Philosophical models of cosmos of Pythagoras and Philolaus: from Antiquity until the beginning of Modern Age

This article determines the fundamental principles of the models of the Cosmos of Pythagoras of Samos (c. 570 – 500 BC) and Philolaus of Croton (c. 470 – c. 388 BC). The perception of Cosmos as “beauty” and “harmony” – one of the basic characteristics of Pythagorean approach towards cognition of the world; it “interweaves” with the rational perception of reality. The harmony of beauty is transformed into the harmony of numerical relations. The achievements of Pythagoreans, subsequently become one of the foundations of Plato's astronomical texts, who describes cosmology as exact scientific discipline. Nicolaus Copernicus resorts to Philolaus as his major predecessor. This article is first to analyze the symbolic elements of Philolaus’ model of Cosmos from the perspective of modern scientific knowledge. Based on the conducted analysis, the author advances a hypothesis on the noematic nature of the elements of Philolaus’ model of Cosmos, as well as indicates the significance of transposing the methods of practical geometry onto the theoretical fields of “celestial” space, independent from the direct measurements. The article describes the key principles of the model of the universe of Pythagoras of Samos and Philolaus of Croton; discusses reconstruction of Philolaus’ model of Cosmos by Ivan Nikolaevich Veselovsky and Sergey Viktorovich Zhitomirsky. Analysis is conducted on the continuity of the principles of the models of Cosmos of Philolaus and Nicolaus Copernicus.

From Practical to Pure Geometry and Back

The purpose of this work is to address the relation existing between ancient Greek (planar) practical geometry and ancient Greek (planar) pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically explicit in definitions, like that of segments (straight lines) in Euclid‘s Elements. Then, we will address how in pure geometry we, so tospeak, ―refer back‖ to practical geometry. This occurs in two ways. One, in the propositions of pure geometry (due to the accompanying figures). The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment.

The Liber Theoreumacie (1214) and the Early History of the Quadrans Vetus

The Liber theoreumacie is a neglected work of practical geometry, written in Strasbourg in 1214, which sheds valuable light on the study and practice of astronomy in early thirteenth-century Europe. In this article, I focus on the first two chapters of Book IV, which both deal with the construction of horary instruments. The first of these chapters contains the earliest known account of the type of universal horary quadrant known as quadrans vetus, which is here given a biblical pedigree by labelling it the “sundial of Ahaz.” The second chapter describes a graphical method of inscribing hour markings on the surface of an astrolabe’s alidade, which appears to have been introduced into Latin Europe by the twelfth-century translator John of Seville. A critical edition and translation of the relevant passages will conclude the article.

From experimental to theoretical geometry in new pedagogical movements at the turn of the 19th and 20th centuries (1872-1906)

There exist many historical works on the new pedagogical movements in the beginning of the 20th century, at the level of one country and at the international level also. Our purpose is to focus on teaching of geometry with comparing situations in four countries: United Kingdom, France, Germany and United States. We show that, behind the agreements, there are deep differences in relation with questions posed by geometrical teaching. We use two kinds of materials, discussions and textbooks, and we specially examine the questions on parallels definitions and their introduction in teaching. Keywords: laboratory method, concrete geometry, experimental geometry, intuitive geometry, practical geometry, rational geometry, Émile Borel, Carlo Bourlet, John Dewey, George Halsted, Julius Henrici, Adelia Hornbrook, Jules Houël, Charles Méray, Eliakim Moore, John Perry, Peter Treutlein.

Sulaymân al-Harâ’irî (1824-1877): his attempts to reconcile the Islamic civilization with modern science and mathematics education

Who was Sulaymân al-Harâ’irî? Little is known about this enigmatic and controversial scholar. Born in Tunis in 1824, he settled in Paris in 1856 and died there in 1877. His unpublished manuscripts contain many translations into Arabic of French books, several of which are pertinent to mathematics. He translated arithmetic textbooks by Hippolyte Vernier, a surveying textbook by G. Frédéric Olivier and La Lande’s survey on uses of logarithms. He also drew up the plan for a comprehensive treatise on practical geometry, thus apparently laying the foundation for a Euro-Islamic hybrid mathematical knowledge. Keywords: Tunisia, Arabic language, translation, hybridization, arithmetic, logarithms, practical geometry

Dwa oświeceniowe wykłady geometrii — dyskursywny obraz świata

Two Enlightenment lectures on geometry: A discursive worldviewThis article focuses on the history of the Polish educational and scientific discourse. The paper examines Polish texts by Józef Czech — Euklidesa początków jeometryi xsiąg ośmioro, to iest sześć pierwszych, jedenasta i dwunasta z dodanemi przypisami i trygonometrią dla pozytku młodzi akademickiey Euclid’s six books on geometry, that is the first six books, the eleventh book and the twelfth book with notes and geometry for the benefit of university youth, 1807 and by Ignacy Zaborowski — Jeometria praktyczna Practical geometry, 1786.The science of the 19th century is an area for gaining empirical, theoretical, and practical knowledge of the world. The paper discusses this problem in the context of methods which can be employed to analyze the image of the world. The author of the article presents the concept of discursive worldview, which is meant to enable the description of dynamic profiling of meanings in educational and scientific discourse.

A Hybrid Obstacle-Avoidance Method of Spatial Hyper-Redundant Manipulators for Servicing in Confined Space

SummaryDue to a large number of redundant degrees of freedom (DOFs), the hyper-redundant manipulator shows outstanding dexterity and adaptability in avoiding the obstacles in confined space. In this paper, a hybrid obstacle-avoidance method of spatial hyper-redundant manipulators is proposed, with both efficiency and accuracy considered. The space around an obstacle is classified into safe, warning, and dangerous zones. A two-level protection strategy is then addressed to handle the obstacle-avoidance problem from qualitative (i.e., pseudo-distance based on super-quadric function) and quantitative (i.e., Euclidean distance based on practical geometry function) perspectives, respectively. The only condition for switching between the two-level protections is the value of pseudo-distance. Then, a modified modal method, which is a trajectory planning method, is presented to plan the collision-free trajectory of the manipulator by maximizing the minimum pseudo-distance or Euclidean distance in different zones. Some parameters, including the arm-angle parameters and the equivalent link length parameters, are defined to represent the manipulator configuration. They are adjusted to avoid the obstacle, singularity, and joint limit. The simulations of 12-DOF manipulator and an experiment of 18-DOF manipulator verify the proposed methods.