41. Arabian Nights: 15 Tale 5 - THE LOVES OF AL-HAYFA AND YUSUF> his rest in haste and anxiety until Allah caused the morn to morrow and break in its sheen and it shone, whereupon the King summoned the mathematicians and the http://www.wollamshram.ca/1001/Sn_5/15tale5.htm

THE LOVES OF AL-HAYFA AND YUSUF. [FN#177] I had a familiar in the Northern region who was called 'Adb al-Jaw d and he was one of the greatest of merchants there and made of money; also he loved voyage and travel, and at whatever time I visited him and we forgathered, I and he, we exchanged citations of poetry. Now one day my heart yearned to visit him, so I repaired to his place and found him there; and as we came together we both sat down in friendly converse, I and he; and he said to me "O my brother, do thou hear what happened and was accomplished for me in these times. I travelled to the land of Al-Yaman and therein met a familiar who, when we sat down to talk, I and he, said, 'O my brother, verily there befel me and betided me in the land of Al-Hind a case that was strange and an adventure that was admirable and it ran as follows. There was erewhile a King of the kings of India and one of her greatest, who was abundant in money and troops and guards and he was called Al-Mihrj n. [FN#178] [FN#179] [FN#180] for her seemlihead. Then he gifted the midwife'"And Shahrazad was surprised by the dawn of day and ceased to say her permitted say. Then quoth her sister Dunyazad, "How sweet is thy story, O sister mine and how enjoyable and delectable!" Quoth she, "And where is this compared with that I would relate to you on the coming night an the King suffer me to survive?" Now when it was the next night and that was The Six Hundred and Sixty-third Night

42. Famous Mathematicians (Reference) The following list contains some of the great mathematicians through history. Al–Khowârizmî, Muhammed (about 780–850); arabian; Algebra. http://www.teachervision.fen.com/lesson-plans/lesson-4360.html

var do_survey = 1; Explore Our Sites... Family Education Network Home PARENTS FamilyEducation MySchoolOnline TEACHERS TeacherVision Quiz Lab MyGradeBook MySchoolOnline REFERENCE Infoplease Fact Monster KIDS FEkids FunBrain Fact Monster Members - Sign In Become a Member Membership Help Teacher Home ... Help Famous Mathematicians From Hands-On Math Projects with Real-Life Applications The following list contains some of the great mathematicians through history. It offers dates, ethnic origins, and major fields of study. Ahmes (about 1650 B.C.); Egyptian; Geometry. Grade Levels: Related Topics: History of mathematics Mathematicians Related Themes: Biographies Math Email This Page Printer-Friendly Version Visit Infoplease FactMonster Rights Privacy

43. The Mathematical Principles Of Urbàntasm star, and the ubiquitous threes that infect everything from The arabian Nights to This is why mathematicians rolled their eyes and grumbled about fractals for http://urbantasm.topcities.com/math.html

Mathematical Principles of Urbàntasm by Connor Coyne (Please reproduce only with permission.) Home. Urbàntasm is a mathematical novel. Urbàntasm is a fractal novel. Urbàntasm is possibly a chaotic system. Urbàntasm is a formal axiom system or, more specifically, eight axiom systems sharing several common terms, rules, and assumptions. Every syllable of Urbàntasm corresponds to a number, and the interaction of these numbers influence everything from the structure of sentences and the ending points of chapters to the stucture of plot, evolution of character, and revelation of secrets. Having said all this, and possibly alienated some aesthetically minded readers with references to math, and scientifically minded readers with my cheek, I will have to spend the rest of this essay explaining what these statements mean, why I am justified in making them, and how such an application of mathematics is essential to this project. The Inadequacy of a Brief Explanation Since I'm always working on this novel, I end up explaining it a lot, and since I've spent as much time on the math as I have the dialogue and plot arc, I end up explaining the math very frequently. Some of these attempts misfire, and I'm writing this essay partly to clarify my ideas to myself. I recall one particularly unsuccessful attempt to describe the math in Urbàntasm . I was talking to a friend I had run into at a coffee shop. He was a Physics major at the University of Chicago, and was preparing to move to Berkeley for grad school. I was asking for help, but after stammering out a few questions, I realized he thought I was wasting my time.

44. In This Problem You Will Use What Mathematicians Call Modular Arithmetic Problem One. In this problem you will use what mathematicians call Modular Arithmetic. A noble arabian king wished to reward his valued Minister of Affairs. http://www.ega.edu/facweb/suellamc/Problems.htm

Problem One In this problem you will use what mathematicians call Modular Arithmetic. That means you may answer the question by thinking about the numbers on a clock face. If it is now 10:45 AM, what time will it be in (a.) 96 hours? (b.) 15 hours? (c.) 32 hours? Problem Two A noble Arabian king wished to reward his valued Minister of Affairs. The Minister of Affairs was modest man, but also a chess player. He knew that there were 64 squares a chessboard. When he asked for a reward, he made the following request. I would like one gold coin on the first square; two gold coins on the second square; four gold coins on the third square; and eight gold coins on the fourth square. Please, continue the pattern until all of the squares are filled? The king accepted his Minister’s request as a reasonable one. How many gold pieces, was the Noble King obligated to give his minister? Problem Three The numbers in this problem form a Magic Square about 30. Notice the sum of the numbers in the fourth row. Continue around the outside of the square. Make the sum of the first column 30. Then complete the first row and the fourth column so that each sum is 30. Complete each remaining row and column. How many different ways can you find 30 in this Magic Square?

45. AWhiteHorse Forums Arabian Horses And Their Friends -> Need Name For My New Foal and scientist of antiquity and one of the three greatest mathematicians of all http://www.awhitehorse.com/horsesmouth/index.php?showtopic=1673&view=getnewpost

46. About Karl J. Smith Hindu and arabian Period 500 to 1199. Indian Mathematics. Al-Khwarizmi http//www-history.mcs.st-and.ac.uk/~history/mathematicians/Alberti.html. http://www.mathnature.com/geometry/links/history.htm

There are multitude of wonderful sources of history on the world wide web. This is one of the best all around sites that I have found. It is titled "The MacTutor History of Mathematics Archive." http://www-groups.dcs.st-and.ac.uk/~history/ Here is another very good historical site: http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html This is an award-winning site that links to all sorts of mathematics information. archives.math.utk.edu/ Earliest uses of various mathematical symbols: http://members.aol.com/jeff570/mathsym.htm History of Mathematics by Region: http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html History of Mathematics Chronologically: http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html#toc We have divided this history of mathematics into six chronological periods, many of which are included at the following world wide web sites Egyptian, Babylonian and Native American Periods - 3000 BC to 601 BC Mesopotamian Mathematics http://www.providence.edu/dwc/mesopot.htm http://aleph0.clarku.edu/~djoyce/mathhist/babylonia.html www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_and_Egyptian.html Egyptology Mathematics www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_and_Egyptian.html

47. Vedic Mathematics Indians used when one of the Indian palmist and fortuneteller happened to visit the arabian lands. So impressed were the Arab mathematicians with Indian http://www.cs.uml.edu/~asaxena/vedic-maths.html

Vedic - Mathematics This interesting article was forwarded by a friend. I wanted to share this with you. Mathematics is the queen of subjects. Rightly so, then, Vedic Mathematics is the glowing crown that adorns its proud forehead. Very few of the masses today are aware today, of Vedic mathematics, the magnanimity of its profound implications, and of its origins which guided the rest of the world towards purer and more intricate branches of mathematics and which, laid the foundations for number theory and arithmetic, the teeny-weeny part of which we are taught during our alma-mater days with hardly any reference made to its rightful owners ? our very own ancestors ?who pursued the study of mathematics with no less finesse than that of a fine art. A well-known fact it is, as every one knows now, for he/she has seen himself/herself or his friend, being answered by his teacher, during the primary years of his education, in response to his/her query full of childish criticism “What has India given to Mathematics? that the numeral was indeed the creation of Indian mathematicians. Introduction of zero brought about a new revolution into the world of mathematics. It was zero that gave rise to the idea of representing numbers using base 10, as it is commonly used today. And it is zero because of which you are able to read this article. But why? How would a computer work without zeros and ones!!! So that’s the zero there, right! Though the Arabs are given the credit of taking mathematics into broader frontiers, they had begun their work with the help of Indian manuscripts. The story goes something like this. It was in 773 that the Arabs were able to set their eyes on the astounding developments of numerical methods Indians used when one of the Indian palmist and fortune-teller happened to visit the Arabian lands. So impressed were the Arab mathematicians with Indian inventions that the Arab mathematician Muhammed-Ibna-Musa-Abu-Jafar-Al-Khwarizmi himself came to India to study Indian mathematics. After stating here for some time after learning the subjects to his satisfaction, he wrote his manuscript “Algebra ’–b-e-Mukabla? This is how ‘Algebra?was born. His works, which were nothing but a translation of his Indian studies, left the European mathematicians spell-bound, especially by the use of base 10 to represent numbers. The idea of representing numbers by base 10, is thus, originally Indian.

48. Somerset Publications : Publishers Of Australia's Crabbet Arabian Horse horses are bred, leaving a legacy that is unmatched by any other arabian Stud in of the poet Lord Byron from a family of writers, mathematicians and artists. http://www.hotkey.net.au/~crabbetarabian/crabbet.html

WHAT IS CRABBET? WHERE IT ALL BEGAN The story of Crabbet Stud would make a fascinating screenplay. It has everything - larger-than-life characters, complicated relationships, exotic settings, travel in distant lands. Against this rich tapestry of colour and excitement was played out a mission of more practical purpose, the acquisition of Arabian horses of the finest types available for a Stud in England that would preserve intact the horse of the desert. That mission was accomplished and the Stud existed for almost 100 years, during which time it bred horses that went to all parts of the world where fine horses are bred, leaving a legacy that is unmatched by any other Arabian Stud in history. The Crabbet Stud was founded by two extraordinary people - Lady Anne and Wilfrid Scawen Blunt. Each was the product of an interesting and accomplished family background. She was the granddaughter of the poet Lord Byron from a family of writers, mathematicians and artists. Her mother was a noted scientist. He was a sensitive, charismatic, if "difficult" man whose interests included politics, art, music and poetry. They married in 1869. "Crabbet", a stately home set in rolling parkland at Crawley in Sussex became their base, but in fact the two spent much of their stormy marriage travelling abroad.

49. Newsletter 50 July, 2002: History And Culture In Mathematics Education Translate this page be included in the maghrebian symposiums on the history of arabian mathematics in investigated and thereby gave rise to some problems for the mathematicians. http://www.hpm-americas.org/nl50/nl50reviews.html

International Study Group on the Relations Between HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER An Affiliate of the International Commission on Mathematical Instruction: No. 50, July 2002 Reviews In you would like to be involved in reviewing books or magazines for this section, please send your contact details and area(s) of interest to the editor who will forward books or magazines for review as and when they become available. If you wish for a book to be reviewed, please send it to the editor who will arrange for it to be reviewed. Tree of Mathematics Reports on Conferences History and Pedagogy of Mathematics in the 7th Maghrebian symposium on the History of Arabic Mathematics 30 May - 2 June 2002 Marrakech, Morocco Abdellah El Idrissi Marrakech, Morocco Isabel Cristina Dias Lisboa, Portugal The Abel-Fauvel conference at Kristiansand, Norway 12-15 June 2002 Most people arrived in the afternoon/late evening at the Gimlekollen Mediasenter on Tuesday 11 June. There was an excellent spread of food available every day, made all the more palatable by the superb companionship of those present. A total of 27 participants represented 11 countries - Norway, Sweden, Iceland, UK, Germany, Italy, France, USA, Peru, Taiwan and Chile. Since 1988, when the "Learn from the Masters" conference was organized, Kristiansand has developed considerably as a centre for the study of relations between education and the history of mathematics. In 1994 Agder University College was founded, a masters degree for mathematical education (including history) was introduced and a doctor program in the field is in progress.

50. All.info: Science And Health / Math / Mathematicians / Sir Isaac Newton s Mathematical Principles of Natural Philosophy Check out the NEW Hotbot Tell me The Electronic Literature Foundation presents The arabian http://all.info/directory/Science_and_Health/Math/Mathematicians/2.html

Search Directory: You are in: Science and Health Math Mathematicians Emmy Noether Emmy Noether Free Web space and hosting - 20megsfree.com Emmy Noether Welcome to EmmyNoether.com a website devoted to: Teaching Symmetry in the Introductory Physics Curriculum Authors: Prof. Christopher T. Hill, Theoretical... http://www.emmynoether.com/ Lanczos Collection Lanczos Collection Announcing the CORNELIUS LANCZOS COLLECTED PUBLISHED PAPERS WITH COMMENTARIES Published by North Carolina State University College of Physical and Mathematical Sciences and Department of Physics Raleigh, North... http://www.physics.ncsu.edu/lanczos/ http://sprott.physics.wisc.edu/pickover/ http://sprott.physics.wisc.edu/pickover/ Thomas H. Cormen Thomas H. Cormen Thomas H. Cormen, Associate Professor Computer Science Undergraduate Advisor Ph.D., Massachusetts Institute of Technology , 1992. Research Interests Parallel computing, out-of-core computing, and analysis of... http://www.cs.dartmouth.edu/~thc/ Charles E. Leiserson : HomePage Charles E. Leiserson : HomePage Charles E. Leiserson Charles E. Leiserson is Professor of Computer Science and Engineering in MIT 's Department of Electrical Engineering and Computer Science . He is a member of MIT 's Computer... http://supertech.lcs.mit.edu/~cel/

51. Stephen Wolfram: A New Kind Of Science -- Relevant Books The Cambridge Library of Ornamental Art arabian Ornament from the 12th to the of Japan Proceedings of the International Congress of mathematicians, Kyoto, 1990 http://www.wolframscience.com/reference/books/t.html

WOLFRAM'S LIBRARY A B C D ... S T U V W X Y Z T Tabor, D. Gases, Liquids, and Solids. Second Edition Cambridge University Press, 1979. [ISBN 0521294665 Taff, Laurence G. Physical Statistics: A Scientific Theory of Statistical Estimation Laurence Taff, 2000 Taha, Hamdy A. Operations Research: An Introduction. Fourth Edition Macmillan, 1987. [ISBN 0024189405 Tait, Peter Guthrie Cambridge University Press, 1900 Tanenbaum, Andrew S. Computer Networks, 2nd Edition Prentice-Hall, Inc., 1989. [ISBN 013162959X Tanford, Charles The Scientific Traveler: A Guide to the People, Places, and Institutions of Europe Tanur, Judith M. et al. Statistics: A Guide to the Unknown, 3rd edition Tarski, Alfred Undecidable Theories North-Holland, 1953 Taylor, Charles Fayette The Internal-Combustion Engine in Theory and Practice, Volume 1: Thermodynamics, Fluid Flow, Performance, Second Edition, Revised MIT Press, 1990. [ISBN 0262700263 Taylor, Charles Fayette The Internal-Combustion Engine in Theory and Practice, Volume 2: Combustion, Fuels, Materials, Design, Revised Edition MIT Press, 1985. [ISBN 0262700271

52. Artlandia Wonderland Symmetry And Pattern Design Resources and terminology of symmetry and pattern analysis for nonmathematicians. Assyrian, Persian, Greek, Pompeian, Roman, Byzantine, arabian, Turkish, Persian, Indian http://www.artlandia.com/wonderland/

"Curiouser and curiouser!" Teaching or studying symmetry? Learning pattern design? Looking for pattern ideas? These tips , interactive tutorials , and books offer help. Wishing to try what you've learned? Fire up your copy of Artlandia SymmetryWorks Symmetry Examples of the seventeen symmetry types (wallpaper groups) Basic Symmetry Operations . An Artlandia SymmetryWorks tutorial. ( PDF file , 232 KB) Symmetry lingo for designers. Figuring out the Type of Symmetry . An Artlandia SymmetryWorks tutorial. ( PDF file , 320 KB) Step-by-step instructions. Creating linear ornaments and borders with Artlandia SymmetryWorks. Gruenbaum, Branko and G.C. Shephard. Tilings and Patterns. New York, NY: W.H. Freeman, 1987. Comprehensive and rigorous mathematical treatment, with excellent, inspiring illustrations. Bix, Robert. Topics in Geometry. San Diego, CA: Academic Press, 1997. A junior-level college geometry course with a chapter on transformation geometry and wallpaper groups. McLenaghan, Ray and Silvio Levy. Geometry, in CRC Standard Mathematical Tables and Formulae.

53. Mathematics Course List Attention will be given to the early EgyptiansBabylonian period, the geometry of Greek mathematicians, the Hindu and arabian contribution, the evolution of http://www.math.hope.edu/courses.html

Hope College Mathematics Department Courses Mathematics Home Faculty Newsletter Problem of the Fortnight ... Natural Science Division America 's Best Colleges 2003." Note: This page is intended to provide an accurate reproduction of the information in the Hope College 2003-04 Catalog , pp. 214-217. Please see the Catalog for Hope College policies relating to changes in this information. 123. A Study of Functions: A study of functions including polynomial, rational, exponential, logarithmic, and trigonometric functions. These will be explored in their symbolic, numerical, and graphic representations, and connections between each of these representations will be made. A graphing calculator is required. A student cannot receive credit for both MA 123 and MA 125. Four Credits, Spring Semester 125. Calculus with Review I: This course covers the material typically taught in the first half of a Calculus I (MA 131) course. The calculus material is supplemented by reviewing topics of high school mathematics as needed. The calculus topics are also taught at a slower pace. Topics include function review, limits and continuity, the concept (and definition) of a derivative, and differentiation rules (product rule, quotient rule, and chain rule are included). A student cannot receive credit for both MA 125 and MA 123. Four Credits, Fall Semester

54. News 9 San Antonio 24 Hour Local News Mike S Notes Yet, had it not been for arabian and Indian mathematicians and their ability to envision the existence of cipher we might never have heard of the http://www.news9sanantonio.com/content/weather/mikes_notes/?ArID=6134&SecID=81

55. 1200.00 NUMEROLOGY 1230.30 Origin of Scheherazade Myth I think the arabian priestmathematicians and their Indian Ocean navigator ancestors knew that the binomial effect of http://www.rwgrayprojects.com/synergetics/s12/p2200.html

56. Princeton Club Of Northern California A Thousand and One arabian Nights is filled with colorful, exotic stories that will as John Nash, one of the most brilliant and haunted mathematicians of his http://www.pcnc.org/newsletters/2002/newsletter082002.shtml

Become a member! The Princeton Club of Northern California is open to all Graduate and Undergraduate Alumni and Princeton Parents. PCNC sponsors events in the San Francisco Bay Area (Peninsula, South Bay, and East Bay), the Monterey Bay Area, and Sacramento. Inquiries about membership and dues can be made by contacting us via e-mail, by phone at (415) 845-8120, or by mail at Cathy Legg '99 PO Box 194752 San Francisco, CA 94119-4752. last updated July 1st, 2002 PCNC Newsletter, August 2002 Event Date Time Location RSVP PCNC Golf Outing San Francisco Michael Culver Golf Tournament for All Abilities Play at one of the Bay Area's prized golf resources — the Presidio Golf Course. It's in great shape, under management by Arnold Palmer Golf Resorts. They're giving the PCNC a tournament for a special rate (golf carts included). Limited to 20 players, so please help by reserving with payment as soon as possible. If you send your check prior to July 15th, you will automatically be entered into a drawing for two complimentary rounds of golf at The Presidio, announced at the conclusion of the event! No matter what your ability, this will be a great day of golf. The Presidio is a magnificent layout that has stood for over 100 years — a beautiful course with beautiful views of San Francisco. You won't be disappointed.

57. COURSE SYLLABUS Course Number MATH3010 Course Title HISTORY OF Chinese, Hindu, and arabian mathematics (before global communication merged them Some mathematicians of this period Fermat, Descartes, Pascal, Leibniz, Newton http://www.auburn.edu/~smith01/txtsyll/syl3010.txt

COURSE SYLLABUS Course Number: MATH3010 Course Title: HISTORY OF MATHEMATICS Credit Hours: 3 Prerequisites: MH 1620 or departmental approval. Corequisite: Objectives: To enhance the student's mathematical perspective through a discussion of the evolution of mathematical concepts and the contributions of outstanding mathematicians, and to enhance the student's appreciation for and facility with deductive reasoning through exercises related to these mathematical concepts and contributions. Course content: Numeral systems , Egyptian and Babylonian mathematics (1 week). Ancient Greek geometry and number theory; deductive reasoning (2 weeks). Chinese, Hindu, and Arabian mathematics (before global communication merged them with European mathematics) (2 weeks). European mathematics in the 12th, 13th, and 14th centuries: translation of Arabic works and the ancient Greek texts; universities established; contributions of Fibonacci. (2 weeks). European mathematics of the 15th, 16th, and 17th centuries: Beginnings of algebraic symbolism, solutions of the general cubic and quartic, logarithms, beginnings of number theory, analytic geometry, projective geometry, and probability, and the discovery of the calculus. Some mathematicians of this period: Fermat, Descartes, Pascal, Leibniz, Newton. (3 weeks). Mathematics of the 18th and 19th centuries: further development of calculus and it's evolution into analysis. Infinite series including Fourier series, the notion of a limit, the notion of a function, the Riemann integral. Non-Euclidean geometry. Abstract algebra and the impossibility of solution by radicals of 5th degree equations. Impossibility of certain constructions by straightedge and compass such as trisecting an angle and squaring a circle. Some mathematicians of this period: Bernoulli brothers, Lagrange, Euler, Gauss, Riemann, Galois, Abel, Cauchy, Fourier. (3 weeks). Mathematics of the 20th century. Evolution of the axiomatic method. Set theory and logic, Russell paradox, Zermelo-Fraenkel axioms, axiom of choice, continuum hypothesis. Godel's incompleteness theorem and other contributions. Topology, measure theory, dynamical systems and chaos, computers and computer science. Solutions of famous problems such as the four color problem and Fermat's Last Theorem. (2 weeks). Text: Howard Eves, An Introduction to the History of Mathematics, 6th Ed. Sample Grading and Evaluation Procedures Students will be expected to have prepared the daily homework assignments. Homework will occasionally be collected. This will be part of the participation grade. Reading the text and working the exercises are an important part of this course. A paper will be assigned; it should be on the history of some mathematician (with emphasis on his mathematical discoveries) or on some mathematical concept; check with the instructor about the topic. Grade Calculation Participation grade (includes: blackboard presentation and classwork, attendance, homework or projects): 10% Reading Quizzes (quizzes are approximately 10-minutes long and may be announced or unannounced): 10% Term paper 15% Hour Tests (three tests): 35% Final Exam: 30% Tentative Test Schedule Hour tests are given at the end of appropriate units and will be announced a week ahead of time. Quizzes may or may not be announced; at least four quizzes will be given in the course of the semester. Friday is typically a good day for quizzes. Sample Statement Re: Accommodations Students who need accommodations are asked to arrange a meeting during office hours the first week of classes, or as soon as possible if accommodations are needed immediately. If you have a conflict with my office hours, an alternate time can, be arranged. To set up this meeting, please contact me by E-mail. Bring a Copy of your Accommodation Memo and an Instructor Verification Form to the meeting. If you do not have an Accommodation Memo but need accommodations, make an appointment with The Program for Students with Disabilities, 1244 Haley Center, 844-2096 (V/TT). (Note: Instructor office room, office hours and email address will be made available on the course syllabus and on the first day of class.) JUSTIFICATION Education majors specializing in mathematics are required to take a History of Mathematics course. This course satisfies this requirement. The course can also be used as a free elective by mathematics majors.

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60. LEONARDO OF PISA Leonardos works are mainly developments of the results obtained by his predecessors; the influences of Greek, arabian, and Indian mathematicians may be clearly http://91.1911encyclopedia.org/L/LE/LEONARDO_OF_PISA.htm

LEONARDO OF PISA LEONARDO OF PISA (LEONARDUS PISANUS or FIB0NAcCI), Leonardos works are mainly developments of the results obtained by his predecessors; the influences of Greek, Arabian, and Indian mathematicians may be clearly discerned in his methods. In his Practica geometriae plain traces of the use of the Roman agrimensores are met with; in his Liber abaci old Egyptian problems reveal their origin by the reappearance of the very numbers in which the problem is given, though one cannot guess through what channel they came to Leonardos knowledge. Leonardo cannot be regarded as the inventor of that very great variety of truths for which he mentions no earlier source. The Liber cibaci, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days. They teach further the solution of problems leading to equations of the first and second degree, to determinate and inde~ terminate equations, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown nul5ntt,j h,-0,~ ,slIs,-i r~ ~rnl ~ As for the influence he exercised on posterity, it is enough to say that Luca Pacioli, about 1500, in his celebrated Summa, leans so exclusively to Leonardos works (at that time known in manuscript only) that he frankly acknowledges his dependence on them, and states that wherever no other author is quoted all belongs to Leonardus Pisanus.

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