Geometry Revisited. 1967
.pdf
H. S. M. Coxeter and S. L. Greitzer
maa review (has remained in print for more than 40 years)
Coxeter (1907 – 2003) Coxeter is regarded as one of the greatest geometers of the 20th century. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. He met M.C. Escher in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works. Publish the full list of uniform polyhedra (1954). He worked for 60 years at the University of Toronto.
Greitze (1905 – 1988) the founding chairman of the United States of America Mathematical Olympiad.
E' una decina d'anni che mi sono reso conto che dimostrazioni che conoscevo, o che facevo con lo stile appreso a scuola (che credo sia alla fine, nonostante gli intermediari, quello degli Elementi di Euclide) venivano semplificate, diventavano lampanti usando le dilatazioni (o omotetie). Quindi ho solidarizzato con
"The modern techniques: the use of transformations such as rotations, reflections, and dilatations, provide shortcuts in proving certain theorems and also relate geometry to crystallography and art." (Dalla prefazione).
Il tono generale della prefazione e' un accorato appello a favore della geometria, ricorrendo a tutte le ragioni che e' riuscito a trovare, con amore ma anche con un velo di tristezza.
Come guerra interna ai matematici
d: quali sono le ragioni della sopraffazione algebra su geometria ?
He who despises Euclidean Geometry is like a man who,
returning from foreign parts, disparages his home.
H. G. Forder
The mathematics curriculum in the secondary school normally includes a single
one-year course in plane geometry or, perhaps, a course in geometry and
elementary analytic geometry called tenth-year mathematics.
This course, presented early in the student's secondary school career, is
usually his sole exposure to the subject. In contrast, the mathematically minded
student has the opportunity of studying elementary algebra, intermediate
algebra, and even advanced algebra. It is natural, therefore, to expect a bias
in favor of algebra and against geometry.
Moreover, misguided enthusiasts lead the student to believe that geometry is
"outside the main stream of mathematics" and that analysis or set theory should
supersede it.
Perhaps the inferior status of geometry in the school curriculum stems from a
lack of familiarity on the part of educators with the nature of geometry and
with advances that have taken place in its development.
These advances include many beautiful results such as Brianchon's Theorem
(Section 3.9), Feuerbach's Theorem (Section 5.6), the Petersen-Schoute Theorem
(Section 4.8) and Morley's Theorem (Section 2.9).
Historically, it must be remembered that Euclid wrote for mature persons
preparing for the study of philosophy. Until our own century, one of the chief
reasons for teaching geometry was that its axiomatic
method was considered the best introduction to deductive reasoning.
Naturally, the formal method was stressed for effective educational purposes.
However, neither ancient nor modern geometers have hesitated to adopt less
orthodox methods when it suited them. If trigonometry,
analytic geometry, or vector methods will help, the geometer will use them.
Moreover, he has invented modern techniques of his own that are elegant and
powerful. One such technique is the use of transformations such as rotations,
reflections, and dilatations, which provide shortcuts in proving certain
theorems and also relate geometry to crystallography and art. This "dynamic"
aspect of geometry is the subject of Chapter 4. Another "modern" technique is
the method of inversive geomtry, which deals with points and circles, treating a
straight line as a circle that happens to pass through "the point at infinity".
Some flavor of this will be found in Chapter 5. A third technique is the method
of projective geometry, which disregards all considerations of distance and
angle but stresses the analogy between points and lines (whole infinite lines,
not mere segments). Here not only are any two points joined by a line, but any
two lines meet at a point; parallel lines are treated as lines whose common
point happens to lie on "the line at infinity".
There will be some hint of the content of this subject in Chapter 6.
Geometry still possesses all those virtues that the educators ascribed to it a
generation ago. There is still geometry in nature, waiting to be recognized and
appreciated. Geometry (especially projective geometry)
is still an excellent means of introducing the student to axiomatics. It still
possesses the esthetic appeal it always had, and the beauty of its results has
not diminished.
Moreover, it is even more useful and necessary to the scientist and practical mathematician than it has ever been. Consider, for instance, the shapes of the orbits of artificial satellites, and the fourdimensional geometry of the space-time continuum.
Through the centuries, geometry has been growing. New concepts and new methods of procedure have been developed: concepts that the student will find challenging and surprising. Using whatever means will best suit our purposes, let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused.
the-foundations-of-euclidean-geometry-by-h-g-forder-1927 credits: >>>
