### Apollonius of Perga

Apollonius of Perga [Pergaeus] (Ancient Greek: Ἀπολλώνιος) (ca. 262 BC–ca. 190 BC) was a Greekgeometer and astronomer noted for his writings on conic 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, are 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.

## 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]

## Other works

Pappus mentions other treatises of Apollonius:
1. Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio")
2. Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area")
3. Διωρισμένη τομή, De Sectione Determinata ("Determinate Section")
4. Ἐπαφαί, De Tactionibus ("Tangencies")
5. Νεύσεις, De Inclinationibus ("Inclinations")
6. Τόποι ἐπίπεδοι, De Locis Planis ("Plane Loci").
Each of these was divided into two books, and—with the Data, the Porisms, and Surface-Loci of Euclid and the Conics of Apollonius—were, according to Pappus, included in the body of the ancient analysis.

### De Rationis Sectione

De Rationis Sectione sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.

### De Spatii Sectione

De Spatii Sectione discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.
In the late 17th century, Edward Bernard discovered an Arabic version of De Rationis Sectione in the Bodleian Library. Although he began a translation, it was Halley who finished it and included it in a 1706 volume with his restoration of De Spatii Sectione.

### De Sectione Determinata

De Sectione Determinata deals with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.[2] The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either (1) to the square on the remaining one or the rectangle contained by the remaining two or (2) to the rectangle contained by the remaining one and another given straight line. Several have tried to restore the text to discover Apollonius's solution, among them Snellius (Willebrord Snell, Leiden, 1698); Alexander Anderson of Aberdeen, in the supplement to his Apollonius Redivivus (Paris, 1612); and Robert Simson in his Opera quaedam reliqua (Glasgow, 1776), by far the best attempt.[3]

### De Tactionibus

De Tactionibus embraced the following general problem: Given three things (points, straight lines, or circles) in position, describe a circle passing through the given points and touching the given straight lines or circles. The most difficult and historically interesting case arises when the three given things are circles. In the 16th century, Vieta presented this problem (sometimes known as the Apollonian Problem) to Adrianus Romanus, who solved it with a hyperbola. Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work Apollonius Gallus (Paris, 1600). The history of the problem is explored in fascinating detail in the preface to J. W. Camerer's brief Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c (Gothae, 1795, 8vo).

### De Inclinationibus

The object of De Inclinationibus was to demonstrate how a straight line of a given length, tending towards a given point, could be inserted between two given (straight or circular) lines. Though Marin Getaldić and Hugo d'Omerique (Geometrical Analysis, Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).[3]

### De Locis Planis

De Locis Planis is a collection of propositions relating to loci that are either straight lines or circles. Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat (Oeuvres, i., 1891, pp. 3-51) and F. Schooten (Leiden, 1656) but also, most successfully of all, by R. Simson (Glasgow, 1749).[3]

Ancient writers refer to other works of Apollonius that are no longer extant:
1. Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola
2. Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus)
3. A comparison of the dodecahedron and the icosahedron inscribed in the same sphere
4. Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements
5. Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of π (pi) than those of Archimedes, who calculated 3+1/7 as the upper limit (3.1428571, with the digits after the decimal point repeating) and 3+10/71 as the lower limit (3.1408456338028160, with the digits after the decimal point repeating)
6. an arithmetical work (see Pappus) on a system both for expressing large numbers in language more everyday than that of Archimedes' The Sand Reckoner and for multiplying these large numbers
7. a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm. on Eucl. x., preserved in Arabic and published by Woepcke, 1856).

## Published editions

The best editions of the works of Apollonius are the following:
1. Apollonii Pergaei Conicorum libri quatuor, ex versione Frederici Commandini (Bononiae, 1566), fol.
2. Apollonii Pergaei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo (Oxoniae, 1710), fol. (this is the monumental edition of Edmund Halley)
3. the edition of the first four books of the Conics given in 1675 by Isaac Barrow
4. Apollonii Pergaei de Sectione, Rationis libri duo: Accedunt ejusdem de Sectione Spatii libri duo Restituti: Praemittitur, &c., Opera et Studio Edmundi Halley (Oxoniae, 1706), 4to
5. a German translation of the Conics by H. Balsam (Berlin, 1861)
6. The definitive Greek text is the edition of Heiberg (Apollonii Pergaei quae Graece exstant Opera, Leipzig, 1891-1893)
7. T. L. Heath, Apollonius, Treatise on Conic Sections (Cambridge, 1896)
8. The Arabic translation of the Books V-VII was first published in two volumes by Springer Verlag in 1990 (ISBN 0-387-97216-1), volume 9 in the "Sources in the history of mathematics and physical sciences" series. The edition was produced by G. J. Toomer and provided with an English translation and various commentaries.
9. Conics: Books I-III translated by R. Catesby Taliaferro, published by Green Lion Press (ISBN 1-888009-05-5).
10. Apollonius de Perge, Coniques: Texte grec et arabe etabli, traduit et commenté (De Gruyter, 2008-2010), eds. R. Rashed, M. Decorps-Foulquier, M. Federspiel. (This is a new edition of the surviving Greek text (Books I-IV), a full edition of the surviving Arabic text (Books I-VII) with French translation and commentaries.)
11. Apollonius of Perga's Conica: Text, Context, Subtext. By Michael N. Fried and Sabetai Unguru (Brill). (This contains, as an appendix, an English translation of Book IV, by M. Fried.)

# Apollonian circles

Figure 1: Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two foci, C and D, and every blue circle separates the two foci.
This article discusses a family of circles sharing a radical axis, and the corresponding family of orthogonal circles. For other circles associated with Apollonius of Perga, please see the disambiguation page, circles of Apollonius.
Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer.

## Definition

The Apollonian circles are defined in two different ways by a line segment denoted CD.
Each circle in the first family (the blue circles in the figure) is associated with a positive real number r, and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r,
$\left\{X\mid \frac{d(X,C)}{d(X,D)} = r\right\}.$
For values of r close to zero, the corresponding circle is close to C, while for values of r close to ∞, the corresponding circle is close to D; for the intermediate value r = 1, the circle degenerates to a line, the perpendicular bisector of CD. The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points.
Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle CXD equals θ,
$\left\{X\mid \; C\hat{X}D \; = \theta\right\}.$
Scanning θ from 0 to π generates the set of all circles passing through the two points C and D.

## Pencils of circles

Both of the families of Apollonian circles are called pencils of circles. More generally, there is a natural correspondence between circles in the plane and points in three-dimensional projective space; a line in this space corresponds to a one-dimensional continuous family of circles called a pencil.
Specifically, the equation of a circle of radius r centered at a point (p,q),
$\displaystyle (x-p)^2 + (y-q)^2 = r^2,$
may be rewritten as
$\displaystyle \alpha(x^2 + y^2) - 2\beta x - 2\gamma y + \delta = 0,$
where α = 1, β = p, γ = q, and δ = p2 + q2 − r2. However, in this form, multiplying the 4-tuple (α,β,γ,δ) by a scalar produces a different four-tuple that represents the same circle; thus, these 4-tuples may be considered to be homogeneous coordinates for the space of circles.[1] Straight lines may also be represented with an equation of this type in which α = 0 and should be thought of as being a degenerate form of a circle. When α ≠ 0, we may solve for p = β/α, q = γ/α, and r =√((−δ − β2 − γ2)/α2); note, however, that the latter formula may give r = 0 (in which case the circle degenerates to a point) or r equal to an imaginary number (in which case the 4-tuple (α,β,γ,δ) is said to represent an imaginary circle).
The set of affine combinations of two circles (α1111), (α2222), that is, the set of circles represented by the four-tuple
$\displaystyle z(\alpha_1,\beta_1,\gamma_1,\delta_1)+(1-z)(\alpha_2,\beta_2,\gamma_2,\delta_2)$
for some value of the parameter z, forms a pencil; the two circles are called generators of the pencil. There are three types of pencil:[2]
• An elliptic pencil is defined by two generators that pass through two points C and D. At these two points, the formula defining the membership in the two generating circles equals zero, and therefore it also equals zero for any affine combination of the two circles. Thus, every two circles of an elliptic pencil pass through the same two points. The red family of circles in the figure form a pencil of this type. An elliptic pencil does not include any imaginary circles.
• A parabolic pencil is defined by two generating circles that are tangent to each other. It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil.
• A hyperbolic pencil is defined by two generators that do not intersect each other. It includes real circles, imaginary circles, and two degenerate point circles C and D, called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil. The blue family of circles in the figure forms a pencil of this type.
A family of concentric circles centered at a single focus C forms a special case of a hyperbolic pencil, in which the other focus is the point at infinity of the complex projective line. The corresponding elliptic pencil consists of the family of straight lines through C; these should be interpreted as circles that all pass through the point at infinity.

## Radical axis and central line

Except for the two special cases of a pencil of concentric circles and a pencil of coincident lines, any two circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxal circles or coaxial circles.[3]
The elliptic pencil of circles passing through the two points C and D (the set of red circles, in the figure) has the line CD as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of CD. The hyperbolic pencil defined by points C and D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD.
The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil. Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear.

## Inversive geometry, orthogonal intersection, and coordinate systems

Circle inversion transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a parabolic pencil is another parabolic pencil.
It is relatively easy to show using inversion that, in the Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a right angle. Inversion of the blue Apollonian circles with respect to a circle centered on point C results in a pencil of concentric circles centered at the image of point D. The same inversion transforms the red circles into a set of straight lines that all contain the image of D. Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a polar coordinate system. Obviously, the transformed pencils meet at right angles. Since inversion is a conformal transformation, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles.
Alternatively,[4] the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point X on the radical axis of a pencil P the lengths of the tangents from X to each circle in P are all equal. It follows from this that the circle centered at X with length equal to these tangents crosses all circles of P perpendicularly. The same construction can be applied for each X on the radical axis of P, forming another pencil of circles perpendicular to P.
More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.