MATHEMATICIAN

Life
Ranked with the great mathematicians Archimedes and Euclid for his contribution to mathematics, Apollonius of Perga (in the Ionian kingdom of Pamphylia) studied at Alexandria with Euclid's successors, and later taught in Ephesus and Pergamum. Pappus (3rd century AD) described him as arrogant and conceited, the title of "Great Geometer" attributed to him during the lifetime of Archimedes having apparently gone to his head. Ptolemy records that "Hipparchus based his theory relating to the orbit of the planets on the work of Apollonius". In Pergamum he met and worked with Eudemus. Many later mathematicians, including Viete, Descartes, Fermat and Desargues, were influenced by his work; and one of the craters on the moon has been named "Apollonius" in his honour.


Work
His principal works are:

"On Proportional Section": 2 books. Cited by Pappus in his "Collection". Survives in Arabic translation, from which Halley translated it into Latin in 1706.

"On Spatial Section": 2 books. Cited by Pappus in his "Collection". Lost. Halley tried to reconstruct it on the basis of references in other writers.

"On Determinate Section": 2 books. Cited by Pappus in his "Collection". Lost.

"Tangencies": 2 books. Cited by Pappus in his "Collection". Lost. French mathematician and algebraist Francois Viete tried to reconstitute this work on the basis of references in other writers.

"Inclinations": The problem studied in this work is cited by Pappus in his "Collection", where he notes that the treatise included 125 theorems and 38 propositions. Halley presented a reconstruction of this work based on references in other writers.

"On Plane Loci": 2 books. According to Pappus, Apollonius described loci as "standing" (not exceeding themselves), "traversant" (related to a class exceeding themselves ) and "conversional" (related to conversion). Lost. Fermat, Soutine and Simpson all tried to reconstitute this work on the basis of references in other writers.

"On the Cylindrical Helix": Cited by Proclus in his "Commentaries on Euclid".

"Comparison of the Dodecahedron and the Icosahedron": This treatise considers the relation between the dodecahedron and the icosahedron, considering the case in which they are inscribed in the same sphere. Hypsicles discussed this work in his commentary on the Elements of Euclid.

"On Unordered Irrationals": Extends the theory of irrational numbers. Cited by Pappus in his "Collection".

"Quick Delivery": Method for abbreviating calculations. Fragments survive. According to Eutocius, who discusses the work in his "Commentary on Archimedes' 'Measurement of the Circle', Apollonius in this work calculates closer limits for the value of ð (pi) than earlier mathematicians.

"The Universal Treatise": Contains the fundamentals of the science of geometry. Cited by Proclus in his "Commentaries on Euclid". Lost.

"On the Burning Mirror": Discusses the focal properties of the parabolic mirror. Survives in fragments.

A treatise on astronomy, whose title has been lost. It explains the anomalies in the movement of the sun through the constellations, on the basis of the theory of epicyclical motion. It also discusses phenomena relating to the moon. Commentaries on this work exist in Hippolytus and Photius.

"Conics": 8 books. This is Apollonius' most important work. Books 1-4 survive in Greek, 5-7 in Arabic. The 8th book is lost, except for fragments preserved in Pappus. On the basis of these precious fragments, Halley attempted to reconstruct the entire work. Eutocius wrote commentaries on books 1-4, which contain a systematic account of the fundamental principles of conics, based on the works of Menaechmus, Aristaeus and Euclid. Initially written in Alexandria, Conics was revised and expanded in Pergamum. This work contains the first systematic study of conic sections and introduces the terms "hyperbola" and "ellipse". Apollonius was the first to consider the two arms of the hyperbola as a curve, gave a method for solving quadratic equations by conic section, constructed a conic section by tangents and determined geometric loci with respect to 3 and 4 lines. The "Conics" is considered one of the most important of the ancient works of geometry, along with the works of Archimedes and Euclid.

Apollonius was also the inventor of the sundial (220 BC). The accurate measurement of time was a perennial preoccupation of the ancient Greeks. Sundials are perhaps the most widely dispersed of the mechanisms and instruments developed in the Hellenistic period, and are often unique architectural devices despite their static form. They consist of a gnomon that casts its shadow over a system of engraved markings that the ancients called an "analemma". The analemma had two curves: that nearest to the gnomon represented the end of the shadow on the day of the summer solstice; while the second, longer, curve represented the path of the shadow on the day of the winter solstice. A straight line between the two curves represented the path of the shadow on the days of each equinox and determined east-west orientation. The two curves were linked by lines that formed eleven straight segments, by which one could tell the hour of the day. The centre segment, where the shadow fell at midday, determined the north-south alignment. These sundials, of course, could only be used locally, unlike the portable sundials developed from them, which could in addition be used as astronomical or navigation instruments.

Apollonius also developed the hydraulis, a musical instrument (an early form of pipe organ, in fact) operated by water power.