Greek Mathematics Research Articles

On the myriad mathematical traditions of ancient greece

To exert one’s historical imagination is to plunge into delicate deliberations that involve personal judgments and tastes. Historians can and do argue like lawyers, but their arguments are often made on behalf of a picture of the past, and these historical images obviously change over time. Why should the history of mathematics be any different? When we imagine the world of ancient Greek mathematics, the works of Euclid (Heath 1926), Archimedes (Heath 1897b), and Apollonius (Heath 1897a) easily spring to mind. Throughout most of the twentieth century, our dominant image of Greek mathematical traditions has been shaped by the high standards of rigor and creative achievement that are purportedly found in extant texts presumed to have been written by these three famous authors. Thanks to the efforts of Thomas Little Heath, the English-speaking world has long enjoyed easy access to this trio’s major works and much else besides (Heath 1921). Yet despite this plentiful source material, our conventional picture of Greek mathematics has largely been sustained by a far smaller corpus of knowledge. Our image of Greek geometry, as conveyed in countless mathematical texts and most books on the history of mathematics, has tended to stress the formal structure and methodological sophistication found in a handful of canonical works or, more accurately, selected portions of the same. Even the first two books of Euclid’s Elements – which concern the congruence properties of rectilineal figures and culminate in theorem II-14 showing how to square such a figure – have often been trivialized. Many writers have distilled their content down to a few definitions, postulates, and elementary propositions intended merely to illustrate the axiomatic-deductive method in classical geometry.

The Philosophy of Greek Mathematics and its Influence on the Development of Arab Civilization and Renaissance

Speaking about classical studies, we usually mean the study of ancient Greek and Latin literature, history, and philosophy that has come to us through the magnificent works of distinguished ancient writers. But a characteristic feature of Greek Philosophy was the consideration of the world and the ideas as one entity that cannot be studied separately. Thus, the Greeks developed Philosophy parallel to Mathematics, since the leading philosophers as Thales, Pythagoras, Plato, Aristotle were pioneers in the development of Mathematics and Physics, as a necessary means for revealing the secrets of Nature and the world around us. These philosophers had traveled to Egypt and Babylon and were acquainted with the knowledge of the civilization thereof. They did not confine themselves to obtaining the results, but they searched for proofs and further development of the received ideas and knowledge. Greeks, spread from the Italian peninsula to Asia Minor, formulated mathematics as a theoretical discipline, built on abstract notions rather than sensible objects, using formal demonstrations to prove the truth of their theorems.

Portraits of science. Proof, amazement, and the unexpected.

Archimedes is now best known for shouting "Eureka" in the bath, and perhaps also for his invention of the cunning machines that helped to defend Syracuse from the Roman fleet during the Punic Wars in 212 BP. He is less famous for his most profound work--the subtle deductions and rigorous, often counterintuitive, mathematical proofs of seemingly prosaic problems. His path-breaking work has reached new prominence among scholars with the rediscovery of a long-lost palimpsest copy of some of his writings, which for the first time shows that Greek mathematicians could work with infinite sets of numbers.

Leopardi and the ancient Greek mathematics

"I consider Leopardi's poetry and pessimism to be the best expression of what a scientist's credo should be". This quotation is from Bertrand Russell, no less. With these very emblematic words, the greatest man of letters, the supreme icon of the Italian Parnasse, the author of such collections of poems as Canti (Poems) and Operette Morali (The Moral Essays) and philosophical thoughts as Zibaldone (Miscellany) has been associated to the world of science. This relationship, very intense and to a certain extent new, was greatly emphasised on the occasion of the poet's birth bicentenary. During the celebration in 1996, an exhibition with the name of Giacomo and Science was organized in his birthplace to underline the close connection between the poet and the scientific culture of his epoch. This point has also been stressed recently.

Greek Mathematical Astronomy Reconsidered

Recent investigations have thrown new light on such topics as the early Greek belief in heliocentricity, the relation between Greek and Babylonian astronomy, the reliability of Ptolemy's Syntaxis, Hipparchus's theory of motion for the sun, Hipparchus's value for the obliquity of the ecliptic, and Eratosthenes' estimate of the size of the earth. Some claims resulting from these investigations are controversial, especially the reevaluation of Ptolemy (though it is notable that no one any longer uses data from the Syntaxis for investigating such things as the spin of the earth). This essay presents the evidence for these claims; it makes no pretense of presenting the evidence against them.

On a Collection of Geometrical Riddles and their Role in the Shaping of Four to Six “Algebras”

For more than a century, there has been some discussion about whether medieval Arabic al-jabr (and hence also later European algebra) has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 BCE, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and later the geometry of Elements II (where the techniques are submitted to theoretical investigation). The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European (Latin and abbaco) mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles.

Are Legal Traditions Incommensurable?

From relatively humble origins in Greek mathematics,' the notion of incommensurability was swept into the philosophy of science in mid-twentieth century and assigned major responsibilities, notably in accounting for significant shifts in scientific understanding.2 Since then it has been taken up more generally in western philosophy (though not, apparently, elsewhere) as an essential means of understanding the world and its values.3 It has now appeared in western legal philosophy4 and comparative law.5 There is apparent paradox

Aristarchus at your service

How to maintain student interest in class? Keep a collection of interesting diversionary topics on hand. One such topic: the surprising astronomical discoveries made more than 2000 years ago by the Greek mathematician, Artistarchus.

The Shaping of Deduction in Greek Mathematics by R. Netz

The Shaping of Deduction in Greek Mathematics by R. Netz

The Nostratic Theory

The Nostratic Theory was first proposed by the Danish scholar, H. Pedersen, at the beginning of this century. He had observed certain similarities between Indo-European and Semitic and saw these two families of languages as belonging to our territory around mare nostrum and hence his name for this group.Speakers of Finno-Ugrian languages were surprised that Indo-Europeanists after working on the comparative philology of IE should turn next to Semitic, but the interest in Arabic as the preserver of Greek mathematics and science and Hebrew as the language of the Old Testament inevitably suggested that Semitic had an interest for Philology as well as Theology. But there is a closer resemblance of Finno-Ugrian to Indo-European than there is between Indo-European and Semitic. Moreover, Finno-Ugrian has been in contact with and a close neighbour to the North of IE since prehistoric times. Burrow lists over 20 words of Sanskrit origin borrowed by Finno-Ugrian languages in the prehistoric period when both families of languages were in contact probably in south Russia. Strangely enough they include the Sanskrit svásar- ‘sister’ giving Finnish sisar, Mordvin sazor ‘younger sister’, Cheremis suzar id. where the survival of the sibilant is a clear indication of borrowing before the Aryan branch split into Indian and Iranian with the characteristic Iranian sound change to h as in Av. xvanhar-. It is surprising to find ‘sister’ among these loan words.

Proclus’ division of the mathematical proposition into parts: how and why was it formulated?

There are a number of ways in which Greek mathematics can be seen to be radically original. First, at the level of mathematical contents: many objects and results were first discovered by Greek mathematicians (e.g. the theory of conic sections). Second, Greek mathematics was original at the level of logical form: it is arguable that no form of mathematics was ever axiomatic independently of the influence of Greek mathematics. Finally, third, Greek mathematics was original at the level of form, of presentation: Greek mathematics is written in its own specific, original style. This style may vary from author to author, as well as within the works of a single author, but it is still always recognizable as the Greek mathematical style. This style is characterized (to mention a few outstanding features) by (i) the use of the lettered diagram, (ii) a specific technical terminology, and (iii) a system of short phrases (‘formulae’). I believe this third aspect of the originality—the style—was responsible, indirectly, for the two other aspects of the originality. The style was a tool, with which Greek mathematicians were able to produce results of a given kind (the first aspect of the originality), and to produce them in a special, compelling way (the second aspect of the originality). This tool, I claim, emerged organically, and reflected the communication-situation in which Greek mathematics was conducted. For all this I have argued elsewhere.

Linguistic formulae as cognitive tools

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a (partial) explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae (expressions used repetitively) in Greek mathematical texts. It is shown that (a) the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that (b) such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level (as against the universal cognitive level) is the best explanatory level for the phenomenon of Greek mathematical deduction.

III. Mathematics

Mathematics probably ranks as the Greeks’ greatest achievement, in the eyes of many modern scientists. And amongst the general public it is fair to say that some Greek mathematicians are better known than any other figures from antiquity, with the possible exception of Alexander the Great. For example, Pythagoras, Euclid, and Archimedes are household names two thousand years after they lived and wrote their mathematics. The Greek alphabet, or some of it anyway, is widely known today because modern mathematics uses it by preference for symbols. For example, α, β, and θ for angles, the amazing π, Σ as the summation sign, and the χ-square test. The Greeks gave us ‘square’ and ‘cube’ numbers, e.g. 22 and 33. ‘Squaring the circle’ is modern English idiom for an impossible task – the original task, set by the Greeks, being the mathematical problem of constructing a square with the same area as a given circle. Aristophanes mentions this problem. Famously, Plato forbade the geometrically-challenged to enter his Academy. He also found mathematically interesting numbers for his ideal Republic. Greek mathematical texts are rich in mathematical concepts, methods, and results. Like Greek plays, they contain timeless insights and truths. Also like literary works they require analysis and interpretation.

To the Scientifically Faint-Hearted Reader

Understanding what is going on in most of Greek science is well within the competence of any intelligent person, as it was in antiquity. Take heart! If you read an ancient scientific text and feel, not that you are out of your depth scientifically, but more or less bewildered by the contents, then you are probably reading it through modern spectacles, so to speak. Assume always that the ancient author makes sense: what he wrote was considered sensible by enough people over enough centuries – more than a thousand years in some cases – for that text to have been repeatedly copied and thus survive for us to read now. The task then is to discover in what sense he makes sense. As Klein put it with respect to mathematics, ‘our task consists precisely in bringing the content of Greek mathematics to light not by externally transposing it into another mode of presentation but rather by comprehending it in the one way which seemed comprehensible to the Greeks’. This search for sense can apply to the structure of a work as well as its contents: our task in that case is to recognize the principle of organization at work in what otherwise appears to be a set of more or less disorganized notes, as for example in the cases of Aristotle’s History of Animals, Theophrastos’ History of Plants, or Dioskorides’ Herbal.

Ancient Chinese mathematics: the ( Jiu Zhang Suan Shu) vs Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge

The following is a preliminary and relatively brief, exploratory discussion of the nature of early Chinese mathematics, particularly geometry, considered largely in terms of one specific example: the ( Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Western traditions, particularly with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematical thought. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that Chinese mathematicians were by nature primarily concerned with practical problems and their solutions, and, therefore, had no interest in developing a highly theoretical mathematics. Others have stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, which, in turn, worked against the development of abstract mathematics of the Greek sort. While both of these views contain elements of truth, and certainly play a role in understanding why the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a different approach is offered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for how the Chinese were able to conceptualize—and think about—mathematics.

Numbers, Magnitudes, Ratios, and Proportions in Euclid'sElements: How Did He Handle Them?

For a century or so much Greek mathematics has been interpreted as algebra in geometric and arithmetical disguise. But especially over the last 25 years some historians of mathematics have raised objections to this interpretation, finding it to be misleading and anachronistic, and even wrong. Accepting these criticisms, I consider Euclid'sElementsin this context: if it cannot be read in this algebraic manner, how did he conceive and handle his various types of quantity? The question is not merely of historical interest, for it raises issues about basic relationships between algebra, arithmetic, and geometry.

Sizing Up Earth: A Universal Method for Applying Eratosthenes' Earth Measurement

Abstract This article is based on an activity used in a team-taught preservice course, “Integrating Geography and History,” at the University of North Alabama. The antecedents to modern descriptions and calculations of earth's shape and size began more than 2,000 years ago. The Greek mathematician and geographer Eratosthenes made the most accurate ancient measurement of Earth based on fundamental math concepts and Earth-Sun relations that provided a basis for determining latitude with greater accuracy. A brief historical statement about Eratosthenes' experiment is followed by an extended explanation of how teachers can use Eratosthenes' method in the classroom to calculate the size of Earth using shadows and simple mathematics independent of noon time observations.

Strategies for conceptual change: Ratio and proportion in classical Greek mathematics

Abstract …all men begin… by wondering that things are as they are…as they do about…the incommensurability of the diagonal of the square with the side; for it seems wonderful to all who have not yet seen the reason, that there is a thing which cannot be measured even by the smallest unit. But we must end in the contrary and, according to the proverb, the better state, as is the case in these instances too when men learn the cause; for there is nothing which would surprise a geometer so much as if the diagonal turned out to be commensurable (Aristotle, Metaph ., 983 a 15).

The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective (review)

BOOK REVIEWS 663 a general schematic view of Greek thought, starting naturally with Homer... and his opinions on the correctness of names, and moving on to other areas of Greek thought" (99). Thus from the etymologies relating to Homer and the nature Qf the gods, we move to those relating to the Presocratics and the nature of the elements and stars, and finally we reach those relating to the Sophists and the nature of the virtues. What is the purpose or purposes of the etymologies? Baxter disagrees with V. Goldschmidt, who held that Plato, in the etymological investigations, is aiming to expose a mistaken view (Heracliteanism) about reality pervading Greek culture and to publicly renounce the influence on him of his former teacher (Cratylus). His own answer stresses the aspect of parody in the etymologies as Plato's way of showing the lack of any seriousness in the Greek practice of etymologizing; and, in partial agreement with R. Brumbaugh, he argues that Plato's main purpose in etymologizing is to show that etymology is an unreliable tool in seeking knowledge about things. This last claim "is the major positive result from the etymological inquiry, and it is here that one should seek the unifying feature of the etymologies. This unifying feature is indeed the exposure of a culturewide error, not concerning flux, but rather language and its relationship to reality, an error which is set in the context of a schematic history of the development of Greek thought" (96). The section on etymologies represents "a full frontal attack by Plato on representative figures in Greek culture, poets, philosophers, 'philologists' and so on" (16o). Heraclitus and his followers are, according to Baxter, major figures in this tradition, with Cratylus being the worst of the lot (162). Hence a refutation of etymologizing as a way of gaining knowledge is an attack on the theory of naming of Cratylus and his method of arriving at the nature of things through that of their names. The final section of the dialogue, the refutation of Cratylus, thus presupposes, according to Baxter, the conclusions of the etymological discussion and is really a continuation of it. Baxter's conclusions regarding the purpose(s) of the discussion on etymologies in the Cratylus may not be acceptable to all. Yet he is quite judicious in discussing the evidence for his conclusions and is aware that his arguments "do not prove it [his interpretation] conclusively... " (162). His conclusions about Plato's mofives in devoting half of the dialogue to etymologies seem to me to have considerable plausibility, and perhaps this is the best one can do here. In addition, I find his detailed discussion of the etymologies quite valuable on its own; it sheds much light on a part of the Cratylusthat to most readers seems inconsequential. GEORGIOS ANAGNOSTOPOULOS Universityof California, San Diego Michael J. White. The Continuousand theDiscrete:Ancient PhysicalTheoriesfrom a ContemporaryPerspective .New York: Oxford University Press, t992. Pp. xiv + 345. Cloth, $72.oo. In a groundbreaking study White reexamines the foundations of Aristotle's conceptions of spatial magnitude, motion, and time, along with two alternative conceptions developed by the atomists and the Stoics. On the basis of a rigorous philosophical 664 JOURNAL OF THE HISTORY OF PHILOSOPHY 39:4 OCTOBER 1994 reading of the relevant texts from the perspective of contemporary mathematics, White offers a cogent, strikingly innovative, reinterpretation that not only sheds new light on a critical ancient philosophical problem, but also brings it into the arena of contemporary philosophical discourse. White's analysis of Aristode, based primarily on discussions in the Physics, drives home the thesis that Aristode held consistently and unremittingly to a strict conception of infinite divisibility, avoiding any concession to or anticipation of a modern set-theoretic ontology, which conceptualizes continuous magnitude as a dense and Dedekind-continuous linear ordering of discrete aaua/elements. Demonstrating that Aristotle's view was shared by Greek mathematicians before and after him, White argues that Aristode's canonization of it effectively prevented subsequent mathematicians from ever entertaining the idea of the sum of an infinite set, even though some (e.g., Archimedes) intuitively came close. Aristotle's grip was broken only in the late nineteenth century by...

A History of Mathematics: An Introduction

Part I. Ancient Mathematics 1. Egypt and Mesopotamia 1.1 Egypt 1.2 Mesopotamia 2. The Beginnings of Mathematics in Greece 2.1 The Earliest Greek Mathematics 2.2 The Time of Plato 2.3 Aristotle 3. Euclid 3.1 Introduction to the Elements 3.2 Book I and the Pythagorean Theorem 3.3 Book II and Geometric Algebra 3.4 Circles and the Pentagon 3.5 Ratio and Proportion 3.6 Number Theory 3.7 Irrational Magnitudes 3.8 Solid Geometry and the Method of Exhaustion 3.9 Euclid's Data 4. Archimedes and Apollonius 4.1 Archimedes and Physics 4.2 Archimedes and Numerical Calculations 4.3 Archimedes and Geometry 4.4 Conic Sections Before Apollonius 4.5 The Conics of Apollonius 5. Mathematical Methods in Hellenistic Times 5.1 Astronomy Before Ptolemy 5.2 Ptolemy and The Almagest 5.3 Practical Mathematics 6. The Final Chapter of Greek Mathematics 6.1 Nichomachus and Elementary Number Theory 6.2 Diophantus and Greek Algebra 6.3 Pappus and Analysis Part II. Medieval Mathematics 7. Ancient and Medieval China 7.1 Introduction to Mathematics in China 7.2 Calculations 7.3 Geometry 7.4 Solving Equations 7.5 Indeterminate Analysis 7.6 Transmission to and from China 8. Ancient and Medieval India 8.1 Introduction to Mathematics in India 8.2 Calculations 8.3 Geometry 8.4 Equation Solving 8.5 Indeterminate Analysis 8.6 Combinatorics 8.7 Trigonometry 8.8 Transmission to and from India 9. The Mathematics of Islam 9.1 Introduction to Mathematics in Islam 9.2 Decimal Arithmetic 9.3 Algebra 9.4 Combinatorics 9.5 Geometry 9.6 Trigonometry 9.7 Transmission of Islamic Mathematics 10. Medieval Europe 10.1 Introduction to the Mathematics of Medieval Europe 10.2 Geometry and Trigonometry 10.3 Combinatorics 10.4 Medieval Algebra 10.5 The Mathematics of Kinematics 11. Mathematics Elsewhere 11.1 Mathematics at the Turn of the Fourteenth Century 11.2 Mathematics in America, Africa, and the Pacific Part III. Early Modern Mathematics 12. Algebra in the Renaissance 12.1 The Italian Abacists 12.2 Algebra in France, Germany, England, and Portugal 12.3 The Solution of the Cubic Equation 12.4 Viete, Algebraic Symbolism, and Analysis 12.5 Simon Stevin and Decimal Analysis 13. Mathematical Methods in the Renaissance 13.1 Perspective 13.2 Navigation and Geography 13.3 Astronomy and Trigonometry 13.4 Logarithms 13.5 Kinematics 14. Geometry, Algebra and Probability in the Seventeenth Century 14.1 The Theory of Equations 14.2 Analytic Geometry 14.3 Elementary Probability 14.4 Number Theory 14.5 Projective Geometry 15. The Beginnings of Calculus 15.1 Tangents and Extrema 15.2 Areas and Volumes 15.3 Rectification of Curves and the Fundamental Theorem 16. Newton and Leibniz 16.1 Isaac Newton 16.2 Gottfried Wilhelm Leibniz 16.3 First Calculus Texts Part IV. Modern Mathematics 17. Analysis in the Eighteenth Century 17.1 Differential Equations 17.2 The Calculus of Several Variables 17.3 Calculus Texts 17.4 The Foundations of Calculus 18. Probability and Statistics in the Eighteenth Century 18.1 Theoretical Probability 18.2 Statistical Inference 18.3 Applications of Probability 19. Algebra and Number Theory in the Eighteenth Century 19.1 Algebra Texts 19.2 Advances in the Theory of Equations 19.3 Number Theory 19.4 Mathematics in the Americas 20. Geometry in the Eighteenth Century 20.1 Clairaut and the Elements of Geometry 20.2 The Parallel Postulate 20.3 Analytic and Differential Geometry 20.4 The Beginnings of Topology 20.5 The French Revolution and Mathematics Education 21. Algebra and Number Theory in the Nineteenth Century 21.1 Number Theory 21.2 Solving Algebraic Equations 21.3 Symbolic Algebra 21.4 Matrices and Systems of Linear Equations 21.5 Groups and Fields - The Beginning of Structure 22. Analysis in the Nineteenth Century 22.1 Rigor in Analysis 22.2 The Arithmetization of Analysis 22.3 Complex Analysis 22.4 Vector Analysis 23. Probability and Statistics in the Nineteenth Century 23.1 The Method of Least Squares and Probability Distributions 23.2 Statistics and the Social Sciences 23.3 Statistical Graphs 24. Geometry in the Nineteenth Century 24.1 Differential Geometry 24.2 Non-Euclidean Geometry 24.3 Projective Geometry 24.4 Graph Theory and the Four Color Problem 24.5 Geometry in N Dimensions 24.6 The Foundations of Geometry 25. Aspects of the Twentieth Century 25.1 Set Theory: Problems and Paradoxes 25.2 Topology 25.3 New Ideas in Algebra 25.4 The Statistical Revolution 25.5 Computers and Applications 25.6 Old Questions Answered