MATH380:433Assignment
MATH380/433 Assignment
Ancient Greece
Submission date: Monday 13th September.
1. Thales’ fifth proposition says that every angle inscribed in a semicircle is a right angle.
Prove this proposition.
(Your proof should be aimed at someone who has done high school mathematics. If
you’re not sure that you have pitched your explanation at the right level, try it out on
one of your class mates, modifying it if necessary.)
2. Give a brief history of the problem of trisecting an angle.
Show that the construction we did in class (see Clemency’s slides 45-46 and the
video of the lecture) does indeed trisect the given angle.
Carry out (and draw) the construction in the case where the given angle is 45 degrees.
(You will need to ‘make’ a marked ruler for this.)
3. Give a brief overview of Proposition 6 from Book II (that is, Book 2) of Euclid’s
Elements.
You should explain what the statement of the proposition actually means (a labelled
diagram might help), and indicate briefly why it is true (you don’t need to go into the
same level of detail as Euclid does).
Find an algebraic equation that is ‘equivalent’ to this proposition.
Give a brief account of the ‘Geometric Algebra’ debate. (Use Proposition 6, or
Proposition 4 that we discussed in class, to illustrate the different points of view.)
Is Book II of the Elements really just algebra disguised as geometry?
4. Proposition IX, 20 of Euclid’s Elements says that ‘Prime numbers are more than any
assigned multitude of prime numbers.’ However, his proof just shows that, if you give
him three prime numbers, then he can find a fourth one.
Modify his proof so that, if someone gives you any list of prime numbers, your proof
shows how to find a prime number not in their list. (You can use modern algebraic
symbols if you like, or you can extra details to Euclid’s ‘wordy’ proof.)
Use your proof to find a prime number not in the following list: 2,3,5,7.
What prime(s) does Euclid’s proof give if you start with the list: 3,5,17?
5. Euclid’s wording of the statement for Proposition IX, 20 (above) could be seen as a
way of avoiding any mention of ‘infinity’ — so, for example, he does not say ‘there are
infinitely many prime numbers’.
Give an example – say, from one of the mathematics courses you have done – where
infinity ( ) is mentioned, and show how to reword your example so that infinity is not
mentioned.
∞