Pappus of Alexandria: Book 4 of the Collection (eBook)

Preface 7
Contents 9
General Introduction 12
1 Life and Works of Pappus of Alexandria 14
1.1 Pappus’ Life and Times 14
1.2 Pappus’ Works 15
1.2.1 Geometry Proper 15
1.2.2 Geography 18
1.2.3 Astronomy/Astrology 18
1.2.4 Mechanics 19
2 Survey of Coll. IV 20
3 Summary: Coll. IV at a Glance 29
I Plane Geometry 29
II Linear Geometry: Symptoma-Mathematics of Motion curves 30
Meta-Theoretical Passage on the Three Kinds of Geometry: Homogeneity criterion 30
III Solid Geometry, Transition “Upward,” Demarcation “Downward” 30
Part I Greek Text and Annotated Translation 31
Introductory Remarks on Part I 32
Tradition, Reception, and Editions of the Text of the Collectio 32
Remarks on the Greek Text Printed Here 33
Remarks on the Translation 34
Remark on the Diagrams 36
List of Sigla and Abbreviations 37
Concordance of Greek Letters (A) and Latin Equivalents (Translation, Commentary, Diagrams) 38
Part Ia Greek Text 39
Part Ib Annotated Translation of Collectio IV 111
Props. 1-3: Euclidean Plane Geometry: Synthetic Style 111
Prop. 1: Generalization of the Pythagorean Theorem 111
Prop. 2: Construction of a Minor6 112
Prop. 3: Construction of an Irration of an Irrational Beyond Euclid 114
Props. 4-6: Plane Analysis Within Euclidean Elementary Geometry 116
Prop. 4: Structure of Analysis-Synthesis 116
Props. 5/6: Reciprocity in Plane Geometry 118
Props. 7–10: Analysis, Apollonian Style (Focus: Resolutio) 120
Prop. 7: Determination of Givens 120
Prop. 8: Analysis, Apollonian Style 122
Prop. 9: Lemma for Prop. 10 125
Prop. 10: Resolutio for a Sub-case of the Apollonian Problem 126
Props. 11 and 12: Analysis: Extension of Configuration3/Apagoge 127
Prop. 11: Chords, Perpendicular, and Diameter in a Circle 127
Prop. 12: Plane Analysis via Apagoge Chords, Parallels, and Angles in a Circle
Props. 13–18: Arbelos Treatise: Plane Geometry, Archimedean Style 131
Prop. 13: Preparatory Lemma: Points of Similarity and Touching Circles 132
Prop. 14: Technical Lemma. Perpendiculars and Diameters in Configurations with Three Touching Circles 134
Prop. 15: Sequence of Inscribed Touching Circles: Induction Lemma 138
Prop. 16: Arbelos Theorem 141
Prop. 17: Lemma Used in Prop. 16, Addition 2 145
Prop. 18: Addition: Progression Theorem, Odd Numbers 146
Props. 19–22: Archimedean Spiral 147
Prop. 19: Genesis and Symptoma of the Spiral 147
Prop. 20: Progression of Spiral Radii2: Proportional to Rotation Angles 149
Prop. 21: Spiral Area4 in Relation to the Circle 149
Addition: Spiral Areas and Circumscribed Circles 151
Prop. 22: Ratio of Spiral Areas as Ratio of Cubes over Maximal Spiral Radii 152
Addition: Measurement of Spiral Quadrants 153
Props. 23–25: Conchoid of Nicomedes/Duplication of the Cube 154
Genesis and Symptoma of the Conchoid 154
Prop. 23: Neusis Construction6 155
Prop. 24: Two Mean Proportionals via Neusis 157
Prop. 25: Cube Duplication, Cube Construction in Given Ratio 158
Props. 26–29: Quadratrix1 159
Genesis and Symptoma of the Quadratrix 159
Criticism of the Quadratrix Under the Description via Motions (Sporus) 160
Prop. 26: Rectification of the Arc of a Quadrant 162
Prop. 27: Squaring the Circle 164
Prop. 28: Analytical Determination of the Quadratrix from an Apollonian Helix 165
Prop. 29: Analytical Determination of the Quadratrix from the Archimedean Spiral 167
Prop. 30: Symptoma-Theorem on the Archimedean Spherical Spiral 169
Prop. 30: Surfaces Cut Off by a Spiral on a Hemisphere 169
Three Kinds of Mathematical Questions, and Their Appropriate Means of Argumentation1 172
Props. 31–34: Angle Trisection 174
Prop. 31: Neusis for Angle Trisection 174
Prop. 32: Trisection of the Angle via Neusis 176
Prop. 33: Analysis-Synthesis for the Hyperbola-Construction in the Trisection Neusis 178
Prop. 34: Alternative Constructions of the Angle Trisection via Solid Loci 4 180
Props. 35–38: Generalization of Solid Problems: Angle Division 183
Prop. 35: General Angle Division 183
Prop. 36: Equal Arcs of Different Circles 185
Prop. 37: Isosceles Triangle with Angles in Given Ratio 185
Prop. 38: Regular Polygon with any Given Number of Sides Inscribed in the Circle 187
Props. 39–41: Constructions Based on the Rectification Property of the Quadratrix4 187
Prop. 39: Converse of Circle Rectification 187
Prop. 40: Arc over Chord in Given Ratio8 188
Prop. 41: Incommensurable Angles 190
Props. 42–44: Analysis of an Archimedean Neusis 191
Prop. 42: Hyperbola for the Archimedean Neusis 191
Prop. 43: Parabola for the Archimedean Neusis 193
Prop. 44: Archimedean Neusis (Following Hultsch’s Partial Restitution1) 194
Part II Commentary 196
Introductory Remarks on Part II 197
II, 1 Plane Geometry, Euclidean Style 199
1 Props. 1–6: Plane Geometry, Euclidean Style 199
1.1 Prop. 1: Generalization of the Pythagorean Theorem 199
1.1.1 Schema of a Classical Apodeixis, According to Proclus2: 200
1.1.2 Proof Protocol Prop. 1 200
1.2 Props. 2 and 3: Construction of Euclidean Irrationals 201
1.2.1 Excursus: Remarks on Elements X (Irrational Lines) 202
1.2.1.1 Survey of Elements X 203
1.2.2 Prop. 2: Construction of a Minor 205
1.2.2.1 Proof Protocol Prop. 2 205
1.2.2.2 Proof Protocol for XIII, 11 206
1.2.3 Prop. 3: Construction of an Irrational Beyond X, with the Notions from X 206
1.2.3.1 Proof Protocol Prop. 3 206
1.3 Props. 4–6: Plane Geometrical Analysis in the Context of Euclidean Geometry 207
1.3.1 Excursus: Greek Geometrical Analysis as a Method: Sketch of the Status Quaestionis 209
1.3.1.1 Sources on Greek Geometrical Analysis 209
1.3.1.2 Pappus’ Outline of Analysis-Synthesis in Coll. VII 210
1.3.1.3 Analysis in Outline 212
1.3.1.4 Examples of Analyses in Coll. IV 214
1.3.1.5 Structural Schema of Analysis-Synthesis1 215
1.3.2 Prop. 4: The Structure of Plane Analysis-Synthesis 216
1.3.2.1 Proof Protocol Prop 4 216
1.3.3 Props. 5/6: Reciprocity in Plane Geometry 217
1.3.3.1 Proof Protocol Prop. 5 218
1.3.3.2 Proof Protocol Prop. 6 218
II, 2 Plane Geometry, Apollonian Style 219
2 Props. 7–10: Plane Geometry, Apollonian Style 219
2.1 Overview Props. 7–10 219
2.1.1 Historical Context for Props. 7–10: Apollonius’ Tangencies and the Apollonian Problem 221
2.2 Prop. 7: Determining Given Features Using the Data 222
2.2.1 Proof Protocol Prop. 7a 225
2.2.2 Proof Protocol Prop. 7b 225
2.3 Prop. 8: Resolutio for an Intermediate Step in the Apollonian Problem 225
2.3.1 Proof Protocol Prop. 8 227
2.4 Prop. 9: Auxiliary Lemma for Prop. 10 227
2.4.1 Proof Protocol Prop. 9 227
2.5 Prop. 10: Resolutio for a Special Case of the Apollonian Problem 227
2.5.1 Proof Protocol Prop. 10 228
II, 3 Plane Geometry, Archaic Style 229
3 Analysis-Synthesis Pre-Euclidean Style 229
3.1 Props. 11 and 12: Chords and Angles in a Semicircle 229
3.2 Prop. 11: Representation of a Chord as Segment of the Diameter 230
3.2.1 Proof Protocol Prop. 11 230
3.3 Prop. 12: Angle Over a Segment of the Diameter 231
3.3.1 Proof Protocol Prop. 12 231
II, 4 Plane Geometry, Archimedean 233
4 Props. 13–18: Arbelos (Plane Geometry, Archimedean Style) 233
4.1 Observations on Props. 13–18 233
4.1.1 Archimedean Character of Props. 13–18 233
4.1.2 Factors that Point Toward an Original Larger Extension of the Arbelos Treatise 237
4.1.3 Arbelos Theorem 238
4.1.4 Structure of Props. 13–18 238
4.2 Prop. 13: Preliminary Lemma 238
4.2.1 Proof Protocol Prop. 13 239
4.2.2 Defense of My Reading of Prop. 13 240
4.3 Prop. 14: Technical Theorem 241
4.3.1 Proof Protocol Prop. 14 242
4.4 Prop. 15: Lemma for Induction 243
4.4.1 Proof Protocol Prop. 15 243
4.5 Prop. 16: Arbelos Theorem 244
4.5.1 Proof Protocol Prop. 16 244
4.6 Prop. 17: Supplementary Auxiliary Lemma for Prop. 16, Corollary 2 245
4.6.1 Proof Protocol Prop. 17 246
4.7 Prop. 18: Analogue to the Arbelos Theorem When the Second Inner Semicircle Is Missing 246
4.7.1 Proof Protocol Prop. 18 246
II, 5 Motion Curves and Symptoma-Mathematics 248
5 Props. 19–30: Motion Curves and Symptoma-Mathematics 248
5.1 General Observations on Props. 19–30 248
5.1.1 Mechanics Versus Geometry and the Problem of Motions 252
5.1.2 “Mechanical” Versus “Instrumental” 253
5.1.3 Survey of Props. 19–30 254
5.2 Props. 19–22: Archimedean Plane Spiral, “Heuristic” Version 255
5.2.1 Relation of Props. 19–22 to SL 255
5.2.2 Heuristic Method in Props. 19–22 256
5.2.3 Survey of Props. 19–22 257
5.2.4 Prop. 19: Genesis and Symptoma of the Archimedean Spiral 258
5.2.5 Prop. 20: Progression of Spiral Radii 258
5.2.6 Prop. 21: Area Theorem 258
5.2.6.1 Argument in Prop. 21 258
5.2.6.2 Proof Protocol of SL 24 259
5.2.7 Prop. 22: Ratio of Spiral Areas and Spiral Segments2 260
5.2.7.1 Argument in Prop. 22 260
5.2.7.2 Addition to Prop. 22: Areas of Spiral Quadrants1 261
5.3 Props. 23–25: Conchoid of Nicomedes 261
5.3.1 General Observations on Props. 23–25 261
5.3.1.1 Nicomedes 262
5.3.1.2 Conchoid 263
5.3.1.4 Neusis 268
5.3.2 Prop. 23: Genesis and Symptoma of the Conchoid 269
5.3.3 Prop. 24: Two Mean Proportionals via Neusis 270
5.3.3.1 Proof Protocol Prop. 24 271
5.3.4 Prop. 25: Cube Multiplication in a Given Ratio 271
5.4 Props. 26–29: Quadratrix/Squaring the Circle 272
5.4.1 General Observations on Props. 26–29 272
5.4.1.1 Structure of Props. 26–29 272
5.4.1.2 Authorship for Props. 26–29 272
5.4.1.3 Quadratrix 274
5.4.1.4 Squaring the Circle 276
5.4.1.5 Circle Quadrature Through the Ages 278
5.4.2 Genesis and Symptoma of the Quadratrix 280
5.4.3 Criticism of the Genesis by Sporus 280
5.4.4 Prop. 26: Rectification Property of the Quadratrix 281
5.4.4.1 Proof Protocol Prop. 26 281
5.4.5 Prop. 27: Squaring the Circle 283
5.4.6 Prop. 28: Geometrical Analysis, Linking the Quadratrix to Loci on Surfaces Through a Cylindrical Helix 283
5.4.6.1 Outline of the Analysis in Prop. 28 283
5.4.6.2 Intersection Plane in Step 2.1: Through EZ or BC? 284
5.4.7 Prop. 29: Geometrical Analysis, Linking the Quadratrix to Loci on Surfaces with Spiral 285
5.4.7.1 Outline of the Analysis in Prop. 29 285
5.4.7.2 Lines, Planes, and Surfaces in Prop. 29 286
5.4.8 Additional Comments on Props. 28 and 29 287
5.4.8.1 Loci on Surfaces 287
5.4.9 Excursus: Speculative Remarks on the Potential of Analysis-Based Symptoma-Characterization of Higher Curves 290
5.5 Prop. 30: Area Theorem on the Spherical Spiral 292
5.5.1 Proof Protocol Prop. 30 294
II, 6 Meta-theoretical Passage 296
6 Meta-theoretical Passage 296
6.1 The Three Kinds of Geometry According to Pappus 296
6.2 The Homogeneity Requirement 298
II, 7 Angle Trisection 301
7 Props. 31–34: Trisecting the Angle 301
7.1 Angle Trisection Through the Ages 302
7.1.1 Attested Ancient Solutions 303
7.1.2 Islamic Middle Ages (Selective3) 304
7.1.3 Occidental Middle Ages (Selective) 304
7.1.4 Some Examples from Renaissance and Early Modern Times 305
7.1.5 Nineteenth Century 305
7.2 Analysis in Props. 31–34 306
7.3 Props. 31–33: Angle Trisection via Neusis 308
7.3.1 Proof Protocol Prop. 31 308
7.3.2 Proof Protocol Prop. 32 309
7.3.3 Proof Protocol Prop. 33 310
7.4 Prop. 34: Angle Trisection Without Neusis 312
7.4.1 Proof Protocol Prop. 34a 313
Corollary (Not in Pappus’ Text) 314
7.4.2 Proof Protocol Prop. 34b 314
II, 8 General Angle Division 316
8 Props. 35–38: General Angle Division and Applications 316
8.1 Prop. 35: Angle Division 317
8.1.1 Proof Protocol Prop. 35 317
8.2 Prop. 36: Equal Arcs on Different Circles 318
8.2.1 Proof Protocol Prop. 36 318
8.3 Props. 37 and 38: Regular Polygon with Any Given Number of Sides 318
8.3.1 Proof Protocol Props. 37 and 38 319
II, 9 Quadratrix, Rectification Property 320
9 Props. 39–41: Further Results on Symptoma-Mathematics of the Quadratrix (Rectification Property1) 320
9.1 Prop. 39: Converse of Circle Rectification 320
9.1.1 Proof Protocol Prop. 39 320
9.2 Prop. 40: Construct a Circular Arc over a Line Segment,in a Given Ratio 321
9.2.1 Proof Protocol Prop. 40 321
9.3 Prop. 41: Incommensurable Angles 323
II, 10 Analysis for an Archimedean Neusis 324
10 Props. 42–44: Analysis for an Archimedean Neusis/Example for Work on Solid Loci 324
10.1 General Observations on Props. 42–44 324
10.1.1 Criticism of Archimedes’ Use of a Neusis in SL 325
10.1.2 Pappus’ Purpose in Props. 42–44, and the Content of 42–44 326
10.1.3 Analysis in 42– 44 as a Criterion for Establishingthe “Solid” Nature of the Neusis 327
10.1.4 Limits and Gaps in the Pappus’ Account 327
10.1.5 Solid Loci in Props. 42–44, in Comparison to Props. 31–34 329
10.1.6 Aristaeus as a Possible Source 330
10.1.7 Alternatives for the Neusis 331
10.2 Props. 42–44: Analysis of an Archimedean Neusis 333
10.2.1 Proof Protocol Prop. 42 333
10.2.2 Proof Protocol Prop. 43 333
10.2.3 Proof Protocol Prop. 44 333
Appendices 335
Index 344

"II, 5 Motion Curves and Symptoma-Mathematics (p. 223-224)

5 Props. 19–30: Motion Curves and Symptoma-Mathematics

5.1 General Observations on Props. 19–30

Props. 19–30 (as well as 35–41) deal with lines and curves that are different both from the circles and straight lines of Euclidean geometry, and from the conic sections. They are generated from moving points, where a rule is given which regulates the “motions” involved. They will be called motion curves here. An example would be the plane spiral of Archimedes, where a point moves along the radius of a circle in uniform speed, and is at the same time carried along on that radius as it rotates the full circle, also in uniform speed.

The point describes a spiral line in the process. Another example, though this is not used in ancient geometry, would be the generation of a circle as the “motion curve” described by the endpoint of a radius as the radius rotates a full 360°. In order to study the mathematical properties of such curves, one has to come to a quantifiable characterization, as a proportion, or an equality that applies to all the points on the curve and only to them. All mathematical properties have to be derived from, or related back to, this original characterizing property.

It is called the symptoma of the curve. It ultimately rests on the motions used to generate the curve, but as they do not appear in the mathematical discourse, the mathematics develops out of the symptoma itself as the starting point. I will call this type of mathematics symptomamathematics. The conchoid of Nicomedes,1 e.g., has the symptoma that all lines drawn from a point of the curve to the pole have a definite neusis property: the segment cut off on it between the canon and the point on the curve has a fixed length.

The curve itself is viewed as the locus for this property, and this is how it is employed in mathematical argumentation. An analogy would be to view the circle as the locus of all points that have a fixed distance to a given point. Arguably this could even be seen as the Euclidean symptoma of the circle. The case of the conics is somewhat similar: they could be viewed (and some scholars think they were) as the symptoma-curves for certain equalities expressible via application of areas, and whether this is their true definition or not, they were often employed this way in mathematical investigation.

The motion curves discussed in Coll. IV are: Archimedean plane spiral (Prop. 19), Nicomedean conchoid (Prop. 23, though defined as quasi-symptoma-curve), quadratrix (Prop. 26, also defined as a symptoma-curve via analysis of loci on surfaces in Props. 28 and 29), Archimedean spherical spiral (Prop. 30) and Apollonian helix (used, not defined in Prop. 28). The account given by Pappus suggests a certain developmental line, which has, on the whole, been tacitly accepted by most scholars, even if they do not think highly of Pappus as a mathematician (e.g., Knorr 1986).

For Props. 19–30 are our main source for this type of “higher” ancient geometry, the basis for its reconstruction.1 Generally, there are two types of motion curves, developing from curves like the Archimedean spiral and the quadratrix. They can be associated with two strategies for dealing with the problem of finding a mathematically acceptable “definition” of the curves."