COVER -- HALF-TITLE -- TITLE -- COPYRIGHT -- DEDICATION -- CONTENTS -- ACKNOWLEDGMENTS -- INTRODUCTION -- 1 goal of the translation -- 2 preliminary notes: conventions -- 2.1 Some special conventions of Greek mathematics -- 2.2 Special conventions adopted in this translation -- 2.3 Purpose and practices of the critical edition of diagrams -- 3 preliminary notes: archimedes ́works -- 3.1 Archimedes and his works -- 3.2 The text of Archimedes -- 3.3 The two books On the Sphere and the Cylinder -- TRANSLATION AND COMMENTARY -- ON THE SPHERE AND THE CYLINDER, BOOK I -- /Introduction: general/ -- textual comments -- general comments -- Introduction: the genre -- Archimedes' audience: conon and dositheus -- /"Axiomatic" introduction/ -- /Definitions/ -- /Postulates/ -- textual comments -- general comments -- Definitions 1-4 -- Postulates 1-2: about what? -- Unpacking Postulate 2 -- Unpacking Postulate 3 -- The overall structure of Definitions 1-4, Postulates 1-4 -- Postulate 5 -- /1/ -- textual comments -- general comments -- The use of the diagram -- The use of the axiomatic introduction -- Generality of the proof -- /2/ -- textual comments -- general comments -- Existence and realism -- Shortcuts used in exposition -- Schematic nature and the intended generality of the diagram -- /3/ -- textual comments -- general comments -- The scholiast's regress -- /4/ -- textual comments -- general comments -- Repetition of text, and virtual mathematical actions -- /5/ -- textual comments -- general comments -- Impatience revealed in exposition -- /6/ -- textual comments -- general comments -- Lack of pedagogic concerns -- Meta-mathematical interests displayed in the text -- /7/ -- textual comments -- general comments -- Equality and identity -- The "grounds" for a claim -- /8/ -- textual comments -- general comments.

The significance of different ways of naming objects -- Generality and the use of letters referring to the diagram -- /9/ -- textual comments -- general comments -- Bifurcating structure of proof -- Ambiguities and their implication for the author/audience attitude -- /10/ -- textual comments -- general comments -- What is the sequence of actions inside a construction? -- The role of the axiomatic discussion -- /11/ -- textual comments -- general comments -- The significance of different ways of using letters -- Operations on phrases as a tool for argumentation -- /12/ -- textual comments -- general comments -- The extent to which words are allowed to shift -- The operation of "imagination" -- A tendency to abbreviate -- /Sequel to sequence 9-12/ -- textual and general comments -- /13/ -- textual comments -- general comments -- Equalities and inequalities -- Arguments based on division of logical space -- Relations between parts of text -- Some notes on operations with ratios -- /14/ -- textual comments -- general comments -- Different layers of imagination -- /15/ -- textual comments -- general comments -- Reference to Proposition 14 -- /16/ -- textual comments -- general comments -- The phenomenon of arguments that are left implicit -- Non-pictorial diagrams -- /Interlude recalling elementary results/ -- textual comments -- general comments -- /17/ -- textual comments -- general comments -- Underspecification of diagram by text -- /18/ -- textual comments -- general comments -- A diagram that is less general than the text -- Overall structure of transformations of proportions -- /19/ -- textual comments -- general comments -- Relations between parts of the proposition -- /20/ -- textual comments -- general comments -- Formulaic expression and definitions -- /21/ -- textual comments -- general comments -- The strange nature of the proposition.

/22/ -- textual comments -- general comments -- Underspecification of objects by the text -- Other aspects of the brevity of the proof -- /23/ -- textual and general comments -- Producing objects by rotation -- The internal structure of the proposition -- /24/ -- textual comments -- general comments -- The absence of algebraic and logical conventions -- /25/ -- textual comments -- general comments -- On "that is," and the nature of the proposition -- /26/ -- textual comments -- general comments -- A move to a more general space -- /27/ -- general comments -- The particular and the general, and the "toy universe" -- The creativity of formulaic language -- /28/ -- textual comments -- general comments -- Proposition I.23 gave rise to a locally established form -- /29/ -- general comments -- Quotations and other textual procedures -- /30/ -- textual comments -- general comments -- Reduced setting-out -- /31/ -- /Corollary/ -- textual and general comments -- /32/ -- textual comments -- general comments -- A fully fledged proposition, finally -- Thinking in ratios and in geometrical relations -- Letters used as indices -- /33/ -- textual comments -- general comments -- A sense of accomplishment -- Elegance of expression -- /34/ -- /Corollary/ -- textual comments -- general comments -- Why state the result in a corollary? -- Strict proposition structure versus ease of geometrical manipulation -- Various ways of "taking as obvious" -- /35/ -- textual comments -- general comments -- The nature of the final "chapter 6" -- Ambiguous language, its significance and its practical resolution -- /36/ -- textual and general comments -- /37/ -- textual comments -- general comments -- Another look at the history of "the obvious" -- /38/ -- /Corollary/ -- textual comments -- general comments -- Archimedean carelessness as regards detail -- /39/ -- /Corollary/.

textual comments -- general comments -- Backwards-looking structure and its significance for how the proposition is to be read -- /40/ -- /Corollary 1/ -- /Corollary 2/ -- textual comments -- general comments -- Limits on visualization -- So-called "corollaries," and the structure of the proposition -- /41/ -- textual comments -- general comments -- A sketchy proposition, with objects sketchily described -- /42/ -- textual comments -- general comments -- Meager diagrams, rich verbal formulations -- Little overarching structure of the proposition -- /43/ -- textual comments -- general comments -- Relations between propositions become even more direct -- /44/ -- textual comments -- general comments -- Haste -- ON THE SPHERE AND THE CYLINDER, BOOK II -- /Introduction/ -- textual comments -- general comments -- Practices of mathematical communication -- /1/ -- textual comments -- general comments -- Does "analysis" find solutions? -- /2/ -- /Corollary/ -- textual comments -- general comments -- The two cones and the generality of the argument -- The operation of "imagination:" the border between the conceptual and spatial -- /3/ -- textual comments -- general comments -- Enumerating problems and the structure of the book -- The strange nature of "being given" -- /4/ -- textual comments -- general comments -- A suggestion on the function of analysis in a complex solution -- The use of interim results -- /5/ -- textual comments -- general comments -- Relying on informal intuitions -- Implicit, non-linear structures of argument -- /6/ -- textual comments -- general comments -- The relation between analysis and synthesis -- /7/ -- textual comments -- general comments -- The nature of numbers in standard Greek geometrical practice -- /8/ -- textual comments -- general comments -- "Exponent" ratios -- Steps f-13: the language and the diagram.

Archimedes takes off -- /In another way/ -- textual comments -- general comments -- Structural properties of the proof -- Extensions of plane geometry to solid figures -- /9/ -- textual and general comments -- The double figure, a preliminary discussion -- Textual and mathematical problems other than the double figure -- How can the proof be seen to apply to both cases? -- conclusion -- EUTOCIUS' COMMENTARY TO ON THE SPHERE AND THE CYLINDER I -- TO BOOK 1 -- To the definitions -- To Theorem 2 -- To 3 -- To 6 -- To 8 -- To 9 -- To 10 -- To 13 -- To 14 -- To 16 -- To 23 -- To 30 -- To 32 -- To 34 -- To 37 -- To 39 -- To 40 -- To 41 -- To 42 -- To 44 -- EUTOCIUS' COMMENTARY TO ON THE SPHERE AND THE CYLINDER II -- To the synthesis of the 1st -- As Plato -- As Hero in the Mechanical Introduction and in the Construction of Missile-Throwing Machines -- As Philo the Byzantine -- As Apollonius -- As Diocles in On Burning Mirrors -- As Pappus in the Introduction to Mechanics -- As Sporus -- As Menaechmus -- In another way -- Archytas' solution, according to Eudemus' History -- As Eratosthenes -- As Nicomedes in On Conchoid Lines -- To the second theorem -- To 3 -- To 4 -- As Dionysodorus -- As Diocles in On Burning Mirrors -- To the synthesis of 4 -- To 5 -- To the synthesis of 5 -- To 6 -- To the synthesis of 6 -- To 7 -- To 8 -- Lemma to the following -- To the alternative of 8 -- To 9 -- BIBLIOGRAPHY -- INDEX.