: Heike Sefrin-Weis
: Heike Sefrin-Weis (Eds.)
: Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis
: Springer-Verlag
: 9781849960052
: 1
: CHF 74.30
:
: Allgemeines, Lexika
: English
: 328
: Wasserzeichen
: PC/MAC/eReader/Tablet
: PDF

Altho gh not so well known today, Book 4 of Pappus' Collection is one of the most important and influential mathematical texts from antiquity. The mathematical vignettes form a portrait of mathematics during the Hellenistic 'Golden Age', illustrating central problems - for example, squaring the circle; doubling the cube; and trisecting an angle - varying solution strategies, and the different mathematical styles within ancient geometry.

This volume provides an English translation of Collection 4, in full, for the first time, including: a new edition of the Greek text, based on a fresh transcription from the main manuscript and offering an alternative to Hultsch's standard edition, notes to facilitate understanding of the steps in the mathematical argument, a commentary highlighting aspects of the work that have so far been neglected, and supporting the reconstruction of a coherent plan and vision within the work, bibliographical references for further study.
"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."
Preface7
Contents9
General Introduction12
1 Life and Works of Pappus of Alexandria14
1.1 Pappus Life and Times14
1.2 Pappus Works15
1.2.1 Geometry Proper15
1.2.2 Geography18
1.2.3 Astronomy/Astrology18
1.2.4 Mechanics19
2 Survey of Coll. IV20
3 Summary: Coll. IV at a Glance29
I Plane Geometry29
II Linear Geometry: Symptoma-Mathematics of Motion curves30
Meta-Theoretical Passage on the Three Kinds of Geometry: Homogeneity criterion30
III Solid Geometry, Transition Upward, Demarcation Downward 30
Part I Greek Text and Annotated Translation31
Introductory Remarks on Part I32
Tradition, Reception, and Editions of the Text of the Collectio32
Remarks on the Greek Text Printed Here33
Remarks on the Translation34
Remark on the Diagrams36
List of Sigla and Abbreviations37
Concordance of Greek Letters (A) and Latin Equivalents (Translation, Commentary, Diagrams)38
Part Ia Greek Text39
Part Ib Annotated Translation of Collectio IV111
Props. 1-3: Euclidean Plane Geometry: Synthetic Style111
Prop. 1: Generalization of the Pythagorean Theorem111
Prop. 2: Construction of a Minor6112
Prop. 3: Construction of an Irration of an Irrational Beyond Euclid114
Props. 4-6: Plane Analysis Within Euclidean Elementary Geometry116
Prop. 4: Structure of Analysis-Synthesis116
Props. 5/6: Reciprocity in Plane Geometry118
Props. 7 10: Analysis, Apollonian Style (Focus: Resolutio)120
Prop. 7: Determination of Givens120
Prop. 8: Analysis, Apollonian Style122
Prop. 9: Lemma for Prop. 10125
Prop. 10: Resolutio for a Sub-case of the Apollonian Problem126
Props. 11 and 12: Analysis: Extension of Configuration3/Apagoge127
Prop. 11: Chords, Perpendicular, and Diameter in a Circle127
Prop. 12: Plane Analysis via Apagoge Chords, Parallels, and Angles in a Circle
Props. 13 18: Arbelos Treatise: Plane Geometry, Archimedean Style131
Prop. 13: Preparatory Lemma: Points of Similarity and Touching Circles132
Prop. 14: Technical Lemma. Perpendiculars and Diameters in Configurations with Three Touching Circles134
Prop. 15: Sequence of Inscribed Touching Circles: Induction Lemma138
Prop. 16: Arbelos Theorem141
Prop. 17: Lemma Used in Prop. 16, Addition 2145
Prop. 18: Addition: Progression Theorem, Odd Numbers146
Props. 19 22: Archimedean Spiral147
Prop. 19: Genesis and Symptoma of the Spiral147
Prop. 20: Progression of Spiral Radii2: Proportional to Rotation Angles149
Prop. 21: Spiral Area4 in Relation to the Circle149
Addition: Spiral Areas and Circumscribed Circles151
Prop. 22: Ratio of Spiral Areas as Ratio of Cubes over Maximal Spiral Radii152
Addition: Measurement of Spiral Quadrants153
Props. 23 25: Conchoid of Nicomedes/Duplication of the Cube154
Genesis and Symptoma of the Conchoid154
Prop. 23: Neusis Construction6155
Prop. 24: Two Me