NOTE: Somehow my “is to” symbol was converted to a Spanish exclaimation point… *shrugs* I guess this is why people don’t use unicode if they can help it…

Classes are still somewhat less than enthralling. We’re working on Lobochievsky’s (sp?) non-Euclidean geometry still, along with preceptorials. Whenever I start work on my preceptorial essay you’ll get a look, if you care to read, into what the text is about. The main question, as far as I can tell (though I’m by no means certain), is about the nature of number. Is a number fundamentally a number *of* something, be it chairs, stars, or indivisible primordial monads, or is it fundamentally an abstraction that only has being insomuch as it is used by us to count. Also, is an operation valid because and only if it produces a counting-number that can be expressed in indivisible monads, or is it such regardless of what kind of odd, obscure, irrational result is produced, so long as all the operations were valid? Is √2 a number? What about 1/3? (Doubts on the former are on account of its irrationality, and on the latter because it is expressing an operation rather than a number still; should the units be multiplied by 3 to produce perfect monads?) Are numbers prior to counting as something that have being in themselves? Yesterday we began by working on Apollonius’ *Conics* 1.11, on the plane of a cone used to produce a parabola. Apparently Mr. V___ was most interested in pointing out the relationship between the line of the upright side and the position of the parabola slice in the cone. Thus:

Given the cone ABC with the axial triangle ABC forming a plane through it, let FG be parallel to AC, and DE perpendicular to BC, and let a plane slice the cone through FDE. Let it be contrived that the square of BC¡the rectangle BA, AC ¡ FH¡ FA. Choose any KL parallel to DE. The result of the proof is that for any KL, the square of KL = the rectangle HF, FL. It’s also demonstrated the Apollonius is the worst writer of annunciations concerning what is to be proved in the history of geometry. Or at least he’s pretty bad — Euclid is the best, as he is at most geometric enterprises. Anyway, there are apparently many possible diameters of the parabola FDE, and that each one has a different upright side, determined by the proportion BC^{2}¡(BA)(AC)¡FH¡FA, and therefore dependent upon the angle of the cone and the placement of point F. This is related (or *is*, I don’t know) to the “ordinate property” and the “limit” of the parabola. You might notice my lack of familiarity with math in my total discomfort with terms like this. Anyway, Mr. V___ stressed that the upright side is a particular line determined by the shape of the cone and the position of F on it. That is for some reason important.

I’m trying to understand what’s at stake here. Does anyone reading who’s taken math in such a way as to be familiar with parabolas know what’s at stake here? I’m, frankly, lost. Not concerning the slicing of cones. Well, OK, the proof involved more composing and decomposing of ratios than I could quite follow, but that was simply sloth: I supposed that A. was correct in his ratios. But the conic section mostly made sense. What didn’t make sense is that we’re apparently comparing the way A does mathematics with the way *we* do mathematics and drawing interesting conclusions regarding conceptual shifts from the one to the other. Unfortunately, *I* do not understand the way “we” do mathematics well enough to have any basis of comparison. This is awkward.

I’m not sure anything is at stake. But I do share your feeling of being lost on conic sections. Ever since the 10th grade, to be exact. Having never heard of Appolonius until 30 minutes ago I can’t help you.

So why are conic sections significant at all? Apollonius was an astronomer, for one, so to him geometry was quite valuable in understanding the movement of the planets. He was also trying to understand mathematics in a somewhat different way than we do today. He didn’t use negative numbers or do things like use equations to plot geometric shapes. That’s about as much as I understand right now. There’s so much history there that to do an exhaustive comparison between him and, say, modern geometry is no small undertaking. Fortunately the information is out there if you know what questions to ask.