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Commentary to Book I of the Elements. Hartshorne and beyond

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Błaszczyk P., Petiurenko A.
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
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2021
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vol. 13
43-99
EN
(Hartshorne, 2000) interprets Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1–34 covering the foundations of Euclidean geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Fifth Postulate, we present a model of non-Euclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the field of hyperreal numbers R*.
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Euclid’s theory of proportion revised

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Błaszczyk P., Petiurenko A.
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
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2019
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vol. 11
37-62
EN
The theory of similar figures, as developed in school mathematics, emulates the theses of Euclid’s propositions included in book VI of the Elements. It does not, however, represent Euclid’s proof technique, i.e. proportions. The theory is usually developed within a metric space, with line segments having lengths, figures having areas, fractions simulating proportions, and the similarity scale being a real number. In like manner, in (Hilbert, 1902, ch. III), David Hilbert develops his own proportion theory to prove Euclid’s propositions VI.2 and VI.4. Yet, Hilbert applies proportion only to line segments, while applying similarity only to triangles. Thus far, no one has managed to develop it further to get Euclid’s proposition VI.31, which crowns the ancient Greek theory of similar figures. Although Robin Hartshorne, in (Hartshorne, 2000, ch. 5), suggests that Hilbert’s project to reinterpret book VI can be completed by applying a concept of the content of a figure, contents of figures are not considered as terms of Hilbert’s proportions. In this paper, we apply the area method as introduced in (Chou, Gao, Zhang, 1994) to reconstruct Euclid’s theory of similar figures, both his propositions, and the proof technique. Our interest is focused on proposition VI.1. It plays a crucial role in Euclid’s system, and yet, it is the most controversial proposition of book VI when viewed from the modern perspective. As the only proposition in book VI, it relies on comparing figures in terms of greater-lesser. Since Euclid’s system does not provide any criteria on how to decide whether one figure is greater then another, this relation relies on diagrammatic evidence rather than explicit mathematical rules. To bypass any reference to the greater-lesser relation, we adopt an axiomatic account of the area method introduced in (Janicic, Narboux, Quaresma, 2012), since it includes proposition VI.1 as an axiom. The plan of this paper is as follows: first, we introduce axioms of the area method, next, we present a model for these axioms. Then, we reconstruct exemplary propositions of book VI within the framework of the area method. Finally, we compare Euclid’s proposition VI.1 with the fundamental theorem of the area method, the so-called Co-side theorem.
PL
Teoria pola po raz pierwszy została opisana w pracy Chou, Gao, Zhang w 1994 roku. W kolejnej pracy (Janicic, Narboux, Quaresma 2012) zaprezentowano nowy system aksjomatów teorii pola i program przeznaczony do automatycznego dowodzenia twierdzen. W artykule chcemy przedstawić interpretację teorii pola w geometrii analitycznej na płaszczyznie kartezjanskiej R×R z porządkiem leksykograficznym. Również pokażemy nową metodę dowodzenia twierdzeń geometrycznych (szczególnie twierdzeń z ksiegi VI Elementów Euklidesa), w której pole trójkąta wystepuje w dowodach (szczególnie w proporcji) jako element pierwotny (wzór na pole trójkąta wprowadza się, jako aksjomat). Podobną metodę stosował Euklides na objektach geometrycznych bez użycia liczb. W omawianej teorii pole trójkąta jest liczbą, a twierdzenie VI.1 Elementów, podstawowe dla teorii Euklidesa, jest przyjmowane jako aksjomat. W artykule również omówimy mało znaną własność, która jest modyfikacją twierdzenia VI.1: w miejsce proporcji trójkątów o wspólnej wysokości, wykorzystuje proporcje trójkątów o wspólnej podstawie.
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Decoding Book II of the Elements

81%
Błaszczyk P., Mrówka K., Petiurenko A.
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
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2020
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vol. 12
39-88
EN
The paper is a commentary to the Polish translation of the Elements Book II, included in this volume.We focus on relations between figures represented and not represented on diagrams and identify rules which enable Euclid to bridge these two kinds of objects. Also, we argue that the main mathematical problem addressed in Book II is constructing a leg of a rightangled triangle, given its hypotenuse and the other leg. In proposition II.14, Euclid solves it through the construction called the geometric mean. We trace the problem in Book III and beyond the Elements: in Heron’s Metrica, Descartes’ La Géométrie, and modern foundations of mathematics. We show that Descartes, by novel interpretation of the Pythagorean theorem, provides a modern solution to this problem.
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