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Euclid’s Geometry
• Definitions
• Postulates
• Common Notions
Math 362 Fall 2006
1
Euclid’s Definitions
• Def. 1.1. A point is that which has no part.
Def. 1.2. A line is a breadthless length.
Def. 1.3. The extremities of lines are
points.
Def. 1.4. A straight line lies equally with
respect to the points on itself.
Math 362 Fall 2006
2
Euclid’s Postulates
•
Let the following be postulated
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a
straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines
make the interior angle 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 two right angles.
Math 362 Fall 2006
3
Euclid’s Common Notions
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Things which are equal to the same thing
are also equal to one another.
If equals be added to equals, the wholes
are equal.
If equals be subtracted from equals, the
remainders are equal.
Things which coincide with one another
are equal to one another.
The whole is greater than the part.
Math 362 Fall 2006
4
Some Problems
• Definitions:
– Euclid never uses the definitions or even
refers to them later in the text.
– Some concepts are never defined, but are
used in proofs and theorems. One example is
what it means for one point to be between
two others.
Math 362 Fall 2006
5