Discussion and Homework
Ninth Reading Assignment

Notes for the Readings

From the Elements of Euclid:

II.6 If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.
(This was in the "Geometric Algebra" homework!)

I.47 Pythagorean Theorem

VI. Def. 2 (Note typo on bottom left of page 168, Def. 3 is cited by mistake)
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so the greater is to the less.

XIII. 9 If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

IV. 15 Corollary If in a given circle an equilateral and equiangular hexagon is inscribed, the side of the hexagon is equal to the radius of the circle.

XIII. 10 If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.

III. 21 In a circle the angles in the same segment are equal to one another.

VI. 4 In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

VI. 6 If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

Questions for Discussion

Reading 44

  1. Write up and be ready to discuss the steps in part 32. of the reading (page 168).
  2. Find a statement of Ptolemy's Theorem and bring it to class.
  3. How can Ptolemy's theorem be considereda generalization of the Pythagorean theorem?
  4. How does our degree notation (of degrees, minutes, seconds: 40° 13' 25'') come from the Babylonians through Ptolemy?

Regiomontanus Reading

  1. What is the Law of Sines?
  2. How is the Law of Sines proven in a modern trigonometry or precalculus book? What are the similarities/differences? Bring a copy of a modern proof to class.

Homework Problems

  1. Use the following diagram to prove that sin a = Crd 2a/120 where the radius of the circle is 60 and Crd 2a is the length of the chord of the central angle 2a.
    graph for the problem
  2. Using sin a = Crd 2a/120 and any trigonometric identities you know, show that (Crd S)2 + [Crd (180-S)]2 = 1202 is equivalent to cos2 x + sin2 x = 1.
  3. Using the chords that Ptolemy has found (Crd 36° and Crd 72°) and problem 1, find sin 18° and sin 36°.
  4. Ptolemy states (on page 169) "We shall explain in due course the manner in which the remaining chords obtained by subdivision can be calculated from these, setting out by way of preface this little lemma which is exceedingly useful for the business in hand." Derive the difference formula sin(a - b) = sin a cos b - cos a sin b using Ptolemy's Theorem. Write in terms of chords first, then use problems 1 and 2 to convert to sines and cosines.
    graph for the problem Note that AD is a diameter of the circle.
  5. Suppose you have a right triangle ABC with angles A, B and right angle C and sides opposite these angles a, b, c. If a and angle A are given, how would you use chords to solve for b and c? Hint: Find a similar triangle with hypotenuse 120p.

A pdf version of this page and the accompanying reading assignment.

Go back to course home page
Go back to my home page.