The Arithmetic of Al Uqlīdisī : The Story of Hindu Arabic Arithmetic as told in Kitāb al Fuṣūl fī al Ḥisāb al Hindī 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. Kitb al Fusl F al isb al Hind by Ab al assan Amad ibn Ibrhm Al Uqldis 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 Uqldis Frontispiece Al Uqldis’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 Uqldis’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 Uqldis’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. 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.