On The Circles Of Apollonius

By Bob Robinson

Apollonius of Perga (260-170 B.C.), called by the ancient Greeks “the great geometer” for his discovery of the concept of “conic sections”, is a much neglected giant in the history of science. According to Pappus (300 A.D.), he traveled to Alexandria from his birthplace in Asia Minor as a young man, attracted by the ideas of Aristarchus of Samos (310-230 B.C.), who discovered the heliocentric principle in astronomy. Apollonius undoubtedly collaborated with Eratosthenes (284-210 B.C), who was the librarian in Alexandria at the time Apollonius was there. Indeed, it would be fair to call him the immediate successor of Archimedes and Eratosthenes in geometry and astronomy. His written works, which except for part of On Conic Sections, have been “lost”, included (according to Pappus) the titles Cutting of an Area, Determinate Section, Tangencies, Inclinations, Plane Loci, and On the Burning Glass. In the latter work, Apollonius demonstrated why only a parabolic, not a spherical, reflector would focus light on a point. He is also known to have developed a sundial with a curved surface to more accurately determine time. How much of his work was destroyed when Julius Caesar burned down the Alexandria library in 48 B.C., we do not know.

Nevertheless, let us attempt to put a “parabolic focus” on the elementary breakthrough contained in Apollonius’ concept of “conic section”. It is not just the fact that circles, ellipses, parabolas, and hyperbolas are all formed by cutting a cone with a plane. Though Apollonius coined the terms ellipse, parabola , and hyperbola, others before him including Archimedes knew these figures were conic sections. Rather, it is the discovery of the significance of the cone, or conical action, itself!

What is the characteristic of the whole cone, as opposed to the characteristics of conic sections? It is the equi-angular, or logarithmic, spiral winding around the entire cone. As far as we know, Apollonius never directly identified the logarithmic spiral as such. But, the intuition Apollonius must have had about the cone was that it gives geometric form, an image, to what we would call the exponential function (raising to higher powers) as the envelope that includes the conic sections. His work is therefore a direct hereditary descendent of the discoveries of Archytas, Menaechmus, and Eratosthenes on the doubling of the cube, and the equally direct hereditary ancestor of Leibniz’ work on the catenary, Gauss work on complex numbers and residues, and Riemann’s work on surface functions of a complex variable.

How can we know this? We have to go outside Apollonius’ work On Conic Sections, and situate that work in the broader corpus of his other titles. Consider, for example, a famous construction, probably derived from Apollonius’ work under the title On Plane Loci, called the Circle of Apollonius. The construction is in two dimensions, and runs as follows.(See http://jwilson.coe.uga.edu/emt725/Apollonius/cir.html for diagrams.)

Construct a triangle with vertices A, B, and C. Bisect the angle at C, and find the point C’ where CC’ intersects AB. Unless the triangle is isosceles with sides AC and BC equal, CC’ will not bisect AB, but AC/BC=AC’/BC’ . Now, extend AC past C to some point E, and bisect angle ECB to intersect the extension of AC’B at some point D. Then, AC/BC=AD/BD. Also, since angle ACE is a straight line (contains 180 degrees), and is equal to angle ACB plus angle ECB, angle C’CD (being 1/2 of angle ACB plus 1/2 of angle ECB) is a right angle (contains 90 degrees).

Next, construct a circle with C’D as diameter. Because C’CD is a right angle, triangle CC’D will be inscribed in the semicircle with diameter C’D, and C will be somewhere on the circumference of that circle. In short, the locus of all the points, whose distances from points A and B have the same proportion as AC to BC, will be a circle, which has come to be known as the Circle of Apollonius

Furthermore, there are three distinct such circles associated with any triangle, depending on whether we bisect the angle at A, B, or C to begin the construction. Those three circles will be collinear at two points, and all three will be orthogonal to (intersect at right angles) the circle circumscribed around triangle ABC!

Thus, Apollonius constructed a planar system of multiply connected circular action, which creates an orthogonal relationship (right angles) at the intersection of those circles. If one were located in the small area around those intersections, one would seem to be surrounded by the square grid of Euclidean space! (This is similar to Riemann’s orthogonal intersection of parabolic surfaces.)

Ironically, Apollonius may have been inspired to understand the significance of the three dimensional cone as the envelope for two dimensional conic sections by geometrical constructions in two dimensions which pointed intuitively toward a domain of multiply connected circular action more powerful than three dimensional space!

Archytas and Apollonius Compared

Take, as a related example, the little model of Archytas’ “doubling of the cube” that I recently constructed, and wrote a pedagogical exercise about in the second week of September. That model, which I know delighted everyone who took the time to look at it, was based on my realization that most demonstrations of Archytas are flawed by failing to rigorously distinguish between two dimensional cross sections of the model, which are constantly changing, and the model as a three dimensional object, which stays the same. The torus, the cylinder, and the circle at the base of the cylinder, and the line of intersection of the torus and the cylinder do not change, but the cone, which is typically lumped together with the torus and cylinder, is not a single cone at all, but is constantly changing! That is, the circle, cylinder, torus, and the curve of their intersection do not change, no matter what chord and diameter we are trying to find two mean proportionals between. They are “integral” features of Archytas model. But the cones do change; they are the “differential” feature of the model!

The only way to truthfully represent the integral of all the possible cones for every possible cross section of the model was with a sphere, having the circle as its equator. That done, any planar cross section passing through the center of the torus of Archytas’ construction, and formed at a vertical right angle to the original horizontal circle of Archytas’ construction, would, when displayed on a suitable planar surface, show the whole construction “clear as day” to any willing student. In the cross section, two circles would appear, the smaller one a cross section of the sphere, and the larger one a cross section of the torus. In cross section, the upright cylinder appears as a vertical straight line. The laser beam I employ forms the (verbal) action in the model by piercing both the torus and the sphere in a locus forming a cone. The laser appears in the planar cross section as a diagonal line cutting both the smaller and the larger circle, as well as the straight line of the cylinder in cross section, in such a way that both extremes, as well as both mean proportionals, “leap out at you”. The student will look at the cross section, then the model, then the cross section, and so back and forth, until the conception is clear.

I am sure Apollonius had a similar experience, when for the first time, he discovered, and then showed students how cross sections of a cone produce conic sections. Take the ellipse, for example. Who would think that a flat plane cutting a cone diagonally, from a point quite close to the vertex of the cone down to a point where the cone’s diameter has become quite wide, would form a perfectly symmetrical ellipse? Or, who would guess that, while hyperbolas and ellipses may have various shapes, that not only the circle, but the parabola (like a catenary) has only one? These things become clear in cross section.

Apart from this, there is a construction of the conic sections in two dimensions based on the locus of points that maintain a constant proportionality of distance to a fixed point and a fixed line, called the “directrix”. Supposedly, Apollonius never investigated this property of conic sections, despite the glaring analogy with the Circle of Apollonius.

So, what is the “spring” in the discovery that Apollonius made concerning conic sections, or that Archytas made with his “conic sections” of the sphere, torus, and cylinder? It is the paradox, that a true understanding of the three dimensional object only occurs when the differential feature of the object, as represented by multiply connected circular action in even “merely two dimensions”, is shown to be the “force” or “vis viva” behind the resolution of visual anomalies associated with the three dimensional realm.

“How can two dimensions be of higher power than three dimensions?”, you ask. That is the wrong question. Ask, “What in the physical universe is of a higher power than the three dimensions of space or the four dimensions of space time, so that it is possible sometimes for a two dimensional picture to be of a higher power than three dimensional space?”

It occurred to Apollonius that multiply connected circular action, portrayed even on a two dimensional planar surface, could represent more degrees of freedom than the three dimensions of visual space, just as conical action is of a higher power than conic sections!