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Euclid's Elements is a work written by an ancient Greek mathematician named Euclid in the 3rd century BC. It is one of history's most famous and influential mathematical works, serving as a significant basis for studying geometry and number theory for centuries. This research aims to describe Euclid's Elements and determine the extent of high school students' knowledge about Euclid and his work and their level of understanding in solving mathematical problems. This qualitative research utilizes case study methods and literature reviews. Data were collected through problem-solving exercises and interviews. Thematic data analysis was conducted to elucidate Euclid's Elements and students' level of knowledge. The case study involved presenting mathematics problems and interviewing five high school students, using assessment instruments incorporating problem-solving and open interviews. The findings reveal that many students lack familiarity with the elements of Euclid's Elements but demonstrate proficiency in solving mathematical problems related to it.

Building on previous scholarly work on the mathematical roots of assertoric syllogistic we submit that for Aristotle, the semantic value of the copula in universal affirmative propositions is the relation of divisibility on positive integers. The adequacy of this interpretation, labeled here 'arithmetical dictum', is assessed both theoretically and textually with respect to the existing interpretations, especially the so-called 'mereological dictum'. [ABSTRACT FROM AUTHOR]

Since recent datings of the Bakhshali manuscript confirm our understanding that the zero is indeed an Indian creation, what is it in Indian culture that could have influenced an invention of this kind? From data relating to the Bakhshali manuscript, it is possible to recognize the influence of Buddhism and, from this, to try to see how this movement could have played a role in the theory leading to the inception of the placeholder zero. Moreover, with Brahmagupta this first form and use of zero was developed to the point of becoming a number, and thus adopting an operative role. While the writer of the Brāhmasphuṭasiddhānta was not a Buddhist, it is still possible to show in that document how Buddhist thinking, particularly its speculative developments in the meaning of śūnyatā as described by the great writers of Mahāyāna Buddhism, may provide a favorable setting for theories about the numerical concept of zero, even to the point where it is perceived as a condition of its 'thinkability' or conceptualization. [ABSTRACT FROM AUTHOR]

In the mid-first century BC Geminus of Rhodes, a scientist and philosopher close to Posidonius, composed a comprehensive Theory of Mathematical Sciences, in the surviving fragments of which the numerous characters are referred to plainly by name, with some of them being namesakes of other, more well-known mathematicians and philosophers. This paper tries to set apart the namesakes of Geminus, of which there are four in his fragments: Theodorus, Hippias, Oenopides, and Menaechmus. [ABSTRACT FROM AUTHOR]

Did science, as we know it, have to be? The article explores a possible response in the negative, organized around a specific contingency: that of Greek mathematics or, even more specifically, that of the mathematics of the generation of Archimedes. The argument is that (1) Greek mathematics, seen against a cross-cultural comparison, is an anomaly, (2) the scientific revolution, as it in fact unfolded, was directly shaped by the anomaly of Greek mathematics, and (3) it is not clear that, absent Greek mathematics, an equivalent scientific revolution would have taken place. The argument is developed in some detail concerning the history of ancient Greek science, but more is said on the inevitably philosophical questions of counterfactuals in history and on the specific question of the contingency of science. [ABSTRACT FROM AUTHOR]

Archimedes' Sand-Reckoner presents a system for naming extraordinarily large numbers, larger than the number of grains of sand that would fill the cosmos. Curiously, Archimedes addresses the treatise not to another specialist but to King Gelon II of Syracuse. While the treatise has thus been seen as evidence for the dynamics of patronage, difficulties in both Archimedes' treatment of Gelon and his discussion of astronomical models make it fit incongruously within contemporary court and scientific contexts. This article offers a new reading of the Sand-Reckoner based on a reconsideration of the relationship between author and addressee: deferring assumptions about the historicity of that relationship, it analyzes Gelon's role in the treatise with respect to both the stylistic features of Archimedes' prose and a broader tradition of narratives about a variety of cultural actors who engage with kings, speaking not so much truth as wit to power. Such a reading resolves the social and scientific difficulties of the treatise, and develops the literary-experimental qualities of Hellenistic science. In turn, the article proposes a revised approach, sensitive to broader patterns of authorship, to understanding ancient scientific authors' relationship to royal authority. It concludes, finally, that the royal patronage seemingly exemplified by the Sand-Reckoner had greater significance as a cultural trope than as a social institution. [ABSTRACT FROM AUTHOR]

Argument: The Fourth Postulate of Euclid's Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. [ABSTRACT FROM AUTHOR]

The article focuses on Euclid's Elements, a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria. It mentions Badiou's theory of points is less a kind of naive liberalism or existentialism; and also mentions Saussure's wafer-shape diagram of the sign is the signifier and signified forming two halves of a circle.

This paper aims to provide an updated synthesis on the works of Archimedes and the fundamental impact these have had on subsequent mathematical practice. The influence his mathematical processes have had on modern mathematics and how these have helped develop the field is discussed in historical perspective. Some of the recent investigations into the Archimedes Palimpsest are discussed and synthesized, namely, how they alter our understanding of some of his earlier works, and how Archimedean principles are seen to have laid the foundations of possible new branches of mathematics. [ABSTRACT FROM AUTHOR]

This article focusses on the generality of the entities involved in a geometric proof of the kind found in ancient Greek treatises: it shows that the standard modern translation of Greek mathematical propositions falsifies crucial syntactical elements, and employs an incorrect conception of the denotative letters in a Greek geometric proof; epigraphic evidence is adduced to show that these denotative letters are 'letter-labels'. On this basis, the article explores the consequences of seeing that a Greek mathematical proposition is fully general, and the ontological commitments underlying the stylistic practice. [ABSTRACT FROM AUTHOR]

The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied by many interpreters in seeking for specific logical rules at work in ancient mathematical texts. [ABSTRACT FROM AUTHOR]

Mathematical writers, above all, Euclid, tend to present their theorems as decontextualized, abstract propositions, which has become the standard modus of textual presentation in theoretical mathematics. Mathematical commentators, however, provide their readers with personal names and historical facts in order to elucidate problems, provide contexts of discovery, or construct doxographies, among other things. Modern readers have used such information for the construction of histories of science. When we look at these passages, however, we see that personal names and information about mathematicians of the past can serve quite a range of different objectives, such as the strategic self-positioning of the commentator vis-à-vis the past or present of mathematics, the education of the reader, mathematical or moral, the construction of the history of the field, etc. Not only does the commentator present a persona of himself to the reader, he can also turn colleagues and predecessors into personae. This paper attempts to elucidate the practice, by offering four examples of such plays of and with personae , in Pappus, Eutocius, al-Nayrīzī, and Proclus. [ABSTRACT FROM AUTHOR]

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disciplines that since the fourth century B.C.E. had been called mathēmata or mathēmatikai epistēmai. Among the mathēmata, which in antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics, and optics, the essay also includes geography. Using a data set based on The Encyclopaedia of Ancient Natural Scientists, it considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for the purposes of this essay), from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal, and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities—ancient scientists and philosophers—allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major sites—Athens, Alexandria, Rhodes, Rome, and Byzantium/Constantinople—that appeared, in succession, as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society. [ABSTRACT FROM AUTHOR]

The first time you encounter a medieval manuscript of a Greek mathematical or astronomical work, like those of Archimedes, Euclid, or Aristarchus, the most impressive feature is the odd configuration many diagrams show. There is a tendency to represent more regularity among the geometric objects than what the argument demands and usually they are not accurate graphical depictions of the mathematical object discussed in the text. Most scholars believe that these tendencies go back to the Greek authors themselves. In this paper, I propose a different explanation: the odd characteristics should not be attributed to Greek authors, but to transmission. [ABSTRACT FROM AUTHOR]

This paper is a contribution to our understanding of the technical concept of given in Greek mathematical texts. By working through mathematical arguments by Menaechmus, Euclid, Apollonius, Heron and Ptolemy, I elucidate the meaning of given in various mathematical practices. I next show how the concept of given is related to the terms discussed by Marinus in his philosophical discussion of Euclid’s Data. I will argue that what is given does not simply exist, but can be unproblematically assumed or produced through some effective procedure. Arguments by givens are shown to be general claims about constructibility and computability. The claim that an object is given is related to our concept of an assignment—what is given is available in some uniquely determined, or determinable, way for future mathematical work. [ABSTRACT FROM AUTHOR]

In this paper, we will examine some modes of reference to mathematical entities used in Greek mathematical texts. In particular, we examine mathematical texts from the Early Greek period, the Euclidean, Neo-Pythagorean, and Diophantine traditions. [ABSTRACT FROM AUTHOR]

We survey the work of Roshdi Rashed, the Egyptian-French historian of mathematics. Surveying Rashed's work gives an overview of the most important part of Greek mathematics that was transmitted to us in Arabic, as well as of the finest pieces of Arabic mathematics that survive. [ABSTRACT FROM AUTHOR]

The aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. [ABSTRACT FROM PUBLISHER]

The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis. [ABSTRACT FROM AUTHOR]

The present study explores reasoning and argumentation in Greek mathematics and physics texts in specific topics related to the notion of periodicity. In our study, argumentation is taken as the sequence of the modes of reasoning (MsoR) that an author develops in a text when organizing and presenting new knowledge. Inductive content analysis was applied on 71 thematic units taken from 4 mathematics and 4 physics textbooks, and a coding system of categories and subcategories of MsoR was produced. Our analysis discerned 4 main categories of MsoR: empirical, logical-empirical, nomological, and mathematical; we argue that each mode of reasoning (MoR) plays a different role in conceptualizing aspects of periodicity. Analysis of the sequence of MsoR in two thematic units raised pragmatic considerations on the text understanding in relation to the scientific argumentation discourse and highlights ontological differences in the two subjects when ascending from observations to generalizations. Educational implications of the findings are discussed. [ABSTRACT FROM AUTHOR]