Euclidean Geometry: A First Course

2. HOW IS THIS TEXTBOOK WRITTEN?

“Getting Tigger down,” said Eeyore, “and Not hurting anybody. Keep those ideas in your head, Piglet, and you’ll be all right.”

A.A. Milne, The House At Pooh Corner.

When writing the text, I attempted my best to make it satisfy the following conditions:

a) The text discusses in consistent and sequential manner the basic principles and results of Euclidean plane geometry; the approach is synthetic, in Euclidean spirit.

b) The discussion is rigorous but not overly formal, comprehensible to secondary school students.

c) A curious young reader will appreciate the beauty of the subject and thus will enjoy learning.

d) As a result of using the textbook, a student will acquire valuable intellectual habits that cannot be obtained otherwise than by studying Euclidean geometry.

The choice of the set of axioms constitutes the most important and difficult part of writing a geometry text. One cannot base an introductory secondary school level discussion of the subject on a set that is rigorous from a modern point of view, such as, for instance, the Hilbert’s set of axioms, which is deemed as the briefest one for a rigorous treatment of the subject: this set is too abstract, and the principle of “Not hurting anybody” would be violated. Some rigor should probably be sacrificed in favour of clarity.

There is an excellent discussion of the problem of excessive rigor in introductory textbooks in the chapter “Rigor” of Morris Kline’s “Why Johnny can’t add”. The author’s opinion has been supported by the ones of great mathematicians such as Blaise Pascal and Henri Poincare. The latter wrote: “…if the demonstration rests on premises which do not appear to him [student] more evident than the conclusion, what would this unfortunate student think? He will think that the science of mathematics is only arbitrary accumulation of useless subtleties; either he will be disgusted with it or he will amuse himself with it as a game and arrive at a state of mind analogous to that of the Greek sophists.”

It is hard to disagree with this opinion when talking about some axioms from Hilbert’s set. I think an introductory text can safely set aside some incidence axioms, such as “There exist at least two points in every straight line” and “There exist three points that do not lie in the same straight line” (will any young student ever question this?!) and all the betweenness axioms. Really, what would any person inexperienced in abstract mathematics think about the postulate claiming that “If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A”?

Morris Kline wrote about these and some other axioms often included in modern texts for beginners: “To ask students to recognize the need for these missing axioms and theorems is to ask for a critical attitude and maturity of mind that is entirely beyond young people. If the best mathematicians did not recognize the need for these axioms and theorems for two thousand years how can we expect young people to see the need for them?” This question concerns all axioms not included in the original text of Euclid’s “The Elements”.

I do concur with this opinion of Morris Kline, whom I hold in great esteem, related to the aforementioned axioms of incidence and betweenness. Yet, based on my personal experience as a student (first and foremost!) and later as an instructor, I would argue about the inclusion of axioms of congruence and continuity.

When I studied geometry in junior high school in Russia in the mid sixties, we followed a great textbook “Geometria” by A.P. Kiselyov (Uchpedgiz, Moscow, 1950, the eleventh edition). The textbook, written somewhere at the end of the 19th century, used the set of the basic axioms equivalent to the original Euclidean postulates. With that, the fifth postulate (Euclidean parallel postulate) was introduced in the discussion after all the basic results following from the first four postulates had been derived. As a result of such a sequence of the presentation, even though the existence of non-Euclidean geometries has not been discussed in the book, the readers could understand that quite a few fundamental results, such as congruence of triangles, did not require the parallel postulate in order to be obtained. The Archimedean continuity axiom was added to the set right before the discussion of proportionality of segments and similarity. Based on this axiom, and the notions of lengths and real numbers were introduced almost rigorously.

Such a discussion seems to be almost perfect. Yet I do remember experiencing a sense of discontent when our teacher proved the first basic theorem: The bisector of the vertical angle of an isosceles triangle is also a median and an altitude, and the base angles of an isosceles triangle are equal. The symmetry of an isosceles triangle seemed to be quite obvious, and still it was being proved by means of superposition performed as a physical motion! The comparison of figures by means of superposition had never been defined or explained by means of postulates, and still we used it for a mathematical proof?! I felt it was unfair, “not playing by the rules”, especially because the rules had been proclaimed just before the proof: the teacher explained that we are going to deal with ideal, i.e. non-physical, objects for which some statements (axioms) are assumed, and then she operated with these ideal objects as if they were made of cardboard! The cardboard-likeness had not been postulated!

Later on, when reading the comments to The Elements in the English translation by Sir Thomas Heath, I was quite happy to learn that neither Euclid nor any of his followers really trusted the superposition principle. This principle had been assumed as one of the axioms or common (not particularly geometrical) notions: Common Notion 4: “The things that coincide with one another are equal to one another”, and then used only twice, in the proofs of congruence tests for triangles. It looks like Euclid shied away from using the superposition, since he never applied it in the theorems where it could simplify the proof (for instance, in Proposition 1.5, which states the equality of the base angles of an isosceles triangle).

Keeping all this in mind, I did postulate the existence of rigid motions (isometries) that are needed for the basic congruence theorems. The motions have been defined as segment-preserving transformations. The congruence (equality) of segments has been introduced as an undefined notion described by two axioms, and the congruence of figures has been defined through superposition by means of rigid motions. This seems to be a natural and almost rigorous way; one can easily remake it into a rigorous one by adding the axioms of addition of segments (I use the Euclidean common notion the sums of equals are equal instead) and the aforementioned “obvious” incidence and betweenness axioms, which I think are not necessary in an introductory text.

The first chapter of the book introduces the undefined terms (point, line, plane, congruence of segments), first definitions and first axioms. It also includes a brief discussion of the importance of theoretical knowledge and of the subject of Euclidean geometry as a geometry in which congruence is defined through superposition by means of rigid motions. The existence of certain types of motions has been explained at the intuitive level, by means of physical motions, and then postulated.

Chapter 2 includes an elementary discussion of the basic rules of logic, specifically applied to conditional statements. Inductive and deductive reasoning are discussed. The role of axioms and derivation of theorems in axiom-based theories is being considered. Special attention has been paid to the proof by contradiction (Reductio ad absurdum). Another important method of proof, The Principle of Math Induction, is inc