Apollonius of perga biography of abraham

His application of reference lines such as a diameter and a tangent is essentially the same as our modern use of a coordinate frame. However, unlike modern analytic geometry, he did not take into account negative magnitudes. Also, he superimposed the coordinate system on each curve after the curve had been obtained. Thus, he derived equations from the curves, but he did not derive curves from equations.

Pappus mentions other treatises of Apollonius. 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 Cutting of a Ratio sought to resolve a certain 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 Cutting of an Area discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle. De Sectione Determinata Determinate Section 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.

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.

De Tactionibus Tangencies 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 sixteenth century, Vieta presented this problem sometimes known as the Apollonian Problem to Adrianus Romanus, who solved it with a hyperbola.

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Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work Apollonius Gallus. The object of De Inclinationibus Inclinations 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.

De Locis Planis Plane Loci is a collection of propositions relating to loci that are either straight lines or circles. Book VI: Equality and similarity of the conic sections. Inverse problem: given the conic, find the cone. There are different hypotheses about what could have been written on it. If there are two lines and each one has a point above them, the problem is to draw another line through another point, so that when cutting the other lines, segments that are within a given proportion are required.

The segments are the lengths located between the points on each of the lines. This is the problem that Apollonius raises and solves in his book About the reason section. The great mathematician Papo of Alexandria was the one who was mainly in charge of apollonius of perga biography of abraham the great contributions and advances of Apollonius of Perga, commenting on his writings and dispersing his important work in a large number of books.

This is how, from generation to generation, Apollonius's work transcended Ancient Greece to reach the West today, being one of the most representative figures in history for establishing, characterizing, classifying and defining the nature of mathematics and geometry in the world. Apollonius of Perga: biography, contributions and writings.

Author: Louise Ward. Video: Problem of Apollonius - what does it teach us about problem solving? 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. Book IV contains 57 propositions. 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, Book V, known only through translation from the Arabic, contains 77 propositions, the most of any book. These terms are not explained. In contrast to Book I, Book V contains no definitions and no explanation.

The ambiguity has served as a magnet to exegetes of Apollonius, who must interpret without sure knowledge of the meaning of the book's major terms. Until recently Heath's view prevailed: the lines are to be treated as normals to the sections. 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. Heath's was accepted as the authoritative interpretation of Book V for the entire 20th century, but the changing of the century brought with it a change of view.

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. Fried and Unguru counter by portraying Apollonius as a continuation of the past rather than a foreshadowing of the future. 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. Book VI, known only through translation from the Arabic, contains 33 propositions, the least of any book. It also has large lacunaeor 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. Book 6 of Euclid's Elements presents similar triangles as those that have the same corresponding angles. In Apollonius' definitions at the beginning of Book VI, similar right cones have similar axial triangles. 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 4—6.

Not every diameter has a conjugate. The topography of a diameter Greek diametros requires a regular curved figure.

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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 kentrontoday called a centroidserving 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. The early printed editions began for the most part in the 16th century. At that time in Europe, scholarly books were expected to be in Neo-Latin.

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As few mathematical manuscripts were written in Latin, the editors of the early printed works translated from Greek or Arabic. Often Greek and Latin were juxtaposed, with the Greek text representing either the original or an editor's restoration. Critical commentary of the time was typically in Latin. Earlier commentaries had been written in ancient or medieval Greek or Arabic.

Only in the 18th and 19th centuries did editions in modern languages begin to appear. A representative list of early printed editions is given below. The difficulty of Conics made an intellectual niche for later commentators, each presenting Apollonius in the most lucid and relevant way for his own times. They use a variety of methods: annotation, extensive prefatory material, different formats, additional drawings, superficial reorganization by the addition of capita, and so on.

There are subtle variations in interpretation. Limited material about Conics was ever written in English, because European mathematicians of the 16th—18th century, including English mathematicians such as Edmund Halley and Isaac Newton, preferred Neo-Latin. In later apollonius of perga biographies of abraham, geometry was re-established using coordinates analytic geometry and synthetic methods fell out of favor, so Conics ' direct influence on mathematical research declined.

His extensive prefatory commentary includes such items as a lexicon of Apollonian geometric terms giving the Greek, the meanings, and usage. His work thus references two systems of organization, his own and Apollonius', parenthetically cross-referenced. Heath was active in the late 19th and early 20th century, passing away inbut meanwhile another point of view developed.

In desperation the board summoned Stringfellow Barr and Scott Buchanan from the University of Chicagowhere they had been developing a new theoretical program for instruction of the Classics. John's, later dubbed the Great Books program, a fixed curriculum that would teach the works of select key contributors to the culture of western civilization.

At St. John's, Apollonius came to be taught as himself, not as some adjunct to analytic geometry. Fried was produced in Conics consisted of 8 books. Books one to four form an elementary introduction to the basic properties of conics. Most of the results in these books were known to EuclidAristaeus and others but some are, in Apollonius's own words 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 asymptotesand he also studies how to draw tangents to given conics.

There are, however, new results in these books in particular in book three. Apollonius writes of book three see [ 4 ] or [ 7 ] Books five to seven are highly original. In these Apollonius discusses normals to conics and shows how many can be drawn from a point. He gives propositions determining the centre of curvature which lead immediately to the Cartesian equation of the evolute.

Heath writes that book five [ 7 ] It deals with normals to conics regarded as maximum and minimum straight lines drawn from particular points to the curve. Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius.

There can be no doubt that the Book is almost wholly original, and it is a veritable geometrical tour de force. The beauty of Apollonius's Conics can readily be seen by reading the propositions as given by Heathsee [ 4 ] or [ 7 ]. However, Heath explains in [ 7 ] how difficult the original text is to read What militates against its being read in its original form is the great extent of the exposition it contains separate propositionsdue partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbersomeness of the enunciations of complicated propositions in general terms without the help of letters to denote particular points and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout.

Pappus gives some indications of the contents of six other works by Apollonius.