Edwin O'Shea

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Description, Syllabus and links to Euclid, for Math 103, Fall 2014

This course is concerned with re-introducing what mathematics is, how mathematicians think and why we should all care about that. In particular, we will study Euclid's Elements, a (if not the) most influential and important of mathematical texts.
A detailed syllabus is here.
This beautiful Euclid's Elements in Color by Oliver Byrne was published in 1847. It bankrupted the publisher.
Here is an online copy of the first few pages of the text, for those of you still waiting for your text in the post.
Here is an online copy of Elements.

Homework

For 8/27: Read Dudley's letter and the excerpt of Lockhart's Lament. With Dudley's letter in mind, summarize the text of the Declaration of Independence. Finally, for fun, watch this.
For 8/29: Re-read and summarize Lockhart's Lament. Come to class with some things from Lockhart that you found surprising and/or you agree/disagree with. Then read Lincoln's Letter to Pierce(1859) and Old-fashioned nationalism. The latter is a challenging read so be sure to leave sufficient time and space to soak it in.
Note on handing in HW: Rather than worrying about typing things up, let me propose instead that you keep a homework journal. If you find yourself worried about ``participation points'' because I sometimes don't see you then see me after class with your HW journal (which you did not write in during class). A "black marble" journal would be perfect for this purpose.
For 9/1: Re-read Lockhart once again. Read, understand ad memorize the 23 definitions of Book 1 of The Elements. In what sense do you think these definitions will be analagous to "life, liberty and the pursuit of happiness"? Bring your compass and straight edge to class for a nose-point on Monday, or on Wednesday at the very latest.
For 9/3: Read and understand the definitions, postulates and common notions of Book I. Pay special attention to writing all descriptions in your own words. Say, for example, what the postulates enable one to do with a straight edge and compass. Also, Common Notion 1 can be read as "If A=B and B=C then A=C." Try to do similarly for the other common notions.
For 9/5: Read and understand Propositions 1,2 and 3 of Book I. Herein, these will be written in the style [I.1], [I.2] and [I.3]. Describe a condition that would enable us to resolve the difficulty of [I.1].
Don't forget: first short test on Friday 9/5, on everything covered so far.
For 9/8: Read and understand Propositions 1,2 and 3 of Book I. Herein, these will be written in the style [I.1], [I.2] and [I.3]. Describe a condition that would enable us to resolve the difficulty of [I.1], that of deciding when two circles cut each other.
For 9/10: Read and understand [I.1]--[I.4] inclusive. Read and understand the statements only of [I.5] and [I.6]. Is [I.4] a "construction" like [I.1], [I.2] and [I.3]? Is there something suspicious about [I.4], in that it is a geometric statement but seems to use none of the geometric postulates? How are [I.5] and [I.6] related to each other?
For 9/12: For Friday's test you are expected to know (with justification) how [I.1], [I.2] and [I.3] work. You are also also expected to say how [I.4] is a statement about congruence of triangles and be able to make a statement with pictures about the statements only (but not the proofs) of [I.5] and [I.6]. There might be questions related to all propositions, similar to how we discussed the circle intersection axiom in class.
For 9/15: Read and understand [I.5]--[I.8] inclusive. Read and understand the statements only of [I.9]--[I.12] inclusive. Note that we presented a different proof of [I.5] in class the last day. Is it surprising that one can prove the truth of a claim in two distinct ways? How do you relate the proofs of [I.6] and [I.7] to be similar to the way in which the Supreme Court rules something to be unconstitutional? How is the statement of [I.8] similar to that of [I.4]?
For 9/22: After the great weeklong discussion on axiomatics, please read [I.6],[I.7] and [I.8] as discussed above.
For 9/24: Read and understand [I.9]--[I.12] inclusive.
For 9/26: Test 3 on [I.5]--[I.10] inclusive.
For 9/29: Read and understand [I.11]--[I.15] inclusive. For [I.13] and [I.14], you must know only the statements. It effectively verifies that the well worn statement that a straight line has an ``angle'' of two right angles.
For 10/1: [I.16]--[I.19] inclusive. Note for [I.16}: We are not yet assuming that the sum of the three angles in a triangle equal two right angles. [I.18 and 19] establish that big angles are opposite big sides, small opposite small, etc.
For 10/3: [I.20]--[I.23] inclusive. Why is the "triangle inequality" a good name for [I.20]? Note that [I.22], given three lengths, tells you how to construct the triangle with those lengths. How is that [I.22] could be considered a converse to [I.20]?
For 10/6: [I.23] and [I.24]. How does [I.23] confirm our suspicion that Euclid does not really believe in motion, like argued in [I.4] (the SAS axiom)?
For 10/8: [I.25] and [I.26]. How does [I.26] fit with the other congruence theorems, and how is it different? Note that the proof of [I.26] has two seperate cases: that when the two angles sandwich the side and when they do not.
For 10/10: Test 4 is today, on all topics covered since last test. That is, [I.11]--[I.26] with related discussion from class. This test covers more than previous tests so while it will have the same format it will have more questions and will be longer than previous tests; 35 minutes, more or less.
After the test I will talk a little about the fifth postulate, which now makes a star return...
For 10/13: Test 4 was pushed forward to today.
For 10/15: Read [I.27]--[I.29] inclusive. How does [I.27] confirm Defn. 23 as a sensible definition and how does it compare to Post. 5? Explain why [I.27] is a converse to Post. 5. With the equator as the third line, how do these propositions hold/not hold on the lines of longtitude on the globe? Read the first four pages (not including the cover page) of History of the parallel postulate and answer the following questions:
- Can you make sense out of Aristotle's definition of axiom? (and who was Aristotle?)
- Indeed, why is it surprising in the context of reaing Euclid that [I.27] should hold but its converse does not?
- Why is it believable, from reading Euclid, that he did not really believe in the postulate (or, at least did not want to believe in it as an axiom, rather than a proposition?).
- In what sense was Gauss afraid to publish his results? (who was Gauss - you might want to look him up!)
For 10/17: No class today.
For 10/20: Read [I.30]--[I.32] inclusive. How do these propositions hold/not hold on the lines of longtitude on the globe? (Disclaimer: I am obligated too to point out at this point that Postulates 2 and 3 do not hold on the sphere but I am trying to convey a sense that it is possible to have surfaces upon which some of the Euclidean postulates hold and others do not, at least do not hold in a general sense)
For 10/22: Read [I.33]--[I.35] inclusive. Note that in the discussion of [I.35] equals now refers to equal areas, not congruency as before. Euclid implicitly equates two areas if they can be both subdivided into matching congruent regions.
For 10/24: Test 5 is today. All topics since last day are fair game, more or less, [I.27]--[I.35] and an understanding of the fifth postulate.
For 10/27: Read [I.36]--[I.38] inclusive.
For 10/29: Read thoroughly and understand [I.39] and [I.46]. Read and understand the statements only of [I.40]--[I.45] inclusive. Purely for pleasure, read [I.47] and [I.48]. Are these last two familiar?
For 10/31: Booo! Read thoroughly and understand [I.47] and [I.48]. Don't forget: There will be a test on Friday 11/21!
For 11/3: Class unexpectedly cancelled.
For 11/5: Again, read thoroughly and understand [I.47] and [I.48].
For 11/7: Test 6 on all topics covered since Test 5.
For 11/10:
For 11/12: [III.1], [III.2], [III.12] and [III.20]. Note that these are all from Book III.
For 11/14: [III.20] and argue that [III.22] and [III.31] follows immediately from [III.20]. Read and understand [VI.1] and [VI.2] too.
For 11/17: First, thanks to all who contributed to the very insightful discussion on Friday.
We'll read parts of this document for the next week. Please bring a printed out copy to class! For what remains, we will discuss various consequences of Thales' Theorem on similar triangles. Chief among them will be line trisection, another proof of the Pyhtagorean Theorem and making sense out of (and proving that) pi (is a constant).
- Look at Section 2.7 for a discussion on [III.20].
- Figure 1.9 and the related discussion is essentially the statement of [VI.1] and [VI.2]. Read and understand Section 2.6 and exercise 1.4.1. This is Thales Theorem.
- Look at [I.9] and [I.10] again, on angle and line bisection. In what sense are they really just the same thing?
- With bisection in mind, read Pages 8-9, starting at "Dividing a line segment..." Using that, trisect (that is, cut into three equal parts) a line segment. Next, understand exercise 1.3.5. Are line and angle trisection effectively the same?
- In class, I will show how to trisect an angle with (whisper it) a marked ruler!
For 11/19: We're continuing with this document
- Read and understand Pages 15 and 16. Give a statement about something being "impossible."
- Read and understand all of Section 2.8.
- In a tip of the hat to Friday's discussion, watch this video on why pi is a constant. This is an alternative (and very nice) proof of [XII.2].
For 11/21: Test 7, on all topics covered since Test 6.
Last week of class: Review week for finals.
Finals week: Offical Exam Schedule is posted here.
As I read it, both sections have their exams on Wednesday, 12/10. Both exams are held in Roop 212, our usual classroom. Section 5 at 10.30-12.30, Section 6 at 1.00-3.00.

The final exam will be comprehensive and will cover all topics in the course. The format will be very similar to the in class tests, with the same ratio of questions to time as the tests.