Geometry is a very important course in the development of mathematics students. The material is ancient, having been first written down by Euclid c. 300 BC. Euclid is to be credited with establishing the method of mathematics used to the present day of organizing theories into sequences of theorems and proofs. A proof is a sequence of statements, each one requiring a reason, ordered so the the reason for each statement is a statement which has been previously established either in the theorem at hand or a previously proven theorem.
However, there are some problems with Euclid's geometry. After he has a sufficient backlog of previously established theorems, the material rolls along brilliantly enough, but he has problems at the outset. For instance, the first theorem on the construction of an equilateral triangle does not follow from Euclid's axioms. While his axioms are all true in a two dimensional Cartesian space over the rational numbers, it can be shown that there is no equilateral triangle which has all rational coordinates. Unfortunately, in such a development, the beginning is the worst place to have logical difficulties.
It should be clear today that Euclid's main problem was that he did not have algebra. The Arabs are credited with inventing algebra in Spain in the eighth century AD, but they did not use the symbology that we employ today. They spelled out the arithmetic to be done with unknowns longhand in prose. Algebra as we know it today was not developed until the end of the sixteenth century AD as a result of work with the radical formulas for the solution of cubic and quartic formulas which was developed by the renaissance Italian mathematicians.
Immediately after the development of what we know of today as algebra, Descartes applied algebra to geometry by coordinatizing the plane and expressing points in the plane with ordered pairs of real numbers, lines with linear equations, and conic sections with quadratic equations. The fundamental theorems of Euclidean geometry can be proven algebraically in the Cartesian plane.
There are thus two approaches to the development of geometry. Proceeding algebraically in the Cartesian coordinate plane is called an analytic approach. Euclid's approach of starting out with axioms and proving theorems without the use of algebra is known as a synthetic approach
In the last hundred years or so, there have been several attempts to establish a system of axioms which would enable one to prove synthetically all of the theorems in Euclid's geometry, and theorems in other similar axiom systems for non-Euclidean geometries. Hilbert's axiom system is probably the best known. The most significant development of twentieth century mathematics is the development of all of mathematics from the empty set using only the axioms of set theory.
After set theory there are two main uses for the term "axiom". In undergraduate courses an axiom is sometimes a theorem which will be needed in the development of the material but whose proof is too difficult to be covered effectively if at all in the course. The other more common use is as part of a definition as in "A group is a set together with a binary operation on the set which satisfies the following axioms".
This is true in Euclidean geometry. The classical axioms are all theorems which can be proven algebraically in the real Cartesian two dimensional plane. The simplest axiomatization of Euclidean geometry requires no axioms at all. One simply defines a Euclidean plane to be a two dimensional real Cartesian space. Not only does this minimize the number of axioms, but the real Cartesian plane is of interest in itself, and it is good to know that Euclidean geometry holds there.
In this approach, the first prerequisites for Euclidean geometry is intermediate algebra. The distance between two points is defined to be the distance which is obtained from the distance formula from intermediate algebra. In doing this we are actually disguising a major axiom as a definition. The distance formula follows from the Pythagorean Theorem. However, we are not assuming the full Pythagorean Theorem. We are only assuming it for right triangles where the perpendicular sides are parallel to the coordinate axes. It is probably not surprising that the Pythagorean Theorem is true if we have assumed it for the special case mentioned above, but the general case of the Pythagorean Theorem still requires a proof. We present three proofs. The first drops out when we are finding the coordinates of the points where a line meets a circle. The second is a straightforward computation, and the third deals with the problem of rotating a general triangle until the perpendicular sides are horizontal and vertical. The second and third proofs are accessed from footnotes to the statement of the theorem in the list of theorems from section 3 on circles.
The other prerequisite is a knowledge of the real number system. A knowledge of the real number system is highly non-trivial, but since one of the main techniques in studying plane figures is to draw lines, a knowledge of one dimensional geometry is necessary to studying two dimensional geometry. When defining angle measurement, since arc length is measured by taking limits of polygonal approximations, just about everything in a first semester course on real analysis is brought to bear. But if one is willing to accept arc length, additivity of angles, and the protractor axiom as axioms, then the only prerequisite necessary is intermediate algebra. Theorem 5.2 proved to be very tricky to properly nail down. If one is willing to accept Theorem 5.2, then the entire section on parametric equations becomes unnecessary
Since geometry was historically developed before geometry, it has become traditional to study Euclidean geometry before introducing the student to intermediate algebra. The fact of the matter is that this practice is logically backward. As a result, there is a great deal of confusion even among professional mathematicians about the correct order of logical dependence of quite a few topics. One of the purposes of this development is to set forth a logical dependence order for the benefit of professional mathematicians.
In the Summer Session of 1998, I was teaching a section of the lower division undergraduate Geometry course, Math 100, at Sonoma State University. I had discovered some time ago that many of the fundamental problems of Euclidean geometry such as transitivity of parallels were rendered free of complications in the Cartesian plane with simple slope considerations. The Math 100 course satisfies a General Education requirement, and is therefore subject to a statewide prerequisite of intermediate algebra. I had had good success basing the course analytically. As I found myself wrestling with the logical dependence order of the fundamental results of Euclidean geometry, I sat down and worked through an analytic development of the basic results of Euclidean geometry.
Trigonometry is not a prerequisite for this material for two reasons. First, trigonometry is not a prerequisite for any lower division courses satisfying the General Education requirement in Mathematics at the California State University. But more importantly, in order to do trigonometry, one needs to know about similar triangles, and that there are 180 degrees in a triangle. These two results are not obtained until the very end of our development. Indeed, one of our purposes is to develop the logical prerequisites to trigonometry.
Students get the most benefit from the experience of formulating synthetic proofs in Euclidean geometry after they have the criteria for congruent triangles at their disposal. At that point I can let them work in groups to prove the results in Math 100 on parallelograms, isosceles triangles, compass and straight-edge constructions, regular polygons, and solid figures, and tessellations.
But the material establishing the congruent triangle criteria is not really appropriate for lower division general education students to work out on their own. In fact, much of the material, like parametric equations, is probably not appropriate to present in class. So I have put all of the details in this web site as reference material for interested students.
Web publishing offers many opportunities and challenges for the presentation of mathematical material. One of the challenges is that it is not easy to put mathematical formulas on the web. My current solution is to put the formulas which cannot be typed into graphics. If your browser is set to a 12 point New York font, then the graphics will appear to be much more seamless than if you use a different font. Moreover, at the present level of development, italics do not come out very well on a web page. For that reason I have abandoned the long standing tradition of italicizing variables. All of the variables are presented in plain text. There is the additional problem that one must be careful not to have the browser mistake mathematical symbols such as a less than sign for an HTML control character. Underlining presents its own problems for web authoring.
Since geometric figures are sets of points, set theory plays an important part in geometry. However, I have found that it is not only possible but preferable to express the set theory that naturally arises in plain English not only to avoid the necessity of having to wrestle with additional mathematical symbols, but also to make the subject more intelligible. The only place where special notation is used instead of plain English is in the notation for the distance between two points where an absolute value notation is used.
It is also not good to have too long of a web page, because it will take too long for the page to load. So the material is broken up into six sections. In each section there is a main page which contains the statements of all the theorems in that section. Each theorem has its own web page,or, in a few cases, several web pages, for its proof. The theorem number in the main list of theorems is the link to the proof. In the proof, the theorem number is a link back to the statement of the theorem in the main page for the section. At the end of the proof, there is a link to the proof of the next theorem. If the top of the web page is not visible from the end of the proof, there is a link back to the top of the page. Each section has a page of definitions of terms which are used in that and later sections.
One advantage of a web format is that wherever there is a reference to a theorem or a definition one can provide a link to it. It is interesting to see how densely terms which have specific definitions appear in the development.