MONT 104Q – Mathematical Journeys Midterm Exam, November 2, 2015 Your Name:

Directions

Do all work on the sheets provided (if you use the back of a sheet, please place a note telling me to look there). There is an extra sheet of scratch paper at the end. There are 100 possible points, distributed as indicated in the questions.

I. (A) (10) G. H. Hardy included two theorems and proofs from Euclid’s Elements as prime

examples of “real, serious, beautiful mathematics” in his book A Mathematician’s Apology. Give the statements of those two theorems. You do not need to supply the proofs for these.

First theorem:

Second theorem:

(B) (5) Hardy says “the mathematics which has permanent aesthetic value ... may continue to cause intense emotional satisfaction to thousands of people after thousands of years.” About how much time (to within 100 years) separates the time of Euclid from Hardy’s own time?

(A) (10) State the 5 Common Notions (Axioms) and 5 Postulates at the start of Book I of the Elements of Euclid.

(B) (5) What is the purpose of the statements from part A in Euclid’s logical scheme? How are the Common Notions different from the Postulates?

III. (10) Proposition 1 in Book I of Euclid’s Elements is a construction for an equilateral triangle with side equal to any given line segment AB. Give the construction and the proof that the construction is valid, with justifications for each of the steps in the proof based on your answers to question II.

IV. Proposition 47 in Book I of the Elements is a famous statement from geometry. A) (10) Give the statement in Euclid’s form and the usual name of this result.

B) (15) Outline Euclid’s proof, indicating where he is using previously proved results from Book I (but it’s not necessary to identify the previous Propositions by number).

IV. Essay. (35) George G. Joseph, the author of an interesting book about the history of mathematics called The Crest of the Peacock, offered this overall evaluation of the ultimate impact of Greek geometry: “There is no denying that the Greek approach to mathematics produced remarkable results, but it also hampered the subsequent development of the subject. ... Great minds such as Pythagoras, Euclid, and Apollonius spent much of their time creating what were essentially abstract idealized constructs; how they arrived at a conclusion was in some way more important than any practical significance.” First, what does the last sentence mean? What is Joseph getting at? Then, based on what he said in A Mathematicians Apology, how would G.H. Hardy respond to Joseph? Finally, which side of this debate do you come down on personally? Should all the mathematics we learn and do have practical usefulness or significance?

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