Heavenly Mathematics Daily Schedule Term 5
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See also Daily Schedule Term 4
Longitude: An Entirely Optional Light Read
For some historical context, consider reading
Longitude by Dava Sobel. It is a
light read, and a bit one-dimensional. However it is impressive and scary to put yourself into a world of sailing ships, often immersed in fog,
whose captains and crews had no idea how far east or west they were. (North and south — e.g., their latitude — was comparatively easy to determine
whenever the sky cleared.) This frightful era of navigation was brought to an end by John Harrison’s perfection of time-keeping at sea with his H4 sea watch which he completed in 1759. It was his fourth design, and the culmination of a lifetime
of dedication to the problem of determining longitude at sea.
Skipping Chapter 4 (The Medieval Approach)
Chapter 4 of Van Brummelen is a short chapter, and it would not take us a great deal of time to work through it, but have no spare time, so we will
be proceeding directly to Chapter 5 in Week 8. Chapter 4 also contains the Medieval method for the determination of the direction of Mecca, which
we advertised on course night as a fundamental and interesting problem. As a similar problem, imagine you are in Vancouver, BC. Which direction
should your plane fly to get to Paris? Is it east? After all, Vancouver and Paris are less than ½° apart in latitude, so going east on
a line of latitude will almost exactly take you there. However, a line of latitude is not a straight line on a sphere (except for the Equator,
which is), and if you look at a globe (not a flat map), it will be obvious that following the line of latitude isn’t the shortest route. Anyway,
if you are interested in the Medieval approach, you will have to work through Chapter 4 on your own.
Week 8 — The Modern Approach
- Preparation for Monday, March 14 — Please remind yourselves of the Menelaus Theorem’s A and B, and how we use them to determine the sun’s right ascension, declination coordinates. Also, please read through pp. 73-79 in the text at least once. We will re-read these pages again later in the week, so do what you can to take notes and generate questions upon a first reading.
- Monday, March 14 — We derived most of the identities for a right triangle on the sphere starting with the Menelaus theorem — Complete the derivations carefully and exhaustively in Problem Set 6 due Thursday
- Reading for Thursday: there is an argument on pages 77-78 (it’s indented) that proves identity I.2. It is called “Geber’s Theorem.” Work through that argument, and be prepared to present it and its variants in class on Thursday.
- Thursday, March 17 — We surveyed the course for the Humanities Chair applicant — We critiqued the course pace and direction — We derived identity I.2 using Geber’s Theorem — We used algebra to derive the same identity from the seven you derived in Problem 2 of Problem Set 6 — Problem Set 6 Solution contains a discussion of the grand synthesis that we have achieved as well as solutions to the three problems
Week 9 — A Series of Applications of the Modern Approach
- Preparation for Monday, March 21: Read text pp. 79-81 (up to the last paragraph) and pp. 51-55 from Chapter 3. We will be working with the figure on p. 53 on Monday, so it will be important to pay attention to all the quantities in that figure, and the steps needed to calculate them (steps 1-4 on page 54). Whereas those steps are completed using Menelaus’s theorems in the text, we will use various of our ten identities to do so on Monday, greatly simplifying the work. So you are welcome to closely follow the Menelaus arguments of those steps in the text or simply skim them; the main thing to note is just what quantity is being solved for at each step. Also, bring a calculator/phone/laptop to Monday’s class. We will be calculating values using trig and inverse trig functions. For practice, verify the calculations on page 81 as you read them in the text.
- Monday, March 21 — We used modern methods to solve the problem of rising times which Van Brummelen solved using medieval methods on pp. 53-55 — A rising times diagram, a rising times photo, and a rising times movie summarizing our methods
- Preparation for Thursday, March 24: Problem Set 7
- Thursday, March 24 — We found the time above the horizon for any star with RA=α and DEC=δ — We found the direction you should fly (if you want to take the shortest route) when going from Vancouver International Airport (in British Columbia) to Charles de Gaulle International Airport (near Paris) which both have latitude 49°, but are separated by 126° of longitude — It is not due east! — And finally, the pièce de résistance: we found the direction of Mecca from Cairo — Problem Set 7 Solution
Week 10 — Revisiting Star Charts — Further Applications of the Modern Approach
Week 11 — From Right Triangles to Oblique Triangles — Law of Sines on the Sphere
Context for Chapter 6 on Oblique Triangles
Our ten identities only apply to right triangles. Therefore finding the direction of Mecca (a problem for which no right triangle is immediately apparent) required us to craftily construct two right triangles (one which involved the “modified longitude” and another which involved the “modified longitude”) and making three crafty applications of the ten identities to these two triangles. We can do some more theoretical work and derive some identities for non-right triangles! These are called oblique triangles. Problems will fall more quickly with these new identities in hand.
- Preparation for Monday, April 4: Read the rest of Chapter 5, pp. 82-91. Do your best to understand Napier’s Pentagramma Mirificum on p. 90
- Monday, April 4 — We constructed the Pentagramma Mirificum on our spheres — We used the Pentagramma Mirificum to derive ten formulas from two — The Pentagramma Mirificum is a formula quintupler
- Problem Set 8 for Thursday, April 7 — Four out of problems #1-5 from the Problems Begun in class March 31 and add Problems #6 and #8 on p. 92 — A Ceylon to Madagascar solution
- Thursday, April 7 — The Locality Principle — The Law of Sines in the plane — The Law of Cosines in the plane — The Law of Sines on the sphere — Problem Set 8 Solution
Week 12 — Oblique Triangles (Continued) — Law of Cosines on the Sphere
- Preparation for Monday, April 11: Read pp. 94-98 from Chapter 6 — Come prepared to either knock out the proof of the Law of Cosines on the sphere or with specific questions about it — Once we have the second of our new tools (the Law of Cosines on the sphere) we will proceed to examples
- Monday, April 11 — Issues with Inverse Trig Functions — The Law of Cosines on the sphere — Using the Principle of Locality to recover the Law of Cosines in the plane
- Preparation for Thursday, April 14: (1) Read the rest of Chapter 6, pp. 98-106. You need to read this closely. You will be responsible for the material, and we will not review it together in class. Questions from the reading can be brought to either of us outside of class, including in the Wednesday night study session 7:30-9:00. (2) Create a one-page sheet for yourself of all the spherical trig identities from the course. This includes the ten fundamental identities on p. 79, the Laws of Sines and Cosines, and Delambre’s and Napier’s analogies (these are found in the new reading above), and anything else that might be considered a formula for the course. You are also welcome to include any planar trig identities from Chapter 1. Keep this sheet — you will use it in class on Thursday and on the Term 5 exam next Thursday.
- Thursday, April 14 — Applications of the Law of Sines on the sphere and the Law of Cosines on the sphere, beginning with a new derivation of the direction of Mecca — Problem Set 9 - Oblique Triangles, begun in class
Week 13 — Review for Term 5 Exam
Week 14 — Celestial Navigation
