Introduction Part I: Algebra and Arithmetic 1 Algebra and its unifying role 2 Algebra and Linguistics 3 The First Classification of Curves 4 Descartes Geometry and the distinction between geometrical and mechanical curves 5 The Ovals of Descartes 6 Descartes and the Infinitely Small 7 Fermat and Algebraic Geometry 8 Euclidean Arithmetic 9 Algorithmic Methods 10 Ibn Qurra and Amicable Numbers 11 Fibonacci and Arabic Mathematics 12 Fibonacci and the Latin Extension of Arabic Mathematics 13 Al-Yazdi 14 Fermat and the Origins of Modern Diophantine Analysis Part II: Geometry 15 The Archemedians and the Problems of Infinity 16 Conics and traditions of the early research projections 17 The Continuous Drawing of Conic Curves and the Classification of Curves 18 The Fifth Postulate of Euclid Part III: The Application of Mathematics ? Astronomy and Optics 19 The Celestial Cinematics of Ibn Al-Haytham 20 Mirrors