c. 262 – 190 BC
The Great Geometer — whose masterwork on conic sections shaped two millennia of mathematics, from ancient optics to planetary orbits
His magnum opus systematically derived all conic sections from a single oblique cone, superseding earlier work by Menaechmus and Euclid. Books I–IV survive in Greek, V–VII in Arabic translation, Book VIII is lost.
Apollonius coined the terms parabola (equal application), ellipse (deficient application), and hyperbola (excessive application) based on the area-application method from Pythagorean geometry.
Ancient catalogues attribute many other treatises: Cutting-off of a Ratio, Cutting-off of an Area, On Determinate Section, Tangencies, Inclinations, and Plane Loci.
Pappus called him "The Great Geometer." His work was considered so definitive that it rendered all prior treatments of conics obsolete and remained the standard reference for 1800 years.
Apollonius showed that a parabola, ellipse, and hyperbola all arise from slicing a single double-napped oblique cone at different angles to its axis.
Apollonius derived the characteristic equation (the symptom) of each curve by relating an ordinate y to the abscissa x measured from the vertex along the diameter:
y² = p · x
"Equal application" — the rectangle of area y² is exactly applied (paraballo) to the parameter p.
y² = p · x − (p/d) · x²
"Deficient application" (elleipsis) — the rectangle falls short by a term proportional to x².
y² = p · x + (p/d) · x²
"Excessive application" (hyperbole) — the rectangle exceeds by a term proportional to x².
Books II–IV of the Conics develop a sophisticated theory of tangent and secant lines, as well as conjugate diameters — pairs of diameters where each bisects all chords parallel to the other.
Apollonius proved that a tangent to a parabola at any point makes equal angles with the focal radius and the axis direction — the principle behind parabolic mirrors and satellite dishes.
All rays parallel to the axis reflect through the focus — the basis of parabolic antenna design
In his lost work Tangencies (De Tactionibus), Apollonius solved a famous problem:
Given three objects, each of which may be a point, a line, or a circle, construct a circle tangent to all three.
Fix a cone and a cutting plane with precise geometric conditions
Use proportions and area-application to extract the "symptom" equation
Establish properties synthetically via reductio ad absurdum
Extend to oblique diameters, general positions, and all conic types
Every argument is phrased in terms of ratios of line segments and areas, never symbolic equations. The symptom y² = px is our modern translation of what Apollonius stated as a proportion between geometric magnitudes.
Apollonius systematically enumerated every case. Book IV alone treats 57 propositions on intersections of conics with each other, cataloguing all possible configurations of two conics meeting in 0, 1, 2, 3, or 4 points.
Ancient sources hint at tension between Apollonius and the legacy of Archimedes, his slightly older contemporary.
"The third book contains many remarkable theorems useful for the synthesis of solid loci... of these, some were worked out by Euclid, though not in the manner Apollonius achieves."
— Apollonius, Conics, Preface to Book ISome ancient accounts claim Apollonius proposed problems specifically designed to stump Archimedes. Whether or not this is true, the two men represent complementary approaches — Archimedes the physicist-calculator, Apollonius the pure geometer.
Descartes and Fermat translated Apollonius's geometric conditions into algebraic equations, creating analytic geometry. The standard equation Ax²+Bxy+Cy²+Dx+Ey+F=0 directly encodes the Apollonian classification.
The harmonic properties Apollonius proved for pole-polar pairs in conics became foundational for Desargues, Pascal, and the 19th-century projective geometers.
When Kepler proved that planetary orbits are ellipses, he relied on the geometric properties catalogued 1800 years earlier by Apollonius.
The Apollonius tangency problem inspired the "Apollonian gasket" — a fractal packing of mutually tangent circles studied in modern number theory and dynamics.
Parabolic reflectors focus electromagnetic waves at the focus — the reflective property proved by Apollonius.
Every orbit in a gravitational field is a conic section: ellipse (bound), parabola (escape at limit), hyperbola (unbound).
Cassegrain and Gregorian telescopes use hyperboloidal and ellipsoidal secondary mirrors based on conic geometry.
LORAN and GPS trilateration use intersections of hyperboloids (surfaces of constant time-difference), descendants of Apollonius's curves.
Charged-particle optics in accelerators uses quadrupole magnets whose fields follow hyperbolic equipotential lines.
Whispering galleries in elliptical rooms exploit the reflective property of ellipses: sound from one focus is heard at the other.
Apollonius of Perga, translated by Michael Fried & Sabetai Unguru (2001). Modern scholarly translation with extensive commentary.
Thomas Heath (1921). The classic survey covering Apollonius's contributions in their full historical context.
T.L. Heath (1896). A "treatise on conic sections" — Heath's earlier edition that remains valuable for its geometric commentary.
Giovanni Ferraro (2008). Places Apollonius's approach in the broader arc of mathematical analysis.
Asger Aaboe (1964). Accessible introduction to Greek mathematics including Apollonius's methods.
Jacob Klein (1968). Explores how Greek geometric reasoning (including Apollonius's) relates to the development of algebra.
"The properties of the conics, set forth in a clear and systematic way by Apollonius, have served mankind for over two thousand years and are, in a sense, the foundation upon which the entire edifice of celestial mechanics has been built."
— Sir Thomas Heath, A History of Greek MathematicsApollonius of Perga · c. 262–190 BC · The Great Geometer