, Brian; Jeppie, Shamil; Sanogo, Salikou; Maiga, Mohammed; Maiga, Ahmed; Dembele, Mamadou; Diakite, Drissa et al. (2008), The Timbuktu Astronomy Project, pp. 179, doi:10.1007/978-1-4020-6639-9_13.
9. ^ "Arabic Star Names". Islamic Crescents' Observation Project. 2007-05-01. http://www.icoproject.org/star.html. Retrieved 2008-01-24.
10. ^ (Ilyas 1997)
11. ^ Dallal (1999), pg. 162
12. ^ This book is not related to al-Khwarizmi's Zij al-Sindh. On zijes see E. S. Kennedy, "A Survey of Islamic Astronomical Tables".
13. ^ Dallal (1999), pg. 162
14. ^ Dallal (1999), pg. 163
15. ^ Dallal (1999), pg. 164
16. ^ Sabra, "Configuring the Universe," p. 322.
17. ^ Dallal (1999), pg. 171
18. ^ David A. King, "Islamic Astronomy," pp. 168-9.
19. ^ David A. King, "Islamic Astronomy," pp. 167-8.
[edit] References
• Abdulhak Adnan, La science chez les Turcs ottomans, Paris, 1939.
• Ahmad Dallal, "Science, Medicine and Technology.", in The Oxford History of Islam, ed. John Esposito, New York: Oxford University Press, (1999).
• Antoine Gautier, L'âge d'or de l'astronomie ottomane, in L'Astronomie, (Monthly magazine created by Camille Flammarion in 1882), december 2005, volume 119.
• Donald R. Hill, Islamic Science And Engineering, Edinburgh University Press (1993), ISBN 0-7486-0455-3
• E. S. Kennedy, "A Survey of Islamic Astronomical Tables," Transactions of the American Philosophical Society, 46, 2 (1956).
• David A. King, "Islamic Astronomy", in Astronomy before the telescope, ed. Christopher Walker. British Museum Press, (1999), pp. 143-174. ISBN 0-7141-2733-7
• A. I. Sabra, "Configuring the Universe: Aporetic, Problem Solving, and Kinematic Modeling as Themes of Arabic Astronomy", Perspectives on Science, 6 (1998): 288-330.
• George Saliba, "Arabic versus Greek Astronomy: A Debate over the Foundations of Science", Perspectives on Science, 8 (2000): 328-41.
[edit] See also
• Islam
• Golden Age of Islam
• History of astronomy
• Hebrew astronomy
• List of Iranian scientists
• List of Muslim scientists
• Islamic astrology
• Arab and Persian astrology
[edit] External links
• "Tubitak Turkish National Observatory Antalya"
• "Scientific American" article on Islamic Astronomy
• The Arab Union for Astronomy and Space Sciences (AUASS)
• King Abdul Aziz Observatory
• History of Islamic Astrolabes
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Abū Ja'far al-Khāzin
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For the 12th century physicist and astronomer, see Al-Khazini.
Abu Jafar Muhammad ibn Hasan Khazini (900–971) was a Persian astronomer and mathematician from Khorasan. He worked on both astronomy and number theory.
Khazini was one of the scientists brought to the court in Ray, Iran by the ruler of the Buyid dynasty, Adhad ad-Dowleh, who ruled from 949 to 983 AD. In 959/960 Khazini was required by the Vizier of Ray, who was appointed by ad-Dowleh, to measure the obliquity of the ecliptic.
One of al-Khazin's works Zij al-Safa'ih ("Tables of the disks of the astrolabe") was described by his successors as the best work in the field and they make many references to it. The work describes some astronomical instruments, in particular an astrolabe fitted with plates inscribed with tables and a commentary on the use of these. A copy of this instrument was made but vanished in Germany at the time of World War II. A photograph of this copy was taken and examined in D.A. King's New light on the Zij al-Safa'ih of Abu Ja'far al-Khazin, Centaurus 23 (2) (1979/80), 105-117.
Khazeni also wrote a commentary on Ptolemy's Almagest in which he gives nineteen propositions relating to statements by Ptolemy. He also proposed a different solar model from that of Ptolemy.
[edit] References
• O'Connor, John J.; Robertson, Edmund F., "Abu Jafar Muhammad ibn al-Hasan Al-Khwarizmi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khazin.html.
• Rashed, Roshdi (1996). Les Mathématiques Infinitésimales du IXe au XIe Siècle 1: Fondateurs et commentateurs: Banū Mūsā, Ibn Qurra, Ibn Sīnān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd. London. Reviews: Seyyed Hossein Nasr (1998) in Isis 89 (1) pp. 112-113; Charles Burnett (1998) in Bulletin of the School of Oriental and African Studies, University of London 61 (2) p. 406.
[edit] External links
• Dold-Samplonius, Yvonne (2008) [1970-80]. "Al-Khāzin, Abū Ja‘far Muḥammad Ibn Al-Ḥasan Al-Khurāsānī". Complete Dictionary of Scientific Biography. Encyclopedia.com.
• O'Connor, John J.; Robertson, Edmund F., "Abu Jafar Muhammad ibn al-Hasan Al-Khwarizmi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khazin.html.
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Ptolemy
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For others with "Ptolemy..." names, and history of those names, see Ptolemy (name).
Ptolemy
An early Baroque artist's rendition of Claudius Ptolemaeus.
Born c. AD 90
Egypt
Died c. AD 168 (aged 77–78)
Alexandria, Egypt
Occupation mathematician, geographer, astronomer, astrologer
Claudius Ptolemy ( /ˈtɒləmi/; Greek: Κλαύδιος Πτολεμαῖος, Klaudios Ptolemaios; Latin: Claudius Ptolemaeus; c. AD 90 – c. AD 168), was a Roman citizen of Egypt who wrote in Greek.[1] He was a mathematician, astronomer, geographer, astrologer, and poet (of a single epigram in the Greek Anthology).[2][3] He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the Thebaid. He died in Alexandria around AD 168.[4]
Ptolemy was the author of several scientific treatises, at least three of which were of continuing importance to later Islamic and European science. The first is the astronomical treatise now known as the Almagest (in Greek, Ἡ Μεγάλη Σύνταξις, "The Great Treatise", originally Μαθηματικὴ Σύνταξις, "Mathematical Treatise"). The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise known sometimes in Greek as the Apotelesmatika (Ἀποτελεσματικά), more commonly in Greek as the Tetrabiblos (Τετράβιβλος "Four books"), and in Latin as the Quadripartitum (or four books) in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day.
Contents
[hide]
• 1 Background
• 2 Astronomy
• 3 Geography
• 4 Astrology
• 5 Music
• 6 Optics
• 7 Named after Ptolemy
• 8 Ptolemy in pop culture
• 9 See also
• 10 Footnotes
• 11 References
o 11.1 Texts and translations
• 12 External links
o 12.1 Primary sources
o 12.2 Secondary material
12.2.1 Animated illustrations
Background
The name Claudius is a Roman nomen; the fact that Ptolemy bore it proves that he was a Roman citizen. It would have suited custom if the first of Ptolemy's family who became a citizen (whether it was he or an ancestor) took the nomen from a Roman called Claudius, who was in some sense responsible for granting citizenship. If, as was not uncommon, this Roman was the emperor, the citizenship would have been granted between AD 41 and 68 (when Claudius, and then Nero, were emperors). The astronomer would also have had a praenomen, which remains unknown. However, it may have been Tiberius, as that praenomen was very common among those whose families had been granted citizenship by these emperors.
Ptolemaeus (Πτολεμαῖος – Ptolemaios) is a Greek name. It occurs once in Greek mythology, and is of Homeric form.[5] It was quite common among the Macedonian upper class at the time of Alexander the Great, and there were several among Alexander's army, one of whom in 323 BC made himself King of Egypt: Ptolemy I Soter; all the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies. There is little evidence on the subject of Ptolemy's ancestry (though see above on his family's Roman citizenship), but most scholars and historians consider it unlikely that Ptolemy was related to the royal dynasty of the Ptolemies.[citation needed]
Beyond his being considered a member of Alexandria's Greek society, few details of Ptolemy's life are known. He wrote in Ancient Greek and is known to have utilized Babylonian astronomical data.[6][7] Although he was a Roman citizen, most scholars have concluded that Ptolemy was ethnically Greek,[8][9][10] while some suggest that he was a Hellenized Egyptian.[9][11][12] He was often known in later Arabic sources as "the Upper Egyptian",[13] suggesting that he may have had origins in southern Egypt.[14] Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic: بطليموس Batlaymus.[15]
Astronomy
Further information: Almagest
The Almagest is the only surviving comprehensive ancient treatise on astronomy. Babylonian astronomers had developed arithmetical techniques for calculating astronomical phenomena; Greek astronomers such as Hipparchus had produced geometric models for calculating celestial motions. Ptolemy, however, claimed to have derived his geometrical models from selected astronomical observations by his predecessors spanning more than 800 years, though astronomers have for centuries suspected that his models' parameters were adopted independently of observations.[16] Ptolemy presented his astronomical models in convenient tables, which could be used to compute the future or past position of the planets.[17] The Almagest also contains a star catalogue, which is an appropriated version of a catalogue created by Hipparchus. Its list of forty-eight constellations is ancestral to the modern system of constellations, but unlike the modern system they did not cover the whole sky (only the sky Hipparchus could see). Through the Middle Ages it was spoken of as the authoritative text on astronomy, with its author becoming an almost mythical figure, called Ptolemy, King of Alexandria.[18] The Almagest was preserved, like most of Classical Greek science, in Arabic manuscripts (hence its familiar name). Because of its reputation, it was widely sought and was translated twice into Latin in the 12th century, once in Sicily and again in Spain.[19] Ptolemy's model, like those of his predecessors, was geocentric and was almost universally accepted until the appearance of simpler heliocentric models during the scientific revolution.
His Planetary Hypotheses went beyond the mathematical model of the Almagest to present a physical realization of the universe as a set of nested spheres,[20] in which he used the epicycles of his planetary model to compute the dimensions of the universe. He estimated the Sun was at an average distance of 1210 Earth radii while the radius of the sphere of the fixed stars was 20,000 times the radius of the Earth.[21]
Ptolemy presented a useful tool for astronomical calculations in his Handy Tables, which tabulated all the data needed to compute the positions of the Sun, Moon and planets, the rising and setting of the stars, and eclipses of the Sun and Moon. Ptolemy's Handy Tables provided the model for later astronomical tables or zījes. In the Phaseis (Risings of the Fixed Stars) Ptolemy gave a parapegma, a star calendar or almanac based on the hands and disappearances of stars over the course of the solar year.
Geography
Main article: Geographia (Ptolemy)
Ptolemy's other main work is his Geographia. This also is a compilation of what was known about the world's geography in the Roman Empire during his time. He relied somewhat on the work of an earlier geographer, Marinos of Tyre, and on gazetteers of the Roman and ancient Persian Empire, but most of his sources beyond the perimeter of the Empire were unreliable.[citation needed]
The first part of the Geographia is a discussion of the data and of the methods he used. As with the model of the solar system in the Almagest, Ptolemy put all this information into a grand scheme. Following Marinos, he assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred in book 8 to express it as the length of the longest day rather than degrees of arc (the length of the midsummer day increases from 12h to 24h as one goes from the equator to the polar circle). In books 2 through 7, he used degrees and put the meridian of 0 longitude at the most western land he knew, the "Blessed Islands", probably the Cape Verde islands (not the Canary Islands, as long accepted) as suggested by the location of the six dots labelled the "FORTUNATA" islands near the left extreme of the blue sea of Ptolemy's map here reproduced.
A 15th-century manuscript copy of the Ptolemy world map, reconstituted from Ptolemy's Geographia (circa 150), indicating the countries of "Serica" and "Sinae" (China) at the extreme east, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Malay Peninsula).
Ptolemy also devised and provided instructions on how to create maps both of the whole inhabited world (oikoumenè) and of the Roman provinces. In the second part of the Geographia he provided the necessary topographic lists, and captions for the maps. His oikoumenè spanned 180 degrees of longitude from the Blessed Islands in the Atlantic Ocean to the middle of China, and about 80 degrees of latitude from Shetland to anti-Meroe (east coast of Africa); Ptolemy was well aware that he knew about only a quarter of the globe, and an erroneous extension of China southward suggests his sources did not reach all the way to the Pacific Ocean.
The maps in surviving manuscripts of Ptolemy's Geographia, however, date only from about 1300, after the text was rediscovered by Maximus Planudes. It seems likely that the topographical tables in books 2–7 are cumulative texts – texts which were altered and added to as new knowledge became available in the centuries after Ptolemy (Bagrow 1945). This means that information contained in different parts of the Geography is likely to be of different date.
A printed map from the 15th century depicting Ptolemy's description of the Ecumene, (1482, Johannes Schnitzer, engraver).
Maps based on scientific principles had been made since the time of Eratosthenes (3rd century BC), but Ptolemy improved projections. It is known that a world map based on the Geographia was on display in Augustodunum, Gaul in late Roman times. In the 15th century Ptolemy's Geographia began to be printed with engraved maps; the earliest printed edition with engraved maps was produced in Bologna in 1477, followed quickly by a Roman edition in 1478 (Campbell, 1987). An edition printed at Ulm in 1482, including woodcut maps, was the first one printed north of the Alps. The maps look distorted as compared to modern maps, because Ptolemy's data was inaccurate. One reason is that Ptolemy estimated the size of the Earth as too small: while Eratosthenes found 700 stadia for a great circle degree on the globe, in the Geographia Ptolemy uses 500 stadia. It is highly probable that these were the same stadion since Ptolemy switched from the former scale to the latter between the Syntaxis and the Geographia, and severely readjusted longitude degrees accordingly. If they both used the Attic stadion of about 185 meters, then the older estimate is 1/6 too large, and Ptolemy's value is 1/6 too small, a difference explained as due to ancient scientists' use of simple methods of measuring the earth, which were corrupted either high or low by a factor of 5/6, due to air's bending of horizontal light rays by 1/6 of the Earth's curvature.[citation needed] See also Ancient Greek units of measurement and History of geodesy.
Because Ptolemy derived many of his key latitudes from crude longest day values, his latitudes are erroneous on average by roughly a degree (2 degrees for Byzantium, 4 degrees for Carthage), though capable ancient astronomers knew their latitudes to more like a minute. (Ptolemy's own latitude was in error by 14'.) He agreed (Geographia 1.4) that longitude was best determined by simultaneous observation of lunar eclipses, yet he was so out of touch with the scientists of his day that he knew of no such data more recent than 500 years before (Arbela eclipse). When switching from 700 stadia per degree to 500, he (or Marinos) expanded longitude differences between cities accordingly (a point first realized by P.Gosselin in 1790), resulting in serious over-stretching of the Earth's east-west scale in degrees, though not distance. Achieving highly precise longitude remained a problem in geography until the invention of the marine chronometer at the end of the 18th century. It must be added that his original topographic list cannot be reconstructed: the long tables with numbers were transmitted to posterity through copies containing many scribal errors, and people have always been adding or improving the topographic data: this is a testimony to the persistent popularity of this influential work in the history of cartography.
Astrology
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The mathematician Claudius Ptolemy 'the Alexandrian' as imagined by a 16th century artist
Ptolemy's treatise on astrology, known in Greek as both the Apotelesmatika ("Astrological Outcomes" or "Effects") and "Tetrabiblios" ("Four Books"), and in Latin as the Quadripartitum ("Four books"), was the most popular astrological work of antiquity and also had great influence in the Islamic world and the medieval Latin West. It was first translated from Arabic into Latin by Plato of Tivoli (Tiburtinus) in 1138, while he was in Spain.[22] The Tetrabiblos is an extensive and continually reprinted treatise on the ancient principles of horoscopic astrology in four books (Greek tetra means "four", biblos is "book"). That it did not quite attain the unrivaled status of the Almagest was perhaps because it did not cover some popular areas of the subject, particularly electional astrology (interpreting astrological charts for a particular moment to determine the outcome of a course of action to be initiated at that time), and medical astrology, which were later adoptions.
The great popularity that the Tetrabiblos did possess might be attributed to its nature as an exposition of the art of astrology and as a compendium of astrological lore, rather than as a manual. It speaks in general terms, avoiding illustrations and details of practice. Ptolemy was concerned to defend astrology by defining its limits, compiling astronomical data that he believed was reliable and dismissing practices (such as considering the numerological significance of names) that he believed to be without sound basis.
Much of the content of the Tetrabiblos was collected from earlier sources; Ptolemy's achievement was to order his material in a systematic way, showing how the subject could, in his view, be rationalized. It is, indeed, presented as the second part of the study of astronomy of which the Almagest was the first, concerned with the influences of the celestial bodies in the sublunar sphere. Thus explanations of a sort are provided for the astrological effects of the planets, based upon their combined effects of heating, cooling, moistening, and drying.
Ptolemy's astrological outlook was quite practical: he thought that astrology was like medicine, that is conjectural, because of the many variable factors to be taken into account: the race, country, and upbringing of a person affects an individual's personality as much if not more than the positions of the Sun, Moon, and planets at the precise moment of their birth, so Ptolemy saw astrology as something to be used in life but in no way relied on entirely.
Music
Ptolemy also wrote an influential work, Harmonics, on music theory and the mathematics of music. After criticizing the approaches of his predecessors, Ptolemy argued for basing musical intervals on mathematical ratios (in contrast to the followers of Aristoxenus and in agreement with the followers of Pythagoras) backed up by empirical observation (in contrast to the overly theoretical approach of the Pythagoreans). Ptolemy wrote about how musical notes could be translated into mathematical equations and vice versa in Harmonics. This is called Pythagorean tuning because it was first discovered by Pythagoras. However, Pythagoras believed that the mathematics of music should be based on the specific ratio of 3:2 whereas Ptolemy merely believed that it should just generally involve tetrachords and octaves. He presented his own divisions of the tetrachord and the octave, which he derived with the help of a monochord. Ptolemy's astronomical interests also appeared in a discussion of the "music of the spheres".
Optics
His Optics is a work that survives only in a poor Arabic translation and in about twenty manuscripts of a Latin version of the Arabic, which was translated by Eugene of Palermo (c. 1154). In it Ptolemy writes about properties of light, including reflection, refraction, and colour. The work is a significant part of the early history of optics.[23]
Named after Ptolemy
There are several characters or items named after Ptolemy, including:
• The crater Ptolemaeus on the Moon;
• The crater Ptolemaeus[24] on Mars;
• the asteroid 4001 Ptolemaeus;
• a character in the fantasy series The Bartimaeus Trilogy: this fictional Ptolemy is a young magician (from Alexandria) whom Bartimaeus loved; he made the journey into "the Other Place" being hunted by his cousin, because he was a magician;
• the name of Celestial Being's carrier ship in the anime Mobile Suit Gundam 00.
• track number 10 on Selected Ambient Works 85–92 by Aphex Twin.
• the Ptolemy Stone used in the mathematics courses at both St. John's College campuses.
• English astronomer and TV presenter Sir Patrick Moore has a cat named Ptolemy.
Ptolemy in pop culture
In Jonathan Stroud's Bartimaeus Trilogy, the djinni Bartimaeus often assumes Ptolemy's form and frequently refers to him as one of his favorite masters. The third and final book in the series is also called Ptolemy's Gate.
See also
Atlas portal
• Pei Xiu
• Ptolemy's Canon – a dated list of kings used by ancient astronomers.
• Ptolemy Cluster – star cluster described by Ptolemaeus
• Ptolemy's theorem – mathematical theorem described by Ptolemaeus
• Ptolemy's table of chords
• Ptolemy's world map – map of the ancient world as described by Ptolemaeus.
• Zhang Heng
Footnotes
1. ^ See 'Background' section on his status as a Roman citizen
2. ^ Select Epigrams from the Greek Anthology By John William Mackail Page 246 ISBN 1406922943 2007
3. ^ Mortal am I, the creature of a day..
4. ^ Jean Claude Pecker (2001), Understanding the Heavens: Thirty Centuries of Astronomical Ideas from Ancient Thinking to Modern Cosmology, p. 311, Springer, ISBN 3-540-63198-4.
5. ^ Πτολεμαῖος, Georg Autenrieth, A Homeric Dictionary, on Perseus
6. ^ Asger Aaboe, Episodes from the Early History of Astronomy, New York: Springer, 2001, pp. 62–65.
7. ^ Alexander Jones, "The Adaptation of Babylonian Methods in Greek Numerical Astronomy," in The Scientific Enterprise in Antiquity and the Middle Ages, p. 99.
8. ^ Enc. Britannica 2007, "Claudius Ptolemaeus" Britannica.com
9. ^ a b Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1.
10. ^ "Ptolemy." Britannica Concise Encyclopedia. Encyclopædia Britannica, Inc., 2006. Answers.com 20 Jul. 2008.
11. ^ George Sarton (1936). "The Unity and Diversity of the Mediterranean World", Osiris 2, p. 406–463 [429].
12. ^ John Horace Parry (1981). The Age of Reconnaissance, p. 10. University of California Press. ISBN 0-520-04235-2.
13. ^ J. F. Weidler (1741). Historia astronomiae, p. 177. Wittenberg: Gottlieb. (cf. Martin Bernal (1992). "Animadversions on the Origins of Western Science", Isis 83 (4), p. 596–607 [606].)
14. ^ Martin Bernal (1992). "Animadversions on the Origins of Western Science", Isis 83 (4), p. 596–607 [602, 606].
15. ^ edited by Shahid Rahman, Tony Street, Hassan Tahiri. (2008). "The Birth of Scientific Controversies, The Dynamics of the Arabic Tradition and Its Impact on the Development of Science: Ibn al-Haytham’s Challenge of Ptolemy’s Almagest". The Unity of Science in the Arabic Tradition. 11. Springer Netherlandsdoi=10.1007/978-1-4020-8405-8. pp. 183–225 [183]. doi:10.1007/978-1-4020-8405-8. ISBN 978-1-4020-8404-1.
16. ^ "Dennis Rawlins". The International Journal of Scientific History. http://www.dioi.org/cot.htm#mjpg. Retrieved 2009-10-07.
17. ^ Bernard R. Goldstein, "Saving the Phenomena: The Background to Ptolemy's Planetary Theory", Journal for the History of Astronomy, 28 (1997): 1–12
18. ^ S. C. McCluskey, Astronomies and Cultures in Early Medieval Europe, Cambridge: Cambridge Univ. Pr. 1998, pp. 20–21.
19. ^ Charles Homer Haskins, Studies in the History of Mediaeval Science, New York: Frederick Ungar Publishing, 1967, reprint of the Cambridge, Mass., 1927 edition
20. ^ Dennis Duke, Ptolemy's Cosmology
21. ^ Bernard R. Goldstein, ed., The Arabic Version of Ptolemy's Planetary Hypotheses, Transactions of the American Philosophical Society, 57, 4 (1967), pp. 9–12.
22. ^ FA Robbins, 1940; Thorndike 1923)
23. ^ Smith, A. Mark (1996). Ptolemy's Theory of Visual Perception– An English translation of the Optics. The American Philosophical Society. ISBN 0-87169-862-5. http://books.google.com/?id=mhLVHR5QAQkC&pg=PP1&dq=ptolemy+theory+of+visual+perception. Retrieved 27 June 2009.
24. ^ Mars Labs. Google Maps.
References
Texts and translations
• Bagrow, L. (January 1, 1945). "The Origin of Ptolemy's Geographia". Geografiska Annaler (Geografiska Annaler, Vol. 27) 27: 318–387. doi:10.2307/520071. ISSN 16513215. http://jstor.org/stable/520071.
• Berggren, J. Lennart, and Alexander Jones. 2000. Ptolemy's Geography: An Annotated Translation of the Theoretical Chapters. Princeton and Oxford: Princeton University Press. ISBN 0-691-01042-0.
• Campbell, T. (1987). The Earliest Printed Maps. British Museum Press.
• Hübner, Wolfgang, ed. 1998. Claudius Ptolemaeus, Opera quae exstant omnia Vol III/Fasc 1: ΑΠΟΤΕΛΕΣΜΑΤΙΚΑ (= Tetrabiblos). De Gruyter. ISBN 978-3-598-71746-8 (Bibliotheca scriptorum Graecorum et Romanorum Teubneriana). (The most recent edition of the Greek text of Ptolemy's astrological work, based on earlier editions by F. Boll and E. Boer.)
• Neugebauer, Otto (1975). A History of Ancient Mathematical Astronomy. I-III. Berlin and New York: Sprnger Verlag.
• Nobbe, C. F. A., ed. 1843. Claudii Ptolemaei Geographia. 3 vols. Leipzig: Carolus Tauchnitus. (The most recent edition of the complete Greek text)
• Ptolemy. 1930. Die Harmonielehre des Klaudios Ptolemaios, edited by Ingemar Düring. Göteborgs högskolas årsskrift 36, 1930:1. Göteborg: Elanders boktr. aktiebolag. Reprint, New York: Garland Publishing, 1980.
• Ptolemy. 2000. Harmonics, translated and commentary by Jon Solomon. Mnemosyne, Bibliotheca Classica Batava, Supplementum, 0169-8958, 203. Leiden and Boston: Brill Publishers. ISBN 90-04-11591-9
• Ptolemy. 1940. Tetrabiblos, compiled and edited by F. E. Robbins , Cambridge, MA: Harvard University Press (Loeb Classical Library).
• Stevenson, Edward Luther (trans. and ed.). 1932. Claudius Ptolemy: The Geography. New York: New York Public Library. Reprint, New York: Dover, 1991. (This is the only complete English translation of Ptolemy's most famous work. Unfortunately, it is marred by numerous mistakes and the placenames are given in Latinised forms, rather than in the original Greek).
• Stückelberger, Alfred, and Gerd Graßhoff (eds). 2006. Ptolemaios, Handbuch der Geographie, Griechisch-Deutsch. 2 vols. Basel: Schwabe Verlag. ISBN 978-3-7965-2148-5. (Massive 1018 pp. scholarly edition by a team of a dozen scholars that takes account of all known manuscripts, with facing Greek and German text, footnotes on manuscript variations, color maps, and a CD with the geographical data)
• Taub, Liba Chia (1993). Ptolemy's Universe: The Natural Philosophical and Ethical Foundations of Ptolemy's Astronomy. Chicago: Open Court Press. ISBN 0-8126-9229-2.
• Ptolemy's Almagest, Translated and annotated by G. J. Toomer. Princeton University Press, 1998
External links
Wikiquote has a collection of quotations related to: Ptolemy
Wikimedia Commons has media related to: Ptolemy
Primary sources
• Ptolemy's Tetrabiblos at LacusCurtius (English translation of a portion of the material, with introductory material)
• Entire Tetrabiblos of J.M. Ashmand's 1822 translation.
• Ptolemy's Geography at LacusCurtius (English translation, incomplete)
• Extracts of Ptolemy on the country of the Seres (China) (English translation)
• Geographia (the Balkan Provinces, with old maps) at Sorin Olteanu's LTDM Project (soltdm.com)
• Almagest books 1–13 The complete text of Heiberg's edition (PDF) Greek.
• Almagest books 1–6 (Greek) with preface (Latin) @ archive.org
Secondary material
• Arnett, Bill (2008). "Ptolemy, the Man". obs.nineplanets.org. http://obs.nineplanets.org/psc/theman.html. Retrieved 2008-11-24.
• Danzer, Gerald (1988). "Cartographic Images of the World on the Eve of the Discoveries". The Newberry Library. http://www.newberry.org/smith/slidesets/ss08.html. Retrieved 26 November 2008.
• Haselein, Frank (2007). "Κλαυδιου Πτολεμιου: Γεωγραφικῆς Ύφηγήσεως (Geographie)" (in German and some English). Frank Haselein. http://wwwuser.gwdg.de/~fhasele/ptolemaeus/index.html. Retrieved 2008-11-24.
• Houlding, Deborah (2003). "The Life & Work of Ptolemy". Skyscript.co. http://www.skyscript.co.uk/ptolemy.html. Retrieved 2008-11-24.
• Toomer, Gerald J. (1970). "Ptolemy (Claudius Ptolemæus)". In Gillispie, Charles. Dictionary of Scientific Biography. 11. New York: Scribner & American Council of Learned Societies. pp. 186-206. ISBN 9780684101149. http://www.u.arizona.edu/~aversa/scholastic/Dictionary%20of%20Scientific%20Biography/Ptolemy%20(Toomer).pdf.
• Sprague, Ben (2001–2007). "Claudius Ptolemaeus (Ptolemy): Representation, Understanding, and Mathematical Labeling of the Spherical Earth". Center for Spatially Integrated Social Science. http://www.csiss.org/classics/content/76. Retrieved 26 November 2008.
Animated illustrations
• Java simulation of the Ptolemaic System – at Paul Stoddard's Animated Virtual Planetarium, Northern Illinois University
• Animation of Ptolemy's Two Solar Hypotheses
• Epicycle and Deferent Demo – at Rosemary Kennett's website at the University of Syracuse
• Flash animation of Ptolemy's universe. (best in Internet Explorer)
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Almagest
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Geometric construction used by Hipparchus in his determination of the distances to the sun and moon.
Ptolemy's Almagest is a 2nd century AD Ancient Greek mathematical and astronomical treatise on the complex motions of the stars and planetary paths. It is one of the most influential scientific texts of all time, with its geocentric model accepted as dogma for more than twelve hundred years from its origin in Hellenistic Alexandria, in the Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus.
The Almagest is a critical source of information on ancient Greek astronomy. It has also been valuable to students of mathematics because it documents the ancient Greek mathematician Hipparchus's work, which has been lost. Hipparchus wrote about trigonometry, but because his works are no longer extant, mathematicians use Ptolemy's book as their source for Hipparchus' works and ancient Greek trigonometry in general.[dubious – discuss]
Contents
[hide]
• 1 Name
• 2 Date
• 3 Contents
o 3.1 Books
o 3.2 Ptolemy's cosmos
o 3.3 Ptolemy's planetary model
• 4 Impact
• 5 Modern editions
• 6 Footnotes
• 7 See also
• 8 References
• 9 External links
[edit] Name
The treatise's original Greek title is Μαθηματικὴ Σύνταξις (Mathēmatikē Syntaxis), and the treatise is also known by the Latin form of this, Syntaxis mathematica. It was later titled Hē Megalē Syntaxis (The Great Treatise), and the superlative form of this (Ancient Greek: μεγίστη, "greatest") lies behind the Arabic name al-majisṭī (المجسطي), from which the English name Almagest derives.
[edit] Date
The date of Almagest has recently been more precisely established. Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148. The late N. T. Hamilton found that the version of Ptolemy's models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence the Almagest cannot have been completed before about 150, a quarter century after Ptolemy began observing.[1]
[edit] Contents
[edit] Books
The apparent movement of sun and planets with earth as center
The Almagest (Almagestum) consisted of thirteen sections, called books. As with many medieval manuscripts that were handcopied or particularly in the early years of printing, there were considerable differences between various editions of the same text, as the process of transcription was highly personal. An example illustrating how the Almagest was organized is given below. It is a 152 page Latin edition printed in 1515 at Venice by Petrus Lichtenstein.[2]
• Book I contains an outline of Aristotle's cosmology: on the spherical form of the heavens, with the spherical Earth lying motionless as the center, with the fixed stars and the various planets revolving around the Earth. Then follows an explanation of chords with table of chords; observations of the obliquity of the ecliptic (the apparent path of the Sun through the stars); and an introduction to spherical trigonometry.
• Book II covers problems associated with the daily motion attributed to the heavens, namely risings and settings of celestial objects, the length of daylight, the determination of latitude, the points at which the Sun is vertical, the shadows of the gnomon at the equinoxes and solstices, and other observations that change with the spectator's position. There is also a study of the angles made by the ecliptic with the vertical, with tables.
• Book III covers the length of the year, and the motion of the Sun. Ptolemy explains Hipparchus' discovery of the precession of the equinoxes and begins explaining the theory of epicycles.
• Books IV and V cover the motion of the Moon, lunar parallax, the motion of the lunar apogee, and the sizes and distances of the Sun and Moon relative to the Earth.
• Book VI covers solar and lunar eclipses.
• Books VII and VIII cover the motions of the fixed stars, including precession of the equinoxes. They also contain a star catalogue of 1022 stars, described by their positions in the constellations. The brightest stars were marked first magnitude (m = 1), while the faintest visible to the naked eye were sixth magnitude (m = 6). Each numerical magnitude was twice the brightness of the following one, which is a logarithmic scale. This system is believed to have originated with Hipparchus. The stellar positions too are of Hipparchan origin, despite Ptolemy's claim to the contrary.
• Book IX addresses general issues associated with creating models for the five naked eye planets, and the motion of Mercury.
• Book X covers the motions of Venus and Mars.
• Book XI covers the motions of Jupiter and Saturn.
• Book XII covers stations and retrograde motion, which occurs when planets appear to pause, then briefly reverse their motion against the background of the zodiac. Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets.
• Book XIII covers motion in latitude, that is, the deviation of planets from the ecliptic.
[edit] Ptolemy's cosmos
The cosmology of the Almagest includes five main points, each of which is the subject of a chapter in Book I. What follows is a close paraphrase of Ptolemy's own words from Toomer's translation.[3]
• The celestial realm is spherical, and moves as a sphere.
• The Earth is a sphere.
• The Earth is at the center of the cosmos.
• The Earth, in relation to the distance of the fixed stars, has no appreciable size and must be treated as a mathematical point.[4]
• The Earth does not move.
[edit] Ptolemy's planetary model
16th-century representation of the Ptolemy's geocentric model
Ptolemy assigned the following order to the planetary spheres, beginning with the innermost:
1. Moon
2. Mercury
3. Venus
4. Sun
5. Mars
6. Jupiter
7. Saturn
8. Sphere of fixed stars
Other classical writers suggested different sequences. Plato (c. 427 – c. 347 BC) placed the Sun second in order after the Moon. Martianus Capella (5th century AD) put Mercury and Venus in motion around the Sun. Ptolemy's authority was preferred by most medieval Islamic and late medieval European astronomers.
Ptolemy inherited from his Greek predecessors a geometrical toolbox and a partial set of models for predicting where the planets would appear in the sky. Apollonius of Perga (c. 262 – c. 190 BC) had introduced the deferent and epicycle and the eccentric deferent to astronomy. Hipparchus (2nd century BC) had crafted mathematical models of the motion of the Sun and Moon. Hipparchus had some knowledge of Mesopotamian astronomy, and he felt that Greek models should match those of the Babylonians in accuracy. He was unable to create accurate models for the remaining five planets.
The Almagest adopted Hipparchus' solar model, which consisted of a simple eccentric deferent. For the Moon, Ptolemy began with Hipparchus' epicycle-on-deferent, then added a device that historians of astronomy refer to as a "crank mechanism":[5] He succeeded in creating models for the other planets, where Hipparchus had failed, by introducing a third device called the equant.
Ptolemy wrote the Almagest as a textbook of mathematical astronomy. It explained geometrical models of the planets based on combinations of circles, which could be used to predict the motions of celestial objects. In a later book, the Planetary Hypotheses, Ptolemy explained how to transform his geometrical models into three-dimensional spheres or partial spheres. In contrast to the mathematical Almagest, the Planetary Hypotheses is sometimes described as a book of cosmology.
[edit] Impact
Ptolemy's comprehensive treatise of mathematical astronomy superseded most older texts of Greek astronomy. Some were more specialized and thus of less interest; others simply became outdated by the newer models. As a result, the older texts ceased to be copied and were gradually lost. Much of what we know about the work of astronomers like Hipparchus comes from references in the Almagest.
Ptolemy's Almagest became an authoritative work for many centuries.
The first translations into Arabic were made in the 9th century, with two separate efforts, one sponsored by the caliph Al-Ma'mun. Sahl ibn Bishr is thought to be the first Arabic translator. By this time, the Almagest was lost in Western Europe, or only dimly remembered in astrological lore. Consequently, Western Europe rediscovered Ptolemy from translations of Arabic versions. In the 12th century a Spanish version was produced, which was later translated under the patronage of Alfonso X. Gerard of Cremona translated the Almagest into Latin directly from the Arabic version. Gerard translated the Arabic text while working at the Toledo School of Translators, although he was unable to translate many technical terms such as the Arabic Abrachir for Hipparchus. This Spanish version was later translated into Latin under the patronage of Frederick II.[citation needed]
Picture of George Trebizond's Latin translation of Almagest
In the 15th century, a Greek version appeared in Western Europe. The German astronomer Johannes Müller (known as Regiomontanus) made an abridged Latin version at the instigation of the Greek churchman Johannes, Cardinal Bessarion. Around the same time, George of Trebizond made a full translation accompanied by a commentary that was as long as the original text. George's translation, done under the patronage of Pope Nicholas V, was intended to supplant the old translation. The new translation was a great improvement; the new commentary was not, and aroused criticism.[citation needed] The Pope declined the dedication of George's work,[citation needed] and Regiomontanus's translation had the upper hand for over 100 years.
During the 16th century, Guillaume Postel, who had been on an embassy to the Ottoman Empire, brought back Arabic disputations of the Almagest, such as the works of al-Kharaqī, Muntahā al-idrāk fī taqāsīm al-aflāk ("The Ultimate Grasp of the Divisions of Spheres", 1138/9).[6]
Commentaries on the Almagest were written by Theon of Alexandria (extant), Pappus of Alexandria (only fragments survive), and Ammonius Hermiae (lost).
[edit] Modern editions
The Almagest was edited by J. L. Heiberg in Claudii Ptolemaei opera quae exstant omnia, vols. 1.1 and 1.2 (1898, 1903).
Two translations of the Almagest into English have been published . The first, by R. Catesby Taliaferro of St. John's College in Annapolis, Maryland, was included in volume 16 of the Great Books of the Western World; the second, by G. J. Toomer, Ptolemy's Almagest in 1998;[3]
An older French translation (facing the Greek text), published in two volumes in 1813 and 1816 by Nicholas Halma, is available online at the Gallica web site [1] and [2]
[edit] Footnotes
1. ^ NT Hamilton, N. M. Swerdlow, G. J. Toomer. "The Canobic Inscription: Ptolemy's Earliest Work". In Berggren and Goldstein, eds., From Ancient Omens to Statistical Mechanics. Copenhagen: University Library, 1987.
2. ^ http://www.univie.ac.at/hwastro/rare/1515_ptolemae.htm
3. ^ a b Toomer, G.J. (1998), Ptolemy's Almagest, Princeton University Press, ISBN 0-691-00260-6
4. ^ Ptolemy. Almagest., Book I, Chapter 5.
5. ^ Michael Hoskin. The Cambridge Concise History of Astronomy. Chapter 2, page 44.
6. ^ Islamic science and the making of European Renaissance, by George Saliba, p.218 ISBN 9780262195577
[edit] See also
• Star cartography
[edit] References
• James Evans, The History and Practice of Ancient Astronomy, Oxford University Press, 1998 (ISBN 0-19-509539-1)
• Michael Hoskin, The Cambridge Concise History of Astronomy, Cambridge University Press, 1999 (ISBN 0-521-57291-6)
• Olaf Pedersen, A Survey of the Almagest, Odense University Press, 1974 (ISBN 87-7492-087-1. A revised edition, prepared by Alexander Jones, is due to be published by Springer on November 29, 2010.
• Olaf Pedersen, Early Physics and Astronomy: A Historical Introduction, 2nd edition, Cambridge University Press, 1993 (ISBN 0-521-40340-5)
[edit] External links
• University of Vienna: Almagestum (1515) PDF:s of different resolutions
• Almagest Planetary Model Animations
• Online luni-solar & planetary ephemeris calculator based on the Almagest
• Ptolemy's Almagest. PDF scans of Heiberg's Greek edition, now in the public domain (Classical Greek)
• A podcast discussion by Prof. M Heath and Dr A. Chapman of a recent re-discovery of a 14th Century manuscript in the university of Leeds Library
Geocentric model
From Wikipedia, the free encyclopedia
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This article is about the historical term. For modern geocentrism, see Modern geocentrism.
"Geocentric" redirects here. For orbits around the Earth, see Geocentric orbit.
Figure of the heavenly bodies — An illustration of the Ptolemaic geocentric system by Portuguese cosmographer and cartographer Bartolomeu Velho, 1568 (Bibliothèque Nationale, Paris)
In astronomy, the geocentric model (also known as geocentrism, or the Ptolemaic system), is the superseded theory, that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece. As such, most Ancient Greek philosophers assumed that the Sun, Moon, stars, and naked eye planets circled the Earth, including the noteworthy systems of Aristotle (see Aristotelian physics) and Ptolemy.[1]
Two commonly made observations supported the idea that the Earth was the center of the Universe. The first observation was that the stars, sun, and planets appear to revolve around the Earth each day, making the Earth the center of that system. Further, every star was on a "stellar" or "celestial" sphere, of which the earth was the center, that rotated each day, using a line through the north and south pole as an axis. The stars closer the equator appeared to rise and fall the greatest distance, but each star circled back to its rising point each day.[2] The second common notion supporting the geocentric model was that the Earth does not seem to move from the perspective of an Earth bound observer, and that it is solid, stable, and unmoving. In other words, it is completely at rest.
The geocentric model was usually combined with a spherical Earth by ancient Greek and medieval philosophers. It is not the same as the older flat Earth model implied in some mythology. However, the ancient Greeks believed that the motions of the planets were circular and not elliptical, a view that was not challenged in Western culture before the 17th century through the synthesis of theories by Copernicus and Kepler.
The astronomical predictions of Ptolemy's geocentric model were used to prepare astrological charts for over 1500 years. The geocentric model held sway into the early modern age, but was gradually replaced from the late 16th century onward by the heliocentric model of Copernicus, Galileo and Kepler. However, the transition between these two theories met much resistance, not only from the Catholic Church and its reluctance to accept a theory not placing God's creation at the center of the universe, but also from those who saw geocentrism as a fact that could not be subverted by a new, weakly justified theory.
Contents
[hide]
• 1 Ancient Greece
• 2 Ptolemaic Model
o 2.1 Ptolemaic system
o 2.2 Geocentrism and Islamic astronomy
• 3 Geocentrism and rival systems
o 3.1 Copernican system
• 4 Gravitation
• 5 Reluctance to change
• 6 Modern geocentrism
o 6.1 Planetariums
• 7 Geocentric models in science fiction
• 8 See also
• 9 Notes
• 10 References
• 11 External links
[edit] Ancient Greece
Illustration of Anaximander's models of the universe. On the left, daytime in summer; on the right, nighttime in winter.
The geocentric model entered Greek astronomy and philosophy at an early point; it can be found in Pre-Socratic philosophy. In the 6th century BC, Anaximander proposed a cosmology with the Earth shaped like a section of a pillar (a cylinder), held aloft at the center of everything. The Sun, Moon, and planets were holes in invisible wheels surrounding the Earth; through the holes, humans could see concealed fire. About the same time, the Pythagoreans thought that the Earth was a sphere (in accordance with observations of eclipses), but not at the center; they believed that it was in motion around an unseen fire. Later these views were combined, so most educated Greeks from the 4th century BC on thought that the Earth was a sphere at the center of the universe.[3]
In the 4th century BC, two influential Greek philosophers wrote works based on the geocentric model. These were Plato and his student Aristotle. According to Plato, the Earth was a sphere, stationary at the center of the universe. The stars and planets were carried around the Earth on spheres or circles, arranged in the order (outwards from the center): Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars, with the fixed stars all being located on the celestial sphere. In his "Myth of Er", a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. Eudoxus of Cnidus, who worked with Plato, developed a less mythical, more mathematical explanation of the planets' motion based on Plato's dictum stating that all phenomena in the heavens can be explained with uniform circular motion. Aristotle elaborated on Eudoxus' system.
In the fully developed Aristotelian system, the spherical Earth is at the center of the universe and all other heavenly bodies are attached to 56 concentric spheres which rotate around the Earth . (The number is so high because several transparent spheres are needed for each planet.) These spheres, known as crystalline spheres, all moved together at varying speeds to create the rotation of bodies around the Earth, and were composed of an intangible substance called aether. Aristotle believed that the moon was in the innermost sphere and therefore touches the realm of Earth, thus contaminating it, causing the dark spots (macula) and the ability to go through lunar phases. It is not perfect like the other heavenly bodies, which shine by their own light.[citation needed] He further described his system by explaining the natural tendencies of earth, water, fire, air, and aether. His system said that all earthly objects (solid objects) had a proclivity to move towards the Earth, water tended to remain on top of earth, air was the firmament between the water and fire, and that fire always tended to move in the opposite direction of earth (striving towards its natural position in the universe.) In addition, all celestial bodies in his system were composed of aether, the same material as the celestial spheres. This system therefore described why the earth was the center of the universe, and explained why the cosmos was arranged the way it seemed to be.
Adherence to the geocentric model stemmed largely from several important observations. First of all, if the Earth did move, then one ought to be able to observe the shifting of the fixed stars due to stellar parallax. In short, if the earth was moving the shapes of the constellations should change considerably over the course of a year. If they did not appear to move, the stars are either much further away than the Sun and the planets than previously conceived, making their motion undetectable, or in reality they are not moving at all. Because the stars were actually much further away than Greek astronomers postulated (making movement extremely subtle), stellar parallax was not detected until the 19th century. Therefore, the Greeks chose the simpler of the two explanations. The lack of any observable parallax was considered a fatal flaw of any non-geocentric theory. Another observation used in favor of the geocentric model at the time was that apparent consistency of Venus' luminosity, thus implying that it is usually about the same distance from Earth, which is more consistent with geocentrism than heliocentrism. In reality, that is because the loss of light caused by its phases compensates for the increase in apparent size caused by its varying distance from Earth. Once again, Aristotle's objections of heliocentrism utilized his ideas concerning the natural tendency of earth like objects. The natural state of heavy earth-like objects is to tend towards the center of the earth and to not move unless forced by and outside object. It was also believed by some that if the Earth rotated on its axis, the air and objects in it (such as birds or clouds) would be left behind.
One major flaw in the Eudoxan and Aristotelian models based on concentric spheres was that they could not explain the changes in brightness of the planets caused by a change in distance.[4] Furthermore, the apparent inaccuracy of these systems became more prevalent over time, causing thinkers such as Ptolemy to posit new ideas and arrangements to better fit newly made observations.
[edit] Ptolemaic Model
Although the basic tenets of Greek geocentrism were established by the time of Aristotle, the details of his system did not become standard. The Ptolemaic system, espoused by the Hellenistic astronomer Claudius Ptolemaeus in the 2nd century AD finally accomplished this process. His main astronomical work, the Almagest, was the culmination of centuries of work by Hellenic, Hellenistic and Babylonian astronomers; it was accepted for over a millennium as the correct cosmological model by European and Islamic astronomers. Because of its influence, the Ptolemaic system is sometimes considered identical with the geocentric model.
Ptolemy argued that the Earth was in the center of the universe from the simple observation that half the stars were above the horizon and half were below the horizon at any time (stars on rotating stellar sphere) and the assumption that the stars were all at some modest distance from the center of the universe. If the Earth was substantially displaced from the center, this division into visible and invisible stars would not be equal.[5]
[edit] Ptolemaic system
The basic elements of Ptolemaic astronomy, showing a planet on an epicycle with an eccentric deferent and an equant point.
Pages from 1550 SACROBOSCO "Tractatus de Sphaera" with the Ptolemaic system.
In the Ptolemaic system, each planet is moved by a system of two or more spheres: one called its deferent, the others, its epicycles. The deferent is a circle whose center point exists halfway between the equant and the earth, marked by the X in the picture to the right where the equant is the solid point opposite the earth. Another sphere, the epicycle, is embedded inside of the deferent and is represented by the smaller dotted line to the right. A given planet then moves along the epicycle at the same time the epicycle moves along the path marked by the deferent. These combined movements cause the given planet to move closer to and further away from the Earth at different points in its orbit, and caused observers to believe that the planet even slowed down, stopped, and moved backward (in retrograde motion). This apparent retrograde motion was one of the largest inconsistencies in Greek cosmological systems, and was one of Ptolemy's main reasons for creating the deferent, epicycle model. The apparent retrograde motion was eventually replaced by the heliocentric model, and dispelled as an observation that is made only from earth bound observers. However, this model of deferents and epicycles made observations and predictions much more accurate than all preceding systems. The epicycles of Venus and Mercury are always centered on a line between Earth and the Sun (Mercury being closer to Earth), which explained why they were always near it in the sky.
The Ptolemaic order of spheres from Earth outward is:
1. Moon
2. Mercury
3. Venus
4. Sun
5. Mars
6. Jupiter
7. Saturn
8. Fixed Stars
9. Sphere of Prime Mover
The deferent-and-epicycle model had been used by Greek astronomers for centuries, as had the idea of the eccentric (a deferent which is slightly off-center from the Earth). In the illustration, the center of the deferent is not the Earth but X, making it eccentric (from the Latin ex- or e- meaning "from," and centrum meaning "center"). Unfortunately, the system that was available in Ptolemy's time did not quite match observations, even though it was considerably improved over Aristotle's system. Sometimes the size of a planet's retrograde loop (most notably that of Mars) would be smaller, and sometimes larger. This prompted him to come up with the idea of an equant. The equant was a point near the center of a planet's orbit which, if you were to stand there and watch, the center of the planet's epicycle would always appear to move at the same speed. Therefore, the planet actually moved at different speeds when the epicycle was at different points on its deferent. By using an equant, Ptolemy claimed to keep motion which was uniform and circular, but many people did not like it because they did not think it was true to Plato's dictum of "uniform circular motion."[citation needed] The resultant system which eventually came to be widely accepted in the west was an unwieldy one to modern eyes; each planet required an epicycle revolving on a deferent, offset by an equant which was different for each planet. But it predicted various celestial motions, including the beginnings and ends of retrograde motion, fairly well at the time it was developed.
[edit] Geocentrism and Islamic astronomy
Main articles: Maragheh observatory, Astronomy in medieval Islam, and Islamic cosmology
Due to the scientific dominance of the Ptolemaic system in Islamic astronomy, the Muslim astronomers accepted unanimously the geocentric model.[6]
In the 12th century, Arzachel departed from the ancient Greek idea of uniform circular motions by hypothesizing that the planet Mercury moves in an elliptic orbit,[7][8] while Alpetragius proposed a planetary model that abandoned the equant, epicycle and eccentric mechanisms,[9] though this resulted in a system that was mathematically less accurate.[10] Fakhr al-Din al-Razi (1149–1209), in dealing with his conception of physics and the physical world in his Matalib, rejects the Aristotelian and Avicennian notion of the Earth's centrality within the universe, but instead argues that there are "a thousand thousand worlds (alfa alfi 'awalim) beyond this world such that each one of those worlds be bigger and more massive than this world as well as having the like of what this world has." To support his theological argument, he cites the Qur'anic verse, "All praise belongs to God, Lord of the Worlds," emphasizing the term "Worlds."[11]
The "Maragha Revolution" refers to the Maragha school's revolution against Ptolemaic astronomy. The "Maragha school" was an astronomical tradition beginning in the Maragha observatory and continuing with astronomers from the Damascus mosque and Samarkand observatory. Like their Andalusian predecessors, the Maragha astronomers attempted to solve the equant problem (the circle around whose circumference a planet or the center of an epicycle was conceived to move uniformly) and produce alternative configurations to the Ptolemaic model without abandoning the geocentric model. They were more successful than their Andalusian predecessors in producing non-Ptolemaic configurations which eliminated the equant and eccentrics, were more accurate than the Ptolemaic model in numerically predicting planetary positions, and were in better agreement with empirical observations.[12] The most important of the Maragha astronomers included Mo'ayyeduddin Urdi (d. 1266), Nasīr al-Dīn al-Tūsī (1201–1274), Qutb al-Din al-Shirazi (1236–1311), Ibn al-Shatir (1304–1375), Ali Qushji (c. 1474), Al-Birjandi (d. 1525), and Shams al-Din al-Khafri (d. 1550).[13] Ibn al-Shatir, the Damascene astronomer (1304–1375 AD) working at the Umayyad Mosque, wrote a major book entitled Kitab Nihayat al-Sul fi Tashih al-Usul (A Final Inquiry Concerning the Rectification of Planetary Theory) on a theory which departs largely from the Ptolemaic system known at that time. In his book, "Ibn al-Shatir, an Arab astronomer of the fourteenth century," E.S.Kennedy wrote "what is of most interest, however, is that Ibn al-Shatir's lunar theory, except for trivial differences in parameters, is identical with that of Copernicus (1473–1543 AD)." The discovery that the models of Ibn al-Shatir are mathematically identical to those of Copernicus suggests the possible transmission of these models to Europe.[14] At the Maragha and Samarkand observatories, the Earth's rotation was discussed by al-Tusi and Ali Qushji (b. 1403); the arguments and evidence they used resemble those used by Copernicus to support the Earth's motion.[15][16]
However, the Maragha school never made the paradigm shift to heliocentrism.[17] The influence of the Maragha school on Copernicus remains speculative, since there is no documentary evidence to prove it. The possibility that Copernicus independently developed the Tusi couple remains open, since no researcher has yet demonstrated that he knew about Tusi's work or that of the Maragha school.[17][18]
[edit] Geocentrism and rival systems
This drawing from an Icelandic manuscript dated around 1750 illustrates the geocentric model.
Not all Greeks agreed with the geocentric model. The Pythagorean system has already been mentioned; some Pythagoreans believed the Earth to be one of several planets going around a central fire.[19] Hicetas and Ecphantus, two Pythagoreans of the 5th century BC, and Heraclides Ponticus in the 4th century BC, believed that the Earth rotated on its axis but remained at the center of the universe.[20] Such a system still qualifies as geocentric. It was revived in the Middle Ages by Jean Buridan. Heraclides Ponticus was once thought to have proposed that both Venus and Mercury went around the Sun rather than the Earth, but this is no longer accepted.[21] Martianus Capella definitely put Mercury and Venus in orbit around the Sun.[22] Aristarchus of Samos was the most radical. He wrote a work, which has not survived, on heliocentrism, saying that the Sun was at the center of the universe, while the Earth and other planets revolved around it.[23] His theory was not popular, and he had one named follower, Seleucus of Seleucia.[24]
[edit] Copernican system
Main article: Copernican heliocentrism
In 1543, the geocentric system met its first serious challenge with the publication of Copernicus' De revolutionibus orbium coelestium, which posited that the Earth and the other planets instead revolved around the Sun. The geocentric system was still held for many years afterwards, as at the time the Copernican system did not offer better predictions than the geocentric system, and it posed problems for both natural philosophy and scripture. The Copernican system was no more accurate than Ptolemy's system because it still used circular orbits. This was not altered until Johannes Kepler postulated that they may be elliptical, a belief that is still held to this day.
With the invention of the telescope in 1609, observations made by Galileo Galilei (such as that Jupiter has moons) called into question some of the tenets of geocentrism but did not seriously threaten it. Because he observed dark "spots" on the moon, craters, he was able to remark that the moon was not a perfect celestial body as had been previously conceived. This was the first time someone was able to see imperfections on a celestial body that was supposed to be composed from perfect aether. As such, because the moon's imperfections could now be related to those seen on Earth, one could argue that neither were unique, rather they were both just celestial bodies made from earth like material. Galileo was also able to see the moons of Jupiter, which he dedicated to Cosimo II de' Medici, and stated that they orbited around Jupiter, not Earth.[25] This was a significant claim because if it were true, it would mean that not everything revolved around Earth, shattering previously held theological and scientific thought. As such, Galileo's theories that challenged the geocentrism of our universe were silenced by the Church and general skepticism surrounding any system that did not place Earth at its center, preserving the thoughts and systems of Ptolemy and Aristotle.
Phases of Venus
In December 1610, Galileo Galilei used his telescope to observe that Venus showed all phases, just like the Moon. He thought that while this observation was incompatible with the Ptolemaic system, it was a natural consequence of the heliocentric system.
However, Ptolemy placed Venus' deferent and epicycle entirely inside the sphere of the Sun (between the Sun and Mercury), but this was arbitrary; he could just as easily have swapped Venus and Mercury and put them on the other side of the Sun, or made any other arrangement of Venus and Mercury, as long as they were always near a line running from the Earth through the Sun, such as placing the center of the Venus epicycle near the Sun. In this case, if the Sun is the source of all the light, under the Ptolemaic system:
If Venus is between Earth and the Sun, the phase of Venus must always be crescent or all dark.
If Venus is beyond the Sun, the phase of Venus must always be gibbous or full.
But Galileo saw Venus at first small and full, and later large and crescent.
This showed that with a Ptolemaic cosmology, the Venus epicycle can be neither completely inside nor completely outside of the orbit of the Sun. As a result, Ptolemaics abandoned the idea that the epicycle of Venus was completely inside the Sun, and later 17th century competition between astronomical cosmologies focused on variations of Tycho Brahe's Tychonic system (in which the Earth was still at the center of the universe, and around it revolved the Sun, but all other planets revolved around the Sun in one massive set of epicycles), or variations on the Copernican system.
[edit] Gravitation
Johannes Kepler, after analysing Tycho Brahe's famously accurate observations, constructed his three laws in 1609 and 1619, based on a heliocentric view where the planets move in elliptical paths. Using these laws, he was the first astronomer to successfully predict a transit of Venus (for the year 1631). The transition from circular orbits to elliptical planetary paths dramatically changed the accuracy of celestial observations and predictions. Because the heliocentric model by Copernicus was no more accurate than Ptolemy's system, new mathematical observations were needed to persuade those who still held on to the geocentric model. However, the observations made by Kepler, using the Brahe's data, became a problem not easily overturned for geocentrists.
In 1687, Isaac Newton devised his law of universal gravitation, which introduced gravitation as the force that both kept the Earth and planets moving through the heavens and also kept the air from flying away, allowing scientists to quickly construct a plausible heliocentric model for the solar system. In his Principia, Newton explained his system of how gravity, previously considered to be an occult force, conducted the movements of celestial bodies, and kept our solar system in its working order. His descriptions of centripetal force[26] were a breakthrough in scientific thought, and finally replaced the previous schools of scientific thought, i.e. those of Aristotle and Ptolemy. However, the process was gradual.
In 1838, astronomer Friedrich Wilhelm Bessel successfully measured the parallax of the star 61 Cygni, disproving Ptolemy's assertion that parallax motion did not exist. This finally substantiated the suppositions made by Copernicus with accurate, dependable scientific observations, and displayed truly how far away stars were from Earth.
A geocentric frame is useful for many everyday activities and most laboratory experiments, but is a less appropriate choice for solar-system mechanics and space travel. While a heliocentric frame is most useful in those cases, galactic and extra-galactic astronomy is easier if the sun is treated as neither stationary nor the center of the universe, but rotating around the center of our galaxy, and in turn our galaxy is also not at rest in the cosmic background.
[edit] Reluctance to change
While the evidence coming from the heliocentric camp was often rejected because its mathematical support was not strong enough, there were a large range of other reasons why people and institutions were not ready for such a monumental change in the way people viewed our universe. The Catholic Church for one found accepting the ideas of heliocentrism to be in conflict with many of their teachings and beliefs. If the Earth were God's creation, man, was not at the center of our universe, what did that mean? When referring to commonly used and believed cosmological arrangements, such as the strongly Christian influenced ones seen in Dante's Paradiso,[27] it was clear that the idea of Earth being the center of our universe was extremely ingrained in society at every level, not just scientific thinkers. The math was almost not even the most important aspect, just the sheer common sense of the system was compelling enough. And when beliefs that seemed so like common sense at the time were challenged by new theories, mainly heliocentrism, that were not even more accurate than the systems of Aristotle and Ptolemy, they were belittled and criticized at every level.
Some thinkers, such as St. Thomas Aquinas, had even melded the ideas of geocentrism into their theology. In his Summa Theologica, he says that the dual nature of Christ's rise to heaven was exhibited by his earthly body remaining on Earth, while his soul made of a different substance rose to heaven.[28] This was an attempt to directly agree with Aristotle's concept that objects composed from earthly substances tend towards the earth, while those composed of other materials, such as fire and aether, directed themselves away from the earth and rose up. Aquinas also discussed the arrangement of the firmament, or sky, compared to the arrangement of water and earth. As such, the Church's greatest thinkers had created a synthesis between theological teachings, and the scientific description of our universe's composition. This made a shift to a completely different system, centering the sun in our solar system, one that many people were not ready to make.
The importance of this resistance cannot be over-stressed. When figures such as Galileo were put on trial several times just to get him to recant, one is able to realize that the Church felt that it had much to lose. Also, Giordano Bruno's cosmological theories went beyond the Copernican model and identified the Sun as just one of an infinite number of independently moving heavenly bodies. He is the first European to have conceptualized the universe as a continuum where the stars we see at night are identical in nature to the Sun; for these ideas he was burned at the stake in 1600 after the Roman Inquisition found him guilty of heresy. After his death he gained considerable fame and in the 19th and early 20th centuries, commentators focusing on his astronomical beliefs regarded him as a martyr for free-thought and modern scientific ideas. Such efforts by the Church to suppress scientific thinking stalled its growth in general and slowed the switch from geocentrism to heliocentrism in particular.
[edit] Modern geocentrism
Main article: Modern geocentrism
The contemporary Association for Biblical Astronomy, led by physicist Dr. Gerardus Bouw, holds to a modified version of the model of Tycho Brahe, which they call geocentricity.[29]
Polls conducted by Gallup in the 1990s has found that 16% of Germans, 18% of Americans and 19% of Britons hold that the Sun revolves around the Earth.[30] A study done in 2005 by Dr. Jon D. Miller of Northwestern University, an expert in the public understanding of science and technology,[31] found that one adult American in five thinks the Sun revolves around the Earth.[32]
[edit] Planetariums
The geocentric (Ptolemaic) model of the solar system is still of interest to planetarium makers, as, for technical reasons, a Ptolemaic-type motion for the planet light apparatus has some advantages over a Copernican-type motion.[33] The celestial sphere, still used for teaching purposes and sometimes for navigation, is also based on a geocentric system.[34]
[edit] Geocentric models in science fiction
Alternate history science fiction has produced some literature of interest on the proposition that some alternate universes and Earths might indeed have laws of physics and cosmologies that are Ptolemaic and Aristotelian in design. This subcategory began with Philip Jose Farmer's short story, Sail On! Sail On! (1952), where Columbus has access to radio technology, and where his Spanish-financed exploratory and trade fleet sail off the edge of the (flat) world in his geocentric alternate universe in 1492, instead of discovering North America and South America.
Richard Garfinkle's Celestial Matters (1996) is set in a more elaborated geocentric cosmos, where Earth is divided by two contending factions, the Classical Greece-dominated Delian League and the Chinese Middle Kingdom, both of which are capable of flight within an alternate universe based on Ptolemaic astronomy, Aristotle's physics and Taoist thought. Unfortunately, both superpowers have been fighting a thousand-year war since the time of Alexander the Great.
[edit] See also
• Claudius Ptolemaeus
• Celestial spheres
• Firmament
• Heliocentrism
• Brahmanda (Earth is in middle planetary system)
• Hindu cosmology
• Religious cosmology#Hindu
• The Flat Earth Society
[edit] Notes
1. ^ Lawson, Russell M. (2004). Science in the ancient world: an encyclopedia. ABC-CLIO. pp. 29–30. ISBN 1851095349. http://books.google.com/?id=1AY1ALzh9V0C&pg=PA30&lpg=PA30&dq=ancient+China+geocentric+astronomy#v=onepage&q=ancient%20China%20geocentric%20astronomy&f=false. Retrieved 2 October 2009.
2. ^ Thomas S. Kuhn, The Copernican Revolution, pp. 5–20
3. ^ Fraser, Craig G. – The Cosmos: A Historical Perspective (2006) – p.14
4. ^ Hetherington, Norriss S. – Planetary Motions: A Historical Perspective (2006) – p.28
5. ^ This argument is given in Book I, Chapter 5, of the Almagest (Crowe, 1990, pp.60–62).
6. ^ A. I. Sabra, "Configuring the Universe: Aporetic, Problem Solving, and Kinematic Modeling as Themes of Arabic Astronomy," Perspectives on Science 6.3 (1998): 288–330, at pp. 317–18:
All Islamic astronomers from Thabit ibn Qurra in the ninth century to Ibn al-Shatir in the fourteenth, and all natural philosophers from al-Kindi to Averroes and later, are known to have accepted ... the Greek picture of the world as consisting of two spheres of which one, the celestial sphere ... concentrically envelops the other.
7. ^ Rufus, W. C. (May 1939). "The Influence of Islamic Astronomy in Europe and the Far East". Popular Astronomy 47 (5): 233–238 [237]
8. ^ Willy Hartner, "The Mercury Horoscope of Marcantonio Michiel of Venice", Vistas in Astronomy, 1 (1955): 84–138, at pp. 118–122.
9. ^ Bernard R. Goldstein (March 1972). "Theory and Observation in Medieval Astronomy", Isis 63 (1): 39–47 [41].
10. ^ "Ptolemaic Astronomy, Islamic Planetary Theory, and Copernicus's Debt to the Maragha School". Science and Its Times. Thomson Gale. 2005–2006. http://www.bookrags.com/research/ptolemaic-astronomy-islamic-planeta-scit-021234. Retrieved 2008-01-22
11. ^ Adi Setia (2004). "Fakhr Al-Din Al-Razi on Physics and the Nature of the Physical World: A Preliminary Survey". Islam & Science 2. http://findarticles.com/p/articles/mi_m0QYQ/is_2_2/ai_n9532826/. Retrieved 2010-03-02
12. ^ George Saliba (1994), A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam, p. 233–234, 240. New York University Press, ISBN 0-8147-8023-7.
13. ^ Ahmad Dallal (1999), "Science, Medicine and Technology", in The Oxford History of Islam, p. 171, ed. John Esposito, New York: Oxford University Press
14. ^ Guessoum, N. (June 2008). "Copernicus and Ibn Al-Shatir: does the Copernican revolution have Islamic roots?". The Observatory 128: 231–239. Bibcode 2008Obs...128..231G.
15. ^ Ragep, F. Jamil (2001a). "Tusi and Copernicus: The Earth's Motion in Context". Science in Context (Cambridge University Press) 14 (1-2): 145–163
16. ^ Ragep, F. Jamil (2001b). "Freeing Astronomy from Philosophy: An Aspect of Islamic Influence on Science". Osiris, 2nd Series 16 (Science in Theistic Contexts: Cognitive Dimensions): 49–64 & 66–71
17. ^ a b Toby E.Huff(1993):The rise of early modern science: Islam, China, and the West[1]
18. ^ N.K. Singh, M. Zaki Kirmani,Encyclopaedia of Islamic science and scientists[2]
19. ^ K. F. Johansen, H. Rosenmeier, A History of Ancient Philosophy: From the Beginnings to Augustine (1998), p.43
20. ^ George Sarton, Ancient Science Through the Golden Age of Greece (1953), p.290
21. ^ Eastwood, B. S. (1992-11-01). "Heraclides and Heliocentrism – Texts Diagrams and Interpretations". Journal for the History of Astronomy 23: 233. Bibcode 1992JHA....23..233E.
22. ^ Lindberg, David C. — The Beginnings of Western Science – p.197 [3]
23. ^ Lawson, Russell M. — Science in the ancient world (2004) – p.19
24. ^ Russell, Bertrand — History of Western Philosophy (2004) – p.215
25. ^ Finocchiaro, Maurice A. The Essential Galileo. Indianapolis: Hackett Publishing Company, 2008. pg 49.
26. ^ Selections from Newton's Principia. ed. Dana Densmore. Green Lion Press, 2004. pg. 12.
27. ^ Alighieri, Dante. "Paradiso". Bantam Classic: New York, 1984.pg. 305
28. ^ Aquinas, St. Thomas. Summa Theologica. Burn Oates & Washbourne Ltd.: London, 1926. Part III (QQ. XXVII. – LIX) pg.432.
29. ^ Geocentricity.com
30. ^ Steve Crabtree (July 6, 1999). "New Poll Gauges Americans' General Knowledge Levels". Gallup. http://www.gallup.com/poll/3742/new-poll-gauges-americans-general-knowledge-levels.aspx.
31. ^ "Jon D. Miller". Northwestern University. http://www.cmb.northwestern.edu/faculty/jon_miller.htm. Retrieved 2007-07-19.
32. ^ Cornelia Dean (30 August 2005). "Scientific Savvy? In U.S., Not Much". New York Times. http://www.nytimes.com/2005/08/30/science/30profile.html?ex=1184990400&en=2fb126c3132f89ae&ei=5070. Retrieved 2007-07-19.
33. ^ William Jillard Hort, A General View of the Sciences and Arts, (1822), Page 182
34. ^ Kaler, James B. – The Ever-changing Sky: A Guide to the Celestial Sphere (2002) – p.25
[edit] References
• Crowe, Michael J. (1990). Theories of the World from Antiquity to the Copernican Revolution. Mineola, NY: Dover Publications, Inc. ISBN 0-486-26173-5. http://books.google.com.au/books?id=IGlhN0MI87oC&printsec=frontcover#v=onepage&q=&f=false.
• Dreyer, J.L.E. (1953). A History of Astronomy from Thales to Kepler. New York, NY: Dover Publications. http://www.archive.org/details/historyofplaneta00dreyuoft.
• Evans, James. The History and Practice of Ancient Astronomy. New York: Oxford University Press, 1998.
• Heath, Thomas. Aristarchus of Samos. Oxford: Clarendon Press, 1913
• Hoyle, Fred, Nicolaus Copernicus, 1973.
• Koestler, Arthur The Sleepwalkers: A History of Man's Changing Vision of the Universe, 1959, Penguin Books, 1986 edition: ISBN 0-14-055212-X, 1990 reprint: ISBN 0-14-019246-8
• Kuhn, Thomas S. The Copernican Revolution. Cambridge: Harvard Univ. Pr., 1957. ISBN 0-674-17103-9
• Linton, Christopher M. (2004). From Eudoxus to Einstein—A History of Mathematical Astronomy. Cambridge: Cambridge University Press. ISBN 978-0-521-82750-8. http://books.google.com.au/books?id=B4br4XJFj0MC&printsec=frontcover#v=onepage&q=&f=false.
• Walker, Christopher, ed. Astronomy before the telescope. London: British Museum Press, 1996. ISBN 0-7141-1746-3
[edit] External links
• Another demonstration of the complexity of observed orbits when assuming a geocentric model of the solar system
• Geocentric Perspective animation of the Solar System in 150AD
• History of the Planetary Systems from Thales to Kepler by J. L. E. Dreyer
• Ptolemy's explanation for retrograde motion
• Ptolemy’s system of astronomy
• The Galileo Project – Ptolemaic System
• Interactive simulation of a heliocentric or geocentric representation of planetary paths
[hide]v • d • eGreek astronomy
Astronomers Acoreus • Aglaonike • Agrippa • Anaximander • Andronicus • Apollonius • Aratus • Aristarchus • Aristillus • Attalus • Autolycus • Bion • Callippus • Cleomedes • Cleostratus • Conon • Eratosthenes • Euctemon • Eudoxus • Geminus • Heraclides • Hicetas • Hipparchus • Hippocrates of Chios • Hypsicles • Menelaus • Meton • Oenopides • Philip of Opus • Philolaus • Posidonius • Ptolemy • Pytheas • Seleucus • Sosigenes of Alexandria • Sosigenes the Peripatetic • Strabo • Thales • Theodosius • Theon of Alexandria • Theon of Smyrna • Timocharis
Works Almagest (Ptolemy) • On Sizes and Distances (Hipparchus) • On the Sizes and Distances (Aristarchus) • On the Heavens (Aristotle)
Instruments Antikythera mechanism • Armillary sphere • Astrolabe • Dioptra • Equatorial ring • Gnomon • Mural instrument • Triquetrum
Concepts Callippic cycle • Celestial spheres • Circle of latitude • Counter-Earth • Deferent and epicycle • Equant • Geocentrism • Heliocentrism • Hipparchic cycle • Metonic cycle • Octaeteris • Solstice • Spherical Earth • Sublunary sphere • Zodiac
Influences Babylonian astronomy • Egyptian astronomy
Influenced European astronomy • Indian astronomy • Islamic astronomy
Retrieved from "http://en.wikipedia.org/wiki/Geocentric_model"
Abū Kāmil Shujāʿ ibn Aslam
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Abū Kāmil, Shujāʿ ibn Aslam
Other names al-ḥāsib al-miṣrī
Born c. 850
Died c. 930
Era Islamic Golden Age
Region Egypt
Main interests Algebra, Geometry
Major works The Book of Algebra
Influenced by[show]
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Influenced[show]
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Abū Kāmil, Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel,[1] Arabic: ابو كامل, also known as al-Ḥasib al-Miṣrī—literally, "the Egyptian calculator") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age. Abu Kamil made important contributions to algebra and geometry,[2] and considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations.[3] His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.[4]
Abu Kamil solved sets of non-linear simultaneous equations with three unknown variables.[5] He was also the first Islamic mathematician to work easily with algebraic equations with powers higher than x2 (up to x8).[6][4] He wrote all problems rhetorically, and some of his books lacked any mathematical notation beside those of integers. For example, he uses the Arabic expression "māl māl shayʾ" ("square-square-thing") for x5 (i.e., ).[7][4]
Contents
[hide]
• 1 Works
o 1.1 Book of Algebra
o 1.2 Book of Rare Things in the Art of Calculation
o 1.3 On the Pentagon and Decagon
o 1.4 Book of Birds
o 1.5 On Measurement and Geometry
o 1.6 Lost works
• 2 Influence
• 3 Sample problems solved by Abu Kamil
o 3.1 Non-linear equations
• 4 On al-Khawarizmi
• 5 Notes
• 6 References
• 7 Further reading
• 8 External links
[edit] Works
[edit] Book of Algebra
In this book, known as Kitāb fi al-jabr wa al-muqābala, Abu Kamil solved systems of equations whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and coefficients to quadratic equations.[3] It is perhaps Abu Kamil's most influential work,[3] which he intended to supersede and expand upon that of Al-Khawarizmi.[8] Whereas the Algebra of al-Khawarizmi was intended towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's Elements.[8]
The first chapter teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots. The second chapter deals with the six types of problems found in Al-Khawarizmi's book,[9] but some of which, especially those of x2, was now worked out directly instead of first solving for x,[6] and now also accompanied with geometrical illustrations and proofs.[9] The third chapter contains examples of quadratic irrationalities as solutions and coefficients.[9] In the fourth chapter, these irrationalities are used to solve problems involving polygons. The rest of the book contains sets of indeterminate equations and systems, problems of application in realistic and unrealistic situations, the latter intended for recreational mathematics.[9]
A number of Islamic mathematicians wrote commentaries on this work, including al-Iṣṭakhrī al-Ḥāsib and ʿAli ibn Aḥmad al-ʿImrānī (d. 955-6),[10] but both commentaries are lost.[2] In Europe, similar material to this book is found in the writings of Fibonacci, and some sections were incorporated and improved upon in the Latin work of John of Seville, Liber mahameleth.[9] A partial translation to Latin was done in the 14th-century by William of Luna, and in the 15th-century the whole work also appeared in a Hebrew translation by Mordekhai Finzi.[9]
[edit] Book of Rare Things in the Art of Calculation
Kitāb al-ṭarā’if fi’l-ḥisāb, describes a number of systematic procedures for finding integral solutions for indeterminate equations.[2] It is also the earliest known Arabic source where solutions are sought to the type of indeterminate equations found in Diophantus's Arithmetica. In this book Abu Kamil explains certain methods not found in any extant copy of the Arithmetica.[4] He also describes one problem for which he found 2,678 solutions.[11]
[edit] On the Pentagon and Decagon
Kitāb … al-mukhammas wa’al-mu‘ashshar. In this treatise algebraic methods are used to solve geometrical problems.[2] Abu Kamil calculated a numerical approximation for the side of a regular pentagon in a circle of radius 10 using the equation x4 + 3125 = 125x2.[12] Some of the calculations uses the Golden Ratio.[11] Fibonacci knew about this treatise and made extensive use of it in his Practica geometriae.[2]
[edit] Book of Birds
Kitāb al-ṭair, a small treatise teaching how to solve indeterminate linear systems with positive integral solutions.[8] The title is derived from a type of problems known in the east which involve the purchase of different species of birds. Abu Kamil wrote in the introduction:
I found myself before a problem that I solved and for which I discovered a great many solutions; looking deeper for its solutions, I obtained two thousand six hundred and seventy-six correct ones. My astonishment about that was great, but I found out that, when I recounted this discovery, those who did not know me were arrogant, shocked, and suspicious of me. I thus decided to write a book on this kind of calculations, with the purpose of facilitating its treatment and making it more accessible.[8]
According to Jacques Sesiano, Abu Kamil remained seemingly unparalleled throughout the middle ages in trying to find all the possible solutions to some of his problems.[9]
[edit] On Measurement and Geometry
Kitāb al-misāḥa wa al-handasa, a manual of geometry for non-mathematicians, like land surveyors and other government officials.[4]
This section requires expansion.
[edit] Lost works
Abu Kamil wrote a now lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭaʾayn)[13]
Another lost work of his is the Book on Augmentation and Diminution (Kitāb al-jamʿ wa al-tafrīq), which gained more attention after historian Franz Woepcke linked it with another Latin work, Liber augmenti et diminutionis.[2]
He also wrote the Book of Estate Sharing using Algebra (Kitāb al-waṣāyā bi al-jabr wa al-muqābala), contains algebraic solutions for problems of Islamic inheritance and discusses the opinions of known jurists.[9]
Ibn al-Nadīm in his Fihrist listed the following additional titles: Book of Fortune (Kitāb al-falāḥ), Book of the Key to Fortune (Kitāb miftāḥ al-falāḥ), Book of the Adequate (Kitāb al-kifāya), and Book of the Kernel (Kitāb al-ʿasīr).[6]
[edit] Influence
Abu Kamil's works influenced other mathematicians, like al-Karaji and Fibonacci, and as such had a lasting impact on the development of algebra.[6][14] Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae.[6][11] Unmistakable borrowings from Abu Kamil, without him being explicitly mentioned and perhaps mediated by lost treatises, are also found in the Liber Abaci.[15]
[edit] Sample problems solved by Abu Kamil
[edit] Non-linear equations
Abu Kamil solved the non-linear simultaneous equations with three unknown variables: (1) x + y + z = 10, (2) x^2 + y^2 = z^2, (3) xy = z^2.[5] He does so by first making an arbitrary non-zero guess, x_0, for x (he chooses x=1), and then solves (2, 3) for corresponding y_0 and z_0. Since 2 and 3 are homogeneous , any solution of (2, 3) will also be a solution if x, y and z are multiplied by any constant. In particular, if they are multiplied by 10 / (x_0 + y_0 + z_0) then they will still solve (2, 3), and will also solve (1) by construction.
[edit] On al-Khawarizmi
Almost nothing is known about the life and career of Abu Kamil except that he was a successor of al-Khawarizmi, whom he never personally met. He was also one of the earliest mathematicians to recognize Al-Khwarizmi's contributions to algebra,[4] and defended him against Ibn Barza who attributed the authority and precedent in algebra to his grandfather, ʿAbd al-Hamīd ibn Turk.[4]
Abu Kamil wrote in the introduction of his Algebra:
I have studied with great attention the writings of the mathematicians, examined their assertions, and scrutinized what they explain in their works; I thus observed that the book by Muḥammad ibn Mūsā al-Khawārizmī known as Algebra is superior in the accuracy of its principle and the exactness of its argumentation. It thus behooves us, the community of mathematicians, to recognize his priority and to admit his knowledge and his superiority, as in writing his book on algebra he was an initiator and the discoverer of its principles, ...[8]
[edit] Notes
1. ^ Rāshid, Rushdī; Régis Morelon (1996). Encyclopedia of the history of Arabic science. 2. Routledge. p. 240. ISBN 9780415124119.
2. ^ a b c d e f Hartner, W. (1960). "ABŪ KĀMIL SHUDJĀʿ". Encyclopaedia of Islam. 1 (2nd ed.). Brill Academic Publishers. pp. 132–3. ISBN 90-04-08114-3.
3. ^ a b c Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000). Mathematics Across Cultures: The History of Non-Western Mathematics. Springer. ISBN 1-4020-0260-2.[1]
4. ^ a b c d e f g O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujāʿ ibn Aslam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Abu_Kamil.html.
5. ^ a b Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. pp. 518, 550. ISBN 978-0-691-11485-9.[2]
6. ^ a b c d e Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0684101149.
7. ^ Bashmakova, Izabella Grigorʹevna; Galina S. Smirnova (2000-01-15). The beginnings and evolution of algebra. Cambridge University Press. p. 52. ISBN 9780883853290.
8. ^ a b c d e Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore. ISBN 9780821844731.
9. ^ a b c d e f g h Sesiano, Jacques (1997-07-31). "Abū Kāmil". Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. pp. 4–5.
10. ^ Louis Charles Karpinski (1915). Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, with an Introduction, Critical Notes and an English Version. Macmillan Co..
11. ^ a b c Livio, Mario (2003). The Golden Ratio. New York: Broadway. pp. 89–90, 92, 96. ISBN 0-7679-0816-3.
12. ^ Ragep, F. J.; Sally P. Ragep, Steven John Livesey (1996). Tradition, transmission, transformation: proceedings of two conferences on pre-modern science held at the University of Oklahoma. BRILL. p. 48. ISBN 9789004101197.
13. ^ Schwartz, R. K (2004). "Issues in the Origin and Development of Hisab al-Khata’ayn (Calculation by Double False Position)". Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia.
14. ^ Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly 21 (2): 37–48. doi:10.2307/2972073. ISSN 00029890. http://www.jstor.org/stable/2972073. Retrieved 2011-03-21.
15. ^ Høyrup, J. (2009). "Hesitating progress-the slow development toward algebraic symbolization in abbacus-and related manuscripts, c. 1300 to c. 1550: Contribution to the conference" Philosophical Aspects of Symbolic Reasoning in Early Modern Science and Mathematics", Ghent, 27-29 August 2009". Preprints. 390. Berlin: Max Planck Institute for the History of Science.
[edit] References
• Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore. ISBN 9780821844731.
• Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0684101149.
• O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujāʿ ibn Aslam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Abu_Kamil.html.
[edit] Further reading
• Yadegari, Mohammad (1978-06-01). "The Use of Mathematical Induction by Abū Kāmil Shujā' Ibn Aslam (850-930)". Isis 69 (2): 259–262. ISSN 00211753. http://www.jstor.org/stable/230435.
• Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly 21 (2): 37–48. doi:10.2307/2972073. ISSN 00029890. http://www.jstor.org/stable/2972073.
• Herz-Fischler, Roger (1987-06). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier Univ Pr. ISBN 0889201528.
• Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001)
[edit] External links
[hide]v • d • eMathematics in medieval Islam
Mathematicians 'Abd al-Hamīd ibn Turk • Abd al-Rahman al-Sufi • Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī • Abū al-Wafā' al-Būzjānī • Abū Ishāq Ibrāhīm al-Zarqālī • Abū Ja'far al-Khāzin • Abū Kāmil Shujā ibn Aslam • Sind ibn Ali • Abu'l-Hasan al-Uqlidisi • Abu-Mahmud al-Khujandi • Abu Nasr Mansur • Abū Rayḥān al-Bīrūnī • Abū Sahl al-Qūhī • Ahmad ibn Yusuf • Al-Abbās ibn Said al-Jawharī • Al-Birjandi • Al-Ḥajjāj ibn Yūsuf ibn Maṭar • Alhazen • Ibn Muʿādh al-Jayyānī • Al-Karaji • Al-Khazini • Al-Kindi • Al-Mahani • Al-Nayrizi • Al-Saghani • Al-Sijzi • Yaʿīsh ibn Ibrāhīm al-Umawī • Alī ibn Ahmad al-Nasawī • Ali Qushji • Avicenna • Banū Mūsā • Brethren of Purity • Hunayn ibn Ishaq • Ibn al-Banna al-Marrakushi • Ibn al-Shatir • Ibn Sahl • Ibn Tahir al-Baghdadi • Ibn Yahyā al-Maghribī al-Samaw'al • Ibn Yunus • Ibrahim ibn Sinan • Jamshīd al-Kāshī • Kamāl al-Dīn al-Fārisī • Kushyar ibn Labban • Muhammad Baqir Yazdi • Muhammad ibn Jābir al-Harrānī al-Battānī • Muḥammad ibn Mūsā al-Khwārizmī • Muhyi al-Dīn al-Maghribī • Nasīr al-Dīn al-Tūsī • Omar Khayyám • Qāḍī Zāda al-Rūmī • Al-Khalili • Shams al-Dīn al-Samarqandī • Sharaf al-Dīn al-Tūsī • Sinan ibn Thabit • Taqi al-Din • Thābit ibn Qurra • Ulugh Beg • Yusuf al-Mu'taman ibn Hud • Na'im ibn Musa • Qotb al-Din Shirazi • Ibn al‐Haim al‐Ishbili
Treatises Almanac • Book of Fixed Stars • Book of Optics • De Gradibus • Encyclopedia of the Brethren of Purity • Tables of Toledo • Tabula Rogeriana • The Compendious Book on Calculation by Completion and Balancing • The Book of Healing • Zij • Zij-i Ilkhani • Zij-i-Sultani
Centers Al-Azhar University • Al-Mustansiriya University • House of Knowledge • House of Wisdom • Istanbul observatory of Taqi al-Din • Madrasah • Maktab • Maragheh observatory • University of Al-Karaouine
Influences Babylonian mathematics • Greek mathematics • Indian mathematics
Influenced Byzantine mathematics • European mathematics • Indian mathematics
Retrieved from "http://en.wikipedia.org/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81%CA%BF_ibn_Aslam"
Geometry
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Geometry (disambiguation).
An illustration of Desargues' theorem, an important result in Euclidean and projective geometry.
Oxyrhynchus papyrus (P.Oxy. I 29) showing fragment of Euclid's Elements
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metri "measurement") "Earth-measuring" is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer.
The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.
While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).[1]
Contents
[hide]
• 1 Overview
o 1.1 Practical geometry
o 1.2 Axiomatic geometry
o 1.3 Geometric constructions
o 1.4 Numbers in geometry
o 1.5 Geometry of position
o 1.6 Geometry beyond Euclid
o 1.7 Dimension
o 1.8 Symmetry
o 1.9 Modern geometry
• 2 History of geometry
• 3 Contemporary geometry
o 3.1 Euclidean geometry
o 3.2 Differential geometry
o 3.3 Topology and geometry
o 3.4 Algebraic geometry
• 4 See also
o 4.1 Lists
o 4.2 Related topics
• 5 References
o 5.1 Notes
o 5.2 Bibliography
• 6 External links
[edit] Overview
Visual proof of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
The recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.
[edit] Practical geometry
Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.
[edit] Axiomatic geometry
An illustration of Euclid's parallel postulate
See also: Euclidean geometry
Euclid took a more abstract approach in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of the 19th century the discovery of non-Euclidean geometries by Gauss, Lobachevsky, Bolyai, and others led to a revival of interest, and in the 20th century David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
[edit] Geometric constructions
Main article: Compass and straightedge constructions
Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are those with compass and straightedge. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
[edit] Numbers in geometry
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favor of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.
[edit] Geometry of position
Main articles: Projective geometry and Topology
Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.
Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.
[edit] Geometry beyond Euclid
Differential geometry uses tools from calculus to study problems in geometry.
For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[2] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
[edit] Dimension
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Exactly why is something to which research may bring a satisfactory geometric answer.
[edit] Symmetry
A tiling of the hyperbolic plane
The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in for instance projective geometry (see Duality (projective geometry)). This is a meta-phenomenon which can roughly be described as: replace in any theorem point by plane and vice versa, join by meet, lies-in by contains, and you will get an equally true theorem. A similar and closely related form of duality appeares between a vector space and its dual space.
[edit] Modern geometry
Modern geometry is the title of a popular textbook by Dubrovin, Novikov and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.
[edit] History of geometry
Main article: History of geometry
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310)
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BCE. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets, and the Indian Shulba Sutras, while the Chinese had the work of Mozi, Zhang Heng, and the Nine Chapters on the Mathematical Art, edited by Liu Hui. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.[3][4]
Until relatively recently (i.e. the last 200 years), the teaching and development of geometry in Europe and the Islamic world was based on Greek geometry.[5][6] Euclid's Elements (c. 300 BCE) was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;[7] Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.[citation needed]
In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry[8][9][unreliable source?] and geometric algebra.[10] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[9] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[11] Omar Khayyám (1048–1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulate contributed to the development of non-Euclidian geometry.[12][unreliable source?] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on elliptical geometry and hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[13]
In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
[edit] Contemporary geometry
[edit] Euclidean geometry
The E8 Lie group polytope Coxeter plane projection
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
[edit] Differential geometry
Differential geometry has been of increasing importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved. Contemporary differential geometry is intrinsic, meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point, and not a priori parts of some ambient flat Euclidean space.
[edit] Topology and geometry
A thickening of the trefoil knot
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.
[edit] Algebraic geometry
Quintic Calabi–Yau threefold
The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.
The study of low dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.
[edit] See also
Geometry portal
[edit] Lists
• List of geometers
o Category:Algebraic geometers
o Category:Differential geometers
o Category:Geometers
o Category:Topologists
• List of geometry topics
• List of important publications in geometry
• List of mathematics articles
[edit] Related topics
• Flatland, a book written by Edwin Abbott Abbott about two and three-dimensional space, to understand the concept of four dimensions
• Interactive geometry software
• Why 10 dimensions?
[edit] References
[edit] Notes
1. ^ It is quite common in algebraic geometry to speak about geometry of algebraic varieties over finite fields, possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary spheres or cones.
2. ^ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
3. ^ The Journal of Egyptian Archaeology. Vol. 84, 1998 Gnomons at Meroë and Early Trigonometry. pg. 171
4. ^ Neolithic Skywatchers. May 27, 1998 by Andrew L. Slayman Archaeology.org
5. ^ Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews (1998). "Fractal geometry in digital imaging". Academic Press. p.1. ISBN 0127039708
6. ^ Amy Shell-Gellasch, Dick Jardine (2005). "From calculus to computers: using the last 200 years of mathematics history". Cambridge University Press. p.59. ISBN 0883851784
7. ^ Boyer (1991). "Euclid of Alexandria". p. 104. "The Elements was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all elementary mathematics-"
8. ^ R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, London
9. ^ a b O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html.
10. ^ Boyer (1991). "The Arabic Hegemony". pp. 241–242. "Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""
11. ^ O'Connor, John J.; Robertson, Edmund F., "Al-Sabi Thabit ibn Qurra al-Harrani", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Thabit.html.
12. ^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html.
13. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:
"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
[edit] Bibliography
• Mlodinow, M.; Euclid's window (the story of geometry from parallel lines to hyperspace), UK edn. Allen Lane, 1992.
[edit] External links
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Integral
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This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation).
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Topics in Calculus
Fundamental theorem
Limits of functions
Continuity
Mean value theorem
[show]Differential calculus
[show]Integral calculus
[show]Vector calculus
[show]Multivariable calculus
Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral
is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.
The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by
Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.
Contents
[hide]
• 1 History
o 1.1 Pre-calculus integration
o 1.2 Newton and Leibniz
o 1.3 Formalizing integrals
o 1.4 Notation
• 2 Terminology and notation
• 3 Introduction
• 4 Formal definitions
o 4.1 Riemann integral
o 4.2 Lebesgue integral
o 4.3 Other integrals
• 5 Properties
o 5.1 Linearity
o 5.2 Inequalities for integrals
o 5.3 Conventions
• 6 Fundamental theorem of calculus
o 6.1 Statements of theorems
• 7 Extensions
o 7.1 Improper integrals
o 7.2 Multiple integration
o 7.3 Line integrals
o 7.4 Surface integrals
o 7.5 Integrals of differential forms
o 7.6 Summations
• 8 Methods
o 8.1 Computing integrals
o 8.2 Symbolic algorithms
o 8.3 Numerical quadrature
• 9 See also
• 10 Notes
• 11 References
• 12 External links
o 12.1 Online books
[edit] History
See also: History of calculus
[edit] Pre-calculus integration
Integration can be traced as far back as ancient Egypt ca. 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. Similar methods were independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.[1] That same century, the Indian mathematician Aryabhata used a similar method in order to find the volume of a cube.[2][verification needed]
The next major step in integral calculus came from the Abbasid Caliphate when the 11th century mathematician Ibn al-Haytham (known as Alhazen in Europe) devised what is now known as "Alhazen's problem", which leads to an equation of the fourth degree, in his Book of Optics. While solving this problem, he performed an integration in order to find the volume of a paraboloid. Using mathematical induction, he was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[3][verification needed] Some ideas of integral calculus are also found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II.
The next significant advances in integral calculus did not begin to appear until the 16th century. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xn up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly by Seki Kōwa.[4] He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion.
[edit] Newton and Leibniz
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.
[edit] Formalizing integrals
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked infinitesimals as "the ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered – particularly in the context of Fourier analysis – to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.
[edit] Notation
Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.
The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359; Leibniz 1899, p. 154). He adapted the integral symbol, ∫, from an elongated letter s, standing for summa (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).
[edit] Terminology and notation
If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. Usually this domain will be an interval, in which case it is enough to give the limits of that interval, which are called the limits of integration. If the integral does not have a domain of integration, it is considered indefinite (one with a domain is considered definite). In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense (such as a sample space in probability theory).
The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted by
The ∫ sign represents integration; a and b are the lower limit and upper limit, respectively, of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval [a,b]; and dx is the variable of integration. In correct mathematical typography, the dx is separated from the integrand by a space (as shown). Some authors use an upright d (that is, dx instead of dx).
The variable of integration dx has different interpretations depending on the theory being used. For example, it can be seen as strictly a notation indicating that x is a dummy variable of integration, as a reflection of the weights in the Riemann sum, a measure (in Lebesgue integration and its extensions), an infinitesimal (in non-standard analysis) or as an independent mathematical quantity: a differential form. More complicated cases may vary the notation slightly.
In the modern Arabic mathematical notation, which aims at pre-university levels of education in the Arab world and is written from right to left, a reflected integral symbol is used (W3C 2006).
[edit] Introduction
Integrals appear in many practical situations. Consider a swimming pool. If it is rectangular with a flat bottom, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)
To start off, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. We ask:
What is the area under the function f, in the interval from 0 to 1?
and call this (yet unknown) area the integral of f. The notation for this integral will be
As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 1⁄5, 2⁄5, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √1⁄5, √2⁄5, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely
Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0.6203, which is too small. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely many fine, or infinitesimal steps.
As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating. Applied to the square root curve, f(x) = x1/2, it says to look at the antiderivative F(x) = 2⁄3x3/2, and simply take F(1) − F(0), where 0 and 1 are the boundaries of the interval [0,1]. So the exact value of the area under the curve is computed formally as
(This is a case of a general rule, that for f(x) = xq, with q ≠ −1, the related function, the so-called antiderivative is F(x) = (xq+1)/(q + 1).)
The notation
conceives the integral as a weighted sum, denoted by the elongated s, of function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx. The multiplication sign is usually omitted.
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.
Differential geometry, with its "calculus on manifolds", gives the familiar notation yet another interpretation. Now f(x) and dx become a differential form, ω = f(x) dx, a new differential operator d, known as the exterior derivative is introduced, and the fundamental theorem becomes the more general Stokes' theorem,
from which Green's theorem, the divergence theorem, and the fundamental theorem of calculus follow.
More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis. Not only do these methods vindicate the intuitions of the pioneers; they also lead to new mathematics.
Although there are differences between these conceptions of integral, there is considerable overlap. Thus, the area of the surface of the oval swimming pool can be handled as a geometric ellipse, a sum of infinitesimals, a Riemann integral, a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.
[edit] Formal definitions
There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.
[edit] Riemann integral
Main article: Riemann integral
Integral approached as Riemann sum based on tagged partition, with irregular sampling positions and widths (max in red). True value is 3.76; estimate is 3.648.
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence
Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left.
This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as
thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xi−xi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1…n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:
For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have
When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.
[edit] Lebesgue integral
Main article: Lebesgue integration
Integral Riemann-Darboux's integration (in blue) and Lebesgue integration (in red).
The Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions are integrable (Rudin 1987). The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.
The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = [a,b] is its width, b − a, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.
To exploit this flexibility, Lebesgue integrals reverse the approach to the weighted sum. As Folland (1984, p. 56) puts it, "To compute the Riemann integral of f, one partitions the domain [a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f".
One common approach first defines the integral of the indicator function of a measurable set A by:
.
This extends by linearity to a measurable simple function s, which attains only a finite number, n, of distinct non-negative values:
(where the image of Ai under the simple function s is the constant value ai). Thus if E is a measurable set one defines
Then for any non-negative measurable function f one defines
that is, the integral of f is set to be the supremum of all the integrals of simple functions that are less than or equal to f. A general measurable function f, is split into its positive and negative values by defining
Finally, f is Lebesgue integrable if
and then the integral is defined by
When the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous functions with compact support. More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure can be defined as any continuous linear functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures.
[edit] Other integrals
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:
• The Riemann–Stieltjes integral, an extension of the Riemann integral.
• The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann–Stieltjes and Lebesgue integrals.
• The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without the dependence on measures.
• The Henstock-Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
• The Itō integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
• The Young integral, which is a kind of Riemann-Stieltjes integral with respect to certain functions of unbounded variation.
• The rough path integral defined for functions equipped with some additional "rough path" structure, generalizing stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
[edit] Properties
[edit] Linearity
• The collection of Riemann integrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration
is a linear functional on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals,
• Similarly, the set of real-valued Lebesgue integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral
is a linear functional on this vector space, so that
• More generally, consider the vector space of all measurable functions on a measure space (E,μ), taking values in a locally compact complete topological vector space V over a locally compact topological field K, f : E → V. Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞,
that is compatible with linear combinations. In this situation the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Qp of p-adic numbers, and V is a finite-dimensional vector space over K, and when K=C and V is a complex Hilbert space.
Linearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See (Hildebrandt 1953) for an axiomatic characterisation of the integral.
[edit] Inequalities for integrals
A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).
• Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that
• Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus
This is a generalization of the above inequalities, as M(b − a) is the integral of the constant function with value M over [a, b].
In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then
• Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then
• Products and absolute values of functions. If f and g are two functions then we may consider their pointwise products and powers, and absolute values:
If f is Riemann-integrable on [a, b] then the same is true for |f|, and
Moreover, if f and g are both Riemann-integrable then f 2, g 2, and fg are also Riemann-integrable, and
This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions f and g on the interval [a, b].
• Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|p and |g|q are also integrable and the following Hölder's inequality holds:
For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
• Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|p, |g|p and |f + g|p are also Riemann integrable and the following Minkowski inequality holds:
An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.
[edit] Conventions
In this section f is a real-valued Riemann-integrable function. The integral
over an interval [a, b] is defined if a < b. This means that the upper and lower sums of the function f are evaluated on a partition a = x0 ≤ x1 ≤ . . . ≤ xn = b whose values xi are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [x i , x i +1] where an interval with a higher index lies to the right of one with a lower index. The values a and b, the end-points of the interval, are called the limits of integration of f. Integrals can also be defined if a > b:
• Reversing limits of integration. If a > b then define
This, with a = b, implies:
• Integrals over intervals of length zero. If a is a real number then
The first convention is necessary in consideration of taking integrals over subintervals of [a, b]; the second says that an integral taken over a degenerate interval, or a point, should be zero. One reason for the first convention is that the integrability of f on an interval [a, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that:
• Additivity of integration on intervals. If c is any element of [a, b], then
With the first convention the resulting relation
is then well-defined for any cyclic permutation of a, b, and c.
Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed of differential forms on oriented manifolds only. If M is such an oriented m-dimensional manifold, and M is the same manifold with opposed orientation and ω is an m-form, then one has:
These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure μ, and integrates over a subset A, without any notion of orientation; one writes to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher dimensional manifolds; see Differential form: Relation with measures for details.
[edit] Fundamental theorem of calculus
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
[edit] Statements of theorems
• Fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is defined for x in [a, b] by
then F is continuous on [a, b]. If f is continuous at x in [a, b], then F is differentiable at x, and F ′(x) = f(x).
• Second fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is a function such that F ′(x) = f(x) for all x in [a, b] (that is, F is an antiderivative of f), then
In particular, these are true whenever f is continuous on [a, b].
[edit] Extensions
[edit] Improper integrals
Main article: Improper integral
The improper integral
has unbounded intervals for both domain and range.
A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.
If the integrand is only defined or finite on a half-open interval, for instance (a,b], then again a limit may provide a finite result.
That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or ∞, or −∞. In more complicated cases, limits are required at both endpoints, or at interior points.
Consider, for example, the function integrated from 0 to ∞ (shown right). At the lower bound, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of . To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, . This has a finite limit as t goes to infinity, namely . Similarly, the integral from 1⁄3 to 1 allows a Riemann sum as well, coincidentally again producing . Replacing 1⁄3 by an arbitrary positive value s (with s < 1) is equally safe, giving . This, too, has a finite limit as s goes to zero, namely . Combining the limits of the two fragments, the result of this improper integral is
This process does not guarantee success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of does not converge; and over the unbounded interval 1 to ∞ the integral of does not converge.
The improper integral
is unbounded internally, but both left and right limits exist.
It may also happen that an integrand is unbounded at an interior point, in which case the integral must be split at that point, and the limit integrals on both sides must exist and must be bounded. Thus
But the similar integral
cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.)
[edit] Multiple integration
Main article: Multiple integral
Double integral as volume under a surface.
Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written:
Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R3. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain. (The same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above mentioned region between the surface and the plane.) If the number of variables is higher, then the integral represents a hypervolume, a volume of a solid of more than three dimensions that cannot be graphed.
For example, the volume of the cuboid of sides 4 × 6 × 5 may be obtained in two ways:
• By the double integral
of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the cuboid. For example, if a rectangular base of such a cuboid is given via the xy inequalities 2 ≤ x ≤ 7, 4 ≤ y ≤ 9, our above double integral now reads
From here, integration is conducted with respect to either x or y first; in this example, integration is first done with respect to x as the interval corresponding to x is the inner integral. Once the first integration is completed via the F(b) − F(a) method or otherwise, the result is again integrated with respect to the other variable. The result will equate to the volume under the surface.
• By the triple integral
of the constant function 1 calculated on the cuboid itself.
[edit] Line integrals
Main article: Line integral
A line integral sums together elements along a curve.
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force, F, multiplied by displacement, s, may be expressed (in terms of vector quantities) as:
For an object moving along a path in a vector field such as an electric field or gravitational field, the total work done by the field on the object is obtained by summing up the differential work done in moving from to . This gives the line integral
[edit] Surface integrals
Main article: Surface integral
The definition of surface integral relies on splitting the surface into small surface elements.
A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:
The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the classical theory of electromagnetism.
[edit] Integrals of differential forms
Main article: differential form
A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.
We initially work in an open set in Rn. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rn, we write it as
(The superscripts are indices, not exponents.) We can consider dx1 through dxn to be formal objects themselves, rather than tags appended to make integrals look like Riemann sums. Alternatively, we can view them as covectors, and thus a measure of "density" (hence integrable in a general sense). We call the dx1, …,dxn basic 1-forms.
We define the wedge product, "∧", a bilinear "multiplication" operator on these elements, with the alternating property that
for all indices a. Note that alternation along with linearity and associativity implies dxb∧dxa = −dxa∧dxb. This also ensures that the result of the wedge product has an orientation.
We define the set of all these products to be basic 2-forms, and similarly we define the set of products of the form dxa∧dxb∧dxc to be basic 3-forms. A general k-form is then a weighted sum of basic k-forms, where the weights are the smooth functions f. Together these form a vector space with basic k-forms as the basis vectors, and 0-forms (smooth functions) as the field of scalars. The wedge product then extends to k-forms in the natural way. Over Rn at most n covectors can be linearly independent, thus a k-form with k > n will always be zero, by the alternating property.
In addition to the wedge product, there is also the exterior derivative operator d. This operator maps k-forms to (k+1)-forms. For a k-form ω = f dxa over Rn, we define the action of d by:
with extension to general k-forms occurring linearly.
This more general approach allows for a more natural coordinate-free approach to integration on manifolds. It also allows for a natural generalisation of the fundamental theorem of calculus, called Stokes' theorem, which we may state as
where ω is a general k-form, and ∂Ω denotes the boundary of the region Ω. Thus, in the case that ω is a 0-form and Ω is a closed interval of the real line, this reduces to the fundamental theorem of calculus. In the case that ω is a 1-form and Ω is a two-dimensional region in the plane, the theorem reduces to Green's theorem. Similarly, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stokes' theorem and the divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration.
[edit] Summations
The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time scale calculus.
[edit] Methods
[edit] Computing integrals
The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. Let f(x) be the function of x to be integrated over a given interval [a, b]. Then, find an antiderivative of f; that is, a function F such that F' = f on the interval. By the fundamental theorem of calculus—provided the integrand and integral have no singularities on the path of integration—
The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals.
The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include:
• Integration by substitution
• Integration by parts
• Changing the order of integration
• Integration by trigonometric substitution
• Integration by partial fractions
• Integration by reduction formulae
• Integration using parametric derivatives
• Integration using Euler's formula
• Differentiation under the integral sign
• Contour Integration
Alternate methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.
Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals.
[edit] Symbolic algorithms
Main article: Symbolic integration
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration presents a special challenge in the development of such systems.
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions exp ( x2), xx and sin x /x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions. Differential Galois theory provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may be still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in Mathematica and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions.
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function, the Incomplete Gamma function and so on - see Symbolic integration for more details). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.
Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. Very complex formulae are unlikely to have closed-form antiderivatives, so how much of an advantage this presents is a philosophical question that is open for debate.
[edit] Numerical quadrature
Main article: numerical integration
The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical methods for approximating integrals, which today use floating-point arithmetic on digital electronic computers. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC created a need for improvements.
The goals of numerical integration are accuracy, reliability, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures (Dahlquist & Björck 2008; Kahaner, Moler & Nash 1989; Stoer & Bulirsch 2002). Consider, for example, the integral
which has the exact answer 94⁄25 = 3.76. (In ordinary practice the answer is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough.) A “calculus book” approach divides the integration range into, say, 16 equal pieces, and computes function values.
Spaced function values
x −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00
f(x) 2.22800 2.45663 2.67200 2.32475 0.64400 −0.92575 −0.94000 −0.16963 0.83600
x −1.75 −1.25 −0.75 −0.25 0.25 0.75 1.25 1.75
f(x) 2.33041 2.58562 2.62934 1.64019 −0.32444 −1.09159 −0.60387 0.31734
Numerical quadrature methods: ■ Rectangle, ■ Trapezoid, ■ Romberg, ■ Gauss
Using the left end of each piece, the rectangle method sums 16 function values and multiplies by the step width, h, here 0.25, to get an approximate value of 3.94325 for the integral. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern. Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3.76001. However, 218 pieces are required, a great computational expense for such little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle.
A better approach replaces the horizontal tops of the rectangles with slanted tops touching the function at the ends of each piece. This trapezium rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width. This immediately improves the approximation to 3.76925, which is noticeably more accurate. Furthermore, only 210 pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy.
Romberg's method builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h0), T(h1), and so on, where hk+1 is half of hk. For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above). But the really powerful idea is to interpolate a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values)! The Lagrange polynomial interpolating {hk,T(hk)}k=0…2 = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76+0.148h2, producing the extrapolated value 3.76 at h = 0.
Gaussian quadrature often requires noticeably less work for superior accuracy. In this example, it can compute the function values at just two x positions, ±2⁄√3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in error analysis, and a little luck. An n-point Gaussian method is exact for polynomials of degree up to 2n−1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.)
Shifting the range left a little, so the integral is from −2.25 to 1.75, removes the symmetry. Nevertheless, the trapezoid method is rather slow, the polynomial interpolation method of Romberg is acceptable, and the Gaussian method requires the least work — if the number of points is known in advance. As well, rational interpolation can use the same trapezoid evaluations as the Romberg method to greater effect.
Quadrature method cost comparison
Method Trapezoid Romberg Rational Gauss
Points 1048577 257 129 36
Rel. Err. −5.3×10−13 −6.3×10−15 8.8×10−15 3.1×10−15
Value
In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulae. Symmetry can still be exploited by splitting this integral into two ranges, from −2.25 to −1.75 (no symmetry), and from −1.75 to 1.75 (symmetry). More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most.
Simpson's rule, named for Thomas Simpson (1710–1761), uses a parabolic curve to approximate integrals. In many cases, it is more accurate than the trapezoidal rule and others. The rule states that
with an error of
The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.
A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. For example, improper integrals may require a change of variable or methods that can avoid infinite function values, and known properties like symmetry and periodicity may provide critical leverage.
[edit] See also
Mathematics portal
• Lists of integrals – integrals of the most common functions
• Multiple integral
• Numerical integration
• Integral equation
• Riemann integral
• Riemann–Stieltjes integral
• Henstock–Kurzweil integral
• Lebesgue integration
• Darboux integral
• Riemann sum
• Symbolic integration
[edit] Notes
1. ^ Shea, Marilyn (May 2007), Biography of Zu Chongzhi, University of Maine, http://hua.umf.maine.edu/China/astronomy/tianpage/0014ZuChongzhi9296bw.html, retrieved 9 January 2009
Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, pp. 125–126, ISBN 978-0-321-16193-2
2. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163-174 [165]
3. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163–174 [165–9 & 173–4]
4. ^ http://www2.gol.com/users/coynerhm/0598rothman.html
[edit] References
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• Saks, Stanisław (1964), Theory of the integral (English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised ed.), New York: Dover, http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=
• Stoer, Josef; Bulirsch, Roland (2002), "Chapter 3: Topics in Integration", Introduction to Numerical Analysis (3rd ed.), Springer, ISBN 978-0-387-95452-3.
• W3C (2006), Arabic mathematical notation, http://www.w3.org/TR/arabic-math/
[edit] External links
Wikibooks has a book on the topic of
Calculus
• Riemann Sum by Wolfram Research
• [1] by Khan Academy
[edit] Online books
• Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
• Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
• Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
• Crowell, Benjamin, Calculus, Fullerton College, an online textbook
• Garrett, Paul, Notes on First-Year Calculus
• Hussain, Faraz, Understanding Calculus, an online textbook
• Kowalk, W.P., Integration Theory, University of Oldenburg. A new concept to an old problem. Online textbook
• Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
• Numerical Methods of Integration at Holistic Numerical Methods Institute
• P.S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) - a cookbook of definite integral techniques
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