In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. The Greek historian Herodotus (5th century B.C.) credits the Egyptians with having originated the subject, but there is much evidence that the Babylonians, the Hindu civilization, and the Chinese knew much of what was passed along to the Egyptians.
The Babylonians of 2,000 to 1,600 B.C knew much about navigation and
astronomy, which required a knowledge of geometry. They also considered
the circumference of the circle to be three times the diameter. Of course, this
would make 
--a small problem. This value for 
carried along to
later times. The Roman architect Vitruvius took . Prior to this it
seems that the Chinese mathematicians had taken the same value for . This
value for was sanctified by the ancient Jewish civilization and
sanctioned in the scriptures. In I Kings 7:23 we find:
He then made the sea of cast metal: it was round in shape, the diameter from rim to rim being ten cubits: it stood five cubits high, and it took a line thirty cubits long to go round it.--The New English BibleRabbi Nehemiah attempted to change the value of
to 
but was
rejected. By 1,800 B.C. the Egyptians, according to the Rhind
papyrus, had the approximation 
. It has been
popularly expressed that in the 1920's the Indiana legislature passed a law
mandating that 
. or even 3. The bill appears to state that it is possible to square the circle. It apparently was tabled and never passed due to the fortuituous chance that a math professor from Purdue happened to be in the Senate that day for another purpose. We assume that it is still tabled, never to be brought up for debate again. In 1789
Johann Lambert proved that is an irrational number, and in 1882 F.
Lindemann proved that is transcendental, i.e., it is not the
solution to any algebraic equation with rational mathfficients.
The ancient knowledge of geometry was passed on to the Greeks. They seemed to be blessed with an inclination toward speculative thinking and the leisure to pursue this inclination. They insisted that geometric statements be established by deductive reasoning rather than trial and error. This began with Thales of Milete. He was familiar with the computations, right or wrong, handed down from Egyptian and Babylonian mathematics. In determining which of the computations were correct, he developed the first logical geometry. This orderly development of theorems by proof was the distinctive characteristic of Greek mathematics and was new.
This new mathematics of Thales was continued over the next two centuries by Pythagoras and his disciples. The Pythagoreans, as a religious sect, believed that the elevation of the soul and union with God were achieved by the study of music and mathematics. Nonetheless, they developed a large body of mathematics by using the deductive method. Their foundation of plane geometry was brought to a conclusion around 400 B.C. in the Elements by the mathematician Hippocrates. This treatise has been lost, but many historians agree that it probably covered most of Books I-IV of Euclid's Elements, which appeared about a century later, circa 300 B.C.
Euclid was a disciple of the Platonic school. Around 300 B.C. he produced the definitive treatment of Greek geometry and number theory in his thirteen-volume Elements. In compiling this masterpiece Euclid built on the experience and achievements of his predecessors in preceding centuries: on the Pythagoreans for Books I-IV, VII, and IX, on Archytas for Book VIII, on Eudoxus for Books V, VI, and XII, and on Theætetus for Books X and XIII. So completely did Euclid's work supersede earlier attempts at presenting geometry that few traces remain of these efforts. It's a pity that Euclid's heirs have not been able to collect royalties on his work, for he is the most widely read author in the history of mankind. His approach to geometry has dominated the teaching of the subject for over two thousand years. Moreover, the axiomatic method used by Euclid is the prototype for all of what we now call pure mathematics. It is pure in the sense of pure thought: no physical experiments need be performed to verify that the statements are correct--only the reasoning in the demonstrations need be checked.
In this treatise, he organized a large body of known mathematics, including discoveries of his own, into the first formal system of mathematics. This formalness was exhibited by the fact that the Elements began with an explicit statement of assumptions called axioms or postulates, together with definitions. The other statements--theorems , lemmæ, corollaries--were then shown to follow logically from these axioms and definitions. Books I-IV, VII, and IX of the work dealt primarily with mathematics which we now classify as geometry, and the entire structure is what we now call Euclidean geometry.
Euclidean geometry was certainly conceived by its creators as an idealization
of physical geometry. The entities of the mathematical system are concepts,
suggested by, or abstracted from, physical experience but differing from
physical entities as an idea of an object differs from the object itself.
However, a
remarkable correlation existed between the two systems. The angle sum of a
mathematical triangle was stated to be 
, if one measured the
angles of a physical triangle the angle sum did indeed seem to be
, and so it went for a multitude of other relations. Because of
this agreement between theory and practice, it is not surprising that many
writers came to think of Euclid's axioms as self evident truths.
Centuries later, the philosopher Immanuel Kant even took the position that the
human mind is essentially Euclidean and can only conceive of space in
Euclidean terms. Thus, almost from its inception, Euclidean geometry had
something of the character of dogma.
Euclid based his geometry on five fundamental assumptions:
that passes through P and Q.
Before we study the Fifth Postulate, let me say a few words about his definitions. Euclid's methods are imperfect by modern standards. He attempted to define everything in terms of a more familiar notion, sometimes creating more confusion than he removed. As an example:
A point is that which has no part. A line is breadthless length. A straight line is a line which lies evenly with the points on itself. A plane angle is the inclination to one another of two lines which meet. When a straight line set upon a straight line makes adjacent angles equal to one another, each of the equal angles is a right angle.Euclid did not define length, distance, inclination, or set upon. Once having made the above definitions, Euclid never used them. He used instead the rules of interaction between the defined objects as set forth in his five postulates and other postulates that he implicitly assumed but did not state.
No one seemed to like this Fifth Postulate, possibly not even Euclid himself--he did not use it until Proposition 29. The reason that this statement seems out of place is that the first 4 postulates seem to follow from experience--try to draw more than one line through 2 different points. The Fifth Postulate is unintuitive. It does come from the study of parallel lines, though. Equivalent to this postulate is :
for our
mathematical system and we can prove that Axiom 
is derivable, or
provable, from the other axioms, then is indeed redundant. In some sense
we are looking for a basis for this mathematical system. Unlike
vector spaces and linear algebra, there is not a unique number of elements in
this basis, for it includes the axioms, definitions, and the rules of
logic that you use.
Many people have tried to prove the Fifth Postulate. The first known attempt to prove Euclid V, as it became known, was by Posidionius(1st century B.C.). He proposed to replaced the definition of parallel lines (those that do not intersect) by defining them as coplanar lines that are everywhere equidistant from one another. It turns out that without Euclid V you cannot prove that such lines exist. It is true that such a statement that parallel lines are equidistant from one another is equivalent to Euclid V.
Ptolemy followed with a proof that used the following assumption:
For every lineWe will show at a later date that this statement is equivalent to Euclid V, and therefore this did not constitute a proof of Euclid V.
Proclus (410-485A.D.) also attempted to prove Euclid V. His argument used a limiting process. He retained all of Euclid's definitions, all of his assumptions except Euclid V, and hence all of his propositions which did not depend on Euclid V. His plan was (1) to prove on this basis that a line which meets one of two parallels also meets the other, and (2) to deduce Euclid V from this proposition. His handling of step (2) was correctly handled. The argument in step (1) runs substantially as follows. Let g and h be parallel lines and let another line k meet h in P. From Q, a point of k situated between g and h, drop a perpendicular to h. As Q recedes indefinitely far from P, its distance QR from h increases and exceeds any value, however great. In particular, QR will exceed the distance between g and h. For some position of Q, then, QR will equal the distance between g and h. When this occurs, k will meet g.
There are a number of assumptions here which go beyond those found in Euclid. I will mention only the following two:
Nasiraddin (1201-1274), John Wallis (1616-1703), Legendre (1752-1833), Wolfgang Bolyai, Girolamo Saccheri (1667-1733), Johann Heinrich Lambert (1728-1777), and many others tried to prove Euclid V, and failed. In these failures there developed a goodly number of substitutes for Euclid V; i.e., statements that were equivalent to the statement of Euclid V. The following is a list of some of these that are more common:
It fell to three different mathematicians to independently show that Euclid V is not provable from the other axioms and what is derivable from them. These were Carl Friedrich Gauss , János Bolyai , and Nicolai Ivanovich Lobachevsky. Once these men broke the ice, the pieces of geometry began to fall into place. More was learned about non-Euclidean geometries--hyperbolic and elliptic or doubly elliptic. The elliptic geometry was studied by Riemann, gave rise to riemannian geometry and manifolds, which gave rise to differential geometry which gave rise to relativity theory, et. al..
This gives us something to anticipate as we learn more about geometry. We will spend our time studying hyperbolic geometry, for it lends itself to better study--not requiring major changes in the axiom system that we have chosen. We may have an opportunity to see that hyperbolic geometry is now lending itself to considerations in the latest research areas of mathematics.