On the evening of July 20th, 1969, two American astronauts took man’s first tentative steps on another world. The traditional story of how humans slipped the surly bonds of Earth has been burned into humanity’s collective memory during the last half-century, but the true story of Project Apollo, and its legacy, begins not with a presidential speech in 1961, but at the hands of Apollo himself in the 4th century BC.

Hi, I’m David Grider, and in this podcast we’re going to explore how humanity went from merely dreaming of flight to exploring worlds beyond our own.

In the year 374 Ab urba condita, which corresponds to 380 BC, the Roman Republic was nearly destroyed by the Praenestines from a nearby city-state, due to political strife within Rome. Titus Quincitius Cincinnatus became dictator and defeated the Praenestines, forcing their surrender, and held a triumph with a statue of Jupiter from Praenste. Incidentally, the city of Cincinnati, Ohio was indirectly named after Titus Quincitius Cincinnatus, and was where Neil Armstrong lived from 1971 until his death in 2012.

Half a world away, from the Roman perspective, a man named Menaechmus was born in Alopeconnesus, a city in Macendonian Thrace, in modern-day Turkey. During his early life he developed a friendship with Plato.

According to legend, in the 350s BC, a plague was sent by the god Apollo to his own birthplace of the island of Delos. Having consulted the oracle at Delphi, the citizens of Delos asked Plato to solve the problem the oracle gave to them: duplicate a cubic alter where the volume of the duplicate was to be twice the volume of the original. Plato then commissioned the greatest mathematicians of the time, including his friend Menaechmus, to solve what we now call the Delian problem: using conventional geometric tools, a compass and a straight-edge, create a cube double the volume of another cube.

In reality, the problem was one that had existed for some time, and was, along with squaring a circle and trisecting an angle, the ancient equivalent to Fermat’s last theorem.

Regardless of the circumstances, Menaechmus set to work to solve the problem. He probably never did come up with a workable geometric solution. Two stories exist, one that no solution was ever found, and another that Menaechmus found a solution, but it was impractical. Either way, no geometric solution exists, but in the process he had discovered something even more important: conic sections.

A conic section is a curve that is created when a plane intersects a cone. They come in three flavors: ellipses, parabolas, and hyperbolas.

Imagine a cone and a plane that cuts the cone in two parts. The angle that the plane is at in relation to the cone determines the conic section. If the plane is parallel to the base of the cone, you get a circle (we’ll talk about circles in a few minutes). If the plane is tilted but still goes from one side of the cone to the other without going through the base, you get an ellipse. If the plane is parallel to one of the two sides of the cone, the plane will go through the base, and you get parabola. If the tilt is even further, you get a hyperbola.

For about a century, those were the working definitions of the three known types of conic section, until a geometer and astronomer named Apollonius Pergaeus worked on the problem.

Apollonius of Perga was born in the Hellenistic city of Perga, Pamphylia, Anatolia circa 240 BC. At some point in his early life he moved to Alexandria, Egypt where he went to school and then taught at a university. While there, he wrote his only surviving treatise, Conics, which over eight books, provided a generalized description of the curves that are obtained from a plane intersecting a cone. Among the advances made were the first uses of the names of the conics that we still use today (ellipse, parabola, and hyperbola), and the definition of the focus and directrix. Also included was the requirement that a conic section should be defined in terms of two cones, one the mirror of the other, touching at their apexes. Another was the special definition of the circle.

Until Apollonius worked on the problem, the circle was not considered a conic section. His new generalized conics added the circle as a special case conic section, an ellipse where the center and both foci all occupy the same point, and the circle can be defined as a locus of points, or set of all points, that are a constant distance, or radius, from the center in any direction.

An ellipse is defined as a closed plane curve surrounding two points called foci, where the sum of the distance between any point and the two foci is a constant. In elementary school, we called this an oval, though the definition of an oval is not nearly as strict. For example, an oval needs only be symmetric across one local axis, while an ellipse must be symmetric across two. This is due to the oval’s definition, which is a closed plane curve that loosely resembles an egg, or ovum; hence “oval”. However, you can best think of an ellipse as a circle that’s been squished on one axis.

An ellipse can be drawn a variety of ways, but one of the best ways follows the exact definition of an ellipse: the gardener’s method. Place two pins through your drawing surface and tie the ends of a string to each one, then take a pencil and use it to pull the string taut. Draw clockwise or counterclockwise with the string taut and you will wind up with an ellipse.

Parabolas, on the other hand, are open curves that are symmetrical across one axis, are U-shaped. Parabolas have only one focus, as opposed to the two of the ellipse. There is also a point called the vertex, which lies where the parabola intersects the axis of symmetry.
The distance from the vertex to the focus is the focal length. On the opposite side of the vertex from the focus is a line that is perpendicular to the axis of symmetry called the directrix. The directrix is the same distance from the vertex as the focus is. From there, a conic section can be defined as the locus of points that are equidistant from both the focus and the directrix. To find a point on the parabola, draw a line segment of any length perpendicular to the directrix and in the direction of the vertex. Then, draw a line segment of the same length from the focus to the end of the first line segment furthest from the directrix. The point where the two meet will be a point on the parabola.

A hyperbola is a two-part curve, ranging from a pair of wide-open U-shapes to near V shapes that mirror each other. The geometric definition as a locus of points is difficult to express verbally, so I will simply leave you with that description and the images on the website to imagine what I’m talking about. The important thing is that a hyperbola is what you get when you slice a double cone from top to bottom without going through the center.

Now you may be asking, David, why are you talking about conic sections and what does this have to do with going to space? Well, it comes down to orbits. As we’ll discuss in an upcoming episode, orbits are now known to be able to be described in terms of conic sections. Normal, closed orbits, in fact, are always ellipses (even circular ones). This applies to spacecraft as well as the moon, the planets, and all minor bodies. It also applies to multiple stars orbiting a barycenter, or the objects of the Milkyway galaxy orbiting the supermassive central black hole Sagittarius A*.

In fact, there is some evidence to suggest that Apollonius hypothesized that elliptical orbits were responsible for observed aberrant planetary motion, a notion that stuck around until the middle ages.

Beyond orbits, many of the things we take for granted in the modern world would not have been possible without the discovery and generalization of conics. In the field of optics and astronomy, reflecting telescopes often use parabolic or hyperbolic mirrors. Eyeglasses rely on parabolic lenses to correct vision. In the field of telecommunications, satellite dishes are usually parabolic. In physics, a ballistic trajectory is parabolic, and in civil engineering, the vertical curves of roads are typically parabolic. The sun itself traces out a hyperbola on the ground throughout the day. In finance, hyperbolas are used to describe optimal investment portfolios. Hyperbolas are even used in biochemistry to describe biological changes to proteins.

The conics of Apollonius of Perga are still defined the same way today, and remained the best understanding of conic sections until the invention of analytical geometry by Rene Decartes and Pierre de Fermat in the 17th century.

In that span, great the discoverers rose and fell, and they each left a piece of the puzzle that took us one step closer to the moon. Next time, we’ll look at another important piece to the puzzle: Archimedes.

Images via Wikipedia