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...because when you are a Bear of Very Little Brain, and Think of Things, you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it. A.A. Milne, The House At Pooh Corner. Writing a book is a personal business, even if you are retelling the stories told by many other people many times: you are retelling them in your way. So it seems natural to write an introduction that explains my motives for writing and the ideas to which I have adhered in the process of doing so. And since nothing is more irritating than hearing or reading a poor interpretation of your favorite story, I cannot refrain from explaining how others misinterpret the story and why you should listen to my telling of it. As a result, this introduction consists of the following sections: Section 1, WHY WRITE ANOTHER TEXT?, explains the necessity to write a decent introductory text on Euclidean Geometry and discusses, based on two popular textbooks, typical flaws that make many (if not all) modern texts unsuitable and even harmful for studying the subject.
It contains three subsections:
Section 2, HOW IS THIS TEXTBOOK WRITTEN?, contains a detailed discussion of the principles to which I adhered when writing the text, and brief summaries of each chapter.
Section 3, LITERATURE, lists the books and papers mentioned in this review. These resources are worth reading, except, of course, for the two modern textbooks criticized in section 1. It has been impossible to avoid some mathematical details in the sections 1 and 2. Skip them at your pleasure and read only the understandable parts of the essay. (Don’t do so however when/if reading the textbook itself: the mathematical details turn out to be important!).
…But the more Tigger put his nose into this and his paw into that, the more things he found which Tiggers didn’t like. A.A. Milne, The House At Pooh Corner.
This section contains some discourse concerning the current situation of geometry instruction in the USA and Canada and the suitability, or rather the unsuitability of modern textbooks. Two very popular textbooks will be discussed. There is also a very brief discussion of “good old” textbooks, which in general are reasonably appropriate resources for teaching the subject. The section turned out to be fairly long, therefore the reader has an option of just reading the epigraphs (they will give you the gist of the contents) and moving to section 2, which discusses the current textbook.
On Tuesday, when it hails and snows, The feeling on me grows and grows That hardly anybody knows If those are these and these are those. A.A. Milne, Winnie-the-Pooh. Some basic results of geometry are mentioned in the majority of modern North American high school textbooks. These results are usually presented as “facts of life”, without proofs or without clear presentations of geometrical and logical underlying principles. It is not only that such texts do not improve students’ understanding of mathematics and their ability to think and discourse on various subjects, which have always been important “by-products” of studying geometry. They also do not promote the knowledge of actual material: the easily acquired (without derivation or substantiation) “facts of life” are mostly forgotten immediately after or even before the diploma exams. Our typical junior undergraduates would only vaguely remember something
about the proportionality in similar triangles and the formula for the
circumference, which they usually confuse with the one for the area of a circle.
Moreover, the majority believes that
The only way to remedy this situation is to teach Euclidean geometry as a separate subject in secondary schools as it had been done until recently for almost 2000 years with comparatively short pauses due to outbreaks of plague, obscurantism, educational reforms, and other natural or not so natural disasters. In the recent times, the periods of obscurantism have been so closely related to educational reforms that it would be hard to tell which of them have followed the others. Both would usually run under slogans that deify the use of technology and the immediate practical usefulness of any knowledge. Of course, classical subjects, including Euclidean geometry, were the first to fall victim to these fads. As a result, it is extremely hard, if not impossible, to find a decent textbook on the subject issued in English in the last approximately 70 years. There are a few good old texts issued almost a century ago, and those will be discussed later on. Those few modern textbooks that claim to be rigorous and axiom-based are usually written in an unsystematic and even illogical manner. A typical example is the textbook “Geometry: tools for a changing world” by Laurie E. Bass, Basia Rinesmith Hall, Art Johnson, and Dorothy F. Wood, with contributing author Simone W. Bess, published by Prentice-Hall, 1998. A detailed reference on this text written by Professor D. Joyce (Clarke University) can be found on line at http://aleph0.clarku.edu/~djoyce/java/elements/geotfacw.html. It is worth reading (the reference, - not the book), and it is really impossible to refrain from quoting at least a few small excerpts, which discuss the typical faults of almost every modern textbook for secondary schools. (The beginning) “This textbook is on the list of accepted books for the states of Texas and New Hampshire. It's a glitzy book filled with pictures to keep the attention of the students. That's fine. It's the content that bothers me, in particular, the lack of logical content. Chapter 1 introduces postulates on page 14 as accepted statements of facts. The four postulates stated there involve points, lines, and planes. Unfortunately, the first two are redundant. Postulate 1-1 says 'through any two points there is exactly one line,' and postulate 1-2 says 'if two lines intersect, then they intersect in exactly one point.' The second one should not be a postulate, but a theorem, since it easily follows from the first. And what better time to introduce logic than at the beginning of the course?! No statement should be taken as a postulate when it can be proved, especially when it can be easily proved. A number of definitions are also given in the first chapter. Later postulates deal with distance on a line, lengths of line segments, and angles. The book does not properly treat constructions. Constructions can be either postulates or theorems, depending on whether they are assumed or proved. For instance, postulate 1-1 above is actually a construction. On pages 40 through 42 four constructions are given: 1) to cut a line segment equal to a given line segment, 2) to construct an angle equal to a given angle, 3) to construct a perpendicular bisector of a line segment, and 4) to bisect an angle. Later in the book, these constructions are used to prove theorems, yet they are not proved here, nor are they proved later in the book. There is no indication whether they are to be taken as postulates (they should not, since they can be proved), or as theorems. At the very least, it should be stated that they are theorems which will be proved later. “ “Chapter 8 finally begins the basic theory of triangles at page 406, almost two-thirds of the way through the book. This chapter suffers from one of the same problems as the last, namely, too many postulates. The three congruence theorems for triangles, SSS, SAS, and ASA, are all taken as postulates. One is enough. The other two should be theorems.” “Final conclusion. Much more emphasis should be placed on the logical structure of geometry. Postulates should be carefully selected, and clearly distinguished from theorems. Every theorem should be proved, or left as an exercise, or noted as having a proof beyond the scope of the course. Very few theorems or none at all, should be stated with proofs forthcoming in future chapters. It should be emphasized that "work togethers" are not a substitute for proofs. They can lead to an understanding of the statement of the theorem, but few of them lead to proofs of the theorem. It must be emphasized that examples do not justify a theorem.” (The end of quotations from the reference).
…then this isn’t an Expo – whatever it is – at all, it’s simply a Confused Noise. That’s what I say.” A.A. Milne, Winnie-the-Pooh.
Another very popular text, “Geometry” by Harold Jacobs, published by W.H. Freeman and Company, 1987 (second edition) cannot be called a better book although at first glance it appears to be the lesser of two evils.
The discussion claims to be axiom-based; at least the author states which of the assertions are postulates and which of them he suggests to be theorems. However the book possesses a really amusing feature: some postulates are derived from the others! Anyone who has ever heard of the axiomatic approach would argue that a derived postulate is an oxymoron: it is common knowledge that postulates (axioms) are statements assumed to be true. – Not in this text. In the lesson “The Volume of a Prism”, Cavalieri’s Principle is introduced as a postulate (Postulate 14). After a short discourse based on this postulate, the author concludes (quotation follows): “Hence, the volume of every prism can be found by the same formula. We will now state this conclusion as a general postulate. Postulate 15: The volume of any prism is Bh, in which B is the area of one of its bases and h is the length of its altitude.” The author truly believes and teaches his students that postulates follow from one another! The aforementioned perversion of the axiomatic method, ridiculous as it sounds, is explicitly proclaimed in Chapter 15; yet a reader can understand that something is wrong in this respect starting with the very first chapters. It is not clear why some of the assumed axioms are not being used to prove theorems. The so-called ruler postulate, which states the existence of lengths and of the coordinates of the points in a line, is assumed to be one of the axioms. The introduction of this postulate in a textbook on Euclidean geometry is already a blunder, and we shall discuss it later. So far one cannot help asking: why has this axiom not been used for proving, for example, the existence of the midpoint of a segment? Such a proof would take no more than one line: the point whose coordinate is the arithmetic mean of the coordinates of the endpoints of the segment is the midpoint: just perform the subtraction and you will see. Instead, the existence and uniqueness of a midpoint of a segment is postulated. Then, why would we not postulate all the results? Why should we prove anything at all?! Even though the ruler postulate is assumed as an axiom, it is not used in all of the proofs in which it could be useful; also, it is never used in constructions. –Why? Do we not introduce axioms in order to obtain results? Or is there a special selection rule discriminating some axioms in favour of others? – Then this rule itself must be postulated as well! Even more mysteries are generated by the introduction of another axiom that has never been used in traditional textbooks on Euclidean geometry: the protractor postulate that states the existence and properties of the angle measure. The postulate is analogous to the ruler postulate for the coordinates on a line. Then, since the existence of the midpoint of a segment has been postulated it seems fair to postulate the existence of a bisector of an angle. Or maybe to state it as a theorem, which would be a good idea since it can be proved?! – Not a word about the existence of bisectors, although their existence is assumed somehow. Instead, the definition of a bisector: “A ray bisects an angle iff it is between the sides of the angle and divides it into two equal angles” is followed by a so-called Theorem 4: “A ray that bisects an angle divides it into angles half as large as the angle”. It looks like the author is telling us that if we divide by two we are getting halves, not one thirds or anything else. Now, a few words about the ruler and protractor postulates, which nowadays are present in almost every geometry text. They were originally introduced by G.D. Birkhoff in his paper “A Set of Postulates for Plane Geometry (Based on Scale and Protractor)”, Annals of Mathematics, 33, 1932, as part of the set of axioms for plane Euclidean geometry. The set includes two more axioms: the incidence postulate (the existence of exactly one line through a pair of points) and a similarity postulate (side-angle-side criterion for similarity of triangles). The four postulates describe the undefined notions of a point, line, distance between two points, and angle formed by three ordered points. Birkhoff’s system of postulates does describe the geometry of the Euclidean plane and it can be used for solving geometric problems. Yet such a choice is not suitable for a textbook on Euclidean geometry, especially for one that introduces readers to the subject. An introductory text ought to have some Euclidean spirit, which is implemented in the so-called synthetic approach (geometry without numbers). The synthetic approach allows the reader to view the subject with maximum generality, without dependence on a number system, and emphasizes the geometric contents of the discipline, in contrast with an approach based on arithmetic or calculus. Another important feature of a synthetic approach is the opportunity or rather the necessity of introducing the notion of continuity, one of the most important notions in mathematics. It was the idea of continuity that made it possible to construct real numbers and to define the notion of the length of a segment. If the existence of lengths is being postulated, the students are missing the whole story including the notion of a real number. Unfortunately, very few resources pay attention to this important issue. For interested readers I would recommend the paper “Teaching Geometry According to Euclid” by Robin Hartshorne in the Notices of the AMS (American Mathematical Society), Volume 47, # 4, April 2000. Let us come back to the H. Jacob’s book. It does contain the ruler and protractor postulates, but the undefined notions and other postulates are neither the same nor equivalent to the ones used by Birkhoff. This is not even the sloppy set of SMSG (School Mathematics Study Group) postulates, fairly criticized in Morris Kline’s bestseller “Why Johnny Cannot Add: The Failure of the New Math”(Vintage Books, New York, 1974). Rather, it is an “Irish soup” of various approaches (Birkhoff, Euclid, Hilbert, - all in one pile, – the more the merrier) complemented with a few superfluous postulates proposed apparently by the author himself in spite of already having too many of them. As in the book discussed above, “Geometry: Tools for a Changing World”, both SAS (side-angle-side) and ASA (angle-side-angle) tests for the congruence of triangles are assumed to be postulates. In the Euclidean approach both tests are theorems; in the Hilbertian – one of them (SAS) is a postulate and the other is proved; in Birkhoff’s system, assuming the ruler and protractor postulates, none of the tests is required as a postulate. In rigorous theories the set of axioms is required to be independent: none of the axioms can follow from others. Then we do know what the underlying statements of the theory are. If the set is overfilled, as in the text discussed above, it is impossible to understand what the theory is “made of”. One could argue that a text for secondary schools cannot be completely rigorous, and I would agree with that. Yet, a textbook should not be based on a mixture of incompatible approaches as it happens with the H. Jacobs’ book. The book reminds me of a children’s story (“Knights” by V. Dragounsky). In this story, a boy, being in need of money in order to buy a present for Mother’s Day, emptied two bottles of his father’s drinks, a sherry and a beer, into a jar, arguing: “Wine and wine mixed together will still be wine.” The only possible reason for making such a cocktail of axioms and redundant propositions is the intention to create the appearance of high-level rigor. This pseudo-rigor does not help anything, - it only generates incomprehensible definitions such as, for instance, the definition of a ray (half-line) by means of betweenness, which does not contain explicitly the notion of the vertex or origin (later called the endpoint) of the ray. In the definition and the subsequent discussion of the notion of an angle not a word is said about its interior, not to mention the fact that angles are introduced after polygons although the latter are actually made of angles and owe them their name (in Greek: poly = many; gonio = angle) . Some axioms of betweenness are introduced (this could be easily avoided in an introductory course, - they have never been substantially used anyway) whereas the others (e.g., Pasch’s axiom) are ignored. The same can be said about some incidence axioms that were implicitly meant by Euclid in his Elements and thus could be omitted without trouble (see the aforementioned Kline’s book about the “new math” and level of rigor). Postulate 2 tells us: If there is a line, there are at least two points on the line. The first part of the phrase instigates the reader to ask: What if there is no line(s)? This formulation is a travesty of a postulate from the system proposed by D. Hilbert: Every line contains at least two points. The next postulate of incidence states that there exist three points not lying in one line. Why has this one been ignored by the author? Moreover, the next axiom, called Postulate 3: If there are three noncollinear points, then there is exactly one plane that contains them, does seem to require the assurance that three noncollinear points do exist! So, why is the axiom of their existence not included? The other numerous flaws of the book are minor comparing to the aforementioned ones. Let us list just a few of them. The notion of the locus, which is crucial for Euclidean geometry, especially for constructions, is not present in the book. Even though the book features a chapter on Non-Euclidean geometries, the author makes no effort to specify which results would still be true if the parallel postulate had not been assumed. Lesson 7 in the Chapter “Parallel Lines” is entitled “Two More Ways to Prove Triangles Congruent” and presents the angle-angle-side (AAS) test for general triangles and the hypotenuse-leg (HL) test for right ones. Thus readers may erroneously think (actually they are encouraged to think so) that these tests follow from the parallel postulate, and this false idea is supported by the proof given in the book. As a matter of fact, these tests are true in a geometry that assumes all Euclidean postulates except the parallel postulate (so-called neutral geometry), thus they would be true, for instance, in hyperbolic geometry. Limits were introduced seemingly to explain the notion of circumference. “Because,” as the author explains, “a mathematically precise definition of the word limit, would be quite difficult to understand, an informal explanation is given instead”. Unfortunately, the given “explanation” is also difficult if not impossible to understand and contains a mistake for good measure (it is the absolute value of the difference, not the difference, between the terms of the sequence and the limit that “can be made as small as we wish”). Even the corrected “explanation” would not help much since the illustrating examples, consisting mostly of another set of “funny” pictures, do not possess the necessary content. The problems of the section also do no explain the mathematical contents of limits but only teach students to write the symbol replacing the words limit when n tends to infinity. An excerpt follows. “Write each of the following statements in symbols. Example:
The limit of the sequence whose nth term is
Answer :
13. The limit of
the sequence whose nth term is 14. The limit of
the sequence whose nth term is The learning of a notion is replaced with a lesson on how to use the notation. The uniqueness
property and the Weierstrass’ theorem, which is really crucial for defining the
circumference, are not even mentioned. As a result, the circumference is defined
as a limit whose existence is vaguely supported by a few numerical experiments.
The number Limits are never used anywhere else although they could help, for example, to provide beautiful proofs for the formulae of the volumes of pyramids and other solids. The problems are trivial and often irrelevant, especially those that were supposed to show a “real-life connection”. The latter is also apparently provided through an enormous number of pictures, cartoons, and photographs, many of them ugly and repulsive (such as a photo of a telephone booth packed with bodies illustrating the section “The Volume of a Prism) with very little relevance to the topic, some of them with erroneous commentaries. Lesson 7, “Spheres”, in the Chapter “Geometric Solids”, starts with a picture of an old man with a huge ball of string. The commentary states: “The photograph shows him [Mr. Roberts] standing with the result of his unusual hobby, a ball of string three feet in diameter! How heavy would a ball of string this size be? To answer this, it would be helpful to know how to find the volume of a sphere. ”(End of quotation). – It will not be helpful: the ball is obviously porous and non-uniform, hence knowing the volume will not help even for a rough estimate of the weight! Does that mean students should not learn about spheres and their volumes? Speaking of the pictures (each chapter starts with one, approximately half a page in size), there is one which is particularly puzzling (not to say – irritating): Lesson 5 “Straight Segments” begins with a picture of Marilyn Monroe presented as a net of straight segments. What was the point of placing this picture? Is there any learning value? It is not easy to understand from it what the late actress looked like: this net of segments may just as well be a picture of Marge Simpson, Euclid, your neighbour Willy, or anyone at all. Does the message of the picture state that everything under the sun can be made of straight segments, and that is why they should be studied? Or is it intended to assert the triumph of science and technology by disfiguring beauty? O’Henry wrote (“Squaring the Circle”, The Complete Works of O’Henry, vol.II): “Beauty is Nature in perfection; circularity is its chief attribute. Behold the full moon, the enchanting gold ball, the domes of splendid temples, the huckleberry pie,…On the other hand, straight lines show that Nature has been deflected. Imagine Venus’s girdle transformed into a ‘straight front!’” So, maybe it would be better not to straighten out naturally occurring curves? There are already too many things to be straightened out in this book: the difference between the axioms and theorems, for instance! Maybe it would be better to leave the pictures beautiful, the axioms believable and independent, the theorems proved, the definitions defining, and the textbooks teaching something useful, – e.g. geometry?
“Rabbit,” said Pooh to himself. “I like talking to Rabbit. He talks about sensible things. He doesn’t use long, difficult words, like Owl”. A.A. Milne, The House at Pooh Corner. The category of “good old textbooks” includes a few geometric texts written approximately a century ago. Usually based on close-to-Euclidean sets of common notions and postulates and written in somewhat simplified language (compared to the English translations of The Elements), they are reasonable textbooks from which students could learn the actual geometric material as well as the methods of proofs and the ways of logic. One of the best of such books is “A School Geometry, Parts I-VI” by H.S. Hall and F.H. Stevens (Macmillan and Co., Limited, London, 1908). The text includes a discussion of both the plane and solid geometries at the secondary school level. There are a few flaws, which are, let us emphasize this again, very minor comparing to the ones in the aforementioned modern texts. The set of postulates contains superfluous statements, such as, for example, the equality (congruence) of right angles, the existence of the bisectors of angles and of the midpoints of segments. The notion of equality (congruence) is not defined and not described as an undefined term (in “The Elements”, it was mentioned among the common notions), which may leave students puzzled about the validity of the proofs of fairly obvious statements whereas less obvious ones are tacitly assumed. Axioms of continuity have not been introduced and thus the notion of the length of a segment has not been discussed even though all kinds of measures, including irrational ones, are used throughout the book. It is a great loss for a school text since the students are losing the only opportunity to be introduced to the theory of real numbers. (The theory of real numbers is never discussed in high school Algebra and Analysis courses, nor in undergraduate Calculus; only a few higher level undergraduate courses include a discussion of Dedekind’s cuts. Thus, students typically spend quite a few years considering functions of real variable without having an idea of what is a real number; it is not surprising then that they encounter great difficulties when dealing with limits and continuity). The results of neutral geometry are not singled out, thus students cannot appreciate the role of the parallel postulate and the idea of generating different geometries by changing some postulates. It is however an important story, both historically and methodologically. The notion of circumference does not receive any consideration. It is only mentioned that the ratio of the circumference to the radius is the same (approximately 3 and one seventh) for all circles. Most of the exercises are technical and trivial, and thus would not excite interest in the subject, especially among strong students. The discussion is very dry, sometimes lacking explanations, such as, for instance, why the constructions should be performed with a straightedge and a compass alone. The book loses the sense of creating and uncovering things that is inherent in “The Elements”. With all these disadvantages, the book by H.S. Hall and H.S. Stevens, in contrast with the aforementioned modern texts, is a decent book from which students can learn the basics of Euclidean geometry.
“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, so that the first eight chapters could be understood by secondary school students. c) A curious young reader will appreciate the beauty of the subject and thus will enjoy working on the course. 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, 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 shyed 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 as well as the general structure of axiom-based fields of knowledge. Inductive and deductive reasoning are discussed. The role of axioms and derivation of theorems in axiom-based theories is being considered. The notions of converse, inverse, and contrapositive statements are introduced and the relations between them are discussed, both theoretically and based on simple examples. Special attention has been paid to the proof by contradiction (Reductio ad absurdum), both the substantiation and the use of the method. Another important method of proof, The Principle of Math Induction, is included in the problems section accompanying the chapter.
In Chapter 3, angles are defined and their basic properties discussed. The generation of angles by the rotation of rays and the generalization of the notion of angle for the case of non-convex interiors has been considered. The existence of right angles and their congruence is proved. Chapter 4 contains the most substantial results of the so-called neutral geometry (Euclidean geometry without the Euclidean parallel postulate): congruence of triangles, existence of the bisectors of angles and midpoints of segments, properties of isosceles triangles, inequality in triangles, loci and axial symmetry.
Chapter 5 discusses constructions. It explains why only a straightedge and a compass are being used, and all the basic classical constructions of neutral geometry are described and substantiated in detail. The general 4-step plan for solving construction problems (analysis, construction, synthesis, and investigation) is given and illustrated in the problems section.
Chapter 6 starts with the notion of parallel lines and a few more results of neutral geometry: the proof of the existence of parallel lines and the tests for parallelism based on the angles. Then the parallel postulate in Playfair’s formulation is stated, and its equivalence to the 5th Euclidean postulate is shown. The congruence/supplementarity of angles with parallel and perpendicular sides has been proved. Applications to constructions are considered.
Chapter 7 contains all the basic classical results of the Euclidean geometry of parallelograms and trapezoids as well as related topics, such as central symmetry and translations and their use in constructions. The discussion closely follows that of the Kiselyov’s “Geometria”.
The basic results of the Euclidean geometry of circles (some of them post-Euclidean, still treated as classical) is discussed in Chapter 8. With a few minor deviations, the text is a translation of the section concerning circles from the Kiselyov’s book. The chapter ends with the discussion of the remarkable points of concurrency: incentre, circumcentre, centroid and orthocentre of a triangle.
Chapter 9 starts with a rigorous discussion of the notion of the measurement of a segment. The continuity is introduced through the Archimedes axiom and Cantor’s principle of nested segments. These axioms are quite natural, and it is easy to visualize them, in contrast with the Dedekind’s postulate, which does not sound very intuitive and is not easy to comprehend for students without experience in abstract mathematics (see the section Principle of Continuity in the comments Other Axioms Introduced After Euclid’s Time of the Heath’s translation of The Elements). Dedekind’s principle is often employed in university level texts for introducing lengths in geometry and real numbers in analysis. One can show the equivalence of this postulate to Archimedes and Cantor’s axioms together, and therefore the latter two, which are more intuitive, are in my opinion a better choice. Irrational numbers are introduced as the measures of segments incommensurable with the unit of length. The existence of incommensurable segments is proved purely geometrically, and another example of such a proof is proposed as a problem for students to solve. The ratio of segments is defined as the ratio of their measures, and it is shown that it does not depend on the choice of the unit length. The similarity of triangles and then of arbitrary polygons is defined through the proportionality of the sides and congruence of angles and the tests are proved for general triangles, then for right triangles. Then, starting with a problem of constructing a polygon similar to a given one, the notions of a similarity transformation, centre of a homothety and magnification ratio have been introduced. These are employed for giving a new definition of similarity that is applicable for all (not necessarily rectilinear) figures and includes the original definition as a particular case. The use of similarity transformations in constructions is discussed. Then follow all the classical results concerning proportional segments. These concern the segments cut by parallel lines on their transversals, on parallel lines by their transversals, the properties of the bisectors of interior and exterior angles in a triangle, proportional segments in circles, and all the metric relations in right triangles, including the Pythagorean theorem, which follows from them. The same section also includes the converse of this famous theorem and its generalizations for the cases of acute-angled and obtuse-angled triangles; the latter are used in the following sections for deriving the cosine theorem for solving triangles. Trigonometric functions of acute angles are introduced and their properties discussed. Then the cosine theorem is proved, the notion of sine is generalized for obtuse angles, and the sine theorem proved. The chapter ends with the discussion of applications of algebra for solving geometric problems, in particular for performing constructions. A few examples, including the golden ratio, the mean proportional, the fourth proportional, and various expressions involving roots are considered. Chapter 9, in contrast with the preceding chapters, does require some rudimentary knowledge of algebra, including solutions of quadratic equations in radicals.
Chapter 10 starts with a discussion of regular polygons, but the main focus of the chapter is the notion of the circumference of a circle. This is an extremely interesting and exciting topic, routinely ignored both in the secondary school and the university level texts. At the same time the notion of circumference is extremely important both from the theoretical and the practical points of view, for instance for such a “minor thing” as defining trigonometric functions on the unit circle. The circumference is introduced based on Cantor’s principle (we knew we would need it again!), and thus does not require the notion of the limit of a sequence. Still, an optional approach, by means of limits is considered, with the limits and their basic properties introduced beforehand. The chapter ends with the introduction of the radian measure of angles.
Chapter 11 deals with areas. A square with a unit side is assigned the unit area; the areas of all the other figures are defined by the principles of the equivalence of congruence figures and equivalence by finite decomposition. The existence of the area of any rectangle (with rational or irrational lengths of its sides) is proved. Various formulae for determining the areas of parallelograms, triangles, trapezoids, and regular polygons are derived. The Pythagorean Theorem is revisited, in terms of areas; two different proofs are proposed. The theorem claiming that the ratio of areas of similar figures is the square of their similarity ratio is proved. The derivation of the formula for the area of a circle concludes the chapter.
There is no cartoons and glitzy pictures in this book: I believe that the subject itself is so beautiful that it does not require any embellishments either in direct or ironical sense of this word. Epigraphs are a different matter. It is an old and noble tradition to start a chapter of a book with a suitable epigraph, which sheds some light on the following contents and thus helps the reader. Since the textbook is written for the children who think that they are almost adults, I found it appropriate to turn their attention to the book which many of them think they have overgrown, but they have not. Thus all the epigraphs have been borrowed from “The Complete Tales of Winnie-the-Pooh”, which is really a book of wisdom for all ages. Some solutions of the problems are also preceded by epigraphs that illustrate the main idea of the solution or give a helpful hint. These are excerpts from various classical books for children and youth.
Every section of each chapter is supplied with a set of problems. Also, each chapter is concluded with a large set of Review problems; the latter are usually the core problems of the course whereas many of the problems concluding sections are simple technical exercises. There are over 1200 problems altogether. Some of them are quite challenging; the most difficult ones are usually marked with an asterisk and supplied with hints in the first part of the Answers, Hints, Solutions section and with a solution (or solutions) in the second part of the section. Also, the problems that demonstrate new methods (e.g., the use of symmetry in constructions) are provided with detailed solutions. The students who use the book should solve problems, - it is absolutely essential. A person who solves a difficult problem experiences a sense of triumph that is unlike anything else. Solving problems and describing their solutions may be difficult in the beginning, but it must become a more customary activity for those who study the methods of proofs and constructions presented in the text. Also, in the course of their studies students will learn how to substantiate their ideas and express them in writing.
Geometry is a difficult but a very rewarding subject. I cannot imagine any other discipline that can arouse a comparable feeling of beauty and harmony. Many of the greatest minds in the history of the humankind considered studying geometry to be the commencement of their intellectual lives. Albert Einstein could never forget the tremendous impression produced on him by his first geometry textbook. Bertrand Russell, an eminent philosopher of mathematics wrote in his autobiography: “At the age of eleven, I began Euclid… This was one of the greatest events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.” Let us hope that modern students will not be deprived from such an experience.
“Owl looked at the notice again. To one of his education the reading of it was easy.” A.A. Milne, The House At Pooh Corner.
This paper can be found on line at http://www.ams.org/notices/200004/fea-hartshorne.pdf
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