Heavenly Mathematics Daily Schedule Term 4
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See also Daily Schedule Term 5
Week 1 — Nightly Celestial Motion — Two Celestial Coordinate Systems: Alt/Az and RA/DEC
Week 2 — Early Trigonometric Achievements
- Monday, Jan. 17 — Prepare for discussion by studying pp. 1-11 of Van Brummelen — Why do the angles of a triangle have to add up to 180°? — What are similar triangles (or why are sine, cosine, and tangent worth tabulating?)
- Thursday, Jan. 20 — Prepare for discussion by studying the remainder of Chapter 1 in Van Brummelen — Detailed examination of the derivation of sin(72°) using the pentagon
- Problem Set 1 due Monday, Jan. 24: Problem Set 1
- Due to cloudy weather (and COVID outages) during the first 1 1/2 weeks of the semester, completion of the Alt/Az Estimation Assignment was indefinitely postponed
Week 3 — Early Trigonometric Achievements (Cont’d)
Week 4 — Exploring the Sphere
- Monday, Jan. 31 — Preparation for Monday: Read Heavenly Mathematics pp. 23-34 — Systematic introduction to the Lenart sphere — We will ask you to present:
- The proof of the lemma on p. 33. You may wish to consult Euclid’s Elements, specifically Book I, Proposition 20
- Problems #1, 3, 6 and 7 on pp. 39-40 — you will not turn the problems in, but you need to be ready to present.
- We will call on individuals to present each of the above in class, so be ready for whichever may come your way.
- Thursday, Feb. 3 — Preparation for Thursday: Continue reading Chapter 2 through page 37. Revisit the ideas from 23-34, now that we’ve had a chance to play a bit with our Lenart Spheres. — As Problem Set 3 for Thursday, turn in Problems 1, 3, 6, 7, 9, and 10 — We will work the proofs of the theorems on pages 34 and 36 in class on Thursday. Be prepared to present. — Problem Set 3 Solution
Week 5 — Exploring the Sphere (Cont’d) — Menelaus Theorems (Started)
- Monday, Feb. 7 — We are in the midst of the proof on p. 36: “The polar triangle of a polar triangle is the original triangle.” Then we have to completely digest the “The Polar Duality Theorem” on p. 38. It is, after all, hailed by our textbook’s author as possibly “the most important theorem of this book.”
- Problem Set 4 due Thursday, Feb. 10: Problems #13, and 14 on page 41, and a better proof of the Theorem on page 45 — Problem Set 4 Solution
- Thursday, Feb. 10 — Reading preparation for class: Start Chapter 3, pp. 42-47 right before Lemma A. Prepare a thorough proof of Menelaus’s Plane Theorem on page 45 to present in class. As noted above, you will turn in your proof of the Theorem on page 45.
Week 6 — Menelaus Theorems (Cont’d) — Term 4 Exam
- Monday, Feb. 14 — Reading preparation for class: continue reading Chapter 3, pp. 46-49, through Menelaus’s Theorem B. In the next class we will ask volunteers to provide proofs of: Lemma A (p. 47), Lemma B (p. 48), Menelaus’s Theorem A (p. 48), and Menelaus’s Theorem B (p. 49)
- Thursday, Feb. 17 — Term 4 Exam Covering Chapters 1 and 2 of Van Brummelen — Term 4 Exam Solution
Week 7 — Menelaus Theorems Applied to the Position of the Sun
- Monday, Feb. 21 — Apply Menelaus’s Theorems to the position of the Sun — Position of the Sun Primer containing results we will need and step-by-step directions for what we need to draw on the Lenart Sphere — Position of the Sun Results — Derive the declination, δ, of the Sun from its ecliptic longitude, λ, and the obliquity of the ecliptic, ε — Derive the right ascension, α, of the Sun from its ecliptic longitude, λ, and the obliquity of the ecliptic, ε — Practice using these results: plug in λ=60° and ε=23.4° and compare with the measured values on your spheres
- Problem Set 5 due Thursday, Feb. 24 — Problem Set 5 Solution
