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Preface; Editorial Considerations; Contents; 1 Ancient Mixed Mathematics; 1.1 Epistemology of Mixed Mathematics; 1.1.1 Aristotelian Subalternate Sciences; 1.2 Geometrical Optics; 1.2.1 Euclid's Optics; 1.2.2 Further Developments; 1.3 Mechanics, Machines, and Equilibrium; 1.3.1 Aristotelian Mechanics; 1.3.2 Hellenistic Science of Equilibrium; 1.3.3 Inversion in the Role of Mathematics; 1.4 The Science of Harmonics; 1.4.1 Fundaments of Greek Harmonics; 1.4.2 Rationalist and Empiricist Theoreticians; 1.4.3 Ptolemy's Harmonica; 1.5 Observational Astronomy; 1.5.1 Astronomical Hypotheses
1.5.2 Ptolemy's Astronomical System1.5.3 Astronomy According to Philosophers; 1.6 Quotations; References; 2 Skills and Mathematics in Renaissance Italy; 2.1 Teaching of Mathematics; 2.2 Treatises of Abacus; 2.2.1 Trattato di Tutta L'arte Dell'Abacho of Paolo dell'Abbaco; 2.2.2 Luca Pacioli's Summa; 2.3 Artists and Engineers; 2.3.1 Leon Battista Alberti; 2.3.2 Leonardo da Vinci; 2.4 Alchemy, Magic, and Medicine; 2.4.1 Alchemy; 2.4.2 Natural Magic; 2.4.3 Medicine; 2.5 Quotations; References; 3 New Forms of Natural Philosophy and Mixed Mathematics; 3.1 Schools of Philosophy in the Renaissance
3.1.1 Humanism and Platonism3.1.2 Evolution of Aristotelianisms ; 3.2 Updating Classical Mixed Mathematics; 3.2.1 A Heated Debate of Philosophers on the Epistemology of Mixed Mathematics; 3.2.2 A Renewed Theory of Proportions; 3.2.3 Mechanical Disciplines; 3.2.4 Harmonics and Acoustics; 3.2.5 Optics: Theories of Vision and Light; 3.3 Mechanical Philosophy; 3.3.1 Early Mechanical Philosophers; 3.3.2 Mechanism and Mathematics; 3.4 The Emergence of Physico-Mathematica; 3.5 Quotations; References; 4 Galilean Epistemology; 4.1 Basic Galilean Ontology and Epistemology; 4.1.1 Definition and Essence
4.2 Method of Scientific Research4.2.1 Possible Hellenistic Influence; 4.2.2 Possible Influences of Jesuit Philosophers; 4.2.3 Possible Influence of Mathematicians; 4.3 Experiment and Experience; 4.3.1 Ascertainment of Empirical Laws; 4.3.2 Overcoming Incompleteness of Mathematics; 4.3.3 Thought Experiments; 4.4 The Role of Causes; 4.4.1 Explanations Through Causes; 4.4.2 Causes and Experiments; 4.5 The Role of Disciples; 4.5.1 Bonaventura Cavalieri; 4.5.2 Evangelista Torricelli; 4.5.3 Benedetto Castelli; 4.5.4 Vincenzo Viviani; 4.6 Quotations; References
5 Post-Galilean Epistemology. Experimental Physico-Mathematica5.1 Galileo's Entourage; 5.1.1 Alfonso Borelli. The Last Heir; 5.1.2 Experiments, Mathematics, and Principles of Natural Philosophy in Giovanni Battista Baliani; 5.1.3 Marin Mersenne's Universal Harmony; 5.2 Jesuitical Tradition; 5.2.1 Jesuit Epistemology; 5.2.2 Production of Experience. Giovanni Battista Riccioli; 5.2.3 Production of Experience. Fancesco Maria Grimaldi; 5.3 Experimental Philosophy; 5.3.1 The Accademia del Cimento; 5.3.2 The Royal Society; 5.3.3 Robert Boyle; 5.4 Quotations; References; 6 Concluding Remarks
1.5.2 Ptolemy's Astronomical System1.5.3 Astronomy According to Philosophers; 1.6 Quotations; References; 2 Skills and Mathematics in Renaissance Italy; 2.1 Teaching of Mathematics; 2.2 Treatises of Abacus; 2.2.1 Trattato di Tutta L'arte Dell'Abacho of Paolo dell'Abbaco; 2.2.2 Luca Pacioli's Summa; 2.3 Artists and Engineers; 2.3.1 Leon Battista Alberti; 2.3.2 Leonardo da Vinci; 2.4 Alchemy, Magic, and Medicine; 2.4.1 Alchemy; 2.4.2 Natural Magic; 2.4.3 Medicine; 2.5 Quotations; References; 3 New Forms of Natural Philosophy and Mixed Mathematics; 3.1 Schools of Philosophy in the Renaissance
3.1.1 Humanism and Platonism3.1.2 Evolution of Aristotelianisms ; 3.2 Updating Classical Mixed Mathematics; 3.2.1 A Heated Debate of Philosophers on the Epistemology of Mixed Mathematics; 3.2.2 A Renewed Theory of Proportions; 3.2.3 Mechanical Disciplines; 3.2.4 Harmonics and Acoustics; 3.2.5 Optics: Theories of Vision and Light; 3.3 Mechanical Philosophy; 3.3.1 Early Mechanical Philosophers; 3.3.2 Mechanism and Mathematics; 3.4 The Emergence of Physico-Mathematica; 3.5 Quotations; References; 4 Galilean Epistemology; 4.1 Basic Galilean Ontology and Epistemology; 4.1.1 Definition and Essence
4.2 Method of Scientific Research4.2.1 Possible Hellenistic Influence; 4.2.2 Possible Influences of Jesuit Philosophers; 4.2.3 Possible Influence of Mathematicians; 4.3 Experiment and Experience; 4.3.1 Ascertainment of Empirical Laws; 4.3.2 Overcoming Incompleteness of Mathematics; 4.3.3 Thought Experiments; 4.4 The Role of Causes; 4.4.1 Explanations Through Causes; 4.4.2 Causes and Experiments; 4.5 The Role of Disciples; 4.5.1 Bonaventura Cavalieri; 4.5.2 Evangelista Torricelli; 4.5.3 Benedetto Castelli; 4.5.4 Vincenzo Viviani; 4.6 Quotations; References
5 Post-Galilean Epistemology. Experimental Physico-Mathematica5.1 Galileo's Entourage; 5.1.1 Alfonso Borelli. The Last Heir; 5.1.2 Experiments, Mathematics, and Principles of Natural Philosophy in Giovanni Battista Baliani; 5.1.3 Marin Mersenne's Universal Harmony; 5.2 Jesuitical Tradition; 5.2.1 Jesuit Epistemology; 5.2.2 Production of Experience. Giovanni Battista Riccioli; 5.2.3 Production of Experience. Fancesco Maria Grimaldi; 5.3 Experimental Philosophy; 5.3.1 The Accademia del Cimento; 5.3.2 The Royal Society; 5.3.3 Robert Boyle; 5.4 Quotations; References; 6 Concluding Remarks