PART IV. COMPLEX RELATIONSHIPS

the last of our discoveries...

I'll admit this section was a little harder to classify than the others, but I think I still managed to scrape together some similarities. The main commonality between all of them is their heavy focus on ratios and relationships between lines and figures.

1. Abscissas

This handful of propositions draws heavily from nearly 20 propositions prior, when abscissas were first defined (I. 20,21). The main goal of these propositions is, by using the abscissa, to create proportions between created lines, the transverse, and the upright.

I. 37 and 38 both create abscissas from ordinate lines created from a tangent on the hyperbola, ellipse, or circle, with the latter prop. also looking at the abscissa's relationship to the second diameter. I. 39 and 40 then take I. 37 and 38 and decide to turn them into compound ratios. (I'm not entirely sure why these are separate -- I suppose because Apollonius is proving that we can add things to the transverse or upright and still have these proportions hold true? They certainly prove useful later.)

2. Figure Equalities

Just as Book VI of Euclid once showed that different figures can be in proportion with one another, Apollonius spends the next several propositions comparing constructed triangles and parallelograms on various lines in conic sections. Once again, this section makes effort to compare constructed lines.

I. 41 essentially takes the compound ratios we just learned about and creates figures using them. The propositions following it compare lines cut off on the diameter, ultimately comparing a triangle to another figure (either by being made equal to a parallelogram, or by being less than another triangle) (I. 42-44). I. 45 does something similar by using the second diameter.

3. More Bisection

Did you miss the time when we were only comparing lines to lines? Then you're in luck! I. 46-48 takes a break from the figures to give examples of cases where lines are equal.

All the propositions are different versions of the same idea, proven on all the conic sections (+ opposite sections). Essentially, a line drawn from a tangent's point of contact will bisect a line parallel to a tangent. The only differences lie in what direction the line need to be drawn, which varies from section to section.

4. The Last Ones With Really Long Enunciations

I. 29 and 30 stick out a little bit, as they seek to make statements about lines cutting things. In the former proposition, it is said if a straight line drawn from the diameter of two opposite sections cuts one section, it must necessarily cut the other. In the latter, it is stated, in the ellipse or opposite sections, that a line drawn from the center of the section to meet the diameter will be bisected at the center.

And so, we approach the end...