Book 1 of the Elements
Euclid's Axioms
Euclid’s axioms: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html
Postulate 1. “To draw a straight line from any point to any point.”
Postulate 2. “To produce a finite straight line continuously in a straight line.”
Postulate 3. “To describe a circle with any center and radius.”
Postulate 4. “That all right angles equal one another.”
Postulate 5. “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
*Notice that the first three postulates (axioms) are essentially Plato's rules for construction.
Proving Euclid's Propositions
Many of Euclid's propositions concern geometric constructions: (a) showing how to complete a construction and then (b) proving that the construction satisfies the specified properties. Use the applet below to (a) complete constructions related to Euclid's propositions, and then use the text box to (b) write up a related proof, explaining how the construction works.
In each case, you should assume only the given postulates and previously proven propositions.
Begin with Euclid's second proposition: "To place a straight line equal to a given straight line with one end at a given point."
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI2.html
Proposition 6
Proposition 6 is the converse of the base angles theorem for isosceles triangles. It says that if two angles in a triangle are congruent then their opposite sides are congruent as well. Read Euclid's argument and explain his approach in your own words.
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI6.html
Propositions 9-12
Propositions 9 through 12 explain how you can complete more advanced constructions, beginning from Plato's rules of construction (using a straightedge and compass). Specifically, they explain how you can bisect a given angle or segment, and how you can construct a line perpendicular to a given line and passing through a given point.
Use the geometry applet to (a) demonstrate each construction and (b) prove that it does what it is supposed to (e.g., bisect an angle).
Proposition 14
Proposition 14 is the first of Euclid's propositions to draw upon the 4th postulate/axiom, that all right angles are equal to each other. Read Euclid's proof of Proposition 14 and explain why he needs the 4th postulate to prove it.
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI6.html
Propositions 15-28
Skim through Propositions 15-28 and try to follow their progression. What major arguments are they building toward?
Proposition 29
Proposition 29 is the first of Euclid's propositions to draw upon the 5th postulate/axiom, "the parallel postulate." Read Euclid's proof of Proposition 29 and explain why he needs the 5th postulate to prove it.
Proposition 32
Proposition 32 states, in part, that the sum of angles in a triangle is π. How does this proposition rely on Proposition 29 and therefore the parallel postulate?
Consider the applet shown below and explain how it might be used to explain the same conjecture without relying on Euclid's postulates or prior propositions.
Propositions 33-46
Skim through Propositions 33-46 and try to following their progression. How are they building toward a proof of the Pythagorean Theorem? What new mental actions are implicitly involved?