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**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.

**Greek** is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

**Greece**, officially the **Hellenic Republic**, also known as **Hellas**, is a country located in Southern and Southeast Europe, with a population of approximately 11 million as of 2016. Athens is the nation's capital and largest city, followed by Thessaloniki.

In geometry, a **spiric section**, sometimes called a **spiric of Perseus**, is a quartic plane curve defined by equations of the form

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

**Proclus Lycaeus**, called **the Successor**, was a Greek Neoplatonist philosopher, one of the last major classical philosophers. He set forth one of the most elaborate and fully developed systems of Neoplatonism. He stands near the end of the classical development of philosophy, and was very influential on Western medieval philosophy.

**Geminus** of Rhodes, was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the *Introduction to the Phenomena*, still survives; it was intended as an introductory astronomy book for students. He also wrote a work on mathematics, of which only fragments quoted by later authors survive.

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.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a *torus of revolution*.

In mathematics, a **plane curve** is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves. Plane curves also include the Jordan curves and the graphs of continuous functions.

In mathematics, a **conic section** is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

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*focal distances, a hyperbola has a constant difference of focal distances and a circle has a constant ratio of focal distances.

A **Cassini oval** is a quartic plane curve defined as the set of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted with an ellipse, for which the *sum* of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.

In geometry, a **locus** is a set of all points, whose location satisfies or is determined by one or more specified conditions.

In mathematics, an **ellipse** is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The elongation of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to but less than 1.

In classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

**Euclid**, sometimes called **Euclid of Alexandria** to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His *Elements* is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the *Elements*, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

A **helix**, plural **helixes** or **helices**, is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the *axis*. Examples of helices are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word *helix* comes from the Greek word *ἕλιξ*, "twisted, curved".

In geometry, the **lemniscate of Bernoulli** is a plane curve defined from two given points *F*_{1} and *F*_{2}, known as **foci**, at distance 2*a* from each other as the locus of points *P* so that *PF*_{1}·*PF*_{2} = *a*^{2}. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from *lemniscatus*, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

**Apollonius of Perga** was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

In geometry, the **Dandelin spheres** are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called **focal spheres**.

In mathematics, the **eccentricity** of a conic section is a non-negative real number that uniquely characterizes its shape.

**Diocles** was a Greek mathematician and geometer.

In geometry, **Villarceau circles** are a pair of circles produced by cutting a torus obliquely through the center at a special angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles. They are named after the French astronomer and mathematician Yvon Villarceau (1813–1883). Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.

**Menaechmus** was an ancient Greek mathematician and geometer born in Alopeconnesus in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and hyperbola.

In geometry, a **hippopede** is a plane curve determined by an equation of the form

A **quartic plane curve** is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:

In algebraic geometry, a **lemniscate** is any of several figure-eight or ∞-shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made.

A **toric section** is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux.

In geometry, the ** n-ellipse** is a generalization of the ellipse allowing more than two foci.

In mathematics, a **generalized conic** is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points – the foci – in the plane is a constant. The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed, finite set of points in the plane is called an *n*–ellipse and can be thought of as a generalized ellipse. Since an ellipse is the equidistant set of two circles, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangular Cartesian coordinates, the equation *y* = *x*^{2} represents a parabola. The generalized equation *y* = *x*^{r}, for *r* ≠ 0 and *r* ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.

- 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 .

**Edmund Frederick Robertson** is a Professor emeritus of pure mathematics at the University of St Andrews.

The **MacTutor History of Mathematics archive** is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics.

The **University of St Andrews** is a British public university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland and the third oldest university in the English-speaking world. St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy.

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