Jan 14, 2016
MAT360 Lecture 1Euclid’s geometry
The origins of geometry
Th 10-11pm, Tu 1-3m. All in 4-103 Math Tower.
Correction in class My questions
Homework due Tuesday Sept 18th Chapter 1: Exercises 1,2,3,4, Mayor Exercise
1. To think (but not submit): Exercises 8 and 11,
and how can “straight” be defined.
Homework 0: Due Sept 12th.
A “jump” in the way of thinking geometry Before Greeks:
After Greeks: Statements should be established by deductive methods.Thales (600 BC)Pythagoras (500 BC)Hippocrates (400 BC)Plato (400 BC)Euclid (300 BC)
The axiomatic method
A list of undefined terms. A list of accepted statements (called axioms
or postulates) A list of rules which tell when one statement
follows logically from other. Definition of new words and symbols in term
of the already defined or “accepted” ones.
What are the advantages of the axiomatic method?
What are the advantages of the empirical method?
point, line, lie on, between, congruent.
More about the undefined terms By line we will mean straight line (when we
talk in “everyday” language”)
How can straight be defined?
Straight is that of which the middle is in front of both extremities. (Plato)
A straight line is a line that lies symmetrically with the points on itself. (Euclid)
“Carpenter’s meaning of straight”
Mistake! The slides of the course are here (remove the dot of the previous address). http://www.math.sunysb.edu/~moira/mat360fall07/slides/
Administrative remarks. Grader: Pedro Solorzano Office Hours.
Tu 4-6pm in 2-119 Math Tower Th 4-6pm in MLC. Contact Pedro for problems with grading, but take
into account that he is not allowed to accept overdue homework.
From Class on: The class will take place in CHECK YOUR EMAIL and/or Blackboard.
Check if you receive an email from me next Thursday. If not, I contact me.
Discussion Board in Blackboard.
Euclid’s first postulate
For every point P and every point Q not equal to P there exists a unique line l that passes through P and Q.
Notation: This line will be denoted by
According with the definitions we made, what is wrong in the previous postulate?
More ways to express “line l passes through point P”.
More undefined terms
Set Belonging to a set, being a member of a set.
We will also use some “underfined terms” from set theory (for example, “intersect”, “included”, etc) All these terms can be defined with the above terms (set, being member of a set).
Given two points A and B, then segment AB between A and B is the set whose members are the points A and B and all the points that lie on the line and are between A and B.
Notation: This segment will be denoted by AB
Second Euclid’s postulate
For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.
Let it be granted that a segment may be produced to any length in a straight line.
Give two points O and A, the set of all points P such that the segment OP is congruent to the segment OA is called a circle. The point O is the center of the circle. Each of the segments OP is called a radius of the circle.
Euclid’s postulate III
For every point O and every point A not equal to O there exists a circle with center O and radius OA.
What terms are defined in the previous slide?
Definition of opposite rays.
Definition of angle
We use the notation
for the angle with vertex A defined previously.
Can we use segments instead of rays in the definition of angles?
Is the zero angle (as you know it) included in the previous definition?
Are there any other angles you can think of that are not included in the above definition?
Definition of right angle.
Euclid’s Postulate IV
All right angles are congruent to each other.
Definition of parallel lines
Two lines are parallel if they do not intersect, i.e., if no point lies in both of them.
If l and m are parallel lines we write l || m
Euclidean Parallel Postulate (equivalent formulation) For every line l and for every point P that
does not lie on l there exists a unique line m through P that is parallel to l.
Euclid’s postulates (modern formulation)I. For every point P and every point Q not equal to
P there exists a unique line l that passes for P and Q.
II. For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.
III. For every point O and every point A not equal to O there exists a circle with center O and radius OA
IV. All right angles are congruent to each otherV. For every line l and for every point P that does not
lie on l there exists a unique line m through P that is parallel to l.
Euclid’s postulates (another formulation)Let the following be postulated: 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.
Concrete plane Abstract plane
Midpoint M of a segment AB Triangle ABC, formed by tree noncollinear
points A, B, C Vertices of a triangle ABC. Define a side opposite to a vertex of a
Warning about defining the altitude of a triangle.
Define lines l and m are perpendicular. Given a segment AB. Construct the
perpendicular bisector of AB.
Prove using the postulates that if P and Q are points in the circle OA, then the segment OP is congruent to the segment OQ.
Things which equal the same thing also equal to each other.
Exercise (Euclid’s proposition 1) Given a segment AB. Construct an
equilateral triangle with side AB.
Exercise. Prove the following using the postulates For every line l, there exists a point lying on l For every line l, there exists a point not lying
on l. There exists at least a line. There exists at least a point.
Second Euclid’s postulates: Are they equivalent? For every segment AB and for every segment
CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE.
Any segment can be extended indefinitely in a line.