Catesby Taliaferro, diagrams by William H. It includes all seven extant books and some very useful notes and analysis. This version has no diagrams, but it refers to diagrams found in other publications. He is believed to have been born in about BC. Among his great works was the eight-volume Conics.
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Catesby Taliaferro, diagrams by William H. It includes all seven extant books and some very useful notes and analysis. This version has no diagrams, but it refers to diagrams found in other publications.
He is believed to have been born in about BC. Among his great works was the eight-volume Conics. Our knowledge of many of his contemporaries is limited to little beyond vague conjecture or inflated stories that challenge credulity.
Apollonius, at least on the subject of conics, can still speak for himself. The first four books of Conics have survived in the original Greek. Books V through VII survived thanks to the efforts of a ninth century AD family of scholars who stepped forward to translate and preserve them in Arabic. All seven surviving books are now available in English. It is a sad fact that so many works of that era are lost and will never be recovered.
There were frequent waves of political turmoil and war, which could wipe out entire civilizations, making the survival of a library unlikely. Also, consider that this was before the development of the printing press. Some of the books of Conics are preceded with personal notes from Apollonius in which he exchanges pleasantries with the recipient. It becomes apparent that he was personally transcribing entire books. Even when they were new they could not have been widely distributed, certainly not by modern standards.
This is not the first great study of conic sections. Euclid, who preceded Apollonius by about two generations, produced a four-volume work on the subject, but it has not survived. Each book has 50 to 60 propositions, most of which are theorems. Certain other propositions are constructions, in which the author takes great pains to address every special case. In II. Apollonius justifies the construction for eleven special cases, and proves the nonexistence of a solution for one other case.
Naturally Book I has most of the definitions. The Apollonius model of conic sections includes oblique cones. It also uses conic surfaces of two nappes. The two disjoint opposite sections are exposed, but they are not included together in the definition of a hyperbola, and are never referred to as a single section. Conjugate opposite sections and the upright side latus rectum are given prominence. Points of application foci do not appear until Book III, and the directrix is absent entirely.
It too is published by Green Lion Press, and as of , they have made the first four books available under one cover. Book IV has been less widely distributed until recently. Special cases and exceptions are addressed perhaps to the point of tedium, making Mr. It begins with properties of poles and polars, which were introduced in Book III.
These properties align with more familiar properties involving circles. Let D be a point outside a conic section. Let tangent lines from D touch the section at A and B.
It is true then that points A, B, and H are collinear. With the more widely accepted modern definitions, the only exceptions more like special cases would arise when D falls on an asymptote of a hyperbola, or when the cutting line DE is parallel to an asymptote. But what Apollonius calls a hyperbola is a single continuous curve. To him the double curve we now call a hyperbola is a pair of opposite sections, and is not classified as a single conic section. That means that Proposition 1, which purportedly applies to all conic sections, actually applies to a hyperbola only under specific conditions.
A rather awkward result is that the first proposition must be qualified by subsequent propositions. Apollonius dutifully considers each of the special conditions, adds cases for opposite sections, considers the cases in which the exterior point falls on an asymptote, and considers cases in which the cutting line is parallel to an asymptote, hence Mr.
The other major concept involves the number of contacts between two conic sections. They can meet at no more than four points. Again, special cases abound. When is the number of contacts fewer than four? Apollonius considers whether intersecting sections have concavity in the same direction or opposing directions, he considers tangency cases, and of course he addresses the many opposite section cases.
Many of the Book IV proofs are indirect proofs. They begin by assuming a geometric relationship that will ultimately be proved impossible. As a compromise, many of the proposition statements are illustrated with no direct connection to the figures in the proof. It should also by noted that some of the proofs are incomplete or flat-out wrong.
Fried suggests that some of the text may have been corrupted in the years of transcriptions and translations. Unlikely as it seems, we must also acknowledge the possibility that Apollonius himself was mistaken. Book V The later books of Conics are handed down to us in a more indirect way. As they were not part of the core of conic section studies, the later books fell into disuse, and nearly disappeared entirely. The brothers were prominant scholars in Baghdad during the ninth century AD.
They directed the collection and Arabic translation of the first seven books. Whether they ever acquired the eighth is unclear. In any case, it is now lost. Toomer Springer-Verlag The first volume has historical background, analysis, and the translation itself. Anyone interested enough to purchase this set should be careful to seek out the original hardcover edition.
There was a softcover edition which inexplicably includes Volume I only, not a single diagram. Book V, to a large extent, concerns normal lines: whether they exist, how many exist, how they might be constructed. They are not actually called normals, although it is proved that each is perpendicular to the tangent line at the point where it meets the section.
In most cases a normal line is called a minimum line, sometimes a maximum line. These designations refer to the distance between a point where a normal meets the section and some certain given point, usually on the axis. See the definitions below. An interesting construction technique also is introduced. In Book II Apollonius showed that he was comfortable with the concept of conic sections as given objects in a construction. For example, in II.
The section itself could not have been constructed under the usual restrictions, but with compass and straightedge it was possible to construct a tangent line through the given point. Here in Book V he has taken it a bit further. The image below is from V. Traditional construction methods are used to define parameters of a hyperbola, and the hyperbola is then used as a construction object, intersecting an ellipse at the endpoints of two minimum lines.
Apollonius could not have had access to any means of representing these curves precisely, and an exact construction is impossible even in theory.
This same proposition has a geometric division step, which would be equivalent to duplicating a cube, yet another impossibility. The construction itself is not the objective. Of greater importance than drawing the curves, Apollonius has proved that they exist, that they intersect, and that the intersections have certain properties.
The images in the Book V Sketchpad document are aligned with the Toomer diagrams, much as the earlier documents were aligned with the Green Lion books.
The point labels are now Greek characters, with no italics. In the previous books most of the sections were left with an oblique orientation in order to discourage any misleading sense of up or down. Most of the propositions of Book V, however, involve relationships with the axes. For that reason, nearly all of the sketches were parked with the axes horizontal and vertical, although they can still be turned by dragging the control points.
Most of the Toomer diagrams show only half of a section, cut along an axis. That convention was not followed in the Sketchpad document. Toomer notes some discontiniuties in the manuscript, which might suggest some missing propositions.
The book begins with several new definitions. The same word is used for both concepts. The volume deals primarily with equality or similarity of conic sections or segments, also symmetries of sections. Proposition 6 states that if any part of a section can be fitted to a second section, then the sections are equal.
Once the concept is proved and accepted, many of the later propositions become intuitively obvious. Even the smallest segment of a section is sufficient for defining the entire section. Two distinct sections can coincide only on a few no more than four discrete points, and not on any continuous segment.
Many of the proposition conclusions again are negatives, making them difficult to illustrate. Several sketches make use of the five-point conic construction, which did not come from Apollonius. The cone itself has been on hiatus since Book I, but now makes a return. The final eight propositions of Book VI are constructions involving the cone and cutting plane.
Most of them apply specifically to a right cone.
Conics of Apollonius
The work of Apollonius of Perga extended the field of geometric constructions far beyond the range in the Elements. He visited both Ephesus and Pergamum , the latter being the capital of a Hellenistic kingdom in western Anatolia , where a university and library similar to the Library of Alexandria had recently been built. In Alexandria he wrote the first edition of Conics, his classic treatise concerning the curves—circle, ellipse , parabola , and hyperbola—that can be generated by intersecting a plane with a cone; see figure. Conic sectionsThe conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse including the circle ; the parabola with one branch ; and the hyperbola with two branches.
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. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all.
Apollonius of Perga
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.