This website uses a format, which is standard for presenting mathematical material: A sequence of theorems. A theorem is a statement, which has a proof. A proof is a sequence of statements, where each statement has a reason, and each reason is a previously established statement. It could be a statement, which has been previously established in the current proof, or it could be a previously proven theorem. The theorems are stated so that they may serve as reasons in subsequent proofs. It also contains definitions, which can also serve as reasons.

The person who gets credit for developing this format is
Euclid, who used it in his *Elements*
(c. 300 BC). One problem with this format is that one cannot have statements
and reasons going back forever. One must have a starting point. Euclid took as
his starting points, in addition to his definitions, some axioms and
postulates. The axioms were what he called "common notions", such as "The whole
is greater than the part." The postulates were more mathematical in nature,
such as "Two points determine a
straight line." Now days, rather than distinguish between axioms and
postulates, we generally just use the term a "axiom" to denote the statements,
we are going to assume without proof, for starting points. The way it actually
works, is that whenever you can show that the axioms, upon which a theory is
founded, are true, then all of the propositions in the theory will be true.

Euclid did not do a good job with his axioms and postulates. His first theorem does not follow from them. The first theorem is to construct an equilateral triangle with a given side. EuclidÕs axioms are true in the set of ordered pairs of real numbers, in the Cartesian coordinate plane, which have rational coordinates, but it is possible to prove that there is no equilateral triangle in the Cartesian coordinate plane, which has all rational coordinates.

EuclidÕs definitions also leave something to be desired. In his definition of "A line is breadthless length." what is "length"? Before getting to what "breadthless" means, what is "breadth"? Some of these definitions, upon which everything rests, use undefined terms.

If one is going to build a towering structure of logic, like EuclidÕs geometry, the worst place to have flaws is in the foundation. Since Euclid, especially in the last several hundred years, there have been many successful attempts to devise a system of axioms which will provide a satisfactory foundation for EuclidÕs geometry.

Euclid did not include some axioms or postulates, which he needed. Two axioms, which have been added, are the so-called ruler axiom and protractor axiom. The ruler axiom states that the distance from the endpoint of a line segment to a point on the segment sets up a one to one correspondence between the points on the segment and the set of real numbers in the interval between 0 and the length of the segment, and the protractor axiom similarly says that the arc length sets up a one to one correspondence between the points on the arc of a circle and the set of real numbers in the interval between 0 and the length of the arc.

Another issue was EuclidÕs fifth or parallel postulate. An
axiom or postulate should be a simple statement, to which everybody would agree.
Where the other axioms and postulates were succinct and seemed obvious, the
fifth postulate was not and struck people as being more like a theorem than an
axiom or postulate. In
1795, John Playfair (1748-1819) proposed the
following substitute:

Given a line and a point not on the
line, there is exactly one line through the given point, which, is parallel to
the given line.

Attempts to prove that the parallel
postulate from the other axioms and postulates came to a halt when, several
years later, Gauss showed that there were geometries where the other axioms
held where the parallel postulate did not. There are two other possibilities.
In an elliptic plane, there are no lines through the given point, which are
parallel to the given line, and, in a hyperbolic plane, there are more than
one. There are examples of all
three types of geometric spaces.
There are many other equivalents to the parallel postulate, such as the
Pythagorean Theorem, and the fact that the angles in a
triangle add up to180^{o}. In an elliptic plane, *a*^{2} + *b*^{2}
> *c*^{2}, and in an
hyperbolic plane, *a*^{2} + *b*^{2} < *c*^{2}. In an elliptic plane, there are more than 180^{o}
in a triangle, and in an hyperbolic plane, there are
less than 180^{o}. Your axiom system will have to include some
equivalent of the parallel postulate in order to specify which type of plane
you have.

Another trouble with EuclidÕs approach,
which is called the synthetic approach, is that while it rolls along quite
nicely when you have a nice backlog of theorems, which are proven and can serve
as reasons, when one is first starting off, one can find oneself rather
cramped.

This website deals with these issues by
using a different approach, which is called the analytic approach and uses
algebra to establish the foundations of geometry. Euclid has to be excused for not using
this approach, because the algebra which it requires was not available in his time. You
will hear that, algebra was developed by the Arabs in
the ninth century AD. Even though this is about 1100 years after Euclid, it was
still not sufficient for our purposes. Algebra is arithmetic with unknowns, but
where we would write

2*x* + 5

the
Arabs of that time would write

Double
your unknown and then add five

long
hand in Arabic. While many people might think it would be easier to express
themselves in their native tongue than to use our algebraic notation, notice
that the second method uses much more ink than the first. Moreover, simple
tasks such as simplifying a linear expression in one unknown, which has many
terms and parenthesis sprinkled liberally throughout, would not be nearly as
easy using long hand as it is using our current short hand. It wasnÕt until
about the end of the sixteenth century that Francois Viete
developed notation along the lines of that which we currently use.

It was very shortly after this, in the
early to middle seventeenth century, nearly two thousand years after Euclid,
that Rene Descartes got credit for developing analytic geometry. In the
Cartesian coordinate plane, geometric objects like lines and circles have algebraic
equations.

In the synthetic approach, some
features for which one looks in the set of axioms are completeness – are
there enough axioms to prove all of the theorems you want, and independence
– can any of the axioms be proved from the other axioms? Another feature,
which follows from independence, is how many axioms are you using? It is not
unusual to find axiomatizations, which use around a
dozen or so axioms. While the more axioms you have, the more statements, which
can serve as reasons in proofs, also the more unproven propositions upon which
your theorems will rest. Using the analytic approach, it is possible to use the
absolute minimum number of axioms: none.

In the analytic approach, we define a
point to be an ordered pair of real numbers. The plane is the set of all
ordered pairs of real numbers. A line is the set of points, which satisfies a
linear equation. There are some axioms lurking around here. They are the axioms
of the real number system. However, these could be taken to be the definition
of real number system, and not axioms at all. They are derived from the axioms
of set theory and the rules of logic. We need to use the real number system for
the ruler axiom and the protractor axiom.

One question, which always comes up, is
what are you using as your equivalent to EuclidÕs parallel postulate. We are
not using any axiom, which is equivalent. We are using the standard distance
formula for defining the Euclidean distance between two points in Cartesian
coordinate plane. This, of course follows from the Pythagorean Theorem, but we are not assuming the full Pythagorean
Theorem. With the distance formula, we are only assuming the Pythagorean Theorem for right triangles where the perpendicular sides
are horizontal and vertical. We will need to prove the full Pythagorean Theorem from that assumption.

This is not intended to eliminate
synthetic approaches. There is a wonderful tradition of synthetic approaches to
geometry going back thousands of years to Euclid. It is also not going to cover
all of geometry. This is intended simply to get enough basic theorems proved
that a student can proceed synthetically. Basically we need to get through
congruent triangles, which is a very ubiquitous and powerful topic. Along the
way we will prove that all of the traditional axioms of Euclidean geometry are
true in the Cartesian coordinate plane

This will prove to be useful is in dealing with questions of completeness and independence for axiom systems. This will show that one way to demonstrate that your axiom system is adequate would be to show that you can use your axioms to coordinatize your plane into a Cartesian plane isomeric to the set of all ordered pairs of real numbers, with the distance function between such ordered pairs given by the distance formula of the Euclidean metric. Any axioms, which are not needed to accomplish this will then be seen to be not independent.

In California, there is a state wide prerequisite of intermediate algebra for any course, which will satisfy the General Education requirement in mathematics for the California State University system. Students will ask, "What is algebra good for?" Algebra is good for geometry. And the algebra is the most incredible algebra you have ever seen. It has been an incredible amount of fun going through it all, and I am very pleased to be able to share it with you.

We only need beginning algebra until we get to arc length, which is a calculus problem. While it is possible to explain to students, who are nowhere near ready for calculus, the process of taking the limit of finer and finer polygonal approximations of the length of an arc, it is a calculus problem.

A website is an excellent venue for such mathematical presentations. When one reads that something is true because of, say, Theorem 3.2, one may need to look up Theorem 3.2. On a web page, the reference to Theorem 3.2 is a link, and, instead of having to flip pages, one needs only click on the link, and Theorem 3.2 will pop up. If it depends on the definition of a word, the word is a link to its definition. At the end of the proof, there is a link to the next theorem.

Another capability of a website is the opportunities for footnotes. In the lists of theorems and the definitions, there are some numbered footnotes. The numbering of the footnotes starts anew with each page, but it really doesn't matter, because the footnote number is a link to the footnote, so one need only click on the footnote number and arrive immediately at the footnote. At the end of the footnote is a link "Return to text" which will take you to the start of the paragraph which contains the footnote. The ends of the footnotes also contain links back to the "Analytic Foundations of Geometry" table of contents homepage for this site, followed by the author's name which will take you back to the author's website.

The first section deals with lines. This is just algebra with first degree polynomials. Finding the point where two lines intersect is a matter of solving a system of simultaneous linear equations. We can show that two lines intersect unless they have the same slope, so we define parallel and perpendicular using slopes. PlayfairÕs Postulate is then the point-slope form of the equations. Things like the transitivity of parallelism are quite immediate using slope considerations.

One point, which comes up in developing the foundations of Geometry, is the Pasch property: if a line goes into a triangle it will come out the other side. If you bisect an angle in a triangle, how do you know that it intersects the opposite side? People often unconsciously assume this without proof, and may authorities take exception when that happens. The second section is devoted to the Pasch property. We use parametric equations of lines, which is a very powerful technique, although it actually only uses first degree linear polynomials. We need it for refining outside approximations when taking arc length. The reason for putting it in a section as early as the second is that it is good for proving the segment addition axiom, which is in Chapter 3.

Chapter 3 deals with circles. At this point we start to use quadratic equations. This is when the algebra displays its incredible intelligence. One direction of the full Pythagorean Theorem falls out as a result of a simple algebra problem. Of course the algebra is rather extensive, but if you enjoy mind spangling walls of algebra, you will enjoy this. Later, the triangle inequality will just drop out of the formula for the points of intersection of two circles.

After we have established the formulas for the intersection
of lines and circles, and parametric equations for lines, The
primary goal is to establish the criteria for congruent triangles. Once you have
congruent triangles, a synthetic approach will work quite nicely. We will
accomplish this by using rigid motions of the plane. Since we will be able to
accomplish everything we want by using translations, reflections, and
rotations, we will not bother to prove that any rigid motion can be obtained in this manner, or that any rigid motion which is not a translation , reflection, or rotation is a glide reflection, *i.e*. a reflection followed by a transaltion. We cover translations and reflections in Chapter 4. By this point, we have a sufficient backlog of previously proven results that a synthetic proof of the fact that a reflection is an isometry is feasible. However, there is also an algebraic proof, which compares with some of our other spectacular algebraic proofs, so we present them both. Rotations will
have to wait until we get to angles, and angle measurement depends upon arc
length, which we cover in Chapter 5. Chapter 6 is congruent and similar
triangles. At this point, in place of a dozen or so axioms, we have eight-two
theorems. I believe you will find that this will serve as a good supply of
reasons for proceeding in a synthetic development, without any unproven
assumptions beyond the axioms of set theory and the rules of logic.