Celestial Geometry: Astronomical Calculations and Conic Sections

Slicing the Cone: Apollonius’s Curves

The conic sections are curves of profound elegance, born from the intersection of a plane with a double cone. Apollonius of Perga systematized their study in his eight-book treatise Conics, written around 225 BCE. I devoted considerable effort to understanding and teaching these curves, for they represent the meeting of pure geometry and cosmic order.

When a plane cuts perpendicular to the cone’s axis, we obtain a circle—all points equidistant from the center. Tilt the plane at an angle less than the cone’s slope, and an ellipse emerges, characterized by two foci such that the sum of distances from any point to both foci remains constant: $PF_1 + PF_2 = 2a$, where $a$ is the semi-major axis. Position the plane parallel to the cone’s slope, and a parabola appears—a curve with one focus where the distance from any point to the focus equals its distance to a fixed line called the directrix. Angle the plane more steeply than the slope, and a hyperbola forms with two branches, where the difference of distances to two foci is constant: $|PF_1 - PF_2| = 2a$.

These curves share a common measure called eccentricity $e$. For a circle, $e = 0$; for an ellipse, $0 < e < 1$; for a parabola, $e = 1$; for a hyperbola, $e > 1$. This single parameter encodes the curve’s elongation and character.

Apollonius proved remarkable properties: light rays emanating from one focus of an ellipse reflect to the other focus, enabling whispering galleries and, centuries later, satellite dishes. The curves possess tangent properties that would later inform calculus, and area formulas that connect geometry to integration. In my lost commentary on Apollonius, I likely extended these applications and connected them to astronomical phenomena—for the wandering planets trace paths that might themselves be conic sections.

Ptolemy’s Circles and Celestial Geometry

The astronomical system I taught was geocentric, following Claudius Ptolemy’s Almagest. In this model, Earth rests at the center while celestial spheres rotate daily. The planets orbit on small circles called epicycles, whose centers themselves orbit Earth on larger circles called deferents. This complex machinery explained the observed retrograde motion of Mars—its occasional backward drift against the background stars—as occurring when Earth overtakes the planet in its orbit.

Ptolemy’s model relied on sophisticated mathematical tools. We used trigonometry extensively, computing chord tables as precursors to the modern sine function. Spherical trigonometry allowed us to calculate celestial positions, predicting where planets would appear months or years hence. The model incorporated eccentric circles, where Earth sat offset from the circle’s center, explaining why the Sun appears to move faster during winter. The equant—a point from which the planet appears to move at constant angular velocity—was an artificial device that violated the principle of uniform circular motion but matched observations remarkably well.

I taught these geometric constructions and calculated planetary tables for practical use in navigation and, yes, astrology, which was widely practiced and respected in my time. The astrolabe, an instrument I likely helped design and certainly used, embodied these principles. Its stereographic projection maps the celestial sphere onto a plane while preserving angles—a conic projection that makes calculations tractable. The device measures star altitudes, solves spherical triangles, and serves as both timekeeper and navigational aid.

Kepler’s Ellipses: Conics in the Sky

Twelve centuries after my death, Johannes Kepler would revolutionize astronomy by discovering that planetary orbits are not circles but ellipses, with the Sun at one focus. His first law shattered the dogma of perfect circular motion. Most planetary orbits have small eccentricity—Earth’s $e \approx 0.017$ makes its orbit nearly circular, while Mars at $e \approx 0.093$ shows more noticeable elongation.

Kepler’s second law states that the line from the Sun to a planet sweeps equal areas in equal times—a consequence of angular momentum conservation. Planets move faster at perihelion (closest approach) and slower at aphelion (farthest point). His third law relates orbital period to semi-major axis: $T^2 \propto a^3$, revealing a harmonic relationship governing all planets.

Isaac Newton later proved that inverse-square central forces produce conic orbits. Universal gravitation, $F = GMm/r^2$, determines the orbit shape through the vis-viva equation: $v^2 = GM(2/r - 1/a)$. If total energy $E < 0$, the orbit is an ellipse—a bound system. If $E = 0$, a parabola—the exact escape velocity. If $E > 0$, a hyperbola—an unbound trajectory, as with interstellar objects passing through our solar system once and never returning.

The eccentricity relates to energy and angular momentum through $e = \sqrt{1 + 2EL^2/(GMm)^2}$. This formula unifies all conic sections under gravitational law. Modern applications abound: satellites orbit Earth in various ellipses, from low Earth orbit to geostationary positions. Hohmann transfer orbits use ellipses to move spacecraft between circular orbits efficiently. Comets follow highly elliptical paths with $e \sim 0.99$, or parabolic and hyperbolic trajectories if unbound. Binary stars orbit their common center of mass in ellipses. Even exoplanets, detected through transits and radial velocity measurements, are assumed to follow Keplerian orbits, allowing us to measure their periods and eccentricities.

Preserving the Light of Reason

I lived and worked in Alexandria during the fourth and fifth centuries—a time of intellectual flourishing but also growing tension. The Museum and Library represented the pinnacle of Hellenistic scholarship, drawing students from across the Roman world. I taught publicly, which was unusual for a woman, lecturing on astronomy, mathematics, and philosophy.

My philosophical orientation was Neoplatonism. I believed that mathematics reveals eternal truths, that the cosmos embodies divine harmony, and that geometric contemplation offers a path toward the transcendent. The perfect curves of conic sections were not merely practical tools for calculation but windows into the structure of reality itself.

My life ended violently in 415 CE, murdered by a mob during political and religious conflicts. I had aligned with the prefect Orestes against the Christian bishop Cyril, and the violence that claimed my life symbolized the declining fortunes of pagan scholarship in Alexandria. The Library would continue its long decline, and much ancient knowledge would be lost or scattered.

Yet my influence survived through channels indirect and delayed. Though my commentaries on Apollonius, Ptolemy, and Diophantus are lost, later scholars cited them. My student Synesius of Cyrene, who became a bishop, preserved some of my teachings in his correspondence, mentioning scientific instruments I designed. Through Arabic scholars who studied Greek texts, some of the knowledge I helped transmit eventually returned to Europe during the Renaissance.

The Moon bears a crater named Hypatia, and asteroid 238 Hypatia orbits the Sun—small monuments to a scholar who bridged classical Greek mathematics and late antiquity. I preserved and transmitted the work of earlier geometers, teaching a generation that would carry that knowledge forward. Though I did not discover that planets follow ellipses, I taught the geometric foundations that would make that discovery possible, and I embodied the conviction that mathematics is a spiritual practice, revealing the harmony underlying the cosmos.