Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things The elements mentioned are those that specify the shape and generation of the figures. Tangents are covered at the end of the book. Apollonius claims original discovery for theorems "of use for the construction of solid loci Analytic geometry derives the same loci from simpler criteria supported by algebra, rather than geometry, for which Descartes was highly praised.
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Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things The elements mentioned are those that specify the shape and generation of the figures.
Tangents are covered at the end of the book. Apollonius claims original discovery for theorems "of use for the construction of solid loci Analytic geometry derives the same loci from simpler criteria supported by algebra, rather than geometry, for which Descartes was highly praised.
He supersedes Apollonius in his methods. The first sent to Attalus, rather than to Eudemus, it thus represents his more mature geometric thought.
The topic is rather specialized: "the greatest number of points at which sections of a cone can meet one another, or meet a circumference of a circle, These terms are not explained. In contrast to Book I, Book V contains no definitions and no explanation. Heath is led into his view by consideration of a fixed point p on the section serving both as tangent point and as one end of the line.
The minimum distance between p and some point g on the axis must then be the normal from p. In modern mathematics, normals to curves are known for being the location of the center of curvature of that small part of the curve located around the foot.
The distance from the foot to the center is the radius of curvature. The latter is the radius of a circle, but for other than circular curves, the small arc can be approximated by a circular arc. The curvature of non-circular curves; e. A map of the center of curvature; i. Such a figure, the edge of the successive positions of a line, is termed an envelope today. Heath believed that in Book V we are seeing Apollonius establish the logical foundation of a theory of normals, evolutes, and envelopes.
These 7 Fried classifies as isolated, unrelated to the main propositions of the book. In his extensive investigation of the other 43 propositions, Fried proves that many cannot be. First is a complete philological study of all references to minimum and maximum lines, which uncovers a standard phraseology. There are three groups of propositions each. Given a fixed point on the axis, of all the lines connecting it to all the points of the section, one will be longest maximum and one shortest minimum.
In the view of Fried and Unguru, the topic of Book V is exactly what Apollonius says it is, maximum and minimum lines. These are not code words for future concepts, but refer to ancient concepts then in use. The authors cite Euclid, Elements, Book III, which concerns itself with circles, and maximum and minimum distances from interior points to the circumference. Given a point P, and a ruler with the segment marked off on it.
In Book V, P is the point on the axis. Rotating a ruler around it, one discovers the distances to the section, from which the minimum and maximum can be discerned. The technique is not applied to the situation, so it is not neusis. The authors use neusis-like, seeing an archetypal similarity to the ancient method. It also has large lacunae , or gaps in the text, due to damage or corruption in the previous texts. The topic is relatively clear and uncontroversial.
Book VI features a return to the basic definitions at the front of the book. They are neither entirely the same nor different, but share aspects that are the same and do not share aspects that are different. Intuitively the geometricians had scale in mind; e. Thus figures could have larger or smaller versions of themselves. The aspects that are the same in similar figures depend on the figure. Similar sections and segments of sections are first of all in similar cones.
In addition, for every abscissa of one must exist an abscissa in the other at the desired scale. Finally, abscissa and ordinate of one must be matched by coordinates of the same ratio of ordinate to abscissa as the other.
The total effect is as though the section or segment were moved up and down the cone to achieve a different scale. These are the last that Heath considers in his edition. In Preface I, Apollonius does not mention them, implying that, at the time of the first draft, they may not have existed in sufficiently coherent form to describe. Whether the reference might be to a specific kind of definition is a consideration but to date nothing credible has been proposed.
Diameters and their conjugates are defined in Book I Definitions Not every diameter has a conjugate. The topography of a diameter Greek diametros requires a regular curved figure. Irregularly-shaped areas, addressed in modern times, are not in the ancient game plan.
A chord is a straight line whose two end points are on the figure; i. If a grid of parallel chords is imposed on the figure, then the diameter is defined as the line bisecting all the chords, reaching the curve itself at a point called the vertex.
There is no requirement for a closed figure; e. A parabola has symmetry in one dimension. If you imagine it folded on its one diameter, the two halves are congruent, or fit over each other.
The same may be said of one branch of a hyperbola. The figures to which they apply require also an areal center Greek kentron , today called a centroid , serving as a center of symmetry in two directions. These figures are the circle, ellipse, and two-branched hyperbola. There is only one centroid, which must not be confused with the foci.
A diameter is a chord passing through the centroid, which always bisects it. For the circle and ellipse, let a grid of parallel chords be superimposed over the figure such that the longest is a diameter and the others are successively shorter until the last is not a chord, but is a tangent point.
The tangent must be parallel to the diameter. A conjugate diameter bisects the chords, being placed between the centroid and the tangent point. Moreover, both diameters are conjugate to each other, being called a conjugate pair. It is obvious that any conjugate pair of a circle are perpendicular to each other, but in an ellipse, only the major and minor axes are, the elongation destroying the perpendicularity in all other cases.
Conjugates are defined for the two branches of a hyperbola resulting from the cutting of a double cone by a single plane. They are called conjugate branches. They have the same diameter. Its centroid bisects the segment between vertices. There is room for one more diameter-like line: let a grid of lines parallel to the diameter cut both branches of the hyperbola.
These lines are chord-like except that they do not terminate on the same continuous curve. A conjugate diameter can be drawn from the centroid to bisect the chord-like lines. These concepts mainly from Book I get us started on the 51 propositions of Book VII defining in detail the relationships between sections, diameters, and conjugate diameters.
As with some of Apollonius other specialized topics, their utility today compared to Analytic Geometry remains to be seen, although he affirms in Preface VII that they are both useful and innovative; i. 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[ edit ] 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.
Although he began a translation, it was Halley who finished it and included it in a volume with his restoration of De Spatii Sectione. De Sectione Determinata[ edit ] 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. 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. The history of the problem is explored in fascinating detail in the preface to J. 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. Schooten Leiden, but also, most successfully of all, by R. Simson Glasgow, As almost no manuscripts were in Latin, the editors of the early printed works translated from the Greek or Arabic to Latin.
The Greek and Latin were typically juxtaposed, but only the Greek is original, or else was restored by the editor to what he thought was original. Critical apparatuses were in Latin. The ancient commentaries, however, were in ancient or medieval Greek.
Only in the 18th and 19th centuries did modern languages begin to appear. A representative list of early printed editions is given below.
The originals of these printings are rare and expensive. For modern editions in modern languages see the references. Pergaeus, Apollonius Conicorum libri quattuor: una cum Pappi Alexandrini lemmatibus, et commentariis Eutocii Ascalonitae. Sereni Antinensis philosophi libri duo Bononiae: Ex officina Alexandri Benatii.
A presentation of the first four books of Conics in Greek by Fredericus Commandinus with his own translation into Latin and the commentaries of Pappus of Alexandria , Eutocius of Ascalon and Serenus of Antinouplis.
Apollonius and conic sections
Definition The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. A conic is the curve obtained as the intersection of a plane , called the cutting plane, with the surface of a double cone a cone with two nappes. It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Apollonius of Perga