La primera sección (páginas 253-260) de
I.M.Yaglom, Elementary Geometry, Then and Now, en
Chandler Davis, Branko Grünbaum and F.A. Sherk (editors)
The Geometric Vein, The Coxeter Festschrift
(Springer, New York, 1981), pp.253-269
Las otras secciones:
2. Discrete Mathematics and Discrete Geometry
3. Combinatorial Geometry: The Elementary Geometry of the Second Half of the 20th Century
1. Elementary Geometry of the 19th Century
What is elementary geometry, and when did it originate? The first of these questions -the content of elementary geometry- is not at all simple, and a clear-cut answer is not possible. The most natural answer for present purposes
would be the following: «Elementary geometry is the collection of those geometric concepts and theorems taken up in secondary school, together with immediate consequences of these theorems.» However, in spite of the seeming simplicity of this answer, it raises at once a host of objections. The appeal to the word «geometric» in the definition is in itself hard to interpret, since the question «what is geometry?» also admits no clear-cut answer (on that, more below); but in any case, the rapid rate of change in school curricula in all countries of the world, currently seeming to reach its maximum, would oblige us if we adopted that definition to accept the existence of indefinitely many elementary geometries. The concept would have to change not merely from country to country, but for each given country also from year to year if not even from school to school. In addition, such a definition clearly refers only to the content of the school subject «elementary geometry,» while we are here asking about the content of the corresponding science or, since the word «science» here may seem pompous, about the corresponding direction of scientific thought.
However, the difficulty of defining the notion of «elementary geometry» does not at all take away our right to use the term. Thus in the first half of this century much discussion surrounded consideration of the term «geometry.» The first general definition of geometry, given in 1872 by the outstanding German mathematician Felix Klein (1849-1925) in his «Erlanger program,» proved not to be applicable to the whole range of subdivisions of geometry -in particular to those which at that period attracted the most attention from mathematicians and physicists. But no substitute for it could be found. In this connection the eminent American geometer Oswald Veblen (1880-1960) proposed in 1932 that geometry be confined by definition to that part of mathematics which a sufficient number of people of acknowledged competence in the matter thought it appropriate so to designate, guided both by their inclinations and intuitive feelings, and by tradition. This «definition» is frankly ironic; yet for many years it stood as the only one generally accepted by scholars, and scientific articles and studies were devoted to defending it (less to analysis of Veblen’s «definition,» of course, than to demonstrating the impossibility of any other). We propose to follow this example, calling elementary geometry that portion of geometry which is recognized by a sufficiently large number of experts and connoisseurs as meriting the title.
With this understanding, it is clear that elementary geometry is the study of a multitude of properties of triangles and polygons, circles and systems of circles -quite nontrivial and in part entirely unexpected properties, set forth in specialized treatises on the subject (for instance, [1]), and well known only to a small number of specialists in the field (among whom, by the way, the author of these lines does not presume to include himself). The specialists are few, just as are the serious specialists in any sufficiently extensive and far advanced domain of knowledge: say, in postage-stamp collecting or algebraic K-theory.
Let us illustrate this for a not too large group of theorems fairly characteristic of «classical elementary geometry» -or, since the adjective «classical» here refers not to musty antiquity but to a relatively recent past on the scale of human history, of «elementary geometry of the 19th century.» Consider an arbitrary quadrilateral Delta not a trapezoid, whose sides are the four lines a1, a2, a3, a4. (Or we may simply mean by Delta a quadruple of lines ai, no two parallel, and no three passing through any single point.) Taken three at a time, these lines form four triangles T1, T2, T3, T4. Then the points of intersection of altitudes (orthocenters) of our triangles Ti lie on a line s (sometimes called the Steiner line of the quadrilateral Delta after the famous Swiss Jacob Steiner (1796-1863); the midpoints of the diagonals of A and the midpoint of the segment joining the points of intersection of its pairs of opposite sides, lie on another line g (this was discovered by the great Karl Friedrich Gauss (1777-1855), in whose honor g is called the Gauss line of Delta; here always s is perpendicular to g. Further, the circles circumscribed about the triangles Ti intersect in a single point C (the letter referring to the Englishman William Kingdon Clifford (1845-1879), in whose honor it would be appropriate to call C the Clifford point of Delta); the feet of the perpendiculars dropped from C on the sides of Delta lie on a line w, which might be called the Wallis line of the quadrilateral after one of Newton’s predecessors, the Englishman John Wallis (1616-1703). Also the nine-point circles of the Ti, which pass through the midpoints of the sides of these triangles, intersect in a point E, which we may call the Euler point of Delta after another Swiss, the renowned Leonhard Euler (1707-1783).
Consider now a pentagon II with sides a1, a2, a3, a4, a5. The five quadruples of lines (a1, a2, a3, a4), . . . , (a2, a3, a4, a5) describe five quadrilaterals A5, A4, A3 A2, A1,. The Gauss lines g5, . . . , g1, of our quadrilaterals intersect in a point G (the Gauss point of III); their Clifford points C5,…,C1, lie on a circle c (the Clifford circle of II); in case the pentagon II is inscribed in a circle one can also define the concepts of Euler point and Wallis line of II (see for example [2, Chapter II, Section 8]); and this array of theorems may be much extended (see [3, Chapter 5]).
Having given this answer to the question of the content of elementary geometry, we may pass to the second of the questions posed, on the date of its origin. To aficionados of elementary geometry the answer to this question is well known, but others may find it a bit surprising: the science of triangles and circles -elementary geometry- was founded in the 19th century. What? no earlier? not in ancient Greece? I hear the doubting questions of the reader not too well informed on the history of mathematics -not by the great Euclid and Archimedes, but by some unknowns or other living no more than a hundred years ago? Yes, is my reply, even less than a hundred years ago; for the central body of elementary geometric theorems known today were discovered in the last third of the 19th century and (to a lesser extent) the first decade of the 20th.(see (1))
The point is that the giants of ancient Greek mathematics (and maybe this is exactly why they deserve to be called giants) seem not to have included anyone seriously concerned with elementary geometry. The great Euclid (around 300 B.C.) was the author of the first textbook of (elementary) geometry that has come down to us (and what a remarkable textbook it is! -which may have something to do with the disappearance of the texts which preceded it). However, Euclid’s personal interests, and apparently also his personal contributions, seem to have lain in other areas (possibly in the study of numbers rather than figures: think of the famous Euclidean proof of the infinitude of primes). So limited was Euclid’s knowledge of the theory of triangles that he did not even know the elementary theorem on the point of intersection of the altitudes, which Albert Einstein so prized for its nontriviality and beauty. The mighty Archimedes (3rd century B.C) was one of the founders of (theoretical or mathematical) mechanics, and one of the progenitors of modern «mathematical analysis» (calculus); but to the triangle, and the points and circles associated with it, he gave little attention. Apollonius of Perga, the younger contemporary of Archimedes, was deeply versed in all possible properties of conic sections (ellipse, parabola, and hyperbola) -but not of triangles and circles. Finally, the last of the great ancient Greek mathematicians, Diophantus of Alexandria (most likely 3rd century A.D.), was interested only in arithmetic and number theory, not in geometry.
Thus in the domain of elementary geometry, as the term is traditionally understood, the knowledge accumulated in ancient Greece was not especially profound; nor was any great progress made there in subsequent centuries, right up to the 19th. In the 19th century, on the other hand, especially the second half, through the work of a multitude of investigators, an appreciable portion of whom were secondary-school mathematics teachers (see 2) a number of striking and unexpected theorems were discovered, an idea of which is given by those set forth above. These theorems were collected in many textbooks of elementary geometry (like the books [1] and [4]) or more narrowly of geometry of the triangle (cf. [5]) or «geometry of the circle» (see [6]), most of which appeared at the end of the 19th or the first third of the 20th century.
The following fact may serve to substantiate this account. At the turn of the 20th century F. Klein conceived the grandiose project of publishing an Enzyklopedie der mathematischen Wissenschaften, which he envisaged as encompassing the whole accumulation to that time of knowledge of pure and applied mathematics. Klein set a lot of activity in motion on the project; he succeeded in enlisting a broad collective of leading scholars from many countries, and in getting out a work of many volumes, which now takes up more than a shelf in many a major library. To be sure, this project was never brought to a conclusion (it grew clearer and clearer that with the passage of time the quantity of material «not yet» included was not diminishing but increasing, for the growth of the «Encyclopedia» was being far outstripped by the progress of science), and now it has long been hopelessly out of date. To prepare the article on elementary geometry for this publication, Klein assigned the German teacher Max Simon, who enjoyed the reputation of being the strongest expert in this area. Subsequently, however, Klein decided against including a section in the Encyclopedia on elementary geometry, rightly considering that this area of knowledge, having more pedagogical significance than scientific, was out of place in a strictly scientific work. As a result, Simon’s survey, which aspired to encyclopedic fullness of coverage of all that was known on elementary geometry at the beginning of the 20th century, had to be published as a separate book; this work [71 is still much prized by specialists and lovers of elementary geometry. In the foreword to his book M. Simon saw fit to emphasize that it dealt with the development of elementary geometry in a single century, the 19th, that it had become clear in the course of preparing the book that a complete survey of all that had been done in elementary geometry essentially coincided with what had been done in the last century.
Thus the 19th is the «golden age» of classical elementary geometry. The flowering of the study of triangles, circles, and their relationships did extend into the beginning of the 20th century, involving some of the prominent mathematicians of that time (for example, Henri Léon Lebesgue (1875-1941), who brought out a book of geometric constructions with circles and lines, and who had curious results on the so-called theorem of F. Morley on the trisectrices of a triangle -on which see [4]). But by about the end of the first quarter of this century one notes a definite falling off of interest in this area. To be sure, broad treatises appear as before on elementary geometry (as on the «geometry of the tetrahedron»), and journals are published devoted entirely or primarily to it (see (3))
Still it becomes noticeable that general interest in this part of geometry is lessening. Witness the almost complete disappearance of publications on this subject in serious mathematical journals, and of talks on the subject by eminent scientists at major conferences and congresses -both of which in the 19th century were almost the norm. And this relative decline of elementary geometry was not at all related to the exhaustion of the subject matter, since new theorems on elementary geometry -frequently still quite striking and unexpected- continued to be discovered; clearly some other, deeper circumstances must be involved.
In order to identify some of the causes both of the flowering and of the subsequent eclipse of classical elementary geometry, we will have to turn to some general laws of scientific development which are sometimes hard to formulate but are easy to observe and in principle fully explainable. It should be noted first of all that the keen interest in the study of triangles, quadrilaterals, and circles which we see throughout the 19th century was by no means an isolated phenomenon; it was intimately related with the flourishing in this period of so-called synthetic geometry, i.e., geometry based not on analytic devices involving the use of one or another system of coordinates, but on sequential deductive inference from axioms (see (4)). Synthetic geometry in this period was not studied just as an end in itself: it stimulated a number of important general mathematical ideas. At the core of this preoccupation was the concept of the non-uniqueness of the traditional (or «school») geometry of Euclid, of the existence of an abundance of in some sense equally deserving geometrical disciplines, such as the hyperbolic geometry of Lobacevskii and Bolyai or projective geometry. They prepared the ground for serious general syntheses such as Klein’s «Erlanger program» mentioned above. All of this facilitated also the serious posing of the question of the logical nature of geometry (or even of all mathematics, since in the 19th century the subject of the foundations of mathematics was analyzed almost exclusively for geometry), giving rise to several systems of axioms for geometry which were elaborated by a number of investigators (foremost among them Italians and Germans: Giuseppe Peano, Mario Pieri, Moritz Pasch, David Hilbert) at the turn of the 20th century; this played a very large role in the development of 20th-century mathematics.
An especially prominent place in the development of synthetic geometry in the
19th century was occupied by projective geometry. I would go so far as to assert that not only did projective geometry in a well-known sense grow out of elementary geometry (this approach to projective geometry is emphasized in the book [9] addressed to beginners), but also 19th-century elementary geometry was in a significant sense produced by projective geometry -a circumstance which Felix Klein liked to point out, and which stands out especially when one analyzes the elementary-geometric work of eminent leaders of 19th-century mathematics like K. F. Gauss or J. Steiner. So, for example, all of the «geometry of the triangle,» with all the properties of the «special points» and «special circles» associated with a triangle, can be fitted neatly into the program of investigating projective properties of pentagons -the first polygons which have distinctive projective properties (quadrilaterals, being all «projectively equivalent,» can’t have individual projective properties). The transition from projective 5-gons to Euclidean 3-gons requires only the identification of two of the five vertices of the 5-gon with the «cyclic (ideal) points» whose fixing in the projective plane converts projective geometry to Euclidean. (See, for example, the classical book [10]; note that the conic section, which is the primary object of investigation in projective geometry, goes into a circle if it is required that it pass through the cyclic points.)
But then in the first half of the 20th century came a very palpable (though perhaps temporary) decline in synthetic geometry. «The Moor has done his work, the Moor may leave»: those general ideas and understandings referred to above, which had grown out of synthetic geometry, were now established, and synthetic geometry was no longer required. It is well known that the history of science exhibits ebbs and flows; if the 19th century was the golden age of geometry, then our times are distinguished by the preeminence of algebra, by the distinctive «algebraization» of all branches of mathematics reflected in the acceptance of Nicolas Bourbaki’s mathematical structures, converting even geometry virtually into a part of algebra In this situation it is not surprising that projective geometry, for instance, while retaining its position as an important part of the school geometry course (see, e.g., the books [111 and [12]), has in the strictly scientific domain undergone inconspicuously such a transformation that today algebraic questions play if anything a bigger role than geometric (see for instance the old but still popular text [13]). Now remembering also that the general «algebraization» of mathematics, putting algebraic structures as much as possible in the foreground, squarely posed the question of revision of school geometry courses, which many mathematicians and educators proposed to base on the (essentially algebraic!) concept of vector space -we see that there has been significant erosion even at the core of the one possible «application» of classical elementary geometry, its use in the teaching of mathematics in secondary school.But
A good illustration of the algebraization of geometry is provided by the popular axiomatic approach to geometry by Friedrich Bachmann [14], which gives priority to purely algebraic concepts (groups generated by their involutory elements). Another clear-cut example is the recent set of axioms of Walter Prenowitz (see [15]), specially suited to the analysis of ideas related to the notion of convexity: it permits the introduction of a novel «multiplication» of points, whereby the product A * B (or A B) of points A and B is to be thought of as the segment with endpoints A and B; this multiplication is commutative, associative, idempotent, and distributive over set-theoretic addition. One might also cite the booklet on polygons [16], so typical of current trends: if its title and the majority of its results make it a work on elementary geometry, yet the tools and methods used identify it rather as a book on general algebra (theory of lattices). I point out also that I was able to fill the book [2] with elementary-geometric material under cover of the purely algebraic nature of the techniques used (the theory of hypercomplex numbers or diverse algebras having elements of the form x + Iy, where x and y belong to R, and I^2=-1,0 or +1); but for that, the beautiful purely geometric constructions appearing in [2] would today interest nobody.
Especially surprising also to mathematicians of my generation is the transformation undergone before our eyes by topology. Whereas in our student years it was regarded as a part of geometry and worked extensively with intuitive visualization, modern (algebraic) topology by its tools and methods belongs not to geometry but to algebra. This revolution has driven out of topology several investigators of a more geometric turn of mind, even producing in isolated cases serious emotion trauma (see (5)).
Interesting in this connection is the position of Jean Dieudonné, one of the authoritative French mathematicians and a leader in the founding of the Bourbaki group. In accordance with the general orientation of this group, Dieudonné is flatly opposed to retaining in school mathematics teaching any trace whatever of classical elementary geometry, i.e., the «geometry of the triangle» and all its relatives and subdivisions. This despite the apparently incontrovertible connection between the high level of French mathematics and the traditions of instruction at the French lycées, where students were trained in the solution of subtle and quite complicated elementary-geometric problems (see for example the classic schoolbooks [17] and [18], the second of which is by one of the greatest mathematicians of the 20th century). Already in 1959, at a conference on the teaching of mathematics in Réalmont, France, Dieudonné rose and hurled the slogans «Down with Euclid!» and «Death to triangles!» -and he maintains his support for these slogans to this day. In numerous speeches on pedagogical topics Dieudonné has repeatedly expressed the wish that secondaryschool students (and teachers) forget as soon as possible the very existence of such figures as triangles and circles. Dieudonné’s idiosyncratically written book [19] (see especially its Introduction) is entirely devoted to advocacy of the following methodological idea: Elementary geometry is nothing but linear algebra -and no other elementary geometry ought to exist (to such a point that the book [19] on elementary geometry is quite without figures, and never mentions the word «triangle»). A different position is taken by A. N. Kolmogorov, whose geometry textbooks are currently in use by all secondary students in the U.S.S.R.; but even these textbooks are arranged in such a way that they almost completely lack substantial geometrical problems. (In this respect they are inferior to American school textbooks, which are generally based on the-also quite ungeometric-axiomatization of G. D. Birkhoff [20].)
Thus today «classical» elementary geometry has distinctly lost status. Note too that the subject has virtually disappeared from the problems proposed to Competitors in the international mathematical olympiads. But «nature abhors a vacuum» -and the place left vacant by «classical» elementary geometry has been taken with alacrity by the «new» or «contemporary» elementary geometry, the «elementary geometry of the 20th century,» to which we now turn.
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(1) Note in this connection that whereas Gauss, Steiner. and Clifford (all mathematicians of the 19th century) really knew the theorems associated with their name, the designations «Wallis line» (Wallis was a l7th-century mathematician) and «Euler point» (Euler lived in the 18th century) are rather a
matter of convention, for the corresponding theorems were not known to these authors (Wallis and Euler knew only simpler assertions related to those we have stated).
(2) Among them may be mentioned especially the short-lived Karl Wilhelm Feuerbach (1800-1834) (whose brother Ludwig became famous as a philosopher). To us today, K. W. Feuerbach appears as the classical representative of this movement. But the greatest scientists of the 19th century, like J. Steiner or even K. F. Gauss, were not at all disdainful of elementary-geometric research. (By the way, Steiner belongs to the intersection of M and T where M is the set of outstanding mathematicians and T is the set of school teachers.)
(3)Perhaps the publication of this type enjoying the greatest reputation was the Belgian journal Mathesis, appearing from 1881 on. This journal maintains its existence to this day, but the general falling off of interest in the subject it champions has taken its toll on the journal, and today few mathematicians and teachers have even heard of its existence. [Incomparably greater popularity is enjoyed at present by another journal, Nico, also published in Belgium and also directed primarily to teachers, which is in every way the exact opposite of Mathesis (the name Nico comes by abbreviation from the name Nicolas Bourbaki).]
(4) Typical of the preferences of that time was the flat prohibition against solving a construction problem by an algebraic method -a prohibition which teachers, preserving attitudes typical of the early 20th century, often took as so self-evident that they didn’t even express it. (I myself recall the time when a construction problem solved algebraically was often regarded as not solved, to the annoyance of pupils.) (For the relation between «geometric» solution of construction problems by such and such a choice of instruments prescribed in advance, and the axiomatic method in geometry, see, for instance, the book [8].)
(5) It is not at all simple to make a neat division between geometry and algebra; but I think it can be stated without qualification on the basis of contemporary physiological data that geometric representations («pictures») are among those which enlist the activity of the right half of the human brain, while (sequential) algebraic formulas are controlled by the left hemisphere. From this point of view, maybe people should be divided into natural geometers and natural algebraists according to the predominance in their intellectual life of one or the other hemisphere. Thus I would count Newton (and Hamilton) among geometers, whereas Leibniz (and still more Grassmann) belong rather to the algebraists. (The philological interests of Leibniz and Grassmann are noteworthy here, for it is known that everything related to speech and language relates to the left hemisphere; by contrast, the extramathematical interests of Newton ran to such sharply visual images of world culture as the Apocalypse.) Thus the simultaneous discovery of the calculus by Newton and Leibniz, or of vectors by Hamilton and Grassmann, were made, so to speak, «from different sides.»
