Prerequisites for Appolonius Conics?

Question

I want to get Thomas Heath's version of Apollonius's Conic Sections. Does anyone know the prerequisites to understand everything in this book? I heard I would need the Euclid's Elements book on Solid Geometry, but I'm not sure. Note: Thomas Heath's version of Apollonius' conics is basically just a translation of the original treatise by Apollonius

Thanks.

Answer 1

My experience is so far this: it is difficult to read, and worth every effort. (Currently in book III).

Advice: Do not delay. Read it.
It is rough going at first, so make a plan. I recommend forcing yourself through the first 16 propositions, building the problems in a 3D-capable Geometry program.

Basically, his first task is to discover a method for navigating the surface and interior of the cone. Propositions 1-16 get from a point and a line to the identification of the sections, each defined by a ratio that can be used to draw it. I started over and over, reviewing definitions and previous propositions. This initial hurdle passed, it gets easier. He gets down to the business of describing what is inside.

I have the newer Green Lion Press edition. I have only worked through the first book of Euclid. The propositions of Euclid used extensively by Apollonius are few in number and mostly identical to their algebraic counterparts: determining equality among a group of ratios (fractions). The translator or editor will indicate outside references. Keep a copy of Euclid handy, or just use the notes in your edition. There is no need to prepare.

For me, a principal difficulty was adjusting to the language and conventions. The distinctions he makes are not in common use today, and I found it hard to grasp simple relationships at first. Geometry is intricate. We define a problem by naming the relationships on the figures themselves, rather than by adherence to a system. And reading $$ KL : LN :: sq. KL : KL \cdot LN$$ is hard at first compared with ${\large {a \over b} = { a^2 \over a b}}$, especially since we must check each relationship against the figure. (Here, we have just multiplied by a/a =1). Many constructions are also long, and it takes time both to complete a single proposition and then digest the result.

I think there are three advantages to Apollonius' approach.

1. Problems are defined and solved directly on the figures, so Apollonius answers the questions which are before him. I wasn't comfortable with the math I learned for conic sections because the proof didn't satisfy me that the underlying assumptions were true. Apollonius. He does not skip steps, and never assumes the answer is already known.

2. The geometric representations and solutions of problems lend a simplicity and elegance that are sorely lacking in, say, homogeneous coordinates. Try locating the axes, given $A x^2 + B x y + C y^2 + D x + E y + F =0$, then compare this with Book II, prop. 47: Given a hyperbola or ellipse, to find the axis. When I get confused with modern representations, now I go back to Apollonius.

3. The subject is self-contained. Apollonius begins by introducing a point and a line. As long as you are able to measure, you know everything you need to know, because he literally works forward step-by-step from that point and line. This isn't how most math books read today; it's how problems are solved. This is an enormous advantage to the student.

I cannot recommend this book more highly.

SUPPLEMENT:

This post was as accepted as the best answer, so I will try to address more clearly the OP's distinct questions, then collect everything into --hopefully-- a more useful post for future readers.

Difficulty
I found the main struggle to be the text (and subject) itself, not its relationship to outside knowledge.

Preparation
I found "solid familiarity with a compass and straightedge" more important than knowing any particular results or theorems. Compared with high school geometry, the first books of Euclid's Elements are much more involved with carrying measure from place to place: e.g. how do I cut a segment into a given ratio, or a given length? or, Draw an angle equal to a given angle, upon an arbitrary line. Most of the first propositions of Apollonius take place in 3-D space and we must picture constructions in the mind's eye. Practice helps, but we can just as easily decide to practice by reading the Conics.

Reliance on Euclid
Apollonius everywhere assumes familiarity with Euclid, but this is rarely an obstacle. The Elements are a system of measure, and most of the results Apollonius uses you already know from the system we call arithmetic and Algebra. In particular, - notation aside - ratio comparisons read the same way. Whether or not to examine the geometric proofs is a matter of personal preference.

That said, there are a few key theorems relating measurements on a circle which may be unfamiliar, and I will make clean, dynamic constructions of those and post links below in the coming weeks. (PLACEHOLDER)

Answer 2

Thomas Heath's version of Apollonius' conics is basically just a translation of the original treatise by Apollonius

Actually, Heath's work, at least the 1896 Cambridge edition (which is available in full at https://archive.org/stream/treatiseonconics00apolrich#page/n9/mode/2up), is much more than that. It comes with extensive introductions both to the earlier history of conic sections among the Greeks (PART I) and to the Conics of Apollonius itself (PART II).

PART I is full of references to Greek mathematicians that studied conic sections and in particular mentions the following, all of which predate Apollonius (262-190 BC):

1. Menaechmus (380–320 BC), "the discoverer of the conic sections"
2. Aristaeus the Elder (370–300 BC)
3. Euclid (ca. 300 BC)
4. Archimedes (287-212 BC)

The (surviving) works of Euclid and Archimedes have also been translated by Heath and are available online while it appears that none of Menaechmus' and Aristaeus' writings survive except through references by other ancient and medieval writers.

There is, however, no need to digest any of these authorities if your only aim is to read and understand the Treatise on Conic Sections: Heath's "introduction" (spanning a total of 150 pages!) is exhaustive and reviews not only the works of Apollonius' predecessors and contemporaries but also (in PART II) Apollonius' own methods and terminology in modern mathematical language. As a result, Heath's Apollonius, when read from start to finish, can comfortably stand on its own. In particular, in the Appendix to Introduction, Heath defines such basic geometric primitives as point and line as employed by the Greeks without relying on any other work, in a manner that can be used as a mini-encyclopedia whenever those terms are encountered in the actual treatise.

(NOTE: In this answer, I have assumed that by "understand everything in this book" you mean its mathematical content. If you are interested in Apollonius' philosophical, in particular metaphysical views, which few classical authors separate fully from their scientific output, you might want to read a biography of Apollonius first, though IIRC, his works are much less metaphysics-heavy than those of some other Greek mathematicians such as Pythagoras.)