From Euclid to Spigotry #2
Ruler and Compass
Last modified: September 15, 2004
References:
Beckmann, A History of Pi, Chapter 4
Bold, Chapters 1 and 2
Euclid, Elements, Book I (on the Web at aleph0.clarku.edu/~djoyce/java/elements/toc.html
Web pages that illustrate the constructions for most of these topics are available on the course Web site. A direct link is http://140.247.147.177/euclid/index.html
Lecture topic 1: Bisection constructions of Euclid
Demonstrate Euclid's constructions for the following:
a. Constructing an equilateral triangle (Book I, Proposition 1)
b. Bisecting a line segment (Book I, Proposition 10)
c. Bisecting an angle (Book I, Proposition 9) 
Lecture topic 2: Constructing congruent segments
Most of us, if asked to construct a line segment with the same length as a given
segment, will use a ruler or will use our compass to "transfer" the length of the given segment to a new location. Remarkably, all of Euclid's constructions can be accomplished even with a "collapsing compass" that can be used only to draw a circle with a given center and a given point on its circumference. Present Book 1, proposition 3, to show how to "transfer a length."
Lecture topic 3: Constructing congruent angles
Just as Euclid did not use a ruler when transferring a length, he did not use a protractor when transferring an angle. Present Book 1, proposition 23, which shows how to make a copy of a specifed angle at a given point on a line. 
Lecture topic 4: Euclid's construction of a parallel line
Euclid is famous for his postulate that given a line and a point P, there is a unique
line through P that is parallel to the given line.  Present his construction of this line,
as described in Book I, Proposition 31.  
Lecture topic 5: Addition of lengths
a. State Euclid, Book I, proposition 34, in modern terms and draw a diagram to illustrate it. Show how you, if you regard construction of a parallel line as a primitive operation, you can use this proposition to make segment AC congruent to OB just by drawing parallel lines.  This is that approach of "affine geometry," in which "parallelism" is a valid concept but "circle" is not. The diagram below illustrates the construction.
b. Following the approach on page 2 of Bold, show how to construct segments of
length a+b and a-b from given segments of length a and b.  This is easier, but it requires the use of a compass to transfer length (proved in lecture topic 2).

Lecture topic 6: Multiplication of lengths
a. State Euclid, Book VI, proposition 2, and draw a diagram to illustrate it.
b. Following the approach on page 2 of Bold, show how to construct segments of
length ab and a/b from given segments of length a, b, and 1.  

Lecture topic 7: Square roots with lengths
Following the approach on page 2 of Bold, show how to construct a segment of
length   from given segments of length a and 1.  Using similar triangles, prove that the construction for  is correct. You may assume Euclid's theorem that an angle inscribed in a right triangle (LPM in the diagram at the bottom of page 2 of Bold) is a right angle.
Lecture topic 8: Solving a quadratic equation by geometry
a. Given that a quadratic equation x2 -ax + b = 0 has two real roots,  show that a2 > 4b.
b. Given a quadratic equation x2 -ax + b = 0 with real roots, show how to use straightedge and compass to construct line segments whose x-coordinates equal the roots of the equation, and prove algebraically that the construction is correct.  You may assume that a point with coordinates (a, b) is available, as is a line segment of unit length.  (Equivalently, you can use a sheet of graph paper.) The construction is on p. 4 of Bold and on the Web site.
Lecture topic 9: Solving a quadratic with roots of opposite sign:
The method presented in topic 8 works even if b is a negative number, but until the Renaissance, numbers were assumed to be positive. So here is a ruler-and-compass construction, appropriate for solving a quadratic equation with one positive root and one negative root, that is expressed in terms only of positive numbers.  This is useful for constructing regular polygons with 5, 17, and 13 sides.
Suppose that the equation x2 = ax + c  has solutions u and -v.
a. Show that u-v = a and that uv = c.
b. Show that a circle centered at x = a/2, y=0, that passes through the point x = 0, y = ,
satisfies the equation x2 + y2 = ax + c.
c. Prove the following construction correct:
	Construct a circle that passes through x = 1, y = 0 and x = -c, y =0. Label
	its point of intersection with the y-axis K.
	Construct a circle, centered at x = a/2, y=0, that passes through the point K.
	The second circle intersects the x axis at the roots x = u and x = -v of x2 = ax + c.