"On the required reading list for all thoughtful students who wish to see mathematics from the 'higher standpoint.' " — American Mathematical Monthly
Elegant and original, this exposition explores the foundations and development of both Euclidean and non-Euclidean geometry, particularly the postulational geometry of planes. Emphasis is placed upon the coordination of affine and projective planes as well as the basic unity of algebra and geometry.
Geared toward undergraduate and graduate students, the treatment begins with a brief but engaging sketch of the historical background of Euclidean geometry and an elementary summary of set theory and propositional calculus. Subsequent chapters explore coordinates in an affine plane, including those with Desargues and Pappus properties, and coordinatizing projective planes. The final two chapters contain detailed developments of simple sets of postulates for the Euclidean and non-Euclidean planes.
About the Author
Leonard M. Blumenthal (1901–84) was Professor of Mathematics at the University of Missouri. In addition to this volume, he wrote Theory and Applications of Distance Geometry.
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A Modern View of Geometry
By Leonard M. Blumenthal
Dover Publications, Inc.Copyright © 1989 Leonard M. Blumenthal
All rights reserved.
Historical Development of the Modern View
I.1. The Rise of Postulational Geometry. Euclid's Elements
It is often said that geometry began in Egypt. The annual inundations of the Nile swept away landmarks and made necessary frequent surveys so that one man's land could be distinguished from that of his neighbors. These surveys did indeed give rise to a collection of geometrical formulas (many of which were merely approximations), but the Egyptian surveyors were no more geometers than Adam was a zoologist when he gave names to the beasts of the field.
For the beginnings of geometry as a deductive science we must go to ancient Greece. Through the efforts of many gifted fore-runners of Euclid, such as Thales of Miletus (640–546 B.C.), Pythagoras (580?–500? B.C), and Eudoxus (408–355B.C.), many important geometrical discoveries were made. Plato was fond of the subject, and though he made few original contributions to it he emphasized the need for rigorous demonstrations and thus set the stage for the role Euclid was to play.
The postulational procedure that stamps so much of modern mathematics was initiated for geometry in Euclid's famous Elements. This epoch-making work, written between 330 and 320 B.C., has probably had more influence on the molding of our present civilization than has any other creation of the Greek intellect. Though far from attaining the perfection it aspired to, the Elements commanded the admiration of mankind for more than two thousand years and established a standard for rigorous demonstration that was not surpassed until modern times.
Euclid was the greatest systematizer of his age. Few, if any, of the theorems established in the Elements are his own discoveries. His greatness lies in his organization, into a deductive system, of all of the geometry known in his day. He sought to select a few simple geometric facts as a basis and to demonstrate all of the remainder as logical consequences of them. The facts he chose as a basis he called axioms or postulates. No proofs are given for them — they were the building blocks used to derive the theorems of his system.
We shall discuss postulational systems in Chapter II, but we may observe at once that every such system has a set of unproved statements as a basis. These statements are called axioms or postulates, or merely assumptions. Though Euclid applied the term axiom to so-called common notions that were not peculiar to geometry (for example, if equals are added to equals, the sums are equal) and reserved the term postulate for assumptions of a geometrical nature (for example, two points may be joined by a line), this distinction is not usually observed today. In this book we shall usually call our assumptions postulates.
Euclid selected five geometrical statements as the basis for his deductive treatment and introduced them in the following manner:
"Let the following be postulated:
"I. To draw a straight line from any point to any point.
"II. To produce a finite straight line continuously in a straight line.
"III. To describe a circle with any center and distance.
"IV. That all right angles are equal to one another.
"V. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
I.2. Some Comments on Euclid's System
The reader will observe that Euclid's postulates involve numerous technical terms (for example, point, straight line, circle, center, right angle) as well as certain operations, such as drawing, producing continuously, and falling (a line that "falls" on two lines is called a transversal). What is the status of these notions?
Though Euclid clearly recognized the necessity for unproved propositions in his scheme, it is doubtful that he realized the necessity for primitive (undefined) notions. And yet it is obvious that the attempt to define everything must result either in a vicious circle or in an infinite regression. Though Euclid's thinking in this respect is not entirely clear to us, it is a fact that, in contrast with his explicitly exhibited set of postulates, his Elements contains no list of undefined terms, but, on the contrary, attempts to define all the terms of the subject. Some of Euclid's definitions (those for circle and right angle, for example) are precise and useful in developing the geometry, whereas others (those for point and line, for example) are vague (point is defined in terms of part, which is itself undefined) and are never used in the sequel. Perhaps they were merely intended to create images in the reader's mind. By invoking such physical notions as "drawing," "producing," and "falling," which are quite out of place in an abstract deductive system, Euclid did not attain to the modern concept of a geometry. But after all, mathematicians should have learned something in two thousand years!
A more serious criticism of Euclid's attempt to establish geometry as a deductive system is that even if he had selected certain of his notions to be primitive and had stated his five postulates without using the physical notions mentioned above, his foundation was simply insufficient to support the lofty edifice he sought to erect on it. He was able to prove many of his theorems because he used arguments that cannot be justified by his postulates. This occurs in the very first proposition, which purports to show that on any given segment A,B an equilateral triangle can be constructed. His proof is invalid (incomplete), inasmuch as it makes use of a point C whose existence is not, nor cannot be, established as a consequence of the five postulates, since the circles with centers A and B and radius AB that he employed in his argument cannot be shown to intersect to give the point C. This difficulty is due to the lack of a postulate that would insure the continuity of lines or circles.
There are other deficiencies in the Elements. In one of his proofs Euclid tacitly assumed that if three points are collinear, one of the points is between the other two, though the notion of betweenness does not appear in his basis. Euclid's concept of congruence of figures was the naive one of superposition, according to which he regarded one figure as being moved or placed on another. It has been conjectured that he disliked the method, but made use of it for traditional reasons or because he was unable to devise a better one.
I. 3. The Fifth Postulate
The fifth (parallel) postulate of Euclid is one of the principal cornerstones on which his greatness as a mathematician rests. In his commentary on Euclid's Elements, Heath remarks, "When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable."
And yet this cornerstone of Euclid's greatness was the basis of the sharpest attacks on his system. The four postulates that precede it are short, simple statements, and it is not surprising that the much more complicated nature of the assertion made by the fifth postulate suggested that it should be a theorem rather than an assumption — a view that unwittingly received some support from Euclid himself, since he did prove the postulate's converse. And so, very soon, attempts were made to remove this flaw from the Elements. These efforts were started in Euclid's lifetime and were continued by reputable mathematicians until the second decade of the nineteenth century, and by cranks even later. The failure of all such attempts securely established Euclid's fame and, what is more important, led to the invention of Non-euclidean geometry.
All attempts to demonstrate the fifth postulate as a logical consequence of the other four postulates (amended, perhaps, to convey Euclid's meaning better than they had in their original forms) introduced, surreptitiously, assumptions that were equivalent to the fifth postulate, and hence, in effect, they assumed what they intended to prove. (For example, neither the infinite extent of the straight line nor the unique extension of segments follows from the second postulate, but Euclid likely intended that both should, since he made frequent use of those properties.) Let us examine an early and a much later example of these efforts.
Proclus (410–485 A.D.) was a very competent mathematician and philosopher who studied in Alexandria when a youth and later went to Athens, where he taught mathematics. He was highly regarded among his contemporaries for his learning and his industry. His commentary on the Elements is one of our chief sources of information concerning early Greek geometry, the original works of the forerunners of Euclid having disappeared. Proclus showed that the fifth postulate can be proved if it is established that (*) if l1, l2 are any two parallel lines and l is any line distinct from and intersecting l1, then l3 intersects l2 also. For, assuming the italicized statement, let m1 and m2 denote two lines, and let m3 denote a transversal such that the sum of the two interior angles α, ß is less than two right angles. Then a line m4 through P exists such that [angle]α' [angle]ß = 2 rt [angle]s, and, on the basis of Proposition 28, Book I of the Elements (whose proof does not involve the fifth postulate), lines m2 and m4 are parallel. (For all references to specific propositions of the Elements, see T. L. Heath, The Thirteen Books of Euclid's Elements, Cambridge, 1908.) Hence line m1, which is distinct from m4 and intersects it at P, intersects line m2 also. Moreover, lines m1, m2 meet on that side of the transversal m3 for which the sum of the interior angles α, ß is less than two right angles, since if it be assumed that they meet on the other side of m3, they would form with m3 a triangle with an exterior angle α that is less than an opposite interior angle ß', which is contrary to Proposition 16, Book I (whose proof does not involve the fifth postulate).
This much having been done in an acceptable way, it remained for Proclus to derive the above proposition (*) from Postulates I–IV. Here is his argument, as translated by Heath.
"Let AB, CD be parallel, and let EFG cut AB) I say that it will cut CD also.
"For, since BF, FG are two straight lines from one point F, they have, when produced indefinitely, a distance greater than any magnitude, so that it will also be greater than the interval between the parallels.
"Whenever, therefore, they are at a distance from one another greater than the distance between the parallels, FG will cut CD."
The assertion made in the second sentence of Proclus' "proof" is essentially an axiom he ascribed to Aristotle. It was not a postulate of the Elements, nor did Proclus attempt to derive it from Postulates I–IV. But a deeper criticism of Proclus' argument concerns the tacit assumption embodied in the phrases "the interval between the parallels" and "the distance between the parallels." Those phrases imply that parallel lines are at a constant distance from one another, but the justification of this implicit assumption is Postulate V itself, which is, indeed, its logical equivalent! For an argument based on Aristotle's axiom it would suffice that the perpendicular distances from points on AB to CD be merely bounded, but this also is equivalent to Postulate V.
Bernhard Friedrich Thibaut (1775–1832) employed a method of "demonstrating" the fifth postulate (published in his book, Grund-riss der reinen Mathematik, 3 ed., Göttingen, 1818) that turned away from properties of parallels (which had been featured in many earlier attempts) and proceeded in a novel manner.
It is shown in the Elements (using the fifth postulate) that the sum of the angles of any triangle equals two right angles, and it was known in Thibaut's day that, conversely, the fifth postulate could be proved if it was assumed that a single triangle exists with angle–sum equal to two right angles. This had been established by the French mathematician, A. M. Legendre, prior to 1823. Thibaut's approach to the fifth postulate was by way of Legendre's angle-sum theorem, and he reasoned as follows:
Rotate the triangle ABC about vertex A through the angle CAC', then rotate triangle AB'C' about point B through the angle ABA?, and finally rotate triangle A"B"C? about point C through angle BCA'".
As a result of these three rotations, triangle ABC has been turned through four right angles, and hence, according to Thibaut,
[MATHEMATICAL EXPRESSION OMITTED]
Since each of the three angles in the left member of this equality is an exterior angle of triangle ABC and is, consequently, equal to two right angles minus the corresponding interior angle of that triangle, substitution in () gives the angle-sum of triangle ABC equal to two right angles.
If, as a result of the three rotations, triangle ABC had been rotated about a single point into itself, Thibaut's demonstration would be acceptable. But the triangle has been translated along line l (through the distance a + b + c) as well as rotated through four right angles and the argument takes no account of this translation. There is no justification for this in Postulates I–IV, and, indeed, it may be shown that assuming every motion may be resolved into a rotation and a translation independent of it is equivalent to assuming Postulate V itself! Thibaut's procedure can be carried out for spherical triangles, and in that case it is clear that the translation must be taken into account, since it is, in fact, a rotation of the triangle about the pole of one of its sides.
I.4. Saccheri's Contribution
The most elaborate attempt to prove the fifth postulate, and the most far-reaching in its consequences, was made by the Italian priest, Girolamo Saccheri (1667–1733), who taught mathematics at the University of Pavia. His great work, Euclides ab omni naevo vindicatus sive conatus geometricus quo stabiliuntur prima ipsa geometriae principia, was published in 1733. It apparently failed to attract much attention, for it was soon forgotten, and there is no evidence that Gauss, Bolyai, or Lobachewsky — the founders of non-euclidean geometry — ever heard of the book or its author. But Saccheri's work establishes him as an important contributor to the development of that subject.
As stated in the title of the book, Saccheri's aim was to free Euclid from all error — in particular (and most important), from the error of having assumed the fifth postulate. His novel procedure for accomplishing this introduced an important figure into geometry: the Saccheri quadrilateral. Saccheri erected, at the endpoints A, B of a segment, equal segments AC and BD, each perpendicular to segment AB, and joined points C, D by a straight line. On the basis of Postulates I–IV it is easily proved that [angle] ACD = [angle] BDC, for if P, Q denote the midpoints of segments AB, CD, respectively, the two right triangles ACP and BDP are congruent (Proposition 4, Book I), so [angle] ACP = [angle] BDP, and side PC = side PD. Then the sides of triangle CPQ are equal, respectively, to the sides of triangle DPQ, and, consequently, these two triangles are congruent (Propositions 4, 8, Book I). (These propositions are proved without the aid of the fifth postulate, which was invoked by Euclid for the first time in the proof of Proposition 29, Book I.) It follows that [angle] PCD = [angle] PDC, and, consequently,
[angle] ACD = [angle] ACP + [angle] PCD = [angle] BDP + [angle] PDC = [angle] BDC.
Calling the equal angles at C and D the summit angles of the Saccheri quadrilateral, the following three possibilities are exhaustive and pairwise mutually exclusive: (1) the summit angles are right angles, (2) they are obtuse angles; (3) they are acute angles. Saccheri called these the right-angle hypothesis, the obtuse-angle hypothesis, and the acute-angle hypothesis, respectively, and he proved that if any one of these hypotheses were valid for one of his quadrilaterals, it was valid for every such quadrilateral. Using the infinitude (unbounded length) of the straight line, he showed that the fifth postulate is a consequence of the right-angle hypothesis and that the obtuse-angle hypothesis is self-contradictory. There remained only to dispose of the acute-angle hypothesis!
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