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Home , Jordanus de Nemore

Raffaele Pisano • Danilo Capecchi

Tartaglia's Science of Weights and Mechanics in the Sixteenth Century

Selections from Quesiti et inventioni diverse: Books VII-VIII

^ Springer Contents

Part I Biographical Sketches & Science in Context

1 Niccolo Tartaglia and the Renaissance Society Between Science and Technique 3 1.1 Niccolo Fontana Called Tartaglia 3 1.1.1 Biographical and Scientific Sketches 4 1.1.1.1 The Roots 12 1.1.1.2 Tartaglia's Education 14

1.1.1.3 Arnaldo Masotti, Tartaglia's Modern Editor. ... 17 1.1.2 Tartaglia's Conceptual Stream in the Renaissance 19 1.1.2.1 : The Third Degree Equations 20 1.1.2.2 On the : 's Elements 31 1.1.2.3 On the : Tartaglia's Triangle 35 1.1.2.4 On Physics: Ballistics 39

1.1.3 Physics and Architecture: On Ballistics & Fortifications . . . 49 1.1.3.1 On Ballistics & Technical Instruments in the Nova Scienta 49

1.1.3.2 The Sesto Libro on Fortifications 71 1.1.3.3 The Gionta del Sesto Libro 75

1.1.3.4 The Gionta del Sesto Libro and Architecture .... 80 1.1.4 On the Opera Archimedis and Archimedis de insidentibus aquae 87 1.1.4.1 On the Opera Archimedis (1543) 87 1.1.4.2 On the Archimedis de insidentibus aquae (1543; 1565) 92

XV xvi Contents

1.1.5 Contents, Former Pupils and Philological Notes 95 1.1.5.1 A Content of Quesiti' et inventiom' diverse 98 1.1.5.2 Scholars, Former Pupils, Correspondence and Commentaries in Quesiti and Around Tartaglia's Science 99 1.1.5.3 Philological Notes and a Historical Hypothesis 102

2 Ancient and Modern Statics in the Renaissance 113 2.1 The Background 113 2.1.1 The Scientia de ponderibus in the Middle Ages 114 2.1.1.1 The Roots in the Arabic Middle Ages 115 2.1.1.1.1 Thabit's Kitab fT il-qarastum 117 2.1.1.2 Continuation in the Latin Middle Ages 119 2.1.1.2.1 Jordanus de Nemore's Liber de ratione ponderis 122 2.1.2 Revival During the Age of Humanism 127 2.1.2.1 A Variety of Approaches to Mathematics 128 2.1.2.2 The Emergence of a New Type of Intellectual Technician: The Engineer 133 2.1.2.3 Leonardo da Vinci's Science of Weights 134 2.1.2.3.1 Powers: Gravity and Force 138 2.1.2.3.2 The Balance and Lever 143 2.1.2.3.3 The Law 149 152 2.1.2.3.4 The Pulley, Block and Tackle . ... 2.1.2.3.5 The Concept of Momento

of a Force 156 2.1.2.3.6 The Law of Composition of Forces 159 2.1.3 Tartaglia's Legacy. A Transition between Science of Weights and Modern Statics 163 2.1.3.1 Statics in Italy During the XVI Century 164 2.1.3.1.1 Giovanni Battista Benedetti 166 2.1.3.1.2 Guidobaldo del Monte 170 2.1.3.1.3 174 2.1.3.2 Stevin's Legacy. The Circulation of Statics in Europe 180

2.1.3.2.1 The Law of the Inclined Plane .... 182 2.1.3.2.2 Forces' Composition: The Rule of Parallelogram 185 Contents xvii

Part II History & Historical Epistemology Analyses

3 The Analysis of Books VII and VIII of Quesiti et in ventioni diverse 191 3.1 A Historical Epistemology Outline on Early Statics in Books VII and VIII 191 3.1.1 The Analysis of Book VII {1554) 193 3.1.1.1 The Aristotelian Mechanical Problems 194 3.1.1.1.1 The Accuracy of Balances 194 3.1.1.1.2 The Stability of Balances 196 3.1.1.2 The First Three Quesiti on Accuracy 198 3.1.1.3 The Last Four Quesiti on Stability 202 3.1.1.3.1 The Balance with Fulcrum Above the Beam 203 3.1.1.3.2 The Balance with Fulcrum Below the Beam 205 3.1.1.3.3 The Balance with Fulcrum Inside in the Beam 206 3.1.2 The Analysis of Book VIII 207 3.1.2.1 The Book VIII and Liber de ratione ponderis. A False Controversy? 209 3.1.2.2 The Liber de ratione ponderis 211 3.1.2.2.1 The Suppositions of Liber de ratione ponderis 212 3.1.2.2.2 The Propositions of Liber de ratione ponderis 218 3.1.2.3 The Structure of Book VIII 225 3.1.2.3.1 On the Roots of Notional

Elements in Tartaglia's Corpus . . . 225 3.1.2.3.2 The Definitions of Book VIII 227 3.1.2.3.3 The Petitions of Book VIII 229 3.1.2.3.4 The Propositions of Book VIII.... 231 3.1.2.4 The Proofs of Propositions 239 3.1.2.4.1 Propositions I-IV: Gravitas Secundum Situm 239 3.1.2.4.2 Propositions V-VI: Balance with Equal Weights and Arms 241 3.1.2.4.3 The Proposition VII: Balance with Equal Weights and Different Arms 248

3.1.2.4.4 Propositions VIII: Law of Lever. . . 249 3.1.2.4.5 Propositions IX-XIII: Balance with Distributed Weights 250 3.1.2.4.6 Propositions XIIII-XV: Law of Inclined Plane 254 xviii Contents

Part III Translations & Transcriptions

4 Translation and Transcription 261 4.1 General Considerations 261 4.1.1 Quesiti et inventioni diverse (1554) 262 4.1.1.1 Tartaglia' s Language 262 4.1.2 Philological Notes on Iordani opusculum de ponderositate (1565) 263 4.1.3 Book VII of Quesiti et inventioni diverse (1554) 265 4.1.3.1 The Facsimile and English Translation 265 4.1.4 Book VIII of Quesiti et inventioni diverse (1554) 284 4.1.4.1 The Facsimile and English Translation 284 4.1.5 The Italian Critical Transcriptions 348 4.1.5.1 Book VII (1554) 348 4.1.5.2 Book VIII (1554) 358 4.1.6 Iordani opusculum de ponderositate (1565) 390 4.1.6.1 The Facsimile and Critical English Translation 390 4.1.6.2 The Latin Critical Transcription 416

Part IV Circulation of Knowledge & Conclusion

5 Foreign Editions of Quesiti et inventioni diverse 457 5.1 An Outline 457 5.1.1 Quesiti Foreign Editions 458 5.1.1.1 The Foreign Editions, 1547-2010 458 5.1.2 Bibliographical Notes 461 5.1.2.1 Uncertain Dates Around Partial and/or Alleged QuesitV's Editions 461

6 Conclusion 465 6.1 Concluding Remarks 465

References 467

Microfilm Tartalea 489

Main Quesiti Accounts 491

Index 493