Greek Text and Annotated Translation.- Greek Text.- Annotated Translation of Collectio IV.- Commentary.- Plane Geometry, Apollonian Style.- Plane Geometry, Apollonian Style.- Plane Geometry, Archaic Style.- Plane Geometry, Archimedean.- Motion Curves and Symptoma-Mathematics.- Meta-theoretical Passage.- Angle Trisection.- General Angle Division.- Quadratrix, Rectification Property.- Analysis for an Archimedean Neusis.
"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."
