# Perseus (geometer)

**Perseus** (c. 150 BC) was an ancient Greek geometer, who invented the concept of spiric sections, in analogy to the conic sections studied by Apollonius of Perga.

## Life

Few details of Perseus' life are known, as he is mentioned only by Proclus and Geminus; none of his own works have survived.

## Spiric sections

The spiric sections result from the intersection of a torus with a plane that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth-order (quartic) plane curves, whereas the conic sections are second-order (quadratic) plane curves. Spiric sections are a special case of a toric section, and were the first toric sections to be described.

### Examples

The most famous spiric section is the Cassini oval, which is the locus of points having a constant *product* of distances to two foci. For comparison, an ellipse has a constant *sum* of focal distances, a hyperbola has a constant difference of focal distances and a circle has a constant ratio of focal distances.

## References

- Tannery P. (1884) "Pour l'histoire des lignes et de surfaces courbes dans l'antiquitÃ©",
*Bull. des sciences mathÃ©matique et astronomique*,**8**, 19-30. - Heath TL. (1931)
*A history of Greek mathematics*, vols. I & II, Oxford. - O'Connor, John J.; Robertson, Edmund F., "Perseus",
*MacTutor History of Mathematics archive*, University of St Andrews.