The History of Mathematics: A Brief Course

6. QUESTIONS AND PROBLEMS 349

FIGURE 14. Greek use of a fundamental inequality. Left: from

Euclid's Optics. Right: from Ptolemy's Almagest.

to the diagram on the left in Fig. 14. Show that BE : EA :: AB : Æ A :: ÃÄ : Æ A

and that this last ratio is larger than ÇÈ.ÆÈ.

11.2. Use the diagram on the right in Fig. 14 to show that the ratio of a larger

chord to a smaller is less than the ratio of the arcs they subtend, that is, show that

ΒÃ : AB is less than BT-.AB, where AT and AZ are perpendicular to each other.

(Hint: BA bisects angle ABT.) Ptolemy said, paradoxically, that the chord of 1°

had been proved "both larger and smaller than the same number" so that it must

be approximately 1; 2,50.) Carry out the analysis carefully and get accurate upper

and lower bounds for the chord of 1°. Convert this result to decimal notation, and

compare with the actual chord of 1° which you can find from a calculator. (It is

120sin(±°).)

11.3. Let A, B, C, and D be squares such that A : Β :: C : £), and let r, s, t, and u

be their respective sides. Show that r : s :: t: u by strict Eudoxan reasoning, giving

the reason for each of the following implications. Let m and ç be any positive

integers. Then

mr >ns=> m^2 A > n^2 B => m^2 C > n^2 D =>mt>nu.

11.4. Sketch a proof of Pappus' theorem on solids of revolution by beginning with

right triangles having a leg parallel to the axis of rotation, then progressing to

unions of areas for which the theorem holds, and finally to general areas that can

be approximated by unions of triangles.

11.5. Explain how Thabit ibn-Qurra's generalization of the Pythagorean theorem

reduces to that theorem when angle A is a right angle. What does the figure look

like if angle A is obtuse? Is there an analogous theorem if BC is not the longest

side of the triangle?

11.6. One form of non-Euclidean geometry, known as doubly elliptic geometry, is

formed by replacing the plane with a sphere and straight lines with great circles,