Book Of Euclid Research Articles

Jerónimo Muñoz’s Reception of Proclus’ In Euclidem: Philosophy of Mathematics and an Attempt to Prove the Parallel Postulate

Abstract The goal of this paper is to show – by way of a case study – how the contents of Proclus’ Commentary on the First Book of Euclid’s Elements were incorporated into university teaching in the sixteenth century. I analyse the impact of Proclus on the works of the Spanish mathematician and university professor Jerónimo Muñoz (ca. 1520–1591). In order to do so, I examine two manuscripts: Adnotationes in commentaria Procli super Euclidem (MS Vat. Lat. 6996), and Astrologicarum et geographicarum institutionum libri sex (MS Vat. Lat. 6998). I show that the contents of Proclus’ commentary pervade Muñoz’s mathematical writings and influence his mathematical ontology, his classification of mathematical disciplines, and the history and terminology of geometry that he adopts. Moreover, I expound on how Proclus’ text inspired Muñoz to maintain that the fifth postulate was a theorem, leading him to attempt a demonstration of it which pre-dates knowledge in the Latin West of Naṣīr ad-Dīn’s (1201–1274) previous attempt.

The Forgotten Gifts of Hermes: The Latin Reception of Proclus’ Commentary on Euclid’s Elements

The present work offers an account of the Latin reception of Proclus’ Commentary in the first book of Euclid’s Elements (In Euclidem). The overall goal of this paper is to offer a unified historical and philological map as a ground-tool for understanding the Latin reception of In Euclidem from the early 16th century to the 17th century, which includes manuscript sources hitherto overlooked. I will deal with the three known complete translations produced by Bartolomeo Zamberti (1473–1543), Giovanni Battista Gabia (c.1500–1590), and Francesco Barozzi (1537–1604), as well as with five fragmentary translations embedded in the works of Giorgio Valla (1447–1500), Jerónimo Muñoz (1520–1591), Conrad Dasypodius (1532–1600), Federico Borromeo (1564–1631), and Johannes Kepler (1571–1630). Each of these translations and the intentions of their authors are discussed in chronological order. Five of them (following the fragment chosen by Kepler) are partially edited in the Appendix.

On the notion of equal figures in Euclid

Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to add fifteen axioms about undefined relations "equal triangles" and "equal quadrilaterals" to replace Euclid's use of the common notions. In this paper, we offer definitions of "equal triangles" and "equal quadrilaterals", that Euclid could have given, and prove that they have the required properties. This removes the need for adding new axioms. The proof uses the theory of proportions. Hence we also discuss the "early theory of proportions", which has a long history.

Geometry and the Gods: Theurgy in Proclus’s Commentary on the First Book of Euclid’s Elements

Abstract The gods that guard the poles have been assigned the function of assembling the separate and unifying the manifold members of the whole, while those appointed to the axes keep the circuits in everlasting revolution around and around. And if I may add my own conceit, the centers and poles of all the spheres symbolize the wry-necked gods (τῶν ίυγγικῶν θεῶν) by imitating the mysterious union and synthesis which they effect; the axes represent the connectors (συνοχάς) of all the cosmic orders … and the very spheres are likenesses of the perfecting divinities (τῶν τελεσιουργῶν θεῶν) … (Proclus 1992a, pp. 74–5; 1873, pp. 90–91)1 Proclus’s Commentary on the First Book of Euclid’s Elements contains one of the most important discussions of the nature of mathematical objects and preserves a wealth of historical information about mathematics from antiquity. However, large sections of the text have been neglected by scholars since the text’s first publication in the sixteenth century. Proclus expounds at length on correspondences between mathematical objects and the gods. At times he uses the language of “theurgy,” the ritual practices that later Neoplatonists insisted were necessary for the human soul to ascend to the intelligibles. The contention of this article is that the Commentary, taken as a whole, concerns the practice of “inner theurgy,” mental rituals used by advanced students to raise the soul beyond its assigned bounds. This practice can be traced back to Plotinus and the very beginnings of Neoplatonism; however, the particular methods presented by Proclus are his own development. This interpretation of the Commentary is supported by Proclus’s works Commentary on the Parmenides and Platonic Theology. Parts of the Commentary that have been studied carefully by modern scholars, especially the theory of the mathematical phantasia or imagination, will be shown to be crucial elements in Proclus’s theory and practice of inner theurgy.

Introduction: The Idiosyncratic Nature of Renaissance Mathematics

Introduction: The Idiosyncratic Nature of Renaissance Mathematics

Newman on Pedagogical Practice

Newman on Pedagogical Practice Jane Rupert (bio) In "Elementary Studies," Newman provided a rare instance of pedagogical practice through dramatized oral exams in Greek and Latin and samples of a student's writing. "Elementary Studies" was first published in 1854 and 1855 in issues of The Catholic University Gazette to indicate to prospective students what they might expect on an entrance exam for admission to the new university's Faculty of Philosophy and Letters. Its placement in The Idea of a University after Newman's discussion of the liberal arts' philosophy of mind serves a new purpose by demonstrating how pedagogical practice is an instrument of this philosophy of mind and teaches students to think. However, Newman also indicated in his discussion of education that traditional pedagogical practice was imperiled by the empirical philosophy of mind of Enlightenment philosophers like Locke and Hume. For them, learning to think meant something quite different from its long tradition in the liberal arts. Newman's observations are of present, urgent interest. In recent decades, an empirical pedagogical practice rooted in this same Enlightenment theory of mind has come to exercise an almost complete hegemony in the elementary and secondary schools of countries like the United Kingdom, the United States, and Canada. As a Canadian high school teacher of French and English, it was the denaturing of pedagogical practice influenced by the reductionist Enlightenment idea of reason that first drew me to the study of Newman's works on education. In particular, I found a clarification of our present dilemma in his exposition of the foundational principle of the liberal arts as an exercise in two particular kinds of reason, a practice exercised in a continuous tradition since antiquity. As I examine Newman's demonstration of pedagogical practice, I shall begin with a consideration of the nature of these two kinds of reason cultivated in the liberal arts and their distinctive uses of language. Then, I shall examine how these two kinds of reason are exercised in "Elementary Studies." Finally, I shall investigate the aberration of reason represented by Locke that had already infiltrated the liberal arts in Newman's time and which now dominates pedagogy in our own time. [End Page 103] The liberal arts defended by Newman in The Idea of a University taught students to think in the two ways conveyed in the title of his University's Faculty of Philosophy and Letters. Students were taught to think through the kind of reason that understands the causes of things, represented in pedagogy in antiquity by Plato, and through the kind of reason essential to matters requiring deliberation, represented by Isocrates, a rhetorician. Plato represented the practice of considering ideas and principles that illuminate large fields of thought: that is, the Greek theoria, which Newman called philosophic or scientific reasoning. In this kind of reason, the principle articulated by Aristotle that justice is both distributive and retributive illuminates the practice of justice. Books of Euclid's geometry, texts in continuous use from 300 BCE to Newman's day, exercised this same kind of deductive reasoning from initial propositions to consequences. The empirical demonstrations that follow a hypothesis in the modern sciences follow the same pattern of reason. Similarly, in the study of languages the rules of grammar illuminate large areas of usage. Language associated with this kind of reason is characterized by a univocal clarity and accuracy of definition, which stand in contrast to the more subtle use of language in the second kind of reason exercised in the liberal arts. Isocrates, the second pillar of the liberal arts, represents the genus of deliberative reasoning as exercised in the art of rhetoric. Rhetoric locates arguments around an idea. It then organizes effectively the many facets of the argument that lead to decision through the consent of both mind and heart in matters like politics and large social questions. This same genus of reason includes personal judgments informed by converging probabilities, a process of reasoning common to pioneering work in science and to decisions in religious belief that Newman described in his Grammar of Assent. The same genus of reason is also implicit in literature in which recurring human...

Re-examination of the different origins of the arithmetical books of Euclid's Elements

A quick examination of the diagrams in the Greek manuscripts of Euclid's Elements shows that VII.28 is the only proposition which has a diagram with both horizontal and vertical lines, and it is quite similar to parallel propositions in Book X (X.15, 16). This leads to an audacious assumption that all the propositions of Book VII after it may have been added later, and their authenticity is examined by logical and linguistic analysis.In many of the propositions of Book VII after VII.28, we have found characteristics or idiosyncrasies which can be explained by the hypothesis of later addition. Moreover, some propositions can be interpreted as lemmata for conspicuous results in Book IX such as IX.20 (the number of prime numbers surpasses any given number), IX.36 (sufficient condition of a perfect number).While the prevalent view is that Book VII is a well-organized basic theory from which various arguments in the subsequent two books are developed, our examination suggests that Book VII itself contains heterogeneous components which are intended for specific arguments in subsequent books. Further investigation is needed to find out whether this heterogeneity in Book VII is the result of later intervention, or that of original compilation.

Proof-checking Euclid

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski’s formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid’s gaps and correcting errors. Euclid Book I has 48 propositions; we proved 235 theorems. The extras were partly “Book Zero”, preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid’s gaps, and partly just variants of Euclid’s propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid’s logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.

An Appreciation of the First Book of Spherics

The first three books of Euclid’s Elements provide the charming classical introduction to deductive mathematics with the geometry of the blackboard. A blackboard with straightedge and a pair of com...

Researchers find history in the diagrams of Euclid’s

The fourth book of Euclid’s Elements, a 2,300-year-old geometry text, includes directions for constructing a 15-sided polygon inside a circle. The first step is familiar to geometry students: Draw an equilateral triangle and a regular pentagon so their vertices touch the circle and the two shapes share one vertex. In addition to text directions, the Elements included diagrams that illustrated the method.

Geometrical First Principles in Proclus’ Commentary on the First Book of Euclid’s Elements

AbstractIn his commentary on Euclid, Proclus says both that the first principle of geometry are self-evident and that they are hypotheses received from the single, highest, unhypothetical science, which is probably dialectic. The implication of this seems to be that a geometer both does and does not know geometrical truths. This dilemma only exists if we assume that Proclus follows Aristotle in his understanding of these terms. This paper shows that this is not the case, and explains what Proclus himself means by definition, hypothesis, axiom, postulate, and the self-evident, and how geometry is a science that receives its principles from dialectic.

Maps of Utopia: H. G. Wells, Modernity, and the End of Culture

Maps of Utopia: H. G. Wells, Modernity, and the End of Culture

NEWTON AND PROCLUS: GEOMETRY, IMAGINATION, AND KNOWING SPACE

abstractI aim to clarify the argument for space that Newton presents in De Gravitatione (composed prior to 1687) by putting Newton's remarks into conversation with the account of geometrical knowledge found in Proclus's Commentary on the First Book of Euclid's Elements (ca. 450). What I highlight is that both Newton and Proclus adopt an epistemic progression (or “order of knowing”) according to which geometrical knowledge necessarily precedes our knowledge of metaphysical truths concerning the ontological state of affairs. As I argue, Newton's commitment to this order of knowing clarifies the interplay of the imagination and understanding in geometrical inquiry and illuminates how geometrical knowledge of space can lead to knowledge that space depends on and is related to God. In general, appreciating the Proclean elements of Newton's argument brings added light to the significance of geometrical inquiry for his general natural philosophical program and grants us insight into the philosophical grounding for the notion of absolute space that is presented in the Principia mathematica (1687).

Proclus

In the history of a specific idea there are moments of reception that radically alter a concept’s functioning, moments where its meaning is transplanted in such a way that an internal rupture emerges. In the case of imagination the Neoplatonist Proclus (410/412-485) seems responsible for such a disruptive reception. Proclus is generally held to be the first to have systematically cut imagination’s exclusive ties with sense-data in his projectionist philosophy of mathematics, presented in his Commentary of the First Book of Euclid’s Elements (Bouriau 2002, 49). Proclus’ theory of geometry consists in a frontal attack on the traditional, Aristotelian viewpoint that geometry deals with sensible objects qua planes, qua lines, etc. From Aristotle’s point of view (Metaph. 1077b24-30), geometry ‘makes abstraction from’ or ‘leaves unconsidered’ the material composition of a specific object, by filtering out the properties of sensible objects that are not mathematically relevant. According to Proclus, in geometry discursive thinking uses imagined representations of innate concepts. Due to and within imagination these unextended ideas are projected as extended and divisible. Therefore the Neoplatonist is called, along with Descartes, a philosopher of the productive imagination (Bouriau 2002, 47; Rabouin 2009, 178), a term hardly compatible with the traditional, Aristotelian vocabulary. The compatibility of this productive geometrical imagination with imagination’s functioning in other contexts of the Neoplatonist’s oeuvre, where imagination is generally understood as the deceitful receptacle of the imprints from sensible objects, is a much debated topic. Nonetheless, imagination’s ambiguous and shifting meaning in the Proclean’ corpus is generally believed to be an interpretative problem. In the present article I will argue that by introducing imagination’s productive side and by simultaneously holding on to its traditional features, Proclus wittingly or unwittingly reveals a shift in meaning and functioning that was always already proper to imagination as a mediating place between sensibility and intelligibility, between productivity and reproductivity. For that reason, imagination’s ambiguous meaning throughout the Proclian corpus should not so much be considered an interpretative problem as a symptomatic manifestation of an underlying structure – a system

Alain-Philippe Segonds (1942–2011)

Alain-Philippe Segonds, honorary director of research at the CNRS and member of the Team for the History of Astronomy at the Paris Observatory, died suddenly on Monday, 2 May 201 1, in his sixty-eighth year.Alain was born in Paris on 1 August 1942. After completing a baccalaureate in philosophy in 1961, he undertook classical studies at the Sorbonne. In parallel with the courses offered by the university, he attended the seminar of Fr Andre-Jean Festugiere, the master of the history of ancient hermeticism and Neoplatonism, at the Ecole Pratique des Hautes Etudes. From this exposure, Alain's passion grew for Neoplatonist authors: Porphyry, Iamblichus, Proclus (for whom he latter contributed critical editions to the Bude collection), and Damascius.But Alain's interests rapidly outgrew the domain of Greek philosophy and philology. In 1963 he was privileged to attend the last course given by Alexandre Koyre, shortly before Koyre's death. Reading the books of this great historian of science and, starting in 1966-67, attending the seminars of Rene Taton and Fr Pierre Costabel at the Centre Alexandre Koyre gave him a taste for the history of science and particularly of astronomy. Taton, thinking that France should embrace a proposal made in 1973 by the Nicolas Copernicus Commitee of the International Union for History and Philosophy of Science, launched the project of a French edition of the works of Copernicus and his disciple Rheticus. With Jean-Pierre Verdet, astronomer at the Paris Observatory, as coordinator and host, a Team was created there consisting of himself, Henri Hugonnard-Roche, former student of the Ecole des Chartes, Alain Segonds, and the author of these lines. After the 1982 publication of Rheticus 's Narrano prima, a critical edition of Copernicus 's De revolutionibus with French translation and commentary became the goal of Alain and the other members of the Team, though each also continued to devote a consistant part of his working time to his personal projects.In 1988, Alain became Director General of Les Belles Lettres, an old and distinguished publishing house famous for its Guillaume Bude series of Greek and Latin classics. There he maintained an astonishing range of activity as an editor and tireless animator of several collections. One of Alain's most notable characteristics was his devotion to young scholars, aiding them in many ways to publish their work and to find, when possible, positions in the academic world. Serious and exigent, but hating pedantry, and also honours - this is the reason why he was member of no academy - Alain had also a lively sense of humour that made collaboration with him very enjoyable.In history of astronomy, the works completed by Alain are impressive, for the Greek period as well as for the Renaissance. His first book was an annotated French translation of the Treatise on the astrolabe of John Philoponus, published in 1981 by Alain Brieux for the International Society for the Astrolabe. Alain's 1987 study on Proclus and astronomy developed an essential aspect of the thought of the Neoplatonic savant. Alain also prepared a translation of Proclus 's Commentary on the first book of Euclid's Elements and his Hypotyposis (or Sketch of planetary theories), which are not yet published. …

Imagination as Self-knowledge: Kepler on Proclus' Commentary on the First Book of Euclid's Elements

AbstractThe Neoplatonist Proclus, in his commentary on Euclid's Elements, appears to have been the first to systematically cut imagination's exclusive ties with the sensible realm. According to Proclus, in geometry discursive thinking makes use of innate concepts that are projected on imagination as on a mirror. Despite the crucial role of Proclus' text in early modern epistemology, the concept of a productive imagination seems almost not have been received. It was generally either transplanted into an Aristotelian account of mathematics or simply ignored. In this respect, Johannes Kepler is a remarkable exception. By rejecting the traditional meta-mathematical framework, Kepler was the first to incorporate the productive side of imagination within an early modern philosophy of mathematics. Moreover, by securing imagination's sensory input, he transformed Proclean imagination into a tool for cosmic self-knowledge.

Clavius, Proclus, and the Limits of Interpretation: Snapshot-Idealization Versus Projectionism

It is hard to overestimate the role played by Christopher Clavius (1538–1612), professor of mathematics at the Collegio Romano from 1565 until his death, in the establishment of mathematics as an incontestable part of the Jesuit curriculum and body of thought. Besides his direct influence on the constitution of the mathematical part(s) of the Ratio studiorum, Clavius’s mathematical writings, ranging from introductory textbooks to commentaries to polemical pamphlets, almost immediately began to serve as standard references, despite their — occasionally — controversial character. Today, Clavius’s contentious stand towards the older school at the Collegio embodied in the person of Benedictus Pereira, who held an extremely negative view on mathematics, is regarded as a decisive factor in the sixteenth-century renaissance of mathematics, both within and outside the Society. Nowadays, most scholars seem to agree that the philosophical framework of Clavius’s institutional struggle can be found in the Prolegomena to his edition of Euclid’s Elements, first published in Rome in 1574. According to some, the general outlines of this preface reflect a philosophy of mathematics that is largely indebted to the one presented by the Neoplatonist Proclus (410/412 – 485) in the double prologue of his commentary on the first book of Euclid’s Elements. Without wanting to obscure the importance of Clavius’s revolutionary project, I will try to sketch a different picture of Clavius’s philosophy of mathematics, both on the ontological and on the epistemological level. In this re-evaluation, the limits of both Proclus’s influence and the interpretation of his commentary on Euclid will be my primary focus. Clavius’s views on the nature of mathematical objects will be shown to be incompatible with the two conventional modes of interpretation found in the ancient commentators, namely the projectionism maintained by Proclus on the one hand, and the constructive abstractionism of writers such as Ammonius and Philoponus on the other. For Proclus, mathematical objects are extended and divisible projections of the unextended and indivisible ideas with which our understanding is equipped. The imagination acts both as the faculty that projects these ideas and as the imaginary screen on which they are projected. The constructive abstractionism read in Ammonius, for example, sees mathematical entities as existing only in the mind of the mathematician who in thought has separated, i.e. abstracted, them from matter. Clavius’s position will lay bare the need to invoke a third category, namely that of a ‘snapshot-idealization’: in mathematics, according to Clavius, sensible objects are indeed considered free from matter, but they are not attributed any properties that they do not perfectly instantiate as sensible objects. As in a photograph, certain

Book X of The Elements: Ordering Irrationals

Book X from The Elements contains more than three times the number of propositions in any of the other Books of Euclid. With length as a factor, anyone attempting to understand Euclidean geometry may be hoping for a manageable subject matter, something comparable to Book VII’s investigation of number theory. They are instead faced with a dizzying array of new terminology aimed at the understanding of irrational magnitudes without a numerical analogue to aid understanding. The true beauty of Book X is seen in its systematic examination and labeling of irrational lines. This paper investigates the early theory of irrationals, the methodical presentation and interaction of these magnitudes presented in The Elements, and the application of Euclidean theory today.

What's wrong with Euclid Book V

Not having the real numbers, Euclid defined ratios abstractly in Book V for use in geometric theorems, but failed to define ratios of ratios, in effect blocking certain further developments of Greek mathematics. We introduce a new axiom that can be used instead of subtraction, not only to prove the propositions of Book V but also to define ratios of ratios.

On the argument of simplicity in Elements and schoolbooks of Geometry

Simplicity arguments are to be found in most geometrical works, from those of Proclus in his Commentaries on the First Book of Euclid’s Elements, up to those of contemporary manuals. Our goal is to read these arguments in their historical contexts to analyze agreements, disagreements and the multiplicity of points of view. For a better apprehension and a better understanding of the different conceptions, we will focus on the notion of angles and their measurements. We will study the notion of ≪ simplicity ≫ in various Elements of Geometry, in particular those of Euclid, Peletier du Mans (1628), Arnauld (1667), Lacroix (1803) and Houel (1867). From there, we will examine French schoolbooks of geometry, beginning from the 1960s up to the 1990s, including those of the so-called period of ≪la reforme des mathematiques modernes≫ in France.