Islamic Mathematics
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Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: Date
Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: ________________________________________ Date: ______________ Block: ________ Identifying Pairs of Lines and Angles Lines that intersect are coplanar Lines that do not intersect o are parallel (coplanar) OR o are skew (not coplanar) In the figure, name: o intersecting lines:___________ o parallel lines:_______________ o skew lines:________________ o parallel planes:_____________ Example: Think of each segment in the figure as part of a line. Find: Line(s) parallel to CD and containing point A _________ Line(s) skew to and containing point A _________ Line(s) perpendicular to and containing point A _________ Plane(s) parallel to plane EFG and containing point A _________ Parallel and Perpendicular Lines If two lines are in the same plane, then they are either parallel or intersect a point. There exist how many lines through a point not on a line? __________ Only __________ of them is parallel to the line. Parallel Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Perpendicular Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Angles and Transversals Interior angles are on the INSIDE of the two lines Exterior angles are on the OUTSIDE of the two lines Alternate angles are on EITHER SIDE of the transversal Consecutive angles are on the SAME SIDE of the transversal Corresponding angles are in the same position on each of the two lines Alternate interior angles lie on either side of the transversal inside the two lines Alternate exterior angles lie on either side of the transversal outside the two lines Consecutive interior angles lie on the same side of the transversal inside the two lines (same side interior) Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. -
Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. -
A Diluted Al-Karaji in Abbacus Mathematics Actes Du 10^ Colloque Maghrebin Sur I’Histoire Des Mathematiques Arabes
Actes du 10 Colloque Maghrebin sur THistoire des Mathematiques Arabes (Tunis, 29-30-31 mai 2010) Publications de 1’Association Tunisienne des Sciences Mathematiques Actes du 10^ colloque maghrebin sur I’histoire des mathematiques arabes A diluted al-Karajl in Abbacus Mathematics Jens H0yrup^ In several preceding Maghreb colloques I have argued, from varying perspectives, that the algebra of the Italian abbacus school was inspired neither from Latin algebraic writings (the translations of al-Khw5rizmT and the Liber abbaci) nor directly from authors like al-KhwarizmT, Abu Kamil and al-KarajT; instead, its root in the Arabic world is a level of algebra Actes du 10*“® Colloque Maghrebin (probably coupled to mu^Smalat mathematics) which until now has not been scrutinized systematically. sur THistoire des Mathematiques Going beyond this negative characterization I shall argue on the present Arabes occasion that abbacus algebra received indirect inspiration from al-KarajT. As it will turn out, however, this inspiration is consistently strongly diluted, (Tunis, 29-30-31 mai 2010) and certainly indirect. 1. Al-KhwSrizml, Abu Kamil and al-KarajI Let us briefly summarize the relevant aspects of what distinguishes al-KarajT from his algebraic predecessors. Firstly, there is the sequence of algebraic powers. Al-KhwarizmT [ed., trans. Rashed 2007], as is well known, deals with three powers only: census (to adopt the translation which will fit our coming discussion of abbacus algebra), roots, and simple numbers. So do ibn Turk [ed., trans. Say_l_ 1962] and Thabit ibn Qurrah [ed., trans. Luckey 1941] in their presentation of proofs for the basic mixed cases, which indeed involve only these same powers. -
The Mathematical Work of John Napier (1550-1617)
BULL. AUSTRAL. MATH. SOC. 0IA40, 0 I A45 VOL. 26 (1982), 455-468. THE MATHEMATICAL WORK OF JOHN NAPIER (1550-1617) WILLIAM F. HAWKINS John Napier, Baron of Merchiston near Edinburgh, lived during one of the most troubled periods in the history of Scotland. He attended St Andrews University for a short time and matriculated at the age of 13, leaving no subsequent record. But a letter to his father, written by his uncle Adam Bothwell, reformed Bishop of Orkney, in December 1560, reports as follows: "I pray you Sir, to send your son John to the Schools either to France or Flanders; for he can learn no good at home, nor gain any profit in this most perilous world." He took an active part in the Reform Movement and in 1593 he produced a bitter polemic against the Papacy and Rome which was called The Whole Revelation of St John. This was an instant success and was translated into German, French and Dutch by continental reformers. Napier's reputation as a theologian was considerable throughout reformed Europe, and he would have regarded this as his chief claim to scholarship. Throughout the middle ages Latin was the medium of communication amongst scholars, and translations into vernaculars were the exception until the 17th and l8th centuries. Napier has suffered badly through this change, for up till 1889 only one of his four works had been translated from Latin into English. Received 16 August 1982. Thesis submitted to University of Auckland, March 1981. Degree approved April 1982. Supervisors: Mr Garry J. Tee Professor H.A. -
Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus -
Parallel Lines and Transversals
5.5 Parallel Lines and Transversals How can you use STATES properties of parallel lines to solve real-life problems? STANDARDS MA.8.G.2.2 1 ACTIVITY: A Property of Parallel Lines Work with a partner. ● Talk about what it means for two lines 12 1 cm 56789 to be parallel. Decide on a strategy for 1234 drawing two parallel lines. 01 91 8 9 9 9 9 10 11 12 13 10 10 10 10 ● Use your strategy to carefully draw 7 67 two lines that are parallel. 14 15 5 16 17 18 19 20 3421 22 23 2 24 25 26 1 ● Now, draw a third line 27 28 in. parallel that intersects the two 29 30 lines parallel lines. This line is called a transversal. 2 1 3 4 6 ● The two parallel lines and 5 7 8 the transversal form eight angles. Which of these angles have equal measures? transversal Explain your reasoning. 2 ACTIVITY: Creating Parallel Lines Work with a partner. a. If you were building the house in the photograph, how could you make sure that the studs are parallel to each other? b. Identify sets of parallel lines and transversals in the Studs photograph. 212 Chapter 5 Angles and Similarity 3 ACTIVITY: Indirect Measurement Work with a partner. F a. Use the fact that two rays from the Sun are parallel to explain why △ABC and △DEF are similar. b. Explain how to use similar triangles to fi nd the height of the fl agpole. x ft Sun’s ray C Sun’s ray 5 ft AB3 ft DE36 ft 4. -
Al-Kindi on Psychology ABSTRACT
pg. n/a Short title of the Thesis: Al-Kindi on Psychology ABSTRACT Author: Redmond Gerard Fitzmaurice. Thesis title: Al-KindI on Psychology. Department: Institute of Islamic Studles. Degree: Master of Arts. This thesis is an examination of the extant psycholog ical treatises of Abu Yusuf YaCqub ibn Ishiq. al-KindI, the ninth century AoD. Arab scholar who was among the first of his race to interest himself in strictly philosophical quest ions. AI-KindI's writings were among the first fruits of the translation of Greek philosophical and scientific works into Arabie. It is under tbat aspect that this thesis approaches his views on soul and intellect - as an instance of the passage of Greek philosophical ideas to the Muslim Arabs. Apart from his specifically Islamic position on the nature and value of divine revelation, nearly aIl of al~indI's ideas on psychol ogy can be traced to Greek sources, and the version of that ~ philosophy with~he was directly familiar was that of the late Greek schools. This thcsis is an attempt to unders:anà and present al~KindI's psychology in the light of the Greek sources ~rom which it was derived. , A 1 - KIN DIO- N P sye H 0 LOG Y AL-KINDI- ON PSYCHOLOGY by Redmond G.' Fitzmaurice A Thesis Submitted to the Faculty of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree of Master of Arts in Islamic Studies. Institute of Islamic Studies McGill University Montrea.l July 1971 ~.' ~ecU.orri G. Fi tz:taurice ACKNOWLEDGEMENTS l wish to express my thanks to the Director and Profess ors of the Institute of Islamic Studies who overthe past three years have introduced me to many aspects of Islamic studles and also express my appreciation of what l have learned from my fellow students. -
General Index
General Index Italic page numbers refer to illustrations. Authors are listed in ical Index. Manuscripts, maps, and charts are usually listed by this index only when their ideas or works are discussed; full title and author; occasionally they are listed under the city and listings of works as cited in this volume are in the Bibliograph- institution in which they are held. CAbbas I, Shah, 47, 63, 65, 67, 409 on South Asian world maps, 393 and Kacba, 191 "Jahangir Embracing Shah (Abbas" Abywn (Abiyun) al-Batriq (Apion the in Kitab-i balJriye, 232-33, 278-79 (painting), 408, 410, 515 Patriarch), 26 in Kitab ~urat ai-arc!, 169 cAbd ai-Karim al-Mi~ri, 54, 65 Accuracy in Nuzhat al-mushtaq, 169 cAbd al-Rabman Efendi, 68 of Arabic measurements of length of on Piri Re)is's world map, 270, 271 cAbd al-Rabman ibn Burhan al-Maw~ili, 54 degree, 181 in Ptolemy's Geography, 169 cAbdolazlz ibn CAbdolgani el-Erzincani, 225 of Bharat Kala Bhavan globe, 397 al-Qazwlni's world maps, 144 Abdur Rahim, map by, 411, 412, 413 of al-BlrunI's calculation of Ghazna's on South Asian world maps, 393, 394, 400 Abraham ben Meir ibn Ezra, 60 longitude, 188 in view of world landmass as bird, 90-91 Abu, Mount, Rajasthan of al-BlrunI's celestial mapping, 37 in Walters Deniz atlast, pl.23 on Jain triptych, 460 of globes in paintings, 409 n.36 Agapius (Mabbub) religious map of, 482-83 of al-Idrisi's sectional maps, 163 Kitab al- ~nwan, 17 Abo al-cAbbas Abmad ibn Abi cAbdallah of Islamic celestial globes, 46-47 Agnese, Battista, 279, 280, 282, 282-83 Mu\:lammad of Kitab-i ba/Jriye, 231, 233 Agnicayana, 308-9, 309 Kitab al-durar wa-al-yawaqft fi 11m of map of north-central India, 421, 422 Agra, 378 n.145, 403, 436, 448, 476-77 al-ra~d wa-al-mawaqft (Book of of maps in Gentil's atlas of Mughal Agrawala, V. -
Mathematics in African History and Cultures
Paulus Gerdes & Ahmed Djebbar MATHEMATICS IN AFRICAN HISTORY AND CULTURES: AN ANNOTATED BIBLIOGRAPHY African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) Mathematics in African History and Cultures Second edition, 2007 First edition: African Mathematical Union, Cape Town, South Africa, 2004 ISBN: 978-1-4303-1537-7 Published by Lulu. Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar Authors Paulus Gerdes Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique E-mail: [email protected] Ahmed Djebbar Département de mathématiques, Bt. M 2, Université de Lille 1, 59655 Villeneuve D’Asq Cedex, France E-mail: [email protected], [email protected] Cover design inspired by a pattern on a mat woven in the 19th century by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96). 2 Table of contents page Preface by the President of the African 7 Mathematical Union (Prof. Jan Persens) Introduction 9 Introduction to the new edition 14 Bibliography A 15 B 43 C 65 D 77 E 105 F 115 G 121 H 162 I 173 J 179 K 182 L 194 M 207 N 223 O 228 P 234 R 241 S 252 T 274 U 281 V 283 3 Mathematics in African History and Cultures page W 290 Y 296 Z 298 Appendices 1 On mathematicians of African descent / 307 Diaspora 2 Publications by Africans on the History of 313 Mathematics outside Africa (including reviews of these publications) 3 On Time-reckoning and Astronomy in 317 African History and Cultures 4 String figures in Africa 338 5 Examples of other Mathematical Books and 343 -
Triangles and Transversals Triangles
Triangles and Transversals Triangles A three-sided polygon. Symbol → ▵ You name it ▵ ABC. Total Angle Sum of a Triangle The interior angles of a triangle add up to 180 degrees. Symbol = Acute Triangle A triangle that contains only angles that are less than 90 degrees. Obtuse Triangle A triangle with one angle greater than 90 degrees (an obtuse angle). Right Triangle A triangle with one right angle (90 degrees). Vertex The common endpoint of two or more rays or line segments. Complementary Angles Two angles whose measures have a sum of 90 degrees. Supplementary Angles Two angles whose measures have a sum of 180 degrees. Perpendicular Lines Lines that intersect to form a right angle (90 degrees). Symbol = Parallel Lines Lines that never intersect. Arrows are used to indicate lines are parallel. Symbol = || Transversal Lines A line that cuts across two or more (usually parallel) lines. Intersect The point where two lines meet or cross. Vertical Angles Angles opposite one another at the intersection of two lines. Vertical angles have the same angle measurements. Interior Angles An angle inside a shape. Exterior Angles Angles outside of a shape. Alternate Interior Angles The pairs of angles on opposite sides of the transversal but inside the two lines. Alternate Exterior Angles Each pair of these angles are outside the lines, and on opposite sides of the transversal. Corresponding Angles The angles in matching corners. Reflexive Angles An angle whose measure is greater than 180 degrees and less that 360 degrees. Straight Angles An angle that measures exactly 180 degrees. Adjacent Angles Angles with common side and common vertex without overlapping. -
1. Quellen 2. Geozentrische Weltbilder 2.1 Heutige Naive Vorstellung 2.2
1. Quellen 2. Geozentrische Weltbilder 2.1 Heutige naive Vorstellung 2.2 Epizykel 2.3 Exzenter Alfred Holl 2.4 Untere Planeten Weltmodelle im Mittelalter 3. Heliozentrisch-geostationär 4. Heliozentrisch-heliostationär Vom geozentrischen zum heliozentrischen Weltbild 5. Astronomische Tafeln (Erfurt CA 2° 19, 78v) Alfred Holl, Weltmodelle 09.03.2013/1 1. Quellen 1.1 Deutsche Sphära Konrad von Megenberg (~1309-1374) Deutsche Sphära wahrscheinlich zwischen 1340 und 1349 in Wien Cgm 156, Cgm 328, Graz II/470 ed. Matthaei 1912 ed. Brévart 1980 (Deutsche Sphaera, Cgm 156, 1r aus Mai, Paul: Ausstellungskatalog, 2009, 155) Alfred Holl, Weltmodelle 09.03.2013/2 1. Quellen Quellen (alle basierend auf Ptolemaios): 1.1 Deutsche Sphära Johannes von Sacrobosco (~1200-~1256) Tractatus de sphaera materiali 1233 Entstehung bis in 16. Jh. universitäre Astronomie-Einführung (ed. Thorndyke 1949, Brévart 1980) auf der Basis von al-Fargani (~863), De scientia astrorum übers. von Johannes Hispalensis (akt. 1135-1153) (ed. Campani 1910, Carmody 1943) al-Battani (-929), Opus astronomicum / Zig übersetzt von Plato von Tivoli (aktiv 1134-1145) (ed. Nallino 1903) Kommentare zu Sacrobosco (z.B. Cecco d’Ascoli) (Cgm 156, 1r) Puechlein von der Spera (~1375) (ed. Brévart 1979) Alfred Holl, Weltmodelle 09.03.2013/3 1. Quellen 1.2 Französische volkssprachliche Sphära Nicole Oresme (~1323-1382) Traité de l’espère (zwischen 1362 und 1377) (ed. Myers 1940, McCarthy 1943) ebenfalls auf der Basis von Johannes von Sacrobosco Nicole Oresme mit Armillarsphäre (www.nicole-oresme.com) (Bibliothèque Nationale, Paris, France, fonds français 565, fol. 1r) Alfred Holl, Weltmodelle 09.03.2013/4 1. Quellen 1.3 Autoren und Übersetzer Adelhard von Bath Adelhard (-1142) Gerhard von Cremona (-1187) Plato von Tivoli / Tiburtinus (-1145) Robert von Chester / Reading (-1150) / Hisp. -
In. ^Ifil Fiegree in PNILOSOPNY
ISLAMIC PHILOSOPHY OF SCIENCE: A CRITICAL STUDY O F HOSSAIN NASR Dis««rtation Submitted TO THE Aiigarh Muslim University, Aligarh for the a^ar d of in. ^Ifil fiegree IN PNILOSOPNY BY SHBIKH ARJBD Abl Under the Kind Supervision of PROF. S. WAHEED AKHTAR Cbiimwa, D«ptt. ol PhiloMphy. DEPARTMENT OF PHILOSOPHY ALIGARH IWIUSLIIM UNIVERSITY ALIGARH 1993 nmiH DS2464 gg®g@eg^^@@@g@@€'@@@@gl| " 0 3 9 H ^ ? S f I O ( D .'^ ••• ¥4 H ,. f f 3« K &^: 3 * 9 m H m «< K t c * - ft .1 D i f m e Q > i j 8"' r E > H I > 5 C I- 115m Vi\ ?- 2 S? 1 i' C £ O H Tl < ACKNOWLEDGEMENT In the name of Allah« the Merciful and the Compassionate. It gives me great pleasure to thanks my kind hearted supervisor Prof. S. Waheed Akhtar, Chairman, Department of Philosophy, who guided me to complete this work. In spite of his multifarious intellectual activities, he gave me valuable time and encouraged me from time to time for this work. Not only he is a philosopher but also a man of literature and sugge'sted me such kind of topic. Without his careful guidance this work could not be completed in proper time. I am indebted to my parents, SK Samser All and Mrs. AJema Khatun and also thankful to my uncle Dr. Sheikh Amjad Ali for encouraging me in research. I am also thankful to my teachers in the department of Philosophy, Dr. M. Rafique, Dr. Tasaduque Hussain, Mr. Naushad, Mr. Muquim and Dr. Sayed.