History of Geometry, Euclid's "Common Understandings", Points Lines & Planes

1. INTRODUCTION TO
GEOMETRY

2. GEOMETRY: A HISTORY
• Said to have been invented over 4,000 years ago by Egyptian pharaoh Sesostris to help
keep track of land ownership and tax its owners
• “Geo” means earth, “metria” means to measure; therefore, Geometry literally means to
measure the Earth
• Greek philosophers built upon the very practical mathematics they learned from the
Egyptians and Babylonians to create a more abstract and general way of thinking of
Earth measures
• The assumptions they developed were minimal which allowed everything else to follow
from these basic assumptions
• Many of the original texts were lost during the fall of the Roman empire, but the
teachings remained prevalent in the Islamic nations.

3. EUCLIDEAN GEOMETRY
• Euclid’s, The Elements, summarized Greek geometry. It is the basis of most
Western mathematics, science, and philosophy
• Euclidean geometry dates back to approximately 400 BC
• Older than algebra and Calculus!
• Many still believe Euclidean geometry is the best introduction to analytic thinking
• We will follow the basic thinking developed by Euclid and attempt to make clear and
distinguish between:
• What we have assumed to be true, and cannot prove
• What follows from what we have previously assumed or proven
• Essentially, we will always question every idea presented.

4. THINK OUTSIDE THE BOX…

6. POINTS, LINES, & PLANES
Unit 1

7. EUCLID’S GEOMETRY
• Euclid’s assumptions are referred to as axioms, postulates, and definitions
• Axioms are very general ideas; postulates and definitions refer to specific ideas
• Definitions are words or terms that have agreed upon meaning; they cannot be
proven or derived
• Major ideas which are proven are called theorems
• Ideas that follow from a theorem are corollaries
• Euclid referred to his five axioms as “Common Understandings”

8. COMMON UNDERSTANDINGS
• Axiom 1: Things that are equal to the same thing are also equal to each other
• Axiom 2: If equals are added to equals, the whole are equal
• Axiom 3: If equals are subtracted from equals, the remainders are equal
• Axiom 4: Things which coincide with one another are equal to one another
• Axiom 5: The whole is greater than the part

9. THE UNDEFINED TERMS
Point
• Simplest figure in Geometry
• Everything else consists of points
• Used for location
• Does not have a “size”
• Represented by a dot
• Labeled with an uppercase non-
cursive letter
Line
• Extends infinitely in two
directions
• Made of an infinite amount of
points
• Does not have “thickness”
• Represented with 2 arrows in
opposite directions
• Labeled with any two points on
the line OR a single lowercase
cursive letter
Plane
• Extends infinitely in all directions
• No edges
• No thickness
• Need at least three points to
create a plane
• Represented with a parallelogram
• Labeled with at least three points
OR a single uppercase cursive
letter

10. MORE IMPORTANT DEFINITIONS
• Collinear
• Points of the same line
• Non-Collinear
• Points not on the same line
• Coplanar
• Figures (points or lines) on the same plane
• Non-Coplanar
• Figures not on the same pane
• Space
• A boundless, three-dimensional set of all points

11. POSTULATES
• 1: Through any two points exists exactly one line
• 2: Through any three non-collinear points exists exactly one plane

12. INTERSECTIONS
• Two Lines
• A Point
• Two Planes
• A Line
• A Lines and a Plane
• A Point (line goes through plane)