An Axiomatic Approach to Geometry (Geometric Trilogy, Volume by Francis Borceux

By Francis Borceux

Focusing methodologically on these historic points which are suitable to helping instinct in axiomatic techniques to geometry, the booklet develops systematic and smooth techniques to the 3 middle points of axiomatic geometry: Euclidean, non-Euclidean and projective. traditionally, axiomatic geometry marks the starting place of formalized mathematical task. it really is during this self-discipline that the majority traditionally recognized difficulties are available, the strategies of that have resulted in numerous almost immediately very energetic domain names of study, in particular in algebra. the popularity of the coherence of two-by-two contradictory axiomatic platforms for geometry (like one unmarried parallel, no parallel in any respect, a number of parallels) has resulted in the emergence of mathematical theories according to an arbitrary approach of axioms, a vital function of latest mathematics.

This is an interesting e-book for all those that train or research axiomatic geometry, and who're drawn to the heritage of geometry or who are looking to see a whole facts of 1 of the well-known difficulties encountered, yet now not solved, in the course of their reports: circle squaring, duplication of the dice, trisection of the attitude, building of normal polygons, building of types of non-Euclidean geometries, and so on. It additionally offers enormous quantities of figures that help intuition.

Through 35 centuries of the background of geometry, become aware of the start and stick to the evolution of these cutting edge principles that allowed humankind to enhance such a lot of features of latest arithmetic. comprehend a number of the degrees of rigor which successively validated themselves during the centuries. Be surprised, as mathematicians of the nineteenth century have been, whilst watching that either an axiom and its contradiction may be selected as a sound foundation for constructing a mathematical concept. go through the door of this exceptional global of axiomatic mathematical theories!

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Extra resources for An Axiomatic Approach to Geometry (Geometric Trilogy, Volume 1)

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4 Squaring the Circle 19 Fig. 10 The tradition also attributes to Hippocrates, around 430 BC, the following theorem, which is essential for studying the circle (see Fig. 10): The areas of two similar circular segments are in the same ratio as the squares of their bases. In view of the mathematical knowledge of the time, it is unlikely that Hippocrates knew a formal proof of this result. Of course the result applies in particular to the case of half circles, in which case the “base” is the diameter, from which we obtain at once: The areas of two circles are in the same ratio as the squares of their diameters.

However, to make this criticism is to miss the point: this proof was written in 350 BC. A more sensible comment is to underline the fact that the mathematicians of the time were already able to recognize the validity of a formal argument, independently of the picture supporting the reasoning. Both parts of Fig. 19 are indeed false, since, a point R distinct from Q, with the property indicated, does not exist. Thus Greek geometers had already discovered that one can write down a correct proof using a false picture.

25 substantial lead. When Achilles starts running, the turtle has already reached some point P1 ; but when Achilles reaches the point P1 , the turtle has advanced to a further point P2 ; Achilles continues his effort but when he reaches the point P2 , the turtle has advanced to a further point P3 ; and so on. So Achilles will never catch the turtle. This paradox, and many others, had deeply puzzled Greek geometers and philosophers. Of course again, the explanation in terms of convergent series was to come two thousand years later.

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