sign in sign up
<<
Home , Conical surface, Parabola, Quadrature (mathematics)

Notes for reading ’ of the

Here is a recipe for reading any piece of . As you look over the material you are trying to read, you should make sure that you are either familiar with—or at least are trying to be familiar with—the basic vocabulary, with the principal “objects of discussion” and, also, with the aim of the discussion. We will not be getting very far, in these notes, in the actual reading of QP, but I want to (at least) set the stage, here, for reading it.

1. Some of the basic vocabulary in QP:

1.1. Vocabulary in Propositions 1-5. • Quadrature • Right circular conical (or right angled ) S • (is the intersection of a P with a right circular S) • the axis of a right circular conical surface

• a Parabola is a conic section whose plane P is to the axis.

• a Segment (of a parabola)

1.2. Vocabulary for lines in the plane of the Parabola. There are three sorts of lines that “stand out.”

• Chords between any two points on the parabola. These define segments. • to any on the parabola. • lines parallel to the axis

1.3. Vocabulary in Propositions 6-13. • Lever • Horizontal 1 2

• Suspended • Attached • Equilibrium • Center of gravity

For people familiar with Archimedes’ The Method here is a question: to what extent does Archimedes use the method of The Method in his proofs in Propositions 6-13?

2. The results

I’ll discuss my “personal vocabulary:” the triangle inscribed in a parabolic segment; and the triangle circumscribed about a parabolic segment. The fundamental result that QP aims toward is

Proposition 16: The of a parabolic segment is 1/3 the area of its circumscribed triangle.

and

Proposition 17, 24: The area of a parabolic segment is 4/3 the area of its inscribed triangle.

3. A modern secret: If you prove these propositions for any single parabolic segment, there is an argument that shows that you’ve done it for all.

4. General questions to think about while you read Propositions 14-16

Question 4.1. What is the basic diagram set up in the first paragraph of each of these propositions? 3

Answer: This diagram which gives the circumscribed triangle of a parabolic segment.

I’ll use the terminology BOΓ to denote the pictured segment of a parabola bounded by the straight BΓ. As in Archimedes, the straight line B∆ is drawn parallel to a diameter, and Γ∆ is to the parabola at Γ. Question 4.2. Return to the basic diagram above. Bisect B∆ at K. Draw the straight line ΓK. 4

(1) Why does ΓK intersect the parabolic arc? (2) Why does it intersect the parabolic arc at exactly one point? (3) Given that ΓK intersects the parabolic arc at one point, call that point O and draw the straight line through O parallel to a diameter, labeling Π and Σ its intersections with BΓ and B∆, respectively. Is this a special case of the diagrams you are asked to draw in Propositions 14,15, and 16?

Question 4.3. Can you go through the proof of Propositions 14, and 15 using the above diagram as a special case? To begin to do this concentrate on the following figures • the rectangle KΠ • the triangles ΣΠΓ, OΠΓ, and BΓ∆ • the segment BOΓ (1) Show ( or rather, note) that

OΠΓ < BOΓ < KΠ + ΣΠΓ. Where, in Archimedes’ text of Props 14, 15 are these inequal- ities acknowledged? (2) Where and how, in Archimedes’ Props 14, 15, do we see the inequalities 1 OΠΓ < B∆Γ < KΠ + ΣΠΓ 3 proved?1 (3) Note that the difference between KΠ + ΣΠΓ and OΠΓ is KΠ + ΣOΓ. Why is this difference2 equal to the triangle BΓK?

Question 4.4. Return to the basic diagram above3 and now trisect B∆ at E and K, so that BE,EK,K∆ all have the same length. Draw

1In this very particular instance, of course, you can say more precisely what OΠΓ and KΠ + ΣΠΓ are, as fractions of B∆Γ. 2i.e., the difference between the upper bound and the lower bound in the chain of inequalities in (1) and (2) above; i.e., the inequalities that sandwich both BOΓ 1 and 3 B∆Γ 3BOΓ is the segment of a parabola bounded by the straight line BΓ, B∆ is parallel to a diameter, and Γ∆ is tangent to the parabola at Γ. 5 the straight lines EΓ, KΓ. Draw the two indicated lines parallel to the diameter of the parabola4.

(1) Which quadrilaterals in this figure can you show have the same area? (2) How can you show this? (3) Which triangles in this figure can you show have the same area? (4) Where in Archimedes’ text is it demonstrated that they do have the same area? How is it demonstrated? (5) What role does it play in the proofs of these propositions? (6) Proposition 14, 15 proves inequalities of of figures. In the special case we are discussing, state explicitly the inequalities that are being proved. (7) Run through the proof—in the text of Archimedes’ Propositions 14,15—for this case.

Question 4.5. Proposition 16 is proved by an indirect argument. Looking at the second paragraph, Archimedes supposes that the equal- ity of areas (asserted by the proposition) is not true and he wishes to (eventually) draw a contradiction. He supposes—first— that the area of the parabolic segment is bigger than one third the triangle B∆Γ. How does this lead him to construct the diagram for Proposition 16?

Question 4.6. Make sure that you know exactly the place(s) where— in Props 14, 15, 16—the hypothesis that the parabolic segment is in- deed a parabolic segment. Where—and exactly how—is it used? Query 4.7. (For people who know )

4These two lines should intersect the parabola at the points at which EΓ and KΓ (respectively) intersect the parabola. 6

(1) Compare (and contrast) Archimedes’ work on this problem with the standard Calculus approach? (2) For an example of some of the many things to note: Archimedes comes at it with an explicit “indirect argument” formulation (discussed in Question 4.5 above). Do you see clearly what this corresponds to in the standard Calculus approach? Query 4.8. As has been already noted in class discussions, Archimedes working with his levers, often replaces a figure (or a part of figures 1 such as 3 B∆Γ )with a “space” (chorion) whose is completely irrelevant but whose area5—abstracted from its shape—is considered. Did the word chorion play—at least a bit— this role (i.e., meaning a space, but where area is of concern and particular shape not) earlier?

5in other contexts he does the same thing with volume, or length