Apollonius was one of the founding fathers of mathematical astronomy in ancient Greece. He also originated the geometric shapes and terms that would become central to Newtonian astronomical physics nearly 20 centuries later. He may have even prefigured Christiaan Huygens' 1673 use of the "evolute," the locus of the centers of curvature in a given curve. Certainly the projective geometry of Gérard Desargues and Blaise Pascal owe their genesis to Apollonius. He also invented his own counting system for very large numbers. Still considered the greatest achievement of Greek geometry, the Conicsearned Apollonius the moniker "The Great Geometer," according to a later mathematician, Eutocius. These books quickly supplanted the works of Euclid as authoritative texts.
Estimations of the time frame in which Apollonius lived have varied over the years, so much so that one reference will only place him in the second half of third century B.C. Others place him according to the reigns of Ptolemy Euergetes, who was king of Egypt beginning around 247 B.C., or of Ptolemy Philopator ending in 210-205 B.C. Apollonius was born in what was then the Greek town of Perga, south Asia Minor, now part of Turkey. He was apparently a second generation Euclidean scholar in Alexandria. Legend has it he was nicknamed "Epsilon" there, because that Greek letter looks like the half moon he studied. Apollonius is also said to have visited Pergamum, where there was a new library and museum like the ones in Alexandria, and traveled to Ephesus. At least one source speculates that he may have been employed as the treasurer-general to Ptolemy Philadelphus. While he lived around the same time as Archimedes, there is no direct proof that the two either influenced each other or had contact. Apollonius did, however, improved upon Archimedes' calculation of the value of .
A New Mind-Set
Apollonius set forth in his eight books on conic sections, along with roughly 400 theorems, a new idea on how to subdivide the cone to produce circles. He also catalogued new kinds of closed curves that he named ellipses, parabolas, and hyperbolas. The Pythagorean distaste for infinities, infinitesimals, and infinite sets was put aside in this new frame of mind, which paved the way for the eventual discovery of the infinitesimalcalculus. Additionally, his epicircles, epicycles, and eccentricsreplaced the concentric spheres of Eudoxusand influenced Ptolemaic cosmology. That framework would stand until Johannes Kepler finally reformed the geometry of astronomical modeling for the current day.
The first half of the Conicssurveys and completes all inherited Greek geometry, including early efforts of Euclid. Apollonius boasted that a Euclidean problem such as finding the locusrelative to three or four lines was completely solvable for the first time, thanks to his new propositions. It is perhaps this style of presentation that led Pappusto accuse him of envy, and for Archimedes' biographer Heracleides to accuse him of plagiarism. The material on conic sections took up the last four volumes, laying the foundation for modern-day astronomy, ballistics, rocketry, and space science. Conic sections, it was only discovered many hundreds of years afterwards, are the shapes formed by the paths or "loci" of projectiles and other objects in orbit.
The Conics cover both pure and applied geometry. In it, Apollonius considered the problem of finding "normals" on points along curves, which involves trigonometry, though he could not apparently figure the focus of a parabolic curvethe way he could for an ellipseor hyperbola. He also presented a method of figuring at what points a planetary orbit takes on apparent retrograde motion. Finally, his still famous "problem of Apollonius" calls for the construction of a circletangent to three given circles. His most important contribution in pure terms was how he generalized the means of production. From one cone he could derive all conic sections, whether perpendicular to it or not. Apollonius used this standard cone in a way that prefigured analytic geometry by splitting it along two fixed lines called the latus transversumand the latus erectum. These "conjugate diameters" became a coordinate system and frame of reference, making geometrydo the work now done by algebra.
The Lost Works
A number of writers have attempted to restore or recreate the lost eighth book of the Conicsor Apollonius' other writings, including Alhazen, Edmond Halley and Pierre de Fermat. The Conics were all that survived, perhaps because most of Apollonius' writings were considered too obscure or outrageous to be worth preserving by his contemporaries. That one masterwork influenced the next generation of mathematicians such as Hipparchus, and later commentators, including Hypatia of Alexandria and Eutocius, reinforced its reputation. Some of Apollonius' ideas and writings are mentioned in other ancient writings, which document some of his other conclusions. In his work on "burning mirrors," for instance, Apollonius disproved the notion that parallel rays of light could be focused by a spherical mirror, and he also noted properties of the parabolic mirror. Titles of his lost works include Quick Delivery, Vergings, and Plane Loci, as well as Cutting-off of a Ratioand Cutting-off of an Area . The subjects of these and some of their formulae and comments were summed up by Pappus.
Although his works were undervalued by many commentators over the years, beginning with his Greek contemporaries, Apollonius has recently undergone a revisionist examination. One academic, Wolfgang Vogel of Massey University in New Zealand, believes that after two millennia the ideas of Apollonius can be applied to current, significant problems related to intersecting conics.
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