Mathematics of medieval europe and north africa

Sourcebook in the Mathematics of Medieval Europe and North Africa Sourcebook in the Mathematics of Medieval Europe and North Africa Edited by V I C T O R J K AT Z MENS O F OLK ERT S BA R N A BA S H U G H E S RO I WAG N E R J LE NNART BERGGRE N PRINCETON UNIVERSITY PRESS • PRINCETON AND OXFORD Copyright c 2016 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved ISBN 978-0-691-15685-9 Library of Congress Control Number: 2016935714 British Library Cataloging-in-Publication Data is available This book has been composed in Times Roman and Arial Printed on acid-free paper ∞ Printed in the United States of America Typeset by Nova Techset Pvt Ltd, Bangalore, India 10 Contents Preface Permissions General Introduction Chapter The Latin Mathematics of Medieval Europe xi xiii Menso Folkerts and Barnabas Hughes Introduction I Latin Schools, 800–1140 I-1 Brief Selections The Quadrivium of Martianus Capella The Quadrivium of Cassiodorus The Quadrivium of Isidore of Seville A New Creation: Rules for Addition of Signed Numbers 6 10 11 I-2 Numbering Roman numerals Finger reckoning Isidore of Seville, Liber numerorum (Book of Numbers) Hindu-Arabic numerals 12 12 13 18 20 I-3 Arithmetic Boethius, De institutione arithmetica (Introduction to Arithmetic) Computus “Gerbert’s jump” Pandulf of Capua, Liber de calculatione 22 22 29 36 39 I-4 Geometry Gerbert, Geometry Gerbert: Area of an equilateral triangle Units of measurement 44 44 48 49 vi Contents Franco of Liège, De quadratura circuli Hugh of St Victor, Practica geometriae I-5 Recreational Mathematics Number puzzles Thought problems The Josephus Problem The most important medieval number game: the Rithmimachia II A School Becomes a University: 1140–1480 II-1 Translations Translators and translations Translations: A practical illustration 50 53 55 55 57 58 59 64 66 66 70 II-2 Arithmetic Al-Khw¯arizm¯ı, Arithmetic Leonardo of Pisa (Fibonacci), Liber abbaci (Book on Calculation) John of Sacrobosco, Algorismus vulgaris Johannes de Lineriis, Algorismus de minuciis Jordanus de Nemore (Nemorarius), De elementis arithmetice artis Combinatorics and probability, De Vetula 75 75 79 85 93 96 100 II-3 Algebra Al-Khw¯arizm¯ı, Algebra Leonardo of Pisa, Liber abbaci (Book on Calculation) Leonardo of Pisa, Book of Squares Jordanus de Nemore, De numeris datis (On Given Numbers) Nicole Oresme, Algorismus proportionum (Algorithm of Ratios) Nicole Oresme, De proportionibus proportionum (On the Ratio of Ratios) 103 103 106 112 116 119 II-4 Geometry Ban¯u M¯us¯a ibn Sh¯akir, The Book of the Measurement of Plane and Spherical Figures Ab¯u Bakr, Liber mensurationum (On Measurement) Leonardo of Pisa, De practica geometrie (Practical Geometry) John of Murs, De arte mensurandi Jordanus de Nemore, Liber philotegni Dominicus de Clavasio, Practica geometriae 123 II-5 Trigonometry Ptolemy, On the Size of Chords in a Circle Leonardo of Pisa, De practica geometrie (Practical Geometry) Johannes de Lineriis, Canones Richard of Wallingford, Quadripartitum Geoffrey Chaucer, A Treatise on the Astrolabe Regiomontanus, On Triangles 148 149 153 155 157 160 162 121 123 126 130 140 143 146 Contents vii II-6 Mathematics of the infinite Angle of contingence Thomas Bradwardine, Tractatus de continuo (On the Continuum) John Duns Scotus, Indivisibles and Theology Does light travel instantaneously or over time? Nicole Oresme, Questiones super geometriam Euclidis (Questions on the Geometry of Euclid) 173 174 178 180 182 II-7 Statics, Dynamics, and Kinematics Robert Grosseteste, De lineis, angulis et figuris (On lines, angles and figures) Jordanus de Nemore, De ratione ponderis (On the Theory of Weights) Thomas Bradwardine, Tractatus de proportionibus William Heytesbury, Regule solvendi sophismata (Rules for Solving Sophisms) Giovanni di Casali, De velocitate motus alterationis (On the Velocity of Motion of Alteration) Nicole Oresme, De configurationibus qualitatum et motuum (On the Configurations of Qualities and Motions) 185 III Abbacist Schools: 1300–1480 184 185 186 189 191 194 197 207 III-1 Foreign Exchange 208 III-2 Geometry 209 III-3 Algebra Gilio da Siena, A Lecture in Introductory Algebra Paolo Girardi, Libro di Ragioni Jacobo da Firenze, Tractatus algorismi Master Dardi, New equations solved 210 210 211 212 213 Sources References Chapter Mathematics in Hebrew in Medieval Europe 216 221 224 Roi Wagner Introduction 224 I Practical and Scholarly Arithmetic 227 227 235 237 239 244 253 Abraham ibn Ezra, Sefer Hamispar (The Book of Number) Aaron ben Isaac, Arithmetic Immanuel ben Jacob Bonfils, On decimal numbers and fractions Jacob Canpant.on, Bar Noten T.a am Elijah Mizrah.i, Sefer Hamispar (The Book of Number) Levi ben Gershon, Ma ase H oshev (The Art of the Calculator) viii Contents II Numerology, Combinatorics, and Number Theory Abraham ibn Ezra, Sefer Ha eh.ad (The Book of One) Abraham ibn Ezra, Sefer Ha olam (Book of the World) Levi ben Gershon, Ma ase H oshev (The Art of the Calculator) Levi ben Gershon, On Harmonic Numbers Qalonymos ben Qalonymos, Sefer Melakhim (Book of Kings) Don Benveniste ben Lavi, Encyclopedia Aaron ben Isaac, Arithmetic III Measurement and Practical Geometry Abraham ibn Ezra (?), Sefer Hamidot (The Book of Measure) Abraham bar H.iyya, H ibur Hameshih.a Vehatishboret (The Treatise on Measuring Areas and Volumes) Rabbi Shlomo Is.h.aqi (Rashi), On the Measurements of the Tabernacle Court Simon ben S.emah., Responsa 165 concerning Solomon’s Sea Levi ben Gershon, Astronomy IV Scholarly Geometry Levi ben Gershon, Commentary on Euclid’s Elements Levi ben Gershon, Treatise on Geometry Qalonymos ben Qalonymos, On Polyhedra Immanuel ben Jacob Bonfils, Measurement of the Circle Solomon ben Isaac, On the Hyperbola and Its Asymptote Abner of Burgos (Alfonso di Valladolid), Sefer Meyasher Aqov (Book of the Rectifying of the Curved) V Algebra Quadratic word problems Simon Mot.ot., Algebra Ibn al-Ah.dab, Igeret Hamispar (The Epistle of the Number) Sources References Chapter Mathematics in the Islamic World in Medieval Spain and North Africa 268 269 271 273 277 283 284 285 286 287 296 313 315 320 326 326 335 337 339 340 345 354 354 358 362 374 376 381 J Lennart Berggren Introduction 381 I Arithmetic 385 385 Ibn al-Bann¯a , Arithmetic Al¯ı b Muh.ammad al-Qalas.a¯ d¯ı, Removing the Veil from the Science of Calculation Muh.ammad ibn Muh.ammad al-Full¯an¯ı al-Kishn¯aw¯ı, On magic squares 398 407 Contents ix II Algebra Ah.mad ibn al-Bann¯a , Algebra Muh.ammad ibn Badr, An Abridgement of Algebra III Combinatorics Ah.mad ibn Mun im, Fiqh al-h.is¯ab (On the Science of Calculation) Ibn al-Bann¯a on Combinatorics, Raising the Veil Shih¯ab al-D¯ın ibn al-Majd¯ı, On enumerating polynomial equations IV Geometry Ab¯u Abd Allah Muh.ammad ibn Abd¯un, On Measurement Ab¯u al-Q¯asim ibn al-Samh., The Plane Sections of a Cylinder and the Determination of Their Areas Ab¯u Abd Allah Muh.ammad ibn Mu a¯ dh al-J¯ayy¯an¯ı, On ratios Al-Mu taman ibn H¯ud, Kit¯ab al-Istikm¯al (Book of Perfection) Muh.y¯ı al-D¯ın ibn Ab¯ı al-Shukr al-Maghrib¯ı, Recension of Euclid’s Elements V Trigonometry Ab¯u Abd Allah Muh.ammad ibn Mu a¯ dh al-Jayy¯an¯ı, Book of Unknowns of Arcs of the Sphere Ab¯u Abd Allah Muh.ammad ibn Mu a¯ dh al-Jayy¯an¯ı, On Twilight and the Rising of Clouds Ab¯u Abd Allah Muh.ammad ibn Mu a¯ dh al-Jayy¯an¯ı, On the qibla Ibr¯ah¯ım ibn al-Zarq¯alluh, On a universal astrolabe Ab¯u Muh.ammad J¯abir ibn Aflah., Correction of the Almagest Sources References 410 410 422 427 427 446 449 452 452 456 468 478 494 502 502 520 530 533 539 544 546 Appendices Appendix Byzantine Mathematics Maximus Planudes, The Great Calculation According to the Indians Manuel Moschopoulos, On Magic Squares Isaac Argyros, On Square Roots Anonymous fifteenth-century manuscript on arithmetic Sources 549 551 554 559 561 562 Appendix Diophantus Arithmetica, Book I, #24 563 ¯ Appendix From the Ganitasarasangraha of Mahavira 563 Appendix Time Line 564 Editors and Contributors Index 567 571 560 Appendices to explain; it comes from the first principles of the ancient methods, but it has been corrected, in order to get the greatest exactitude possible, and with the help of God we used these methods themselves This method is the following: For every given number which is not a perfect square, we find the square root this way We take the nearest square number, either more than the given one or less The root of this square number is clear, since it is an integer; we take this value and write it down Next, we multiply by two the root obtained and make this result the denominator of a fraction We add to the root written down a fraction of which the numerator is the number of units by which the number whose root we are finding exceeds the nearest square number, if that number is greater If it smaller, we subtract the fraction Suppose, for example, that we are looking for the square root of The number is between and 9, but it is closer to than to It exceeds that one, in fact, by two units, and it is short of the other by three If we had wanted to find the root of 7, we would have been closer to than to To find the root of 6, we have taken the square as the closest square, and we write down the root, that is, Next, we multiply the by two, giving 4, and this value gives to the fractions the value of quarters Since exceeds by two units, we add to the root written down, that is to say, to two units, two quarters, that is, a half You should know, in fact, that if the added or subtracted fractions have a numerator greater than a unit and if it is possible to reduce this to a unit fraction, as here we have reduced two quarters to a unit fraction, that is, to a half, we must always that We find thus as a first approximation greater than the root of six, units and a half This approximation is greater, since if one multiplies two units by itself and by a half, and then by a half again, one obtains the number But it is also necessary to multiply a half by itself In performing this multiplication, one gets a quarter, and one obtains a number that exceeds We have established this coarse method as a foundation; it will be corrected, with the help of God, by another method we have derived and one of the greatest exactitude We must now describe this method Recording the number from which is derived the denominator of the fraction or fractions, that which exceeds the number for which one is finding the root, we begin with the method previously explained Multiply by two the root obtained and write down the result of the multiplication by two, then multiply the numbers together Take the number obtained as the denominator of a fraction, then subtract from the aforesaid root a unit fraction with this number as denominator After this subtraction, we have a new root of the number for which we are finding the root, closer to the exact root We know definitively that this root will be closer, since if one multiplies by itself this root that we have called exact, a part of the number obtained adjusted exactly with the proposed number, but the fraction subtracted according to the proposed method multiplied by itself forms a fraction infinitely smaller than the preceding, which one finds newly in excess by the ratio to the number for which we are seeking the root Since this fraction is an essentially negligible quantity because of its smallness, we not consider it This calculation will be clearer by an example To find the root of 6, we have found this to be units and a half, and if one multiplies this by itself, it makes 14 This 1/4 is the amount that the square exceeds We write the number 4, which is the denominator of the fraction one quarter We multiply by two the [first] root 12 and we write down the result, We multiply the by the to get 20, and we set this number as the denominator of a fraction 1/20 We subtract this fraction Appendices 561 from 12 The root nearer to the exact root is two units and twentieths The half of a unit is in fact 10 twentieths If one subtracts one twentieth, there remains twentieths The product of this number by itself is and one four hundredth We can neglect this fraction as very small and imperceptible But if we consider that it is not sufficient to stop here, and we want to continue, we may find a more precise value by beginning again in the same fashion by writing 400 Then, we multiply the second root by two, that is, the units and twentieths, and then multiply the result, and 18 twentieths by 400, and we set the result 1960 as the denominator of the fraction one nineteen hundred sixtieth We subtract this from the second root, that is, from two and twentieths, and we find a value more precise than the second, since that one had an excess [when squared] of one four hundredth, which, we have said, may be neglected because of its smallness, but this latter one has an excess of one nineteen hundred sixtieth of one nineteen hundred sixtieth, that is, approximately one three thousand eight hundred thousandths Since this value will continue to increase if we continue the process further, this is why in beginning our proposal we have said that it is impossible to find the exact root of a number which is not a perfect square ANONYMOUS FIFTEENTH-CENTURY MANUSCRIPT ON ARITHMETIC This arithmetic manuscript was brought to Vienna from Constantinople in the midsixteenth century, but the evidence indicates that it was written in the Byzantine Empire before the fall of the capital in 1453 The book is a collection of problems, many with solutions, accomplished by various methods Many of the problems are old and have appeared in other problem compilations from such diverse times and locales as China at the beginning of our era, fifth-century India, and medieval Islam 18 Two people each want to buy silk fabric They walked together to the market and found two single pieces at a merchant, for 40 florins for both They negotiated to buy one piece without the other, but he would only sell as a whole He told them that one was worth a certain amount and the other 12 times that and still florins more You now want to know how much each piece is worth If one subtracts the from the 40, there remains 32; then use the rule of three The first amount and the second (the 12 times) together give 12 Then say, if the 12 gives 32, what will 12 give? It gives 19 15 florins Now add the 8, and it gives you that one piece of silk fabric was worth 27 15 , the other just 12 45 florins Check that there are 40 florins in total 45 A man became ill and he made a will, and his fortune was found to be in the amount of 1000 florins His wife was pregnant, and he said, if my wife bears a boy, then the child shall have one share of my fortune and my wife two shares; if she, however, bears a girl, then my wife shall have one share of my fortune and the daughter two She gave birth to twins, one boy and one girl I want you to find the share of the inheritance coming to each, according to the available information It is necessary that for the boy, you make a half of the share of the mother and for the daughter twice as much as the mother Therefore take for the share of the son 1, and of the mother and of the daughter Add these, the and and 4, and you have parts, and this is the divisor Multiply the parts with the sum, namely the 1000 florins, and it will be 2000 and divide by the 7, and thus the mother’s share is 285 57 Multiply 1000 by the shares of the daughter and you get 4000; 562 Appendices divide by 7, and there will be 571 37 Finally, multiply the one share of the son by 1000, and it remains as 1000 Then divide this by 7, and the son gets 142 67 46 In a meadow there were girls dancing, and a man went by and greeted them and said: There are 100 girls dancing And one of those answered and said: We are not 100, but if we were again as much as we are, and then a half and a quarter more, and with you together, there would be 100 I want to know how many there were Do it thus: Take the smallest number, which contains a half and a quarter, and that is Now make the basis Also, since one said again as many as we are and the half and the quarter, so this with the and take another and half of and a quarter of 4, and this gives 11 But she also said, with him there will be 100, so there remain 99, and then you will later add him So divide the 99 by 11, and there results Then multiply the by the 4, and there results 36, so there were 36 girls 53 A man told his servant that he should buy birds of three types, pigeons, turtledoves, and sparrows Each pigeon cost coins, each turtledove coins, and one can get three sparrows for one coin He gave him 100 coins that he should buy 100 birds, neither more nor fewer I want to know, how many of each type of bird he will buy Do it thus: begin with the birds of smallest value, namely the sparrows, and say: Suppose that he took 100 sparrows for 33 13 coins, and there remain to 100, 66 23 coins Make everything in thirds, and there are 200/3 Look at how much money more the turtledoves cost than the sparrows, and you need 5/3 more for each, and these are the pigeons Find now also the turtledoves Divide the 200 by 5, and this is 40 Then look at how much more the pigeons cost than the sparrows, and they are 11/3 more, and these 11 must be subtracted from 40, and there remain 29, and these 29 are the turtledoves; and there remain 66 sparrows 59 A man wishes to build a house, and a builder said that he could this in 24 days A second builder said that he could it in 12 days; another one said, I can build it in days In how many days can the three builders this together? The first builds a house in 24 days, the second can build houses in 24 days, and the third can in 24 days build houses It follows that they can build the house together in 37 days Do it thus: If they can build houses in 24 days, in how many days can one build house? Thus it happens that the three builders can this in 37 days 71 We want to find a number, that if you multiply it by 12 and then divide it by 36, then the result is 64 Do it thus: Multiply the 36 by the 64, and you get 2304, and this number is such that if you divide it by 36, then the quotient is 64 Further you say, that you divide this by 12 Thus divide the 2304 by 12 and the result is 192, and this is the very number which you can multiply by 12 and then divide by 36, and get the quotient 64 SOURCES Maximus Planudes, The Great Calculation According to the Indians, translation in Peter G Brown 2006 “The Great Calculation According to the Indians, of Maximus Planudes,” Convergence, 3, http://www.maa.org/publications/periodicals/convergence/the-great-calculation-according-to-the-indiansof-maximus-planudes-introduction Appendices 563 Manuel Moschopoulos, On Magic Squares, translation in Peter G Brown 2005, “The Magic Squares of Manuel Moschopoulos,” Convergence 2, http://www.maa.org/publications/periodicals/convergence/ the-magic-squares-of-manuel-moschopoulos-introduction Isaac Argyros, On Square Roots, translated by Victor Katz from the French translation in André Allard 1978, “Le petit traité d’Isaac Argyre sur la racine carrée,” Centaurus 22, 1–43 Anonymous fifteenth-century Byzantine manuscript, translated by Victor Katz from the German translation in Herbert Hunger and Kurt Vogel 1963 Ein Byzantinisches Rechenbuch des 15 Jahrhunderts: Text, Übersetzung und Kommentar, Vienna: Herman Böhlaus Nachf APPENDIX Diophantus Arithmetica, Book I, #24 To find three numbers such that, if each receives a given fraction of the sum of the other two, the results are equal Let the first receive 1/3 of (second + third), the second 1/4 of (third + first); and the third 1/5 of (first + second) Assume the first is x, and, for convenience sake, take the sum of the second and third as a number of units divisible by 3, say Then the sum of the three is x + 3, and the first +1/3 (second + third) is x + Therefore, the second + 1/4(third + first) is x + 1; hence times the second plus the sum of all is 4x + 4, and therefore the second is x + 1/3 Lastly, the third +1/5(first + second) is x + 1, or times the third plus the sum of all is 5x + 5, and the third is x + 1/2 Therefore x + (x + 1/3) + (x + 1/2) = x + 3, and x = 13/12 The numbers, after multiplying all by the common denominator, are 13, 17, 19 Source From Thomas L Heath 1964 Diophantus of Alexandria: A Study in the History of Greek Algebra New York: Dover (reprint of 1910 edition published by Cambridge University Press) APPENDIX ¯ From the Ganitasarasangraha of Mahavira From Chapter VI The rule for arriving at the value of the money contents of a purse which, when added to what is on hand with each of certain persons, becomes a specified multiple of the sum of what is on hand with the others: 233–235: The quantities obtained by adding one to each of the specified multiple numbers in the problem, and then multiplying these sums with each other, giving up in each case the sum relating to the particular specified multiple, are to be reduced to their lowest terms by the removal of common factors These reduced quantities are then to be added Thereafter, the square root1 of this resulting sum is to be obtained, from which one is This makes no sense mathematically In terms of the problem in verses 236–237, the value needed is 564 Appendices to be subsequently subtracted Then the reduced quantities referred to above are to be multiplied by this square root as diminished by one Then these are to be separately subtracted from the sum of those same reduced quantities Thus the moneys on hand with each of the several persons are arrived at These quantities measuring the moneys on hand have to be added to one another, excluding from the addition in each case the value of the money on the hand of one of the persons; and the several sums so obtained are to be written down separately These are then to be respectively multiplied by the specified multiple quantities mentioned above; from the several products so obtained the already found out values of the moneys on hand are to be separately subtracted Then the same value of the money in the purse is obtained separately in relation to each of the several moneys on hand An example in illustration thereof: 236–237: Three merchants saw dropped on the way a purse containing money One of them said to the others, “If I secure this purse, I shall become twice as rich as both of you with your moneys on hand.” Then the second of them said, “I shall become three times as rich.” Then the other, the third, said, “I shall become five times as rich.” What is the value of the money in the purse, as also the money on hand with each of the three merchants? Source From Mah¯av¯ıra, 1912, Ganit¯asarangraha, edited and translated by M Rang¯ac¯arya Madras: Government Press APPENDIX Time Line What follows is a listing of the dates of every identifiable author whose work we have used in this sourcebook Note that most dates are approximate Claudius Ptolemy Martianus Capella Boethius Aurelius Cassiodorus Isidore of Seville Bede the Venerable Alcuin of York Muh.ammad al-Khw¯arizm¯ı Ban¯u M¯us¯a ibn Sh¯akir Ab¯u Abdallah ibn Abdun Gerbert of Aurillac Ibn al-Samh Ibn Mu a¯ dh al-Jayy¯an¯ı Al-Mu taman ibn H¯ud 100–170 370–440 480–524 490–583 560–636 672–735 735–804 780–850 ninth century 923–976 945–1003 984–1035 mid-eleventh century d 1085 Appendices Franco of Liége Pandulf of Capua Ibrahim ibn al-Zarq¯alluh Shlomo Ishaqi Abraham bar H.iyya Adelard of Bath Plato of Tivoli Abraham ibn Ezra Hugh of St Victor Hermann of Carinthia Gerard of Cremona Robert of Chester Domingo Gundisalvo Jabir ibn Aflah Ah.mad ibn Mun im al- Abdari Leonardo of Pisa Robert Grosseteste John of Sacrobosco Jordanus de Nemore William of Lunis William of Moerbeke Muh.y¯ı al-D¯ın ibn Ab¯ı al-Shukr al-Maghrib¯ı Roger Bacon Campanus of Novara Witelo Ibn Badr Maximus Planudes Ah.mad ibn al-Bann¯a Manuel Moschopoulos John Duns Scotus Abner of Burgos Giovanni Villani Qalonymos ben Qalonymos Levi ben Gershon Thomas Bradwardine Richard of Wallingford Isaac Argyros Immanuel ben Jacob Bonfils Johannes de Lineriis John of Muris William Heytesbury Nicole Oresme Dominicus de Clavasio Giovanni di Casali Paolo Girardi d 1083 late eleventh century 1029–1099 1040–1105 1065–1145 1080–1152 early twelfth century 1089–1167 1096–1141 1100–1160 1114–1187 mid-twelfth century mid-twelfth century mid-twelfth century d 1228 1170–1240 1175–1253 1195–1236 early thirteenth century early thirteenth century 1215–1286 d 1290 1219–1292 1220–1296 1230–1300 thirteenth century(?) 1255–1305 1256–1321 1265–1315 1266–1307 1270–1348 1278–1348 1287–1329 1288–1344 1290–1349 1291–1336 1300–1375 1300–1377 early fourteenth century early fourteenth century 1313–1373 1320–1382 mid-fourteenth century mid-fourteenth century mid-fourteenth century 565 566 Appendices Jacobo da Firenze Master Dardi Geoffrey Chaucer Isaac ibn al-Ah.dab Simon ben S.emah Don Benveniste ben Lavi Jacob Canpant.on Ibn al-Majd¯ı Aaron ben Isaac Ali al-Qalas.a¯ d¯ı Simon Mot.ot Regiomontanus Elijah Mizrah.i Solomon ben Isaac Muh.ammad al-Kishn¯aw¯ı mid-fourteenth century mid-fourteenth century 1343–1400 1350–1430 1361–1444 late fourteenth century late fourteenth century d 1447 mid-fifteenth century mid-fifteenth century mid-fifteenth century 1436–1476 1450–1526 early sixteenth century d 1741 Editors and Contributors EDITORS VICTOR J KATZ is Professor of Mathematics Emeritus at the University of the District of Columbia He has long been interested in the history of mathematics and its use in teaching His most recent book, written with Karen Parshall, is Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century, which appeared in 2014 The third edition of his well-regarded textbook, A History of Mathematics: An Introduction, appeared in 2008 Katz is also the editor of The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, published in 2007 Professor Katz has edited three books dealing with the use of the history of mathematics in the teaching of the subject as well as two collections of historical articles taken from journals of the Mathematical Association of America in the past 90 years He has directed two NSF-sponsored projects to help college teachers learn the history of mathematics and its use in teaching and also involved secondary school teachers in writing materials demonstrating this use in the high school curriculum These materials, Historical Modules for the Teaching and Learning of Mathematics, were published on a CD in 2005 Professor Katz was also the founding editor of Convergence, the Mathematical Association of America’s online magazine on the history of mathematics and its use in teaching MENSO FOLKERTS studied mathematics and classical philology at the University of Göttingen and received his Ph.D in 1967 with a dissertation on a medieval mathematical text In 1973 he received his “Habilitation” in the history of exact sciences and technology at the Technical University of Berlin He was Associate Professor of Mathematics and Its History at the University of Oldenburg (1976–1980) and Full Professor of History of Science at the University of Munich (1980–2008) His main research area is mathematics in the Middle Ages and in early modern times Folkerts has edited numerous medieval mathematical texts in Latin and has published ten books and about 200 articles in the history of science He is the editor of two series and co-editor of six journals in the history of science He has been the president of the German Society of the History of Medicine, Science and Technology, member of the German National Committee in the International Union of the History and Philosophy of Science, and member of the Executive Committee of the International 568 Editors and Contributors Commission on the History of Mathematics He is member of four German academies, including the German National Academy of Sciences Leopoldina, and of the Académie Internationale d’Histoire des Sciences In 2013 he was awarded the Kenneth O May Medal and Prize for outstanding contributions to the history of mathematics BARNABAS B HUGHES, O.F.M., began his interest in the history of mathematics as an aid to the teaching of mathematics to high school students Both they and he found the subjects more interesting because of historical anecdotes from the history of mathematics and its originators During his 40 years at California State University, Northridge, he published numerous articles and half a dozen books on topics from its history, the last being this cooperative work on sources in medieval mathematics ROI WAGNER has a mathematics Ph.D (1997) and a philosophy Ph.D (2007) from Tel Aviv University He taught mathematics and philosophy in Tel Aviv University, the Hebrew University of Jerusalem, and the Academic College of Tel Aviv Jaffa, and held postdoc and visiting positions in Paris 6, Cambridge University, the Max Planck Institute for the History of Science, and the Buber Society at the Hebrew University He publishes on the history and philosophy of mathematics (usually applying semiotic perspectives) as well as political philosophy (focusing on resistance studies) His first book, S(zp,zp): Post Structural Readings of Gödel’s Proof, was published in 2009, and his new book, Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice will be published in 2016 by Princeton University Press Roi Wagner is currently Professor for the History and Philosophy of Mathematical Sciences at the ETH Zurich J LENNART BERGGREN is Emeritus Professor in the Department of Mathematics at Simon Fraser University, Canada, where he taught mathematics and its history for forty years He has held visiting positions in the Mathematics Institute at the University of Warwick, and the History of Science Departments at Yale and Harvard Universities He has published sixty refereed papers and authored or co-authored eight books on the history of mathematics His special interest is the history of mathematical sciences in ancient Greece and medieval Islam, including cartography, spherical astronomy, and such mathematical instruments as sundials and astrolabes Among his books are Episodes in the Mathematics of Medieval Islam (1986), which has been translated into Farsi and German, Euclid’s Phænomena (with Robert Thomas, 1996; reprinted, 2006), and Pi: A Sourcebook (third edition, 2004, with J Borwein and P Borwein) He is a member of the Canadian Society for the History and Philosophy of Science (of which he has served three two-year terms as President) CONTRIBUTORS IMMO WARNTJES is lecturer in Irish Medieval History at Queen’s University Belfast, United Kingdom He is a graduate of the University of Göttingen (Germany) in both History and Mathematics and holds a Ph.D in Medieval Studies from the National University of Ireland, Galway His primary research interest is early medieval scientific thought, and he is a specialist in the science of computus (medieval time-reckoning) in the Latin West, ca AD 400–1200 Editors and Contributors 569 NAOMI ARADI received her Ph.D in 2016 from the Hebrew University of Jerusalem, Department of History and Philosophy of Science Her dissertation is on the Arithmetic of the Jews in the Middle Ages AVINOAM BARANESS was born and lives in Israel His closest family members serve as models and inspiration for his studies, especially his grandfather, who dedicated himself to the Torah and meditated on the Bible and Talmud day and night; his mother, who devoted herself to the Bible and Hebrew literature; and his father, who studies and teaches mathematics and Hebrew grammar Avinoam attended the Hebrew University of Jerusalem and Lifschitz College and earned his bachelor’s degree in Talmud and mathematics, with a minor in musicology He holds an M.S in mathematics teaching from the Hebrew University, in which he is currently a Ph.D student in the department of the history and philosophy of science His doctoral research is focused on Medieval and early Modern Hebrew treatises dealing with geometry He has a number of publications on the topic of Talmud, which appear in Ha’Me’ir La’aretz, the yearbook of Ha’me’iry Yeshiva Currently he is working with Professor Ruth Glasner on an English commented translation of Rectifying the Curve by Abner of Burgos, a section of which appears in this Sourcebook DAVID GARBER graduated from Bar-Ilan University in 1994, and finished his Ph.D in Mathematics in Bar-Ilan University in 2001 He is now a Senior Lecturer at Holon Institute of Technology, Israel His main interests are geometric topology (combinatorics and topology of plane curves), algebraic combinatorics, and algebraic cryptography His additional area of interest is mathematical discussions in Rabbinical writings, and he has written numerous papers on this topic in both Hebrew and English STELA SEGEV studied Mathematics and Science Education at the Hebrew University of Jerusalem (B.S and M.S.) She taught mathematics in high school, instructed mathematics teachers, and wrote several brochures and manuals on mathematics for secondary high schools Later she became interested in the history of mathematics and in particular in Hebrew mathematicians in the Middle Ages She received a Ph.D from the Hebrew University of Jerusalem The subject of her dissertation was The Book of the Number, written by Elija Mizrah.i at the beginning of the sixteenth century in Constantinople Currently she is the head of the Mathematics Department at Herzog College (near Jerusalem) She is also continuing her research with the goal of publishing a critical edition of Mizrah.i’s book SHAI SIMONSON is currently professor of computer science at Stonehill College His research interests are in theoretical computer science (algorithms and complexity theory, grammars, and automata), computer science education, math education, and history of mathematics Shai is the author of the text Java, From the Ground Up and the director of ArsDigita University, which offers free computer science lectures to students all over the world, especially in developing countries Shai has a particular interest in mathematics education at the middle school and high school levels He designed an innovative mathematics curriculum for the South Area Solomon Schechter Day School in Norwood, which led to his book Rediscovering Mathematics Sponsored by the NSF, Shai spent 1999 at the Hebrew University working on Ralbag and Ma’ase Hoshev 570 Editors and Contributors ILANA WARTENBERG studied mathematics (B.S., M.S.), linguistics (M.A.), as well as history and philosophy of science (Ph.D.) in Tel Aviv and Paris She taught mathematics and logic at universities in Israel She also worked as a linguist in the hi-tech industry in Israel In the past fourteen years she has specialized in medieval Hebrew science, mainly in mathematics and the reckoning of the Jewish and other calendars She also researches the connection between Hebrew and Arabic science and the creation and evolution of the Hebrew scientific terminology Her past and future publications focus on the treatises of Isaac ibn al-Ah.dab (Iggeret ha-Mispar/The Epistle of the Number, composed in Sicily at the end of the fourteenth century), Jacob ben Samson (Sod ha-Ibbur/The Calculation of the Jewish Calendar, composed in northern France in 1123), Abraham bar H.iyya (Sefer ha-Ibbur/The Book on the Jewish Calendar, composed in southern France in 1123), Isaac Israeli (Yesod Olam/The Foundation of the World, composed in Toledo in 1310), and various shorter tracts and Genizah fragments in Hebrew and Judaeo-Arabic She works as a research associate at the Department of Hebrew and Jewish Studies at University College London Index Page numbers in italics refer to figures and tables Aaron ben Isaac, 225, 227, 235–237, 285–286 abacus table, 21–22, 23, 39–44 abbacist schools, 5–6, 207–216 Abbo of Fleury, 13 ‘Abd al-Rah.m¯an ibn Sayyid, 383 Abner of Burgos, 225, 326, 345–353 Ab¯u al-Q¯asim al-Qurash¯ı, 384 Ab¯u al-Salt, 384 Ab¯u Bakr, 68, 126–130 Ab¯u Sahl of Qairaw¯an, 384 Adelard of Bath, 67, 71–73 Alcuin of York, 30, 34–36, 57–59 Alfonso di Valladolid See Abner of Burgos algebra: linear equations in, 80–83, 106–107, 251–253, 266–268, 289, 419–420, 425–427 (see also false position, method of); operations in, 109–110, 120–121, 210, 215–216, 369–374, 413–414, 416–418, 422–424; quadratic equations in (see quadratic equations in mathematical texts); terms of, 104, 358–359; 367–369, 412; use of, in solving triangles, 165–166 algebra in mathematical texts: by Gilio da Siena, 210–211; by ibn al-Ah.dab, 362–374; by ibn Badr, 422–427; by ibn al-Bann¯a’, 410–422; by Jacobo da Firenze, 212–213; by Jordanus de Nemore, 116–119; by al-Khw¯arizm¯ı, 104–106; by Leonardo of Pisa, 106–112, 131–134; by Master Dardi, 213–216; by Nicole Oresme, 119–123; by Paolo Girardi, 211–212; by Simon Mot.ot., 358–362 Almagest of Ptolemy, 68, 149–153, 254, 526 amicable numbers, 268, 283–286 angle of contingence, 174–178 Apollonius, 326, 349, 460, 486–487, 497, 537 Archimedes, 50, 67, 69, 124, 135, 140–141, 143–144, 179, 186, 457–458, 465, 471 Argyros, Isaac, 550, 559–561 Aristotle, 50, 67, 69, 178, 181, 183–184, 189, 336, 471 arithmetic (calculations): addition in, 87, 389–390, 405–406; division in, 90, 396, 400–405; fractions in, 12–15, 93–96, 230–231, 234, 237–239, 248–251, 260–261, 397–398, 406–407; multiplication in, 40–44, 89, 245, 261–262, 394–395, 400–402; progressions in, 91; root extraction in, 91–93, 233–235, 240–243, 552–554; of signed numbers, 11–12; squares in, 245–247; subtraction in, 87–88, 260–261, 393–394, 399–400; sums of sequences in, 247–248, 262–265, 390–392 arithmetic (theory): amicable numbers in, 268, 283–286; even and odd numbers in, 25–26; multiplication in, 255–256; perfect numbers in, 24–25, 27–29, 392–393; prime numbers in, 27; ratio and proportion in, 60–65, 98–100, 119–123, 256–257; sequences in, 257–259, 277–286, 428–432, 446–447; square numbers in, 97–98 astrolabe, 54–56; 160–162, 293–296, 533–539 Averroes (ibn Rushd), 178, 189 Bacon, Roger, 183–184 Ban¯u M¯us¯a, 68, 123–126, 339 Bar H.iyya, Abraham, 129–130, 135, 225, 286, 296–313, 315–316 Bede the Venerable, 13–14, 17–18, 30, 43, 55–57 Benveniste ben Lavi, Don, 284 Bible: Deuteronomy, 298; Exodus, 236, 254, 314; Habakkuk, 298; Isaiah, 20, 297–298, 319, 345; Kings 1, 316–317, 319; Leviticus, 297, 299, 319; Numbers, 298; Wisdom of Solomon, 6, 10, 40 572 Index Boethius, Anicius Manlius Severinus, 4, 6, 12, 18, 44–45, 49, 60, 66, 92; Arithmetic, 22–29, 36–39, 96 Bonfils, Immanuel, 225, 227, 237–239, 326, 339–340 Bradwardine, Thomas, 119, 176–180, 189–194 Byzantine mathematics, 549–562 Campanus of Novara, 72, 75–76, 121, 142, 174–176 Canpanton, Jacob, 225, 227, 239–243 Capella, Martianus Felix, 6–9 Casali, Giovanni di, 194–197 Cassiodorus, M Aurelius, 6, 9–10, 30–34 Ceva, Giovanni, 482–484 Charlemagne, 4, 57 Chaucer, Geoffrey, 160–162 combinations and permutations, 100–103, 201, 271–277, 434–448 computus, 29–36 conchoid of Nicomedes, 347–353 Cosines, 170–171; law of, for right triangles, 532, 540; plane law of, 129–130; spherical law of, 172–173 counts: polynomial equations, 449–451; words in Arabic alphabet, 434–446 cubic equations, 214–215 Dardi, Master, 213–216 De Vetula,100–103 Dionysius Exiguus, 30–34 Diophantus, 3, 82, 259, 354, 549, 563 division of areas, 135–139, 143–144, 310–313 Dominicus de Clavasio, 146–148 Duns Scotus, John, 180–182 Easter calculation, 31 See also computus Efrayim, Enbellshom, 315 Elements of Euclid, 50, 67, 70, 116, 135, 178, 184, 244, 549; Book I, 71–75, 125, 136–139, 142, 146–147, 164, 167, 175, 180–181, 291, 311–313, 326–336, 341, 488, 493, 495–496; Book II, 111, 129–132, 146, 150, 158–159, 245–246, 297, 299–303, 305, 324, 341–342, 360; Book III, 152, 158, 174, 297, 304, 342, 487, 489, 491, 526, 529; Book IV, 124, 151, 479, 488; Book V, 126, 158, 327–328, 468–478, 487–488, 492, 500, 524; Book VI, 144, 150, 152, 164, 167, 341–342, 353, 473, 490, 500, 524; Book VII, 96, 255, 387–388, 392, 394; Book VIII, 245, 255; Book IX, 23, 255, 392; Book X, 121, 181; Book XI, 167, 353, 462; Book XII, 500; Book XIII, 150–151, 337, 497–498, 500, 502; Book XIV, 337, 497; Book XV, 497 false position, method of, 80, 107–110, 231–232, 235, 251, 321–323, 363–367, 410–412 Fibonacci See Leonardo of Pisa figured numbers, 427–432, 433 finger reckoning, 13–14, 16, 17–18, 43 Franco of Liège, 51–53 geometry (measurement), 47–48, 287; of circles, 50–53, 134–135, 291–292, 306–308, 316–320, 339–340, 455–456; of ellipses, 467–468; of heights and distance, 53–55, 146–147, 209–210, 293–296; of quadrilaterals, 126–128, 131–134, 289–291, 300–304, 452–453; of sloping terrain, 307–310; of solid figures, 140–143, 147–148, 209, 293, 316–320, 454; of triangles, 48–49, 128–130, 209–210, 287–289, 304–306, 453–454 geometry (theory): of Alhazen’s problem, 485–494; of angle trisection, 351; of circles, 339–340; of the conchoid, 347–351; definitions, 44–48; of the ellipse, 457–468; of the heptagon, 144–145; of the hyperbola, 340–344; of lunes, 346–347; of mean proportionals, 351–353, 484–485; of optics, 485–494; of the parallel postulate, 326–336, 495–496; of polyhedra, 140–141, 337–339, 353, 497–502; of quadrilaterals, 135–139, 310–313; of ratios, 468–478, 480–482; of triangles, 123–126, 143–144, 478–480, 482–484 Gerard of Cremona, 66–68, 71, 73–74, 103, 124, 126, 149, 468 Gerbert of Aurillac (Pope Sylvester II), 21–22, 36–39, 44–50 Gilio da Siena, 210–211 Girardi, Paolo, 211–212 graphs, 196–207 Grosseteste, Robert, 70–71, 179, 185–186 Gundisalvo, Domingo, 66–67 al-H.akam, 382, 456 al-H.as.s.a¯ r, Ab¯u Bakr Muh.ammad, 385 heat, 194–197 Hermann of Carinthia, 66, 71, 73 Heron’s theorem, 124–126, 129, 287, 478–480 Heytesbury, William, 191–194 Hindu-Arabic number system, 20–22, 21, 23–24, 39–44, 40, 76–78, 86–87, 228–229, 236–239, 387–389, 551–552 Hippocrates, 346 Hugh of St Victor, 11, 18, 53–55, 64–65 ibn Abd¯un, Ab¯u ‘Abd Allah Muh.ammad, 130, 452–456 ibn Aflah., Ab¯u Muh.ammad J¯abir, 68, 539–544 ibn al-Ah.dab, Isaac, 225, 354, 362–374 ibn Badr, Muh.ammad, 422–427 ibn al-Bann¯a’, Ah.mad, 225, 354; Fundamentals and Preliminaries for Algebra, 384, 415–422; Raising the Veil on the Various Procedures of Calculation, 385–398, 446–448; A Summary Index 573 Account of the Operations of Computation, 362–374, 385–399, 410–414 ibn Ezra, Abraham, 59, 130, 225, 236, 550; Book of Measure, 286–296; Book of Number, 227–235, 245, 287; Book of One, 268–271; Book of the World, 271–273 ibn al-Haytham (Alhazen), 182–183, 408, 485–494 ibn H¯ud, al-Mu’taman, 383, 478–494 ibn al-Jayy¯ab, 384 ibn al-Majd¯ı, Shih¯ab al-D¯ın, 383, 449–451 ibn Mu’¯adh al-J¯ayy¯an¯ı, Abu ‘Abd Allah Muh.ammad: Book of Unknowns of Arcs of the Sphere, 502–520; On the Qibla, 530–533; On Ratios, 468–478; On Twilight and the Rising of Clouds, 520–530 ibn Mun’im, Ah.mad, 383, 385, 427–446 ibn al-Samh, Ab¯u al-Q¯asim, 382, 456–468 ibn al-Y¯asam¯ın, 383, 385 ibn al-Zarq¯alluh, Ibr¯ah¯ım, 533–539 indivisibles, 178–182, 306 infinite series, 184, 204–207 infinitesimals, 176–177 infinity, 182, 184 Isidore of Seville, 10–11, 18–21, 64 Jacobo da Firenze, 212–213 Johannes de Lineriis, 93–96; 155–157 John of Murs, 140–143 John of Palermo, 112 John of Sacrobosco, 85–93 Jordanus de Nemore, 2, 96; De elementis arithmetice artis, 96–100, 254; De numeris datis, 116–119; De ratione ponderis, 186–188; Liber philotegni, 143–145 Josephus problem, 58–59 mathematical induction, 255–256, 258–259, 273–275, 446–448 mathematics, reasons for studying: for Abraham bar H.iyya, 297–299; for ibn al-Bann¯a’, 386–387, 415; for Levi ben Gerson, 254 mean speed theorem, 191–197, 203–204 Menelaus’s theorem, 482, 503, 539 Mizrahi, Elijah, 83, 225, 227, 244–253, 357–358 Moschopoulos, Manuel, 550, 554–559 motion, 189–194, 197–207 Mot.ot., Simon, 225, 340, 354, 358–362 number theory, 112–116, 277–286, numerology, 18–20; 269–271 optics, 485–494 Oresme, Nicole, 2, 146, 194; Algorithm of Ratios, 119–121; On the Configurations of Qualities and Motions, 197–207; Questions on the Geometry of Euclid, 184; On the Ratio of Ratios, 121–123 Pachymeres, George, 549 Pandulf of Capua, 39–44 Pappus, 348–349, 480, 485 Pascal triangle, 98, 441 Philippe de Vitry, 119, 277 Pitiscus, Bartholomew, 162 Planudes, Maximus, 549, 551–554 Plato, 178, 180 Plato of Tivoli, 66, 296 polyhedra, 337–339 probability, 100–103 Ptolemy’s theorem, 152–153, 345–346 law of the lever, 186–188 Leonardo of Pisa, 2, 11, 57, 124, 149, 207, 216, 550; Book of Squares, 96, 112–116; De Practica Geometrie, 130–139, 153–155, 310; Liber Abbaci, 79–85, 103, 106–112, 208 Levi ben Gershon, 82, 119, 225, 227, 268, 283, 316; Astronomy, 320–326; Commentary on Euclid’s Elements, 326–335; On Harmonic Numbers, 277–283; Ma’ase H oshev, 253–268, 273–277, 354–355; Treatise on Geometry, 335–336 al-Qalas.a¯ d¯ı, ‘Al¯ı b Muh.ammad, 384, 386 Qalonymos ben Qalonymos, 225, 283, 326, 337–339, 457, 497 quadratic equations in mathematical texts: by Abraham bar H.iyya, 300–303; by Abraham ibn Ezra, 289–291; by Ab¯u Bakr, 126–128; by anonymous Hebrew author, 355–357; by Elijah Mizrah.i, 357–358; by ibn Abd¯un, 452–453; by ibn al-Ah.dab, 374; by ibn Badr, 424–427; by ibn al-Bann¯a’, 412–413, 418–422; by Jacobo da Firenze, 212–213; by Jordanus de Nemore, 117–119; by al-Khw¯arizm¯ı, 104–106; by Leonardo of Pisa, 109–112, 131–134; by Levi ben Gershon, 354–355; by Master Dardi, 214; by Paolo Girardi, 211–212; by Simon Mot.ot., 359–362 quadrature of the lune, 346–347 quadrivium, 4, 6–11, 65–66 magic squares, 407–409, 554–559 al-Maghrib¯ı, Muhy¯ı al-D¯ın, 337, 383 Mahavira, 3, 81, 563–564 Maimonides, 340, 539 Rashi (Rabbi Shlomo Ishaqi), 225, 313–314 recreational mathematics, 55–64, 251–253, 561–562; buying a horse, 82–83, 252–253, 259, 419–420; container and holes, 266; finding a al-Khw¯arizm¯ı, Muh.ammad ibn M¯us¯a, 2, 20, 68, 354, 550; Algebra, 103–106; Arithmetic, 76–79 al-Kishn¯aw¯ı, Muh.ammad ibn Muh.ammad, 383, 407–409 574 Index purse, 81–82, 107, 267–268; tree problem, 80–81 Regiomontanus (Johannes Müller), 2, 25, 162–173, 539 Richard de Fournival, 100–103 Richard of Wallingford, 157–160 Rithmimachia (game), 59–64 Robert of Chester, 66, 71, 74, 103 Roman numerals, 12–15, 36, 56–57 rule of four quantities, 166–167, 539–541 rule of three, 79–81, 208, 231–233, 265–267, 364, 561 Simon ben S.emah (Duran), 225 Sines: calculation of, 151–152, 154, 156, 156–160, 292–293, 307–308, 320–321; plane law of, 163–164; spherical law of, 168–170, 512–514, 532–533, 540–543 Solomon ben Isaac, 225, 340–344 Talmud, 225, 227, 232, 240, 298–299, 314, 316–320, 345 Th¯abit ibn Qurra, 67–68, 283, 458, 487, 504, 536 Theon of Smyrna, translations: from Arabic to Hebrew, 225, 284, 296, 337, 354, 362, 457, 497, 520, 539; from Arabic to Latin, 67–68, 70–75, 103, 123–124, 126, 149, 173, 468, 520, 539; from Greek to Arabic, 70; from Greek to Latin, 69–70; from Hebrew into Latin, 237, 244, 277, 287, 296, 320; from Latin to Hebrew, 237 trigonometry: applications, 520–533; determining spherical arcs, 504–509; solving plane triangles in, 164–166, 323–326; solving spherical triangles in, 171–173; 509–512, 515–520; theorems of, 163–164, 166–171, 503, 512–515, 540–544 See also Cosines; Sines trigonometry in mathematical texts: by ibn Mu’¯adh al-J¯ayy¯an¯ı, 502–520; by J¯abir ibn Aflah., 539–544; by Johannes de Lineriis, 155–157; by Leonardo of Pisa, 153–155; by Levi ben Gershon, 320–326; by Ptolemy, 149–153; by Regiomontanus, 162–173; by Richard of Wallingford, 157–160 universities: Bologna, 5, 66, 155; Cambridge, 66; Oxford, 66, 157–158, 176–180, 185–186, 188–194; Paris, 5, 66, 93, 146 velocity of light, 182–184 Villani, Giovanni, 5, 207 William de Lunis, 103 William of Moerbeke, 68–69 Witelo, 182–183 ...Sourcebook in the Mathematics of Medieval Europe and North Africa Sourcebook in the Mathematics of Medieval Europe and North Africa Edited by V I C T O R J K AT Z MENS... Herbert Hunger and Kurt Vogel (1963), by permission of J.B Metzler’sche Verlagsbuchhandlung Sourcebook in the Mathematics of Medieval Europe and North Africa General Introduction Medieval Europe, from... Hebrew, and Arabic had usually studied classical Greek mathematics, mostly in translation, and were trying both to understand that mathematics and, often, to develop it further However, they also often