Apollonius of Perga

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Apollonius of Perga [Pergaeus] (ca. 262 B.C.E.–ca. 190 B.C.E.) was a Greek geometer and astronomer of the Alexandrian school, 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 12.1. Apollonius also researched the lunar theory, for which he is said to have been called Epsilon (ε). The Apollonius crater on the Moon was named in his honor.

Life and major work

Apollonius was born circa 262 B.C.E., some 25 years after Archimedes. He flourished in the reigns of Ptolemy Euergetes and Ptolemy Philopator (247-205 B.C.E.). His treatise on conics earned him fame as "The Great Geometer," an achievement that has assured his fame ever since.

Of all his treatises, only Conics survives. Of the others, we have titles and some indication of their contents thanks to later writers, especially Pappus. After the first edition of the eight-book Conics, Apollonius brought out a second edition at the suggestion of one Eudemus of Pergamum. As he revised each of the first three books, Appolonius sent Eudemus a copy; the most considerable changes came in the first two books. Eudemus died before the completion of the rest of the revision, so Appolonius dedicated the last five books to King Attalus I (241-197 B.C.E.). Only four books have survived in Greek; three more are extant in Arabic; the eighth has never been discovered.

Although a fragment has been found of a thirteenth-century Latin translation from the Arabic, it was not until 1661 that Giovanni Alfonso Borelli and Abraham Ecchellensis made a translation of Books v-7 into Latin. Although they used Abu 'l-Fath of Ispahan's Arabic version of 983, which was preserved in a Florentine manuscript, most scholars now agree that the best Arabic renderings are those of Hilal ibn Abi Hilal for Books 1-4 and Thabit ibn Qurra for Books 5-7.

In 1710, Halley used for his version of Conics an Oxford University copy of the Borelli-Ecchellensis translation; only for correcting his version did he look at the best (Arabic) manuscript (Bodl. 943). Thus the only part of the Arabic manuscript for Books V-VII to be published remains the 1889 L. Nix Arabic and German edition (publ. Drugulin, Leipzig) of a fragment of Book V. However, Halley also tried to reconstruct Book 8, his task guided partly by lemmas "to the seventh and eighth books" that Pappus included in his own writings and partly by Apollonius's statement that problems solved in the eighth book illustrated the content of the seventh.

Apollonius was concerned with pure mathematics. When he was asked about the usefulness of some of his theorem in Book 4 of Conics he proudly asserted that "they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that "the subject is one of those which seems worthy of study for their own sake."[1]


Conics

The degree of originality of the Conics can best be judged from Apollonius's own prefaces. In books 1-4, he describes an "elementary introduction" containing essential principles, while the other books are specialized investigations in particular directions. He then claims that, in Books 1-4, he only works out the generation of the curves and their fundamental properties presented in Book 1 more fully and generally than did earlier treatises, and that a number of theorems in Book 3 and the greater part of Book 4 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 5-7 are clearly original.

Apollonius's genius reaches its highest heights in Book 5. 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. Apollonius' me in the Conics in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use fo a coordinate frame, whether rectangular or, more generally, oblique.

Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed a posteriori upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down a priori for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed.

Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation;...That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.

Other works

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

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

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

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

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

De Inclinationibus

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

De Locis Planis ("Plane Loci") is a collection of propositions relating to loci that are either straight lines or circles.

Legacy

Known as "The Great Geometer," Apollonius' works had a very great influence on the development of mathematics and his famous book Conics introduced the terms parabola, ellipse, and hyperbola. 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.

Notes

  1. Boyer, Carl B. (1991). "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 152. ISBN 0471543977. “It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, as in ours, there were narrow-minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: "They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." (Heath, 1961).
    The preface to Book 5, relating to maximum and minimum straight lines drawn to a conic, again argues that "the subject is one of those which seems worthy of study for their own sake." While one must admire the author for his lofty intellectual attitude, it may be pertinently pointed out that in his day it was a beautiful theory, with no prospect of applicability to the science or engineering of his time, has since become fundamental in such fields as terrestrial dynamics and celestial mechanics.”
     

References
ISBN links support NWE through referral fees

  • Boyer, Carl B. Apollonius of Perga, A History of Mathematics, John Wiley & Sons, 1991. ISBN 978-0471543977
  • Fried, Michael N. and Unguru, Sabetai. Apollonius of Perga’s Conica: Text, Context, Subtext, Brill, 2001. ISBN 978-9004119779, ISBN 978-9004119772
  • Toomer, G.J. "Apollonius of Perga," Dictionary of Scientific Biography (Vol. I, pp. 179-193), Charles Scribner's Sons, 1970. ISBN 978-0684101149

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