Why Studying Euclidean Geometry?

• It is well known that most students find it difficult to learn to do proofs. Geometry, especially Euclid, provides an excellent setting for students to improve their proof skills.

• Secondary school teachers are going to be teaching Euclidean geometry so they need to know it.

• We believe that a student who does not have a good background in Euclidean geometry is not in a position to get the point of non-Euclidean geometry.

Significances of the discovery of non-Euclidean Geometry

• Most of people are unaware that around a century and a half ago a revolution took place in the field of geometry that was scientifically profound as the Copernican revolution in astronomy and, in its impact, as philosophically important as the Darwinian theory of evolution.

• The effect of the discovery of hyperbolic geometry on our ideas of truth and reality has been so profound, writes the great Canadian geometer H.S.M. Coxeter, “that we can hardly imagine how shocking the possibility of a geometry different from Euclid’s must have seemed in 1820.”

• Albert Einstein stated that without this new conception of

geometry, he would not have been able to develop the theory of relativity.

• Euclidean geometry is the kind of geometry you learned in high school, the geometry most of us use to visualize the physical universe.

• It comes from the text by Greek mathematician Euclid, the Elements, written around 300 B.C.

• Our picture of the physical universe based on this geometry was painted largely by Isaac Newton in the late seventeenth century.

• Geometries that differ from Euclid’s own arose out of a deeper study of parallelism.

• Consider this diagram of two rays perpendicular to segment PQ:

P

Q

• In Euclidean geometry the perpendicular distance between the rays remains equal to the distance from P to Q as we move to the right.

• However, in the early nineteenth century two alternative geometries were proposed. In hyperbolic geometry the distance between the rays increases.

• In elliptic geometry the distance decreases and the rays eventually meet.

• These non-Euclidean geometries were later incorporated in a much more general geometry developed by C.F. Gauss and G. F. B. Riemann.

• Now, we continue with a brief history of geometry in ancient times, and emphasize the development of the axiomatic method by the Greeks. And finally we will present Euclid’s five postulates.

A Quick Review of the History of Euclidean Geometry

Plato 427-347 BCE Let no one ignorant of geometry enter this door. Entrance to Plato’s Academy

Euclidean 325-265 BCE

• The word “geometry” comes from the Greek geometrien (geo: earth, and metrein: to measure); geometry was originally the science of measuring land.

• The Greek historian Herolotus (5th century B.C.) credits Egyptian surveyors with having originated the subject of geometry, but other ancient civilizations (Babylonian, Hindu, Chinese) also possessed much geometric information.

• Egyptian geometry was not a science in the Greek sense, only a grab bag of rules for calculation without any motivation or justification.

• The Babylonians were much more advanced than the Egyptians in arithmetic and algebra. Moreover, they knew the Pythagorean theorem – in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs – long before Pythagoras was born.

• However, the Greeks, beginning with Thales of Milete, insisted that geometric statements be established by deductive reasoning rather than that by trial and error.

• The orderly development of theorems by proof was characteristic of Greek mathematics and entirely new.

• The systematization begun by Thales was continued over the next two centuries by Pythagoras and his disciples.

• Pythagoras was regarded as a religious prophet. The Pythagoreans differed from other religious sects in their belief that elevation of the soul and union with God are achieved by the study of music and mathematics.

• The Pythagoreans were greatly shocked when they discovered irrational lengths, such as . 2

Pythagoras of Samos 569BC - 475BC

Thales of Miletus 624BC - 547BC

• Later, Plato repeatedly cited the proof for the irrationality of the length of a diagonal of the unit square as an illustration of the method of indirect proof.

• The point is that this irrationality of length could never have been discovered by physical measurements, which always include a small experimental margin of error.

1862 - 1943

• Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published Grundlagen der Geometrie in 1899 putting geometry in a formal axiomatic setting. The book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics which has been one of the major characteristics of the subject throughout the 20th century.

• Hilbert's famous 23 Paris problems challenged (and still today challenge) mathematicians to solve fundamental questions. Hilbert's famous speech The Problems of Mathematics was delivered to the Second International Congress of Mathematicians in Paris. It was a speech full of optimism for mathematics in the coming century and he felt that open problems were the sign of vitality in the subject.

We must know, we shall know

• A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin,P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr,I. Langmuir, M. Planck, Mme. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson

Chapter 0 : Notations and Conventions

• We will introduce our basic terminology for geometric objects and the relations between them.

• Our main concern is the theorems of Euclidean Geometry rather than the abstract construction of this geometry as an axiom system.

• Points : will be denoted by capital letters A,B,…

• Lines : the line containing A and B will be denoted by ; however sometimes we will use script letters like ,…

• Three or more points that lie on the same line are said to be collinear.

• Three or more lines which meet at the same point are said to be concurrent at that point.

• A line divides the plane into two half planes, each of which may be described by specifying the line and any one point in the half plane.

AB

• Rays (half lines) : The ray with initial point A and containing point B will be denoted .

• Line Segments : The line segment with end points A and B will be denoted ; and the length of will be denoted AB.

• A point P on that is distinct from A and B is called an interior point of . P is the midpoint of if .

• Angles : The angle with vertex A and sides and will be denoted or . The same symbols are also used to denote the sizes of angles.

AB

AB AB

ABAB PBPA

AB ACA BAC

AB

• We will use degrees rather than radians for angle measurements.

• A right angle is one whose measurement is ; an acute angle has measure less than ; while an obtuse angle has measure greater than .

• If then these angles are said to be

complements of each other, whereas they are called supplements of each other if their sum is .

• The interior of is the set of points common to the half plane on the C side of and the A side of , excluding the points on and .

90

90

90

90 BA

180

ABCAB

BCBABC

• When P is interior to , then is a bisector of this angle if .

• Congruence : Two line segments of the same length are said to be congruent. is congruent to is shown by

• Note that is not the same as , which means that and are the same line segment.

• We also say that two angles are congruent if they have the same size. In this case and mean the same thing.

PBCABP BPABC

AB CD

CDAB CDAB

AB CD

BA BA

CDAB

• Triangles : The triangle with vertices A, B, and C is denoted .

• An equilateral triangle has all three sides the same length. An obtuse triangle has an obtuse angle at some vertex, where as an acute triangle has no obtuse or right angles.

• A right triangle has a angle. The side opposite this angle is called the hypotenuse.

ABC

90

The Axiomatic Method

• Mathematicians can make use of trial and error, computation of special cases, inspired guessing, or any other way to discover theorems. The axiomatic method is a method of proving that results are correct.

• So proofs give us assurance that results are correct. In many cases they also give us more general results.

• For example, the Egyptians and Hindus knew by experiment that if a triangle has sides of lengths 3, 4, and 5, it is a right triangle.

• But the Greeks proved that if a triangle has sides of lengths a, b, and c, and if , then the triangle is a right triangle.

222 cba

• It would take an infinite number of experiments to check this result (and, besides, experiments only measure things approximately).

• There are two requirements that must be met for us to agree that a proof is correct:

• REQUIREMENT 1. Acceptance of certain statements called “axioms,”, or “postulates” without further justification.

• REQUIREMENT 2. Agreement on how and when one statement “follows logically” from another, i.e., agreement on certain rules of reasoning.

• Euclid’s monumental achievement was to single out a few simple postulates, statements that were acceptable without further justification, and then to deduce from them 465 propositions, many complicated and not at all intuitively obvious, which contained all the geometric knowledge of his time.

• One reason the Elements is such a beautiful work is that so much has been deduced from so little.

• We have been discussing what is required for us to agree that a proof is correct. Here is one requirement that we took for granted:

• REQUIREMENT 0: Mutual understanding of the meaning of the words and symbols used in the discourse.

• If, for example, I define a right angle to be a angle, and then define a angle to be a right angle, I would violate the rule against circular reasoning.

90

90

• Also, we cannot define every term that we use. In order to define one term we must use other terms, and to define these terms we must use still other terms, and so on. If we were not allowed to leave some terms undefined, we would get involved in infinite regress.

• To apply the Axiomatic System Method in geometry, we have to consider the following objects:

1. Undefined Terms or Primaries

2. Defined Terms

3. Axioms or Postulates

4. Logic or Rules of Reasoning

5. Theorems

• Here are the five undefined geometric terms that are the basis for defining all other geometric terms in plane Euclidean geometry:

• Point

• Line

• Lie on (as “two points lie on a unique line”)

• Between (as in “point C is between points A and B”)

• Congruent

A B

C D

lM

• The mentioned list of undefined geometric terms is due to David Hilbert (1862 - 1943). His treatise, The Foundations of Geometry (1899), not only clarified Euclid’s definitions but also filled in the gaps in some of Euclid’s proofs.

• Some other mathematicians who worked to establish rigorous foundations for Euclidean geometry are: G. Peano, M. Pieri, G. Veronese, O. Veblen, G. de B. Robinson, E. V. Huntington, and H. G. Forder.

• These mathematicians used lists of undefined terms different from the one used by Hilbert. Pieri used only two undefined terms (as a result, however, his axioms were more complicated).

Euclid’s First Four Postulates

• Euclid based his geometry on five fundamental assumptions, called axioms or postulates.

• EUCLID’S POSTULATE I. For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.

• EUCLID’S POSTULATE 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 segment CD is congruent to segment BE.

C D

A B E

• 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.

• EUCLID’S POSTULATE IV. All right angles are congruent to each other.

• Euclid’s first postulates have always been readily accepted by mathematicians. The fifth (parallel) postulate, however, was highly controversial until the nineteenth century.

• In fact, consideration of alternatives to Euclid’s parallel postulate resulted in the development of non-Euclidean geometries.

OA

• THE EUCLIDEAN PARALLEL POSTULATE. For every line and for every point P that does not lie on there exists a unique line m through P that is parallel to .

P m

l

• Why should this postulate be so controversial?

• It may seem “obvious” to you, perhaps because you have been conditioned to think in Euclidean terms.

• However, if we consider the axioms of geometry as abstractions from experience, we can see a difference between this postulate and the other four.

• The first two postulates are abstractions from our experiences drawing with a straightedge; the third postulate derives from our experiences drawing with a compass. The fourth postulate also derives from our experiences measuring angles with a protractor.

• The fifth postulate is different in that we cannot verify empirically whether two lines meet, since we can draw only segments, not lines.

Attempts to Prove the Parallel Postulate

• Euclid himself recognized the questionable nature of the parallel postulate, for he postponed using it for as long as he could.

• Remember that an axiom was originally supposed to be so simple and intuitively obvious that no one could doubt its validity.

• From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption.

• For two thousand years mathematicians tried to derive it from the other four postulates or to replace it with another postulate, one more self-evident.

• All attempts to derive it from the first four postulates turned out to be unsuccessful because the so-called proofs always entailed a hidden assumption that was unjustifiable.

• The substitute postulates, purportedly more self-evident, turned out to be logically equivalent to the parallel postulate, so that nothing was gained by the substitution.