Perseus (geometer)

Perseus (Greek: Περσεύς; 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[edit]

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[edit]

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[edit]

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[edit]

* Tannery P. (1884) "Pour l'histoire des lignes et de surfaces courbes dans l'antiquité", Bull. des sciences mathématique 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

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Mathematicians
(timeline)|

* Anaxagoras
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* Aristaeus the Elder
* Aristarchus
* Aristotle
* Apollonius
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* Autolycus
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* Zenodorus

Treatises|

* Almagest
* Archimedes Palimpsest
* Arithmetica
* Conics (Apollonius)
* Catoptrics
* Data (Euclid)
* Elements (Euclid)
* Measurement of a Circle
* On Conoids and Spheroids
* On the Sizes and Distances (Aristarchus)
* On Sizes and Distances (Hipparchus)
* On the Moving Sphere (Autolycus)
* Optics (Euclid)
* On Spirals
* On the Sphere and Cylinder
* Ostomachion
* Planisphaerium
* Spherics (Theodosius)
* Spherics (Menelaus)
* The Quadrature of the Parabola
* The Sand Reckoner

Problems|

* Constructible numbers
* Angle trisection
* Doubling the cube
* Squaring the circle
* Problem of Apollonius

Concepts
and definitions|

* Angle
* Central
* Inscribed
* Axiomatic system
* Axiom
* Chord
* Circles of Apollonius
* Apollonian circles
* Apollonian gasket
* Circumscribed circle
* Commensurability
* Diophantine equation
* Doctrine of proportionality
* Euclidean geometry
* Golden ratio
* Greek numerals
* Incircle and excircles of a triangle
* Method of exhaustion
* Parallel postulate
* Platonic solid
* Lune of Hippocrates
* Quadratrix of Hippias
* Regular polygon
* Straightedge and compass construction
* Triangle center

Results|

| In Elements|

* Angle bisector theorem
* Exterior angle theorem
* Euclidean algorithm
* Euclid's theorem
* Geometric mean theorem
* Greek geometric algebra
* Hinge theorem
* Inscribed angle theorem
* Intercept theorem
* Intersecting chords theorem
* Intersecting secants theorem
* Law of cosines
* Pons asinorum
* Pythagorean theorem
* Tangent-secant theorem
* Thales's theorem
* Theorem of the gnomon

|

Apollonius|

* Apollonius's theorem

Other|

* Aristarchus's inequality
* Crossbar theorem
* Heron's formula
* Irrational numbers
* Law of sines
* Menelaus's theorem
* Pappus's area theorem
* Problem II.8 of Arithmetica
* Ptolemy's inequality
* Ptolemy's table of chords
* Ptolemy's theorem
* Spiral of Theodorus

Centers|

* Cyrene
* Mouseion of Alexandria
* Platonic Academy

Related|

* Ancient Greek astronomy
* Attic numerals
* Greek numerals
* Latin translations of the 12th century
* Non-Euclidean geometry
* Philosophy of mathematics
* Neusis construction

History of|

* A History of Greek Mathematics
* by Thomas Heath
* algebra
* timeline
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