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MARCH ’10
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MATH 28A — LECTURE 25 — OUTLINE NOTES
§6. The problems of antiquity— We discussed a little history about Euclid of Alexandria, and we
stated his postulates. It is given:
• To draw a straight line from any point to any point.
• To produce a finite straight line continuously in a straight line.
• To describe a circle with any center and radius.
• That all right angles are equal to one another.
• 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.
In particular, we are able to construct four kinds of things: points, line segments, lines and circles.
The three great problems of antiquity are:
• Given a circle, to construct a rectangle with the same area.
• Given two lines which meet at some angle (θ, say), to construct two more lines which meet at an
angle θ/3.
• Given a pair of line segments, say s0 and s3 , to construct two more line segments s1 and s2 such that
all the ratios:
len(s0 ) : len(s1 ),
len(s1 ) : len(s2 ),
len(s2 ) : len(s3 ),
and
len(s3 ) : len(s4 )
are equal. (Here len(s1 ) refers to the length of s1 , and so on...)
We then talked about Descartes, who invented a cunning technique for turning problems in geometry
into problems in algebra: the Cartesian plane. Each point in the plane is labelled with a pair of (real)
numbers—the x coordinate, and the y coordinate. Having labelled points, we label line segments by giving
the two points at the two ends of the line segment (the order in which we give the points doesn’t matter).
Lines are a little more tricky: we represent those as equations: for any given line, we can find numbers
a, b, c such that the points of the line are precisely the points with coordinates (x, y) satisfying the equation
ax + by + c = 0.
We regard the numbers a, b, c as being like ’coordinates’ for the line, but we have to remember that if we
multiply all three of them by the same number, we are talking about the same line. (We also have to keep
at least one of a and b nonzero, otherwise we’re not talking about much of a line.)