Ancient Mathematics Research Articles

Effects of an Ancient Chinese Mathematics Enrichment Programme on Secondary School Students’ Achievement in Mathematics

Effects of an Ancient Chinese Mathematics Enrichment Programme on Secondary School Students’ Achievement in Mathematics

Ancient Egyptian Mathematics: New Perspectives on Old Sources

Ancient Egyptian Mathematics: New Perspectives on Old Sources

Letters

42 Historically Speaking · May/June 2006 Letters Narrative and Popular History Writing a response to an essay becomes more of a challenge when the responder fundamentally agrees with it. Paul Jankowski's essay "Guilt byAssociation" {Historically Speaking, March/April 2006) is a first-rate discussion of narrative history. There is also a certain irony that Jankowski and I share, writing analytically in praise of narrative. Narrative and popular history are often linked. Many historians find popular history suspect. Look at another field, say, music, and this argument seems silly. The Beatles are the best-selling recording artists ever. They are also among the best. Being good and popular is hard, but why not try? Some first-rate historical works, such as those by James McPherson, sell very well. Good historical narrative is harder to write than narrowly focused analysis. Too many historians do not wish to make the effort to relate their little picture to the context provided by the ever-present big picture. Narrowness, too, is often a product of political correctness. Some historians consciously try to restore a balance by shedding light on groups ignored in the past. This is good. It is not good, however , when distortions creep in, perspective is lost, and narrow segments are inflated far past their real importance. Like it or not, for most of the past white guys were running things. Why pretend otherwise? Narrative, when taken to mean good storytelling , is the best way to get people interested in history. I am a writer, not an academic historian , and writers must use storylines to hook readers just as a carnival uses hucksters to attract customers. The best information, the greatest insights, the newest interpretations are useless unless disseminated. Bruce L. Brager Arlington, Virginia Goldstone and Ancient Mathematics Both Jack Goldstone and readers of Historically Speaking should be informed that at least one statement in his article "Knowledge—Not Capitalism" {Historically Speaking, March/April 2006) is very much Claudius Ptolemy, 1886 print. Library of Congress, Prints and Photographs Division [reproduction number , LC-USZ62-92884]. incorrect: he writes "Greek mathematics remained weak on arithmetic and never developed algebra (the study of solving equations with unknown quantities)" or (the next sentence implies) "trigonometry." The exact opposite is the case. The Greeks had developed algebra and advanced it as far as quadratic equations well before the time Diophantus wrote a practical textbook on the subject sometime before the 4th century A.D. (evidence that places him in the 3rd century is highly conjectural, but on no account did he write later than that). The Arabs didn't invent algebra. Arabic writers learned ofand commented and improved upon algebra from Greek sources like this. Goldstone's naive folk argument that the word "Algebra" being of Arabic derivation is evidence that Arabs invented algebra is as faulty as arguing that Ptolemy must have been an Arab because the title of his Almagest is Arabic. The Greeks also invented trigonometry. Some basic trigonometric ideas may have originated in India, but the first theoretical treatises in the field begin with Apollonius of Perga and Hipparchus, both in the 2nd centu- . ry B.C. Hipparchus composed the first known trigonometric tables and originated the modern system of degrees, minutes, and chords. Spherical trigonometry was then developed by Menelaus in the 1st century A.D. Hence, again, the Arabs didn't invent trigonometry. Arabic writers learned of and commented and improved upon trigonometry from Greek sources like this. On both facts, see "Diophantus," "Apollonius," "Hipparchus," and "Menelaus" in the Dictionary of Scientific Biography (including supplemental material in entries added in volume 15 ofthat work). For context see Serafina Cuomo, Ancient Mathematics (2001). In fact, I strongly urge all scholars and readers to make a more thorough effort to consult qualified sources on the actual achievements of ancient Greek mathematicians and scientists before issuing blanket statements about what they supposedly failed to do. Rodney Stark, for example, is thoroughly corrected on the subject ofHellenistic science by works like Lucio Russo's The Forgotten Revolution: How Science Was Born in 300 B.C. and Why It Had to Be Reborn (1996). Richard C. Carrier Columbia University Goldstone's Reply to Carrier Richard Carrier...

A Humanist History of Mathematics? Regiomontanus's Padua Oration in Context

In the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Muller (1436-76), known as Regiomontanus (a Latinization of the name of his hometown, Konigsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history and utility of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenthand-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various algorisms (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). …

Of Orbits, Conics, and Grammar

In the half-dozen or so years leading up to the publication of the Principia, Isaac Newton observed the comets of 1680 and 1682 and wrestled with the extent to which his law of gravitation could be applied. In time, he would see the connections between the four possible orbits of a satellite (circular, elliptical, parabolic, and hyperbolic) and the four curves produced by the careful carving of a cone. But if we look a little further into the conic sections, we find some interesting connections among the natural orbit of a satellite, ancient mathematics, and the roots of familiar words. Illuminating these connections for introductory physics students may help them to better understand the role of language and mathematics in the descriptions of science.

Using Ancient Egyptian Fractions to Review Fraction Concepts

Much of What We Know Today About the mathematics of ancient Egypt is contained in a papyrus scroll that was copied from an earlier scroll by the scribe Ahmes in about 1650 BME (before the modern era) (Boyer 1968). A fascinating feature of ancient Egyptian mathematics is its treatment of common fractions. In most cases, the Egyptians used only unit fractions, that is, fractions with numerators of 1. The one common exception is 2/3, and they would occasionally use fractions of the form n/(n + 1). However, both forms are complements of unit fractions.

Completing Book II of Archimedes’s On Floating Bodies

One need only glance at Archimedes’s Proposition 8 above to see thatOn Floating Bodies is several orders of magnitude more sophisticated than anything else found in ancient mathematics. It ranks with Newton’sPrincipia Mathematica as a work in which basic physical laws are both formulated and accompanied by superb applications.

Sherlock Holmes in Babylon and other tales of mathematical history

Part I. Ancient Mathematics: Foreword Sherlock Holmes in Babylon Words and pictures: new light on Plimpton 322 Mathematics 600 BC-600 AD Diophantus of Alexandria Hypatia of Alexandria Hypatia and her mathematics The evolution of mathematics in ancient China, Liu Hui and the first golden age of Chinese mathematics Number systems of the North American Indians The number systems of the Mayas Before the conquest Afterword Part II. Medieval and renaissance mathematics Foreword The discovery of the series formula for pi Ideas of calculus in Islam and India Was calculus invented in India? An early iterative method for the determination of sin 1 * Leonardo of Pisa and his liber quadratorum The algorists vs. the abacists: an ancient controversy on the use of calculators Sidelights on the Cardan-Tartaglia controversy Reading Bombelli's x-purgated algebra The first work on mathematics printed in the New World Afterword Part III. The Seventeenth Century: Foreword An application of geography to mathematics: history of the integral of the secant Some historical notes on the cycloid Descartes and problem-solving Rene Descartes' curve-drawing devices: experiments in the relations between mechanical motion and symbolic language Certain mathematical achievements of James Gregory The changing concept of change: the derivative from Fermat to Weierstrauss The crooked made straight: Roberval and Newton on tangents On the discovery of the logarithmic series and its development in England up to Cotes Isaac Newton: man, myth, and mathematics Reading the master: Newton and the birth of celestial mechanics Newton as an originator of polar coordinates Newton's method for resolving affected equations A contribution of Leibniz to the history of complex numbers Functions of a curve: Leibniz's original notion of functions and its meaning for the parabola Afterword Part IV. The Eighteenth Century: Foreword Brook Taylor and the mathematical theory of Linear Perspective Was Newton's calculus a dead end? The continental influence of Maclaurin's treatise of fluxions Discussion of fluxions: from Berkeley to Woodhouse The Bernoulli and the harmonic series Leonhard Euler 1707-1783 The number Euler's vision of a general partial differential calculus for a generalized kind of function Euler and the fundamental theorem of algebra Euler and the differentials Euler and quadratic reciprocity Afterword Index About the editors.

Introduction: The History of Modern Mathematics – Writing and Rewriting

The present issue of Science in Context comprises a collection of articles dealing with various, specific aspects of the history of mathematics during the last third of the nineteenth century and the first half of the twentieth. Like the September issue of 2003 of this journal (Vol. 16, no. 3), which was devoted to the history of ancient mathematics, this collection originated in the aftermath of a meeting held in Tel-Aviv and Jerusalem in May 2001, under the title: “History of Mathematics in the Last 25 Years: New Departures, New Questions, New Ideas.” Taken together, these two topical issues are meant as a token of appreciation for the work of Sabetai Unguru and his achievements in the history of mathematics.

The Influence of Ancient Egypt on Greek and Other Numeration Systems

You may have learned how the ancient Egyptians wrote numbers. For example, for the number 600, you would write a symbol for a scroll six times. Actually, ancient Egypt had two main systems of writing: hieroglyphic and hieratic. Hieroglyphics, dating back over 5,000 years, were used mainly for inscriptions on stone walls and monuments. Hieratic writing was a cursive script suitable for writing on papyrus, the Egyptian form of paper. Much of our knowledge of ancient Egyptian mathematics comes from a papyrus written by the scribe Ahmose around 1650 B.C.E. Although he wrote in hieratic script, recent historians transcribed this document and others into hieroglyphics, giving readers the impression that all Egyptian writing was in hieroglyphics, the system that you may have learned.

Ancient Rhetoric and Greek Mathematics: A Response to a Modern Historiographical Dilemma

ArgumentIn this article, I compare Sabetai Unguru’s and Wilbur Knorr’s views on the historiography of ancient Greek mathematics. Although they share the same concern for avoiding anachronisms, they take very different stands on the role mathematical readings should have in the interpretation of ancient mathematics. While Unguru refuses any intrusion of mathematical practice into history, Knorr believes this practice to be a key tool for understanding the ancient tradition of geometry. Thus modern historians have to find their way between these opposing views while avoiding an unsatisfactory compromise. One approach to this, I propose, is to take ancient rhetoric into account. I illustrate this proposal by showing how rhetorical categories can help us to analyze mathematical texts. I finally show that such an approach accommodates Knorr’s concern about ancient mathematical practice as well as the standards for modern historical research set by Unguru 25 years ago.

Some interpolation formulas in Chinese ancient mathematics

When Newton found his interpolation formula in 1670, ancient Chinese mathematicians had used the formula in more sophisticated forms for more than one millennium. A new proof to those formulas are geometrically illustrated.

How a Scribe Learned Math, ca. 1800 B.C.

JOINT MATHEMATICS MEETINGSBALTIMORE, MARYLAND-- Earlier this month, 5000 mathematicians converged here for the joint annual meetings of the American Mathematical Society and the Mathematical Association of America. Highlights included an unexpected new insight into the practice of ancient mathematics.

On Heiberg's Euclid

In two articles published in 1881 and 1884, the young academics, Martin Klamroth and Johan L. Heiberg, engaged in a brief debate on the textual choices that should govern the publication of a new critical edition of Euclid's Elements. This short debate seemed to settle the problem in Heiberg's favor as to what should be taken as the definitive text of Euclid's Elements. But the issue ought to be considered once again for there are good reasons for the claim that Klamroth was right, and that Heiberg was wrong. If so, we have been consulting, and continue to consult, the wrong text in our efforts to interpret the Euclidean tradition. In order to substantiate this claim, the textual issue debated by Klamroth and Heiberg is rehearsed again, and the principal reasons brought forth by Heiberg against Klamroth's position are reconstructed. Specimens from three broad areas of evidence – structural, linguistic, and technical – will be considered. They reveal how the medieval tradition of the text advocated by Klamroth displays textual superiority to the Greek tradition promoted by Heiberg. Such a reorientation of the texts has the potential to change significantly our understanding of ancient mathematics.

Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Marshall Clagett

<i>Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics</i>. Marshall Clagett

Ancient Egyptian Science: A Source Book, Vol. 3: Ancient Egyptian Mathematics

Ancient Egyptian Science: A Source Book, Vol. 3: Ancient Egyptian Mathematics

The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion

Nicomachus of Gerasa has gained a position of importance in the history of ancient mathematics due in great measure to his Introduction to Arithmetic, one of the only surviving documentations of Greek number theory. Jay Kappraff discusses a pair of tables of integers found in the Arithmetic and shows how they lead to a general theory of proportion, including the system of musical proportions developed by the neo-Platonic Renaissance architects Leon Battista Alberti and Andrea Palladio, the Roman system of proportions described by Theon of Smyrna, and the Modulor of Le Corbusier.

A history of the circle: mathematical reasoning and the physical universe

I found the book to be fascinating, and the author's presentations and illustrations of the contrasts bewtween mathematical reasoning and scientific reasoning were especially appealing. The book is rich in history, which is carefully integrated into the discussions, and it includes wonderful illustrations and stories. --The Mathematics Teacher is a wonderful storyteller, and his choices of topics reveal not only the depth of explanation afforded by the available mathematics but the beauty in the explanations; he succeeds in keeping the explanations accessible to the most general audience. --Choice The concept of the circle is ubiquitous. It can be described mathematically, represented physically, and employed technologically. The circle is an elegant, abstract form that has been transformed by humans into tangible, practical forms to make our lives easier. And yet no one has ever discovered a true mathematical circle. Rainbows are fuzzy, car tires are flat on the bottom, and even the most precise roller bearings have measurable irregularities. Ernest Zebrowski, Jr., discusses how investigations into the circle have contributed enormously to our current knowledge of the physical universe. Beginning with the ancient mathematicians and culminating in twentieth-century theories of space and time, the mathematics of the circle has pointed many investigators in fruitful directions in their quests to unravel nature's secrets. Johannes Kepler, for example, triggered a scientific revolution in 1609 when he challenged the conception of the earth's circular motion around the sun. Arab and European builders instigated a golden age of mosque and cathedral building when they questioned the Roman structural arches that were limited to geometrical semicircles. Throughout his book, Zebrowski emphasizes the concepts underlying these mathematicians' calculations, and how these concepts are linked to real-life examples. Substantiated by easy-to-follow mathematical reasoning and clear illustrations, this accessible book presents a novel and interesting discussion of the circle in technology, culture, history, and science. Ernest Zebrowski, Jr., hold professorships in science and mathematics education at Southern University in Baton Rouge, and in physics at Pennsylvania College of Technology of the Pennsylvania State University. He is the author of Perils of a Restless Planet: Scientific Perspectives on Natural Disasters and The Last Days of St. Pierre: The Volcanic Disaster That Claimed 30,000 Lives.

What is ancient mathematics?

What is ancient mathematics?

The Language in Chinese Mathematical Texts Before AD 1300: A Case for Logical Abstraction

While there seems to be a general consensus that literacy should include the ability to deal with basic reading, writing, and arithmetic, a view is growing that literacy is in fact more than equipping learners with sets of essential skills (Wells, 1981; Schoenfeld, 1985; Bereiter & Scardamalia, 1987; Cumming, 1990; Wells & Chang-Wells, 1992). Olson (1994) reviews in his book The World on Paper a large number of studies in the field of psychology, sociology, history, and linguistics and concludes that modern literacy in the West is characterized by a new way of looking at written texts and the representation the texts project. In the contemporary East, while many countries including China are striving for economic development and modernization, the question of why science and modernity did not first develop in some oriental cultures such as China, which used to be the most advanced, remains interesting but unanswered. In the light of Olson's discussion regarding literacy and Western modernity, this paper adopts a cognitive-linguistic perspective and examines the social conditions in which ancient Chinese mathematics struggled to develop. Examples and experiences in ancient Chinese mathematical texts are presented to reveal the close relationship between modernity, literacy, and written language. In response to a new interest in a context-rich approach to mathematics teaching, the paper also discusses its implications for the mathematics curriculum.