Apollonius of PergaFrom Wikipedia, the free encyclopedia

Apollonius of Perga.

Apollonius of Perga [Pergaeus] (Ancient Greek: Ἀπολλώνιος) (ca. 262 BC – ca. 190 BC)

was a Greek geometer and astronomer noted for his writings onconic sections. His

innovative methodology and terminology, especially in the field of conics, influenced many

later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes.

It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which

we know them. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles,

to explain the apparent motion of the planets and the varying speed of the Moon, is also

attributed to him. Apollonius' theorem demonstrates that the two models are equivalent

given the right parameters. Ptolemy describes this theorem in the Almagest XII.1.

Apollonius also researched the lunar history, for which he is said to have been

called Epsilon (ε). The crater Apollonius on the Moon is named in his honor.

Contents

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1 Conics

2 Other works

o 2.1 De Rationis Sectione

o 2.2 De Spatii Sectione

o 2.3 De Sectione Determinata

o 2.4 De Tactionibus

o 2.5 De Inclinationibus

o 2.6 De Locis Planis

o 2.7 Additional works

3 Published editions

4 See also

5 Notes

6 References

7 The Works of Apollonius of Perga online

8 External links

[edit]Conics

The degree of originality of the Conics can best be judged from Apollonius's own prefaces.

Books i–iv he describes as an "elementary introduction" containing essential principles,

while the other books are specialized investigations in particular directions. He then claims

that, in Books i–iv, he only works out the generation of the curves and their fundamental

properties presented in Book i more fully and generally than did earlier treatises, and that a

number of theorems in Book iii and the greater part of Book iv are new. Allusions to

predecessor's works, such as Euclid's four Books on Conics, show a debt not only to Euclid

but also to Conon and Nicoteles.

The generality of Apollonius's treatment is indeed remarkable. He defines the fundamental

conic property as the equivalent of the Cartesian equation applied to oblique axes—i.e.,

axes consisting of a diameter and the tangent at its extremity—that are obtained by cutting

an oblique circular cone. The way the cone is cut does not matter. He shows that the

oblique axes are only a particular case after demonstrating that the basic conic property can

be expressed in the same form with reference to any new diameter and the tangent at its

extremity. It is the form of the fundamental property (expressed in terms of the "application

of areas") that leads him to give these curves their names: parabola, ellipse, and hyperbola.

Thus Books v–vii are clearly original.

Apollonius's genius reaches its highest heights in Book v. Here he treats of normals as

minimum and maximum straight lines drawn from given points to the curve (independently

of tangent properties); discusses how many normals can be drawn from particular points;

finds their feet by construction; and gives propositions that both determine the center of

curvature at any point and lead at once to the Cartesian equation of the evolute of any

conic.

Apollonius in the Conics further developed a method that is so similar to analytic

geometry that his work is sometimes thought to have anticipated the work of Descartes by

some 1800 years. His application of reference lines, a diameter and a tangent is essentially

no different than our modern use of a coordinate frame, where the distances measured

along the diameter from the point of tangency are the abscissas, and the segments parallel

to the tangent and intercepted between the axis and the curve are the ordinates. He further

developed relations between the abscissas and the corresponding ordinates that are

equivalent to rhetorical equations of curves. However, although Apollonius came close to

developing analytic geometry, he did not manage to do so since he did not take into

account negative magnitudes and in every case the coordinate system was superimposed

upon a given curve a posteriori instead of a priori. That is, equations were determined by

curves, but curves were not determined by equations. Coordinates, variables, and

equations were subsidiary notions applied to a specific geometric situation.[1]

[edit]

Apollonius of Perga

The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga, whose

eight volume Conic Sections summarized the existing knowledge at the time and greatly extended it.

Apollonius's major innovation was to characterize a conic using properties within the plane and intrinsic to

the curve; this greatly simplified analysis. With this tool, it was now possible to show that any plane cutting

the cone, regardless of its angle, will produce a conic according to the earlier definition, leading to the

definition commonly used today.

Pappus is credited with discovering importance of the concept of a focus of a conic, and the discovery of

the related concept of a directrix.

The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

  Apollonius of Pergaby Alicia Schamburg

 

Apollonius of Perga, also known as “The Great Geometer” was born around262 BC in Perga, Pamphylia. We know Perga today as Murtina, in Antalya, Turkey. In Apollonius's earlier years, he went to Alexandria where he studied under the followers of Euclid. Little information is known on the details of Apollonius of Perga's life, but what is certain is the great influence he had on the development of early mathematics. Apollonius is so greatly remembered because during his lifetime he was able to write more than 20 books. His works included Cutting of a ratio , Cutting an area , On determinate section , Tangencies , Plane lociand On verging constructions . The most famous of Apollonius's works however were the Conics . Conics was written in eight books. Unfortunately only the first seven survive today. To get an understanding of what Conics is, in definition it means of or relating to a cone. Also, Conic sections are the curves formed when a plane intersects the surface of a cone.    In the eight books of which Conics consisted, books one to four form an introduction to the basics of conics. In book one the relations satisfied by the diameters and tangents of conics are studied while in book two Apollonius investigates how hyperbolas are related to their asymptotes, and he also studies how to draw tangents to given conics. In his third book, Apollonius told how it would not have been possible for the synthesis to be completed without the additional theorems that he discovered. Books five to seven discuss normals to conics and shows how many can be drawn from a point. Apollonius credited Conon of Samos (c 280-c 220 BC) and Euclid of Alexandria(c 325-c 265 BC) with the original work on conical sections that inspired his work. Apollonius's contributions to the development of mathematics are countless. Apollonius showed how to construct the circle, which is tangent, to three given circles.   He extended Euclid's theory of irrationals and improved Archimedes's approximation of ‘pi.' Apollonius showed that parallel rays of light are not brought to a focus by a spherical mirror and discussed the focal properties of a parabolic mirror. In his mathematical astronomy studies he found the point where a planet appears stationary, namely the points where the forward motion change to a retrograde motion or the converse. To top everything off, Apollonius developed the hemicyclium, a sundial which has the hour lines drawn on the surface of a conic section. In conclusion, it is clear that Apollonius of Perga was able to contribute very much to the early development of mathematical geometry. One of the most significant factors to us today was the introduction of the Parabola, Ellipse and Hyperbola. 

This is an Illustration of the Conic Sections.

conic section 1: 1 straight lines, 2 circle, 3 ellipse, 4 parabola, 5 hyperbola  

Intersections of parallel planes and a double cone, forming ellipses, parabolas, and hyperbolas respectively. 

The Hemicyclium Sundial created by Apollonius

     

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