1. A BRİEF HİSTORY OF THE CONİC SECTİON
by Eugenio
CONICS
A BRİEF HİSTORY OF THE CONİC SECTİON

2. WHAT İS A CONİC?
A conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane.
(demonstration of a conic section)

3. History of the conics…
Great ancient mathematicians who worked onto the
conic section:
Manaecmus
Euclid
Archimedes of Syracuse
Apollonius
Aristeus
Let’s talk about some of the most important of these: T

4. Manaecmus ( bC)
We don’t have lots of infos about his life and his work and our sources are an epigram by Eratosthenes, some writings of Proclus and an episode of Plutarch’s opera about the relation between that matematician and one of his teachers, Plato. Menaechmus was said to have been the tutor of Alexander the Great.
He was the first to investigate curves that would come to be known as the ellipse, the parabola and the hyperbola as sections of a cone. For that reason these curves were called for a long time the Maenechmian triads.
He discovered that curves as a by-product of his attempt to solve the “Delian Problem”, one of the three most famous geometry problems, also known as doubling a cube.

5. Euclid (around bc)
The real inventor of the modern geometry, probably knew Manaechmus as an other probable pupil of Plato’s Academy in Athen. He wrote a foundamental opera of geometry called “Elements”, a collection of definitions, postulates, propositions and mathematical proofs of the propositions (The last book of it cover the Euclidean geometry).
Between his lost works you can find a treatise that cover deeply the conic sections: “Conics”. In this book was probably explained by Euclid all the previous knowledge about this topic in his tipical way of writing. These contents were directly connected with the homonymous work of Apollonius where was discovered , that if one allowed the cutting plane to vary its angle with respect to the side of the cone, then any cone would produce all three types of section. He also provided solutions to the tangent and normal problems for each of these curves.

6. Archimedes of Syracuse (c. 287 BC – c. 212 BC)
One of the most eclectic thinker in history he worked on
engeenering, mechanic, physics, mathematic and
geometry. He wrote a big number of treatises and essays.
In relation to the conic section he wrote: “On Conoids and
Spheroids” wherein there were explicated ways to calculate
the areas and volumes of section of cones, spheres, and
paraboloids; “The Quadrature of the Parabola”, a work that
presents 24 propositions regarding parabolas, culminating
in a proof that the area of a parabolic segment is 4/3 that
of a certain described triangle.

7. Evolution of conics after the Greek period
The theories about conic section, discovered and esplicated by the ancient Greek
mathematicians, remained unimproved until the studies of two great thinkers of
the 17° centurie:
Friedrich Johannes Kepler Renè Descartes:
Using the ellipse for
planet’s orbit.
Making a parabola representable on a plane .

8. Kepler’s ellipse:
In the revolutionary keplerian way of looking to the Universe, the ellipse became the basical curve that determin all the rotations of planets and celestial bodies. His famos three laws were the sum of all his work on the ellipse and the discover of the astronomical mechanism:
The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semi-major axes.

9. Descartes’ coordinates:
“Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistance from them is a parabola.”
This definition of parabola appears really well comprehensible just thank to Descartes and the invention of the cartesian plane and of the Analytic geometry. From it we can find the formulas used to describe this curve:
Equation of a parabola:
Vertex:
Focus:
Directrix: