PART III. LINES ON SECTIONS

the longest section of all...

These next few sections will be mostly, but not wholly, chronological, as the propositions are not grouped as closely by concept in this part of the book as they are in others.

1. Special Note: Abscissas

Proposition 20 and 21 are the black sheep of this part of the book. While all the props. preceding and following them are simply concerned with where lines fall, these props. are the first since the last part setting out to prove a certain proportion exists.

This relationship is as follows:

Parabola: sq. on 1st ord. line : sq. on 2nd ord. line :: line cut by 1st ord : line cut by 2nd ord.
Hyperbola/Ellipse/Circle: sq. on ord. line : rect. of line cut by ord. :: upright : transverse and sq. on 1st ord. line : sq. on 2nd ord. line :: rect. of lines cut by 1st ord : rect. of lines cut by 2st ord

Pictured Above: Propositions I. 20 (left) and I. 21 (right).

These lines are also known as the abscissa.

I would not normally give so much detail to just a pair of propositions, but these are yet more props. of critical importance. Many more complex relationships, which will be seen later, rely on the abscissa in some capacity.

2. Lines Which Meet the Section

This combination is a bit odd -- Propositions I. 18 and 19, alongside 26-28, fit together by virtue of investigating lines which meet the section at a point (which are NOT necessarily tangent, as will be seen later).

The first pair of propositions describe circumstances in which lines within any conic section would meet the section (either parallel to a line meeting the section or to an ordinate). The following three are a little more complicated. I. 26 and 27 both focus on the parabola, describing circumstances in which a line somehow related to the diameter would touch the section. I. 28 instead looks at opposite sections, though, like I. 27, looks at lines which would meet the section if produced both ways.

3. Lines Which Meet the Diameter

We next look at propositions I. 22-25. (Thankfully, this grouping is straightforward!)

These are pretty simple. Essentially, if, in any conic section, a straight line cuts the section at two points (and, in the ellipse, falls within the conjugate diameters) and does not touch the diameter inside, then it will meet the diameter outside of the section (I. 22, 23). Similarly, if a line falls outside any section and touches it at a point (and, in the ellipse, falls between both diameters), it will meet the diameter(s).

Pretty short and sweet stuff. Savor it while you can.

4. Lines Which Cut

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.

5. Lines Which Are Tangent to the Section

I. 17, is the only proposition from this section not from the same “cluster” (I. 31-36). It is related in that it explains that a line drawn the vertex of a section will necessarily fall outside the section. Importantly, it is the first reference (to my knowledge) to the tangent, which will show up both here and quite often later in the book.

Props. I. 33 and 34 demonstrate how to find the tangent of any conic section, and the rest of the props. outline two properties of the tangent. Namely, a tangent will fall inside the section if produced beyond the point it touches the section. (I. 31), and no line will fall between it and the section. (I. 32, 35, 36)