81. Ivars Peterson's MathLand In the ninth century, thabit ibn qurra (826901) discovered a remarkable formulafor finding amicable numbers if p, q, and r are prime numbers, and if they http://www.maa.org/mathland/mathland_1_20.html

Ivars Peterson's MathLand January 20, 1997 Fragments of the Past The early history of mathematics is like a jigsaw puzzle missing many of its pieces. Historians and mathematicians have been painstakingly filling in the blanks, gradually constructing a richer, more complete story of how and where mathematical thought originated and spread. One period of considerable interest is that between the decline of Greek mathematics, coinciding with the collapse of the western Roman Empire in the fifth century, and the rise of European mathematics in the fifteenth century. Mathematics professor Morris Kline of New York University's Courant Institute of Mathematical Sciences expressed a common view of that period in his 1972 book Mathematical Thought from Ancient to Modern Times . "The Arabs made no significant advance in mathematics," he wrote. "What they did was absorb Greek and Hindu mathematics, preserve it, and ultimately, ... transmit it to Europe." In other words, Islamic scholars did little more than put Greek mathematics into cold storage until Europe was ready to accept it. Historian George G. Joseph challenged that view in his provocative book

82. Publications: 56—90. The Book of Assumptions by thabit ibn qurra (836—901) .History of Mathematics States of the Art. J. Dauben, M. Folkerts http://www.iwr.uni-heidelberg.de/~Yvonne.Dold-Samplonius/YDoldPublications.html

"Die Konstruktion des regelmäßigen Siebenecks nach Abu Sahl al-Quhi Waijam ibn Rustam". Janus "Die Handschriften der Amsterdamer mathematischen Gesellschaft". Janus "Archimedes. Einander berührende Kreise". Sudhoffs Archiv "Archimedes. Über einander berührende Kreise ". Archimedes Opera Omnia, Band IV. In collaboration with Heinrich Hermelink and Matthias Schramm. (Stuttgart, Teubner, 1975). Book of Assumptions by Aqatun (PhD Thesis Amsterdam, 1977). "Some Remarks on the Book of Assumptions by Aqatun". Journal History of Arabic Science "On al-Khwarizmi’s algebraical equation case IV, ". (Russian) Commemoration Volume for the 1200th Anniversary of Muhammad ibn Musa al-Khwarizmi "The Solution of Quadratic Equations according to al-Samaw’al". "Developments in the Solution to the Equation from al-Khwarizmi to Fibonacci". From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E.S. Kennedy "The Evolution in the Solution to the Quadratic Equation as Seen in the Work of al-Samaw’al".

83. L'infini En Mathématiques Translate this page Plus tard, Pappus (vers 300 après JC), Proclus (5e s. après J. C), Al Kindi (9es.) Al Nayrizi (9e -10e s), thabit ibn qurra (9e s.), Avicenne (10e s.), et d http://www.reunion.iufm.fr/recherche/irem/histoire/l'infini_en_mathématiques.

Accueil Histoire des mathématiques Philosophie des sciences Axiomatiques ... Informations - Contacts Nous remercions vivement M. Hourya Sinaceur qui nous a généreusement transmis son article sur l'infini mathématique paru dans le "Dictionnaire de Philosophie et d'Histoire des Sciences" et qui nous a accordé l'autorisation de le publier sur notre site. Infini mathématique La préhistoire L’infini offre peu de prise à l'expérience immédiate. Des myriades de brins d'herbe dans un pré, c'est un nombre très grand, mais pas infini. On trouverait une image plus suggestive dans les figures en abyme ou deux miroirs face à face. Mais même dans ces cas on poursuit en imagination un processus dont on ne perçoit effectivement que les premières étapes. L'infini cependant est présent dès qu'il y a mathématique. Les Grecs déjà l'avaient rencontré. Par exemple, Zénon d’Elée (V e siècle avant J. C) avec ses paradoxes sur la divisibilité à l’infini d’un segment de droite, les Pythagoriciens (VI e siècle avant J. C.) avec leur découverte de l’incommensurabilité de la diagonale du carré, Eudoxe (début du IV

84. Jabir_ibn_Aflah Both may be based on the work of thabit ibn qurra, or the work of ibn Aflah, Abu'lWafa,and thabit ibn qurra may all be based on some still unknown source. http://homepages.compuserve.de/thweidenfeller/mathematiker/Jabir_ibn_Aflah.htm

al-Ishbili Abu Muhammad Jabir ibn Aflah Born: about 1100 in possibly Seville Died: about 1160 Jabir ibn Aflah is often known by the Latinised form of his name, namely Geber. Although not he was not in the first rank of Arabic mathematicians, he is important in the development of mathematics since his works were translated into Latin, and so became available to European mathematicians, whereas the work of some of the top rank Arabic mathematicians such as Abu'l-Wafa were not translated into Latin. Very little information is available regarding Jabir ibn Aflah's life. That he came from Seville is known from two sources. Firstly he is described as "al-Ishbili" in manuscripts containing his treatises; this means "from Seville". The other source gives us not only the information that he came from Seville, but also a good estimate for the period in which he lived. The information comes from Maimonides. Moses Maimonides, whose Arabic name was Abu 'Imran Musa ibn Maymun ibn 'Ubayd Allah, was a Jewish philosopher, jurist, and physician who was born in Córdoba in 1135. Among many important works he wrote The Guide of the Perplexed in Arabic in which he writes of:- ... ibn Aflah of Seville, whose son I have met ...

85. Le Maghreb  Dans Les Sciences Sociales Translate this page La première est due à al- Hajjaj ibn Matar. La seconde est l'œu­vre d'ishaqibn Hunayn. La traduction d'Ishaq a été éditée par thabit ibn qurra. http://www.fondation.org.ma/References/cooper.htm

La Fondation du Roi Abdul Aziz une institution scientifique et de documentation spécialisée dans les études maghrébines et dans l'Occident musulman Renforcement du réseau de coopération et d’échange Accord Cadre de Coopération avec la Bibliothèque Générale et Archives L’intérêt tout particulier que la Fondation porte à l’espace maghrébin, et à l’Occident musulman en géné­ral, n’a pas cessé de s’affirmer depuis 1988-1989. Cet intérêt s’est manifesté à travers la recherche et l’acquisition du maximum possible de docu­ments relatifs à cet espace, sur leur traitement et la diffusion de l’information, comme il s’est décliné dans l’animation culturelle ainsi que dans le soutien accordé à la recherche scientifique. Un autre pas vient d’être franchi dans le sens de la consolidation du projet maghrébin de la Fonda­tion. Il s ’agit d e la signature d’un accord cadre de coopération, entre le ministère de la Culture et de la Communica­tion et la Fondation. Cet accord prévoit l’élaboration et la mise en œuvre de projets documentaires annuels communs. L’année 2002 a ainsi été mar­quée par la contribu­tion de la

86. Beth Mardutho Meltho Fonts In addition to translating and revising the translations of others, many translatorsgraduated at his hands. Another translator is thabit ibn qurra (d. 901). http://www.bethmardutho.org/AboutSyriac/civilization.html

87. Introduction thabit ibn qurra He established a school of translators in whichhe and his family worked. Astronomical translation within Islam http://www.peddie.org/princip/espana97/web/Sp-PMP/2)f.htm

The Principio Project AT THE PEDDIE SCHOOL Medieval Islamic Astronomy The Spanish Connection An overview Introduction My initial search for information regarding the topic of "Medieval Islamic Astronomy," was almost entirely fruitless. Every single text I looked into referred to a "general lack of innovation and intellectual discourse" during the medieval period. Historian after historian described this time as unscientific and unenlightened; an epoch best forgotten by any self-respecting scholar. They went on to discuss the contrast between the medieval period and the golden ages of Rome and Greece as one would discuss the difference between a fallow field and a field at harvest. The apparent intellectual gap that these texts addressed extinguished any hope I had of finding information on my topic. However, even after this initial disappointment, one question remained: I knew that the Islamic world, especially in Spain, was flourishing during this era. How, then, is this gap possible? Did Islam contribute nothing to the course of scientific learning? Upon Further Inspection. . .

88. The Virtual Pomegranate The translation of hanputho as 'Pagan' was not mine. The passage in questionis a text by thabit ibn qurra preserved in Bar Hebraeus' Chronography. http://www.uscolo.edu/natrel/pom/old/POM11a4.html

The Pomegranate Permission to reprint must be granted by these writers, and we will be happy to forward your requests The following letters were written in response to Donald H Frew's article Harran: Last Refuge of Classical Paganism in The Pomegranate #9 (Aug '99). Dear Pomegranate Editor, I have read the article on Harran by Donald Frew in The Pomegranate 9 with great interest. Mr Frew has correctly referred to the group known as the Sabi'ah (Sabaeans) whose exact identity or qualification remains wide open to scholarship. The Qur'an lists them among those who were religiously acceptable in the eyes of God: Jews, Christians, Sabaeans, and of course Muslims. Th. Ibn Qurrah, it is reported by Ibn an-Nadim, claimed that he and his community were Sabaeans, and were thus exempt from being considered enemies of Islam, coming under the same protection ( dhimmah ) that is accorded to Christian and Jewish subjects. What exactly was the religion of the town and its people remains open to question. The Syriac term with which he described himself (and his community) may be related to the enigmatic Arabic term Hanif.

89. Biography-center - Letter Q Quinquaud, Charles Eugene www.whonamedit.com/doctor.cfm/1105.html; qurra, thabitibn wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/thabit.html. http://www.biography-center.com/q.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish Q 18 biographies Qalasadi, Abu'l al- www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Qalasadi.html Quasimodo, Salvatore www.nobel.se/literature/laureates/1959/quasimodo-bio.html Queen Mum, Elizabeth www.tsmanxman.org.im/Queenmum.htm Queen Victoria, www.worldwar1.com/bioevic.htm Quercia, Jacopo della www.kfki.hu/~arthp/bio/q/quercia/biograph.html Quervain, Fritz de www.whonamedit.com/doctor.cfm/1106.html Quetelet, Lambert www-history.mcs.st-and.ac.uk/~history/Mathematicians/Quetelet.html Quezon, Manuel L. www.pbs.org/wgbh/pages/amex/macarthur/peopleevents/pandeAMEX108.html Quhi, Abu al- www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Quhi.html Quick, Armand James www.whonamedit.com/doctor.cfm/2386.html Quidde, Ludwig

90. Uczony Heretyk - Nowinki Matematyczne - Wirtualny Wszech¶wiat Tabit ibn qurra (ok. 826901). Dokladna data urodzin Tabita ibn Qurry (Thabitibn qurra) nie jest znana; miesci sie w przedziale lat 824-836. http://www.wiw.pl/nowinki/matematyka/200102/20010219-001.asp

W iw.pl Na bie¿±co: I nformacje C o nowego Matematyka i przyroda: A stronomia B iologia ... odelowanie rzeczywisto¶ci Humanistyka: F ilozofia H istoria ... ztuka Czytaj: B iblioteka D elta ... ielcy i wiêksi Przydatne: S ³owniki C o i gdzie studiowaæ ... szech¶wiat w obrazkach Jeste¶ tutaj: Wirtualny Wszech¶wiat Informacje Nowinki 2000-2002 Matematyka Jeste¶ tutaj nowinka: Uczony heretyk autor: Jaros³aw W³odarczyk z dnia: Szukacz Przeszukaj Wirtualny Wszech¶wiat: Jak zadawaæ pytania? Uczony heretyk Tysi±c sto lat temu zmar³ w Bagdadzie wszechstronny uczony islamu, Tabit Ibn Qurra. Pozostawi³ blisko 100 (choæ nie wszystkie siê zachowa³y) traktatów naukowych, z których za najwa¿niejsze uwa¿a siê prace po¶wiêcone matematyce i astronomii. Tabit Ibn Qurra (ok. 826-901) Dok³adna data urodzin Tabita Ibn Qurry (Thabit ibn Qurra) nie jest znana; mie¶ci siê w przedziale lat 824-836. Wiadomo natomiast, ¿e Tabit pochodzi³ z Harranu w Górnej Mezopotamii (obecnie Turcja), gdzie podobno w m³odo¶ci para³ siê wymian± pieniêdzy. Miasto to by³o o¶rodkiem kultu astralnego: cz³onkowie tamtejszej sekty sabijczyków utrzymywali, ¿e jako pierwsi uprawiali ziemiê, budowali miasta i... rozwinêli naukê. Dzieje Harranu tak siê potoczy³y, ¿e jego mieszkañcy przyswoili sobie jêzyk grecki w epoce hellenistycznej, a po podboju przez Arabów - arabski, zachowuj±c jednak ojczysty aramejski wraz z religi± przodków. Niemniej wolnomy¶licielskie pogl±dy Tabita sprawi³y, ¿e popad³ w konflikt z sabijczykami i opu¶ci³ Harran. Wêdruj±c spotka³ na swej drodze matematyka Muhammada Ibn Musê Ibn Shakira (jednego ze s³ynnych trzech braci Banu Musa), na którym g³êbia wiedzy matematycznej i filozoficznej Ibn Qurry, jak równie¿ jego bieg³o¶æ w jêzykach wywar³y olbrzymie wra¿enie. Muhhamad zaprosi³ go do Bagdadu, gdzie pod rz±dami dynastii Abbasydów rozkwita³a nauka. Najwybitniejszym jej patronem by³ kalif Al-Mamun, który za³o¿y³ Dom M±dro¶ci (

91. Uczony Heretyk - Nowinki Matematyczne - Wirtualny Wszechœwiat Tabit ibn qurra ok. 826901 Dokladna data urodzin Tabita ibn Qurry Thabitibn qurra nie jest znana; miesci sie w przedziale lat 824-836. http://www.wiw.pl/fiszki/nowinki-matematyka-200102-20010219-001.html

W³aœciwa strona - http://www.wiw.pl/nowinki/matematyka/200102/20010219-001.asp W³aœciwa strona - http://www.wiw.pl/nowinki/matematyka/200102/20010219-001.asp

92. Scientists & Scholars Mohammad Bin Musa alKhawarizmi died 840. Yaqub ibn Ishaq al-Kindi 800. Thabitibn qurra 836. Ali ibn Rabban al-Tabari 838. Abu Abdullah al-Battani 858. http://www.islamia.com/History/muslim_scientists_&_scholars.htm

In the Name of God, Most Gracious, Most Merciful The following is taken from and linked to http://www.ummah.net Jabir Ibn Haiyan died 803 Mohammad Bin Musa al-Khawarizmi died 840 Yaqub Ibn Ishaq al-Kindi Thabit Ibn Qurra Ali Ibn Rabban al-Tabari Abu Abdullah al-Battani ... Abul Hasan Ali al-Masu'di died 957 Abu al-Qasim al-Zahrawi Abul Wafa Muhammad al-Buzjani Abu Ali Hasan Ibn al-Haitham Abu al-Hasan al-Mawardi ... Ibn al-Baitar died 1248 Nasir al-Din al-Tusi Jalal al-Din Rumi Ibn al-Nafis Ibn Khaldun ... Ibn Sina - doctor of doctors El Zahrawi - father of surgery Ibn Battuta - the great traveller

93. History 935 B.C. philosophy. . Author References Ibrahim, ibn Sinan ibn Thabitibn qurra http//www.cwi.nl/~keesh/Iran/Maths/qurra.htm. Mac http://faculty.oxy.edu/jquinn/home/Math490/Timeline/935BC.html

935 B.C. At the age of twenty-seven, Ibrahim ibn Sinan, was the only known mathematician in the year 935 BC. He was born in the city of Baghdad in 908 BC, where he also died at the age of thirty-eight. Ibrahim ibn Sinan’s interests were in geometry, especially tangents to circles, astronomy, and mathematical philosophy. He also wrote several books on geometry, including On Drawing the Three Conic Sections , which explains the constructions of the ellipse, hyperbola, and parabola. By studying the geometry of the shadows of the sun, Sinan tried to describe what he thought was the motion of the sun. The most famous work of Ibrahim ibn Sinan was the quadrature of the parabola. From this problem, Sinan developed a method of integration that was more general than the previously defined technique by Archimedes. His book, On the Measurement of the Parabola , introduces a theorem that states that the area of a segment of a parabola is four-thirds times the area of the triangle inscribed in that parabola. Ibrahim ibn Sinan translated many Greek mathematical and philosophical works. Because of his work in mathematical philosophy, he has been labeled the "foremost Arab mathematician to treat mathematical philosophy." Author References: Ibrahim, ibn Sinan ibn Thabit ibn Qurra

94. The Mathematics Of Islam, Part 2 In this lecture, the topics consisted of alKhwarizmi (750-850), thabit ibnQurra (830-890), Abu-Sahl al-Kuhi (early 900s), ibn al-Haytham (965-1039 http://public.csusm.edu/public/DJBarskyWebs/330CollageOct01.html

The Mathematics of Islam, Part 2 The presentation given today by Dr. Barsky generated an overall theme that the works of the mathematicians of Islam during the period of (965-1039) seemed to show traces of calculus, despite the fact that calculus came about later in time. In this lecture, the topics consisted of al-Khwarizmi (750-850), Thabit ibn Qurra (830-890), Abu-Sahl al-Kuhi (early 900s), ibn al-Haytham (965-1039), Mohammed's Flight from Mecca (622), the Battle of Tours (732), the period of Caliphates, the Fall of Baghdad to Seljuk Turks (1055), the beginning of the first Crusade (1096), the arrival of the Mongols under Ghengis Khan (early 1200s), al-Khwarizmi's truncated pyramid problem, Mishnat ha-Middot, and ibn al-Haytham's volume of a paraboloid calculations. We did not have a mathematician of the day, instead we talked about our papers that will be due October 8. We then covered more of the history of Islamic mathematics. We focused on al-Khwarizmi (750-850), Thabit ibn Qurra (830-890), Abu-Sahl al-Kuhi (early 900s), and ibn al-Haytham (965-1039). We concentrated on problems from al-Khwarizmi and ibn al-Haytham, which mainly dealt with finding volumes. I really enjoyed the lecture about the volume of a parabola from ibn al-Haytham. I have previously seen the symbol for (the sum of), but I usually stopped at that point because I did not understand, or I felt like it was too complicated. I understand the logic behind finding the circumscribed volume, and inscribed volume of the parabola. I can see that the difference between the two is the volume of the bottom disk of the circumscribed volume. I see that the (sum symbol) is included in equations that are interested in finding the sum of the differences between two estimates which will give you the solution to a problem. I also see that as you take the sum of the differences you are reaching the limit which is related to the volume of the parabola.

95. Teoremadepitagoras Translate this page La segunda escena se basa en la demostración que Meavilla (1989) atribuye a ThabitIbn qurra, matemático árabe del s.IX y se caracteriza por ser el puzzle http://almez.pntic.mec.es/~jdec0000/geometria_dinamica_del_triangulo/teorema_de_

Teorema de Pitágoras E n un triángulo rectángulo, la suma de los cuadrados de los catetos (b y c) es igual al cuadrado de la hipotenusa (a): a =b +c Los números a b y c que verifican esta relación se llaman ternas pitagóricas o números pitagóricos en alusión al estudio que de ellos hicieron Pitágoras y sus discípulos. Los antecedentes históricos de este teorema se remontan a las civilizaciones babilónica y egipcia en el segundo milenio a.J.C. El papiro Rhind y el de Moscú confirman la existencia de tablas de número pitagóricos en esa época. Tras las inundaciones del Nilo, los agrimensores egipcios construían triángulos rectángulos de catetos 3 y 4 y de hipotenusa 5, mediante una cuerda de 12 nudos para parcelar el terreno. Euclides demuestra el Teorema de Pitágoras en la proposición 47 del Libro I de los Elementos En los triángulos rectángulos el cuadrado sobre el ángulo opuesto al ángulo recto es equivalente a los cuadrados sobre los lados que forman el ángulo recto En la proposición 48 demuestra que si el cuadrado construido sobre uno de los lados de un triángulo es equivalente a los cuadrados, juntos, de los otros dos lados, el ángulo formado por esos dos lados es recto, es decir, el recíproco de la Proposición 47.

96. îÏ×ÁÑ áÓÔÒÏÌÏÇÉÞÅÓËÁÑ üÎÃÉËÌÏÐÅÄÉÑ The summary for this Russian page contains characters that cannot be correctly displayed in this language/character set. http://encyclopedia.astrologer.ru/cgi-bin/guard/S/Sabit.html

97. The History Of Mathematics - Library Center For E-courses The summary for this Hebrew page contains characters that cannot be correctly displayed in this language/character set. http://www-lib.haifa.ac.il/www/mesila/math/sites.htm

The History of Mathematics Trinity College, Dublin:á åôñàðù íåçúá íéøúà David R. Wilkins éãé ìò The History of Mathematics David R. Wilkins : é"ò êøòð History of mathematics resources Indexes of Biographies MacTutor History of Mathematics archive:êåúî Mathematicians of the Seventeenth and EigHteenth Centuries Mathematics Genealogy Project Mathematical Journey through Time The Mactutor History of Mathematics archive University of st Andrews Scotland,School of Mathematics and Statistics:êåúî Philosophy and History of Science Kyoto University World of Scientific Biography Erics Treasure Trove of Scientific Biography Arabic mathematics : forgotten brilliance? Doubling the cube History Topics: Babylonian mathematics History Topics: Ancient Egyptian mathematics ... udoxus of Cnidus The Mac Tutor History of Mathematics Archive, University of St. Andrews êåúî Eudoxus of Cnidus An Introduction to the works of Euklid with an Emphasis on the Elements Euclid of Alexandria The Mac Tutor History of Mathematics Archive University of St. Andrews:êåúî

98. CONTENTS The summary for this Gujarati page contains characters that cannot be correctly displayed in this language/character set. http://www.ias-worldwide.org/contents_noble.htm

Academy Publishes " Personalities Noble" CONTENTS Abu Abdullah al-Battani Abu Raihan al-Biruni Abu Wafa Muhammad al-Buzjani Abu al-Naser al-Farabi Al-Farghani Abu Hamid al- Ghazali Al- Idrissi Ibn al-Bitar Abu Ali Hassan Ibn al-Haitham Ibn Al-Nafis Ibn Khaldun Ibn Rushd Ibn Sina Abu Marwan Ibn Zuhr Jabir Ibn Haiyan Mohammad Bin Musa al-Khawarizmi Omer al-Khayyam Yaqub Ibn Ishaq al-Kindi Abu al-Hassan Ali al-Mas'udi Abu al-Hassan al-Mawardi Mohammad Ibn Zakariya al- Razi Jalal Al- Din Rumi Ali Ibn Rabban al-Tabari Thabit Ibn Qurra Nasir al-Din al- Tusi Abu al-Qasim al-Zahrawi About Us IAS Structure Programme Activities ... Home

99. What Is Islam This picture indicates Imam Khomeini in a certain place before hisreturn to Iran in 1979.What is the exact name of this place? http://www.tebyan.net/english/tebyan.htm

Who was the architecture of Ortakoy mosque in Istanbul ( Turkey ) ? Articles Islam - What Does it Mean Science and civilization in Islam Reflections on Islam and Modern Life Islam and the Question of Violence Islamic sects Isma'ilism Shi'ism (1) Shi'ism (2) Shi'ism (3) ... Shi'ism (8) History of Islam A Brief Chronology Duties Hajj International Day for the Elimination of Racial Discrimination Happy Nowrooz THE HIJRA CALENDAR ... Quranic Stories Articles Protection of Qur'an from any Alteration The koran Collection of Quran Study Of Quran ... The Qur'an and Modern Science Tajweed Text of Quran in different language English German French Bosnian ... Turkish History Of Iran Geography of Iran Iranian Cities Isfahan Shiraz Literature Mowlana Jalaluddin Rumi Attar Language Iranian Scientists Abu l'Hasan Ali ibn Ahmad Al-Nasawi Ibn Sina Biruni Biography of Prophet (1) ... Ahl-ul-bayt Companions Abu Dharr Al-Ghifari Salman Jafar ibn Abi Talib Ibn Abbas Islamic Countries BAHRAIN Republic of Albania Afghanistan Algeria ... Chad Islamic cities

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

Page 5     81-99 of 99    Back | 1  | 2  | 3  | 4  | 5