The Arithmetic of Al-Uqlidisi

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The purpose of presenting this book to the scholarly world is twofold. In the first place, I wish to provide for the English reader a translation of the earliest extant Arabic work of Hindi arithmetic. It shows this system at its earliest stages and the first steps in its development, a subject not yet well known except for readers of some Arabic publications by the present writer. This book is therefore of particular importance for students of the history of mathematical techniques. The medieval author, AI-UqHdisI, was, it seems, not noticed by bibliographers; neither was his work, which lay hardly noticed by modern scholars until 1960 when I happened to see a microfilm copy of it in the Institute of Arabic Manu- scripts in Cairo. A steady labour immediately followed to make a comparative study of the text together with over twenty other texts, some of them not yet known to scholars. This pursuit resulted in (i) a doctoral degree awarded to me in 1966 by the University of Khartoum, (ii) the publication of several texts in Arabic including the text here translated, and (iii) the publication of several articles in Arabic and English on the history of arithmetic in the Middle Ages. The second purpose of this book is to make the main results of my study available to the English reader.

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I - Introduction.- 1.1. Past Attempts in the Study of Islamic Arithmetic.- 1.2. The Present Work.- 1.3. Transliteration.- 1.4. Outline of Islamic Contact with Indian Science.- 1.5. Arithmetic Systems in Arabic Texts; General Remarks.- 1.6. The Sexagesimal Scale.- 1.7. Finger-Reckoning.- 1.8. Indian Arithmetic.- 1.9. Greek Arithmetica.- 1.10. Types of Texts.- 1.11. Hindi and Hindu.- 1.12. Kit?b al Fus?l F? al-?is?b al-Hind?, by Ab? al-?assan, A?mad ibn Ibr?h?m, Al-Uql?dis?.- 1.13. The Subsidiary Texts: A-Type.- 1.14. Subsidiary Texts: H-Type.- 1.15. Subsidiary Texts: HA-Type.- 1.16. Subsidiary Texts: Islamic Type.- 1.17. Subsidiary Texts to Hindu Sources.- II - The Text The Arithmetic of Al-Uql?dis?.- Frontispiece.- Al-Uql?dis?'s Introduction and Table of Contents of Book I.- Book I.- 1. On the Nine Letters and the Places.- 2. On Doubling and Halving.- 3. On Increasing and Decreasing.- 4. On Multiplication.- 5. On Division.- 6. Multiplication of Fractions by Numbers.- 7. Combining Fractions and Parts.- 8. On Multiplying a Number with Fractions by a Number.- 9. Multiplying Fractions by Fractions.- 10. On Multiplying a Number with Fractions by Fractions.- 11. Multiplying a Number with Fractions by a Number with Fractions.- 12. On Extracting Roots of Numbers, Open and Asamm.- 13. On Extracting the Roots of Fractions, Isolated or Joined to Units.- 14. On Division in General.- 15. The Arithmetic of Degrees and the Concepts It Involves.- 16. Halving the Degree and Doubling It.- 17. On Increasing and Decreasing.- 18. Multiplication of Degrees and Minutes.- 19. On Division of Degrees and Minutes.- 20. On Extracting the Roots of Degrees and Minutes.- 21. Extracting the Roots of A$amm Numbers.- Book II.- Al-Uql?dis?'s Introduction and Table of Contents of Book II.- 1. On Doubling.- 2. On Halving.- 3. Halving where Fractions Occur, and Doubling - All Methods.- 4. Increasing, Unlike the Preceding (Type).- 5. On Decreasing of the Preceding Type.- 6. On the Preceding Type of Increasing and Decreasing; Minute (Problems).- 7. Exhausting One Number by Another.- 8. Queries on the Curiosities of Casting Out.- 9. Multiplication of Integers - All Methods: The Upright and the Branching.- 10. On Multiplication by (the Method of) Houses.- 11. Curiosities in Multiplication.- 12. On Division.- 13. On the Balances.- 14. On Multiplying Integers and Fractions by Integers and Fractions, Upright and Branching.- 15. On Which Is More, Such Parts of This Number or Such Parts of That?.- 16. On Multiplying Fractions of Fractions of Fractions.- 17. On Making Perfect Squares to Extract Their Roots.- 18. On Extracting the Root of the Asamm.- 19. Extracting the Roots of the Asamm and Testing Them.- 20. On Degrees and Minutes and the Concepts They Involve.- Book III.- Table of Contents of Book III.- 1. Justifications of the Hindi (Arithmetic) and Its Whys and Hows.- 2. Queries on Doubling and Halving.- 3. Queries on Increasing and Decreasing.- 4. Queries on Multiplication.- 5. Queries on Division.- 6. Queries on Roots.- 7. On Checking by Balancing and Its Justifications.- 8. Balances of Fractions.- 9. Queries on Doubling and Halving.- 10. Queries on Increasing and Decreasing.- 11. Queries on Adding Different Kinds.- 12. Queries on Increasing and Decreasing (Coming) Together.- 13. Curiosities in the Multiplication of Nines.- 14. Extracting Roots by Zeros Where Fractions Occur.- 15. On Converting Fractions and Parts, One to Another.- 16. On Decreasing Fractions and Parts, One from the Other.- 17. Queries on Minutes and Their Minute Parts.- 18. Division by the Upright (Method).- 19. Queries on Root (Extraction) by the Upright (Method).- 20. Queries About Profit, Involving a Curiosity.- 21. On the Justification of (the Method of) Extracting the Side of the Cube.- Book IV.- Al-Uql?dis?'s Introduction and Table of Contents of Book IV.- 1. The Method of Addition.- 2. On Doubling.- 3. On Halving.- 4. On Increasing.- 5. On Decreasing.- 6. On Multiplication.- 7. On Division.- 8. On Multiplying Fractions by Numbers.- 9. On Adding Fractions and Parts.- 10. On Multiplying a Number with Fractions by a Number.- 11. On Multiplying Fractions by Fractions.- 12. On Multiplying a Number with Fractions by Fractions.- 13. On Multiplying Integers and Fractions by Integers and Fractions.- 14. General, on Division.- 15. On Differentiating Between Fractions.- 16. On Converting Fractions and Parts from one Number to Another Number.- 17. On Extracting Roots of Numbers.- 18. On Extracting Roots Where Fractions Occur.- 19. On Degrees and Their Schemes.- 20. On Adding Degrees, Minutes and Lower Orders.- 21. Decreasing Degrees, Minutes and Lower Orders from Each Other.- 22. Reducing Degrees and Whatever Minutes and Seconds Follow to Their Lowest Places.- 23. Multiplication of Degrees, Minutes, Seconds, and Lower Orders, by Equal or Different Orders.- 24. On the Division of Degrees and Minutes and Lower Orders by Degrees and Minutes and Lower Orders.- 25. On Extracting the Roots of Degrees, Minutes and Lower Orders.- 26. On Working Out All That Is Needed of Hindi Arithmetic Without the Nine Letters.- 27. By Hindi: Writings, Names, Calculations - Everything We Want.- 28. A Calculating (Tool) for the Blind and the Weak-Sighted.- 29. On Extracting the Side of the Cube.- 30. On Knowing Fractions That Are Cubic and Extracting Their Sides, Single or Combined to Units.- 31. On Extracting the Sides of Numbers Which Have No Roots.- 32. On Doubling One, Sixty-Four Times.- III - Remarks and Comparative Studies.- Unit 1. Aims and Contents of the Four Books (The Introductions to the Four Books).- Unit 2. The Numerals.- Unit 3. Duplation and Mediation.- Unit 4. Addition and Subtraction.- Unit 5. Multiplication.- Unit 6. Division.- Unit 7. Fractions.- Unit 8. Square Roots.- Unit 9. Cube Roots.- Unit 10. Checking.- Unit 11. Profit and Loss.- Unit 12. Remarks on Decimal Fractions.- A Recapitulation.- Index of Auxiliary Works and Authors.- Table of Important Remarks.

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