The History of the Priority Di Pute Between Newton and Leibniz Mathematics in History and Culture
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The Discovery of the Series Formula for Π by Leibniz, Gregory and Nilakantha Author(S): Ranjan Roy Source: Mathematics Magazine, Vol
The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha Author(s): Ranjan Roy Source: Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690896 Accessed: 27-02-2017 22:02 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine This content downloaded from 195.251.161.31 on Mon, 27 Feb 2017 22:02:42 UTC All use subject to http://about.jstor.org/terms ARTICLES The Discovery of the Series Formula for 7r by Leibniz, Gregory and Nilakantha RANJAN ROY Beloit College Beloit, WI 53511 1. Introduction The formula for -r mentioned in the title of this article is 4 3 57 . (1) One simple and well-known moderm proof goes as follows: x I arctan x = | 1 +2 dt x3 +5 - +2n + 1 x t2n+2 + -w3 - +(-I)rl2n+1 +(-I)l?lf dt. The last integral tends to zero if Ix < 1, for 'o t+2dt < jt dt - iX2n+3 20 as n oo. -
GEMS of NORTHERN EUROPE Hamburg
T HE M ETROPOLITAN M USEUM OF A RT 1000 Fifth Avenue New York, NY 10028 T HE M ETROPOLITAN M USEUM OF A RT T HE M ETROPOLITAN M USEUM OF A RT PRSRT STD U.S. Postage PAID Dear Members and Friends of The Metropolitan Museum of Art, BURLINGTON, VT PERMIT NO. 601 G EMS OF N ORTHERN E UROPE Major cities in Northern Europe have had a dazzling history. Trade, freedom, and the occasional dynastic rule combined to produce great wealth, and perhaps more surprisingly, a love of the arts and Hamburg, Hannover & The Hague the garden. Traveling with us is art historian Olivier Bernier whose lectures at The Metropolitan Museum of Art delight an enthusiastic audience. Mr. Bernier’s memories and experiences of travel to Northern Europe date back to trips with his grandparents, and he has designed this trip to include some of his favorite places. Hamburg, one of the great Hanseatic cities, flourished for centuries and even the destruction of World War II served in the end to rejuvenate this bustling city. Its museums are justly famous; the restoration and conversion of its waterfront are great examples of post-war Renaissance. Hannover has a very different history. Its dynasty built palaces and gardens before it was taken over by the English throne. As a result, this city has interesting architecture and two fascinating museums, as well as a beautifully restored Baroque garden. On our way to The Hague, the Kröller-Müller Museum with its extraordinary collections of Van Gogh and Mondrian will offer the perfect introduction to the almost limitless art treasures of the Netherlands. -
Gregory's Sixth Operation
Found Sci (2018) 23:133–144 https://doi.org/10.1007/s10699-016-9512-9 Gregory’s Sixth Operation 1 2 3,4 Tiziana Bascelli • Piotr Błaszczyk • Vladimir Kanovei • 5 5 6 Karin U. Katz • Mikhail G. Katz • Semen S. Kutateladze • 5 7 8 Tahl Nowik • David M. Schaps • David Sherry Published online: 20 December 2016 Ó Springer Science+Business Media Dordrecht 2016 Abstract In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathe- matics under scrutiny. Following Ian Hacking, we point out that in the history of mathe- matics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of p. Here Gregory referred to the last or & Mikhail G. Katz [email protected] Tiziana Bascelli [email protected] Piotr Błaszczyk [email protected] Vladimir Kanovei [email protected] Karin U. Katz [email protected] Semen S. Kutateladze [email protected] Tahl Nowik [email protected] David M. Schaps [email protected] David Sherry [email protected] 1 Lyceum Gymnasium ‘‘F. Corradini’’, Thiene, Italy 2 Institute of Mathematics, Pedagogical University of Cracow, Cracow, Poland 3 IPPI, Moscow, Russia 4 MIIT, Moscow, Russia 123 134 T. Bascelli et al. ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an ‘‘unrigor- ous’’ practice, in fact finds close procedural proxies in modern infinitesimal theories. -
Annual Report 2017−2018
ROYAL COLLECTION TRUST ANNUAL REPORT REPORT COLLECTION TRUST ANNUAL ROYAL 2017−2018 www.royalcollection.org.uk ANNUAL REPORT 2017−2018 ROYA L COLLECTION TRUST ANNUAL REPORT FOR THE YEAR ENDED 31 MARCH 2018 www.royalcollection.org.uk AIMS OF THE ROYAL COLLECTION TRUST In fulfilling The Trust’s objectives, the Trustees’ aims are to ensure that: ~ the Royal Collection (being the works of art ~ the Royal Collection is presented and held by The Queen in right of the Crown interpreted so as to enhance public and held in trust for her successors and for the appreciation and understanding; nation) is subject to proper custodial control and that the works of art remain available ~ access to the Royal Collection is broadened to future generations; and increased (subject to capacity constraints) to ensure that as many people as possible are ~ the Royal Collection is maintained and able to view the Collection; conserved to the highest possible standards and that visitors can view the Collection ~ appropriate acquisitions are made when in the best possible condition; resources become available, to enhance the Collection and displays of exhibits ~ as much of the Royal Collection as possible for the public. can be seen by members of the public; When reviewing future plans, the Trustees ensure that these aims continue to be met and are in line with the Charity Commission’s general guidance on public benefit. This Report looks at the achievements of the previous 12 months and considers the success of each key activity and how it has helped enhance the benefit to the nation. -
Ca Atalogu Ue 48: P Portra Aits
Grosvenor Prints 19 Shelton Street Covent Garden London WC2H 9JN Tel: 020 7836 1979 Fax: 020 7379 6695 E-mail: [email protected] www.grosvenorprints.com Dealers in Antique Prints & Books Catalogue 48: Portraits October 2014 Item 11. Rubens with his Wife and Child. This catalogue is illustrated in full at www.groosvenorprints.com Registered in England No. 1305630 Registered Office: 2, Castle Business Villlage, Station Roaad, Hampton, Middlesex. TW12 2BX. Rainbrook Ltd. Directors: N.C. Talbot. T.D.M. Rayment. C.E. Elliis. E&OE VAT No. 217 6907 49 1. [Portrait of Joshua Barnes] Vera Collection. Lot 2369 & Ex: Collection of the Hon. Effigies Joshuae Barnes S.T.B. [...] Christopher Lennox-Boyd. R. White ad vivum fecit [1694] Stock: 34967 Line engravinng, small margins on 3 sides; platemark 295 x 190mm (11½ x 7½"). Glued at corners to 5. Andrea Vannucchii, d.o And.a del Sarto. backing sheet. £95 Pittore nato in Firenze nel 1488. n. del 1530. Joshua Barnes (1654-1712), Greek scholar and [n.d., c.1795.] antiquary. In 1679 his 'History of Esther' rendered the Coloured engraving. Sheet 285 x 185mm (11¼ x 7¼"). Book of Esther in Homeric hexameters, complete with Trimmed in two sections and laid on card, some commentary in Greek. He later turned to English rubbing. £70 history with a book on Edward III A self-portrait of Andrea d'Agnolo di Francesco di Stock: 34871 Luca di Paolo del Migliore (11486-1530), Florentine paainter better known as 'del Sarto' ('tailor's son'). High- 2. Gulielmus King LLD LLD. Aulæ Beatæ regarded during his lifetime, he was eclipsed after his Mariæ Virginis P Ætatis Anno LXV. -
Printing Natural Knowledge
Chapter 3: Printing Natural Knowledge “The purpose of writing may be to persuade the wise and intelligent; or else it may be to persuade everyone in town.” So wrote Giovanni Alfonso Borelli, pro- fessor at the university of Pisa and already the author of an important work on Euclid, to his ex-colleague Marcello Malpighi, just starting out on the faculty at the university of Bologna and about to publish his fundamental work revealing the microstructure of the lungs. Deciding what to publish, and for whom, was no easy matter, still some thirty years after the twin trials of Orazio Morandi and Galileo Galilei. “But if you choose to write to the common citizens of Bo- logna to persuade them about the calumnies made against truth,” Borelli con- tinued, “I conceive that in this case you may use every effort to deride any per- son who peddles a pile of idiocies just to offend virtuous and meritorious people.” 1 Of course, for Borelli, “virtuous and meritorious people” meant per- sons who followed empirical method for the study of nature, exemplified by Galileo; and “truth” meant the truth drawn from the use of that method. “Idio- cies” meant ideas conceived by those whom the Galileians conceived as dogma- tists and mystifiers. Still in the 1660s, whether Galileo’s disciples and their followers would triumph against the dogmatists and mystifiers was by no means clear. The story of their campaign to capture public attention, with all its fits and starts, is far less well known than the content of their ideas. It will occupy us here. -
Jürgen LIT Fischer
For the past three years, Robert Simon, gallery owner and Seit drei Jahren zeigt Robert Simon als Partner der Celler art collector from Hanover, has shown his collection in Kunst-Stiftung seine Sammlung im dortigen Bomann- collaboration with the Art Foundation of Celle at the Museum. Hier entstand jetzt das erste 24-Stunden-Kunst- Bomann-Museum in Celle. This is where recently the museum der Welt. Am Tag, nämlich zu den normalen Öffnungszeiten des world’s first 24-hour-art museum was opened. Museums, geht der Besucher ins Haus. Er zahlt seinen Ein- During the day and normal opening hours, a visitor tritt und kann sich in diesem breitgefächerten Haus von der can enter the building. After paying an entrance fee, the Landes- und der Stadtgeschichte über die Volkskunde bis zur visitor can move freely among the wide range of exhibits Moderne in der Kunst, die im Anbau des Museums unterge- on Lower Saxony’s and Celle’s history and an anthropo- bracht ist, bewegen. Er sieht in den Räumen der Kunst-Stif- logical section to the exhibits on modernism in art which tung die Sammlung Robert Simon mit Kabinetten von Joseph have been housed in the annex of the museum. In the Beuys und Peter Basseler, er sieht Bilder von Roland Dörfler, Ralph Fleck, Dieter Krieg, Lienhard von Monkiewitsch, rooms of the Art Foundation, the visitor can see Robert Hartmut Neumann, Giso Westing, Ben Willikens und Alfred Simon’s collection with cabinets containing Joseph Beuys Winter-Rust und nimmt nur partiell etwas von dem wahr, was and Peter Basseler and he can view pictures by Roland ihn bei Nacht oder besser gesagt nach der Schließung des Dörfler, Ralph Fleck, Dieter Krieg, Lienhard von Monkie- Museums erwartet. -
Cavalieri's Method of Indivisibles
Cavalieri's Method of Indivisibles KIRSTI ANDERSEN Communicated by C. J. SCRIBA & OLAF PEDERSEN BONAVENTURA CAVALIERI(Archivio Fotografico dei Civici Musei, Milano) 292 K. ANDERSEN Contents Introduction ............................... 292 I. The Life and Work of Cavalieri .................... 293 II. Figures .............................. 296 III. "All the lines" . .......................... 300 IV. Other omnes-concepts ........................ 312 V. The foundation of Cavalieri's collective method of indivisibles ...... 315 VI. The basic theorems of Book II of Geometria .............. 321 VII. Application of the theory to conic sections ............... 330 VIII. Generalizations of the omnes-concept ................. 335 IX. The distributive method ....................... 348 X. Reaction to Cavalieri's method and understanding of it ......... 353 Concluding remarks ........................... 363 Acknowledgements ............................ 364 List of symbols ............................. 364 Literature ................................ 365 Introduction CAVALIERI is well known for the method of indivisibles which he created during the third decade of the 17 th century. The ideas underlying this method, however, are generally little known.' This almost paradoxical situation is mainly caused by the fact that authors dealing with the general development of analysis in the 17 th century take CAVALIERI as a natural starting point, but do not discuss his rather special method in detail, because their aim is to trace ideas about infinites- imals. There has even been a tendency to present the foundation of his method in a way which is too simplified to reflect CAVALIERI'S original intentions. The rather vast literature, mainly in Italian, explicitly devoted to CAVALIERI does not add much to a general understanding of CAVALIERI'Smethod of indivi- sibles, because most of these studies either presuppose a knowledge of the method or treat specific aspects of it. -
How a Dangerous Mathematical Theory Shaped the Modern World
it. As Clavius points out in the “Prolegomena,” it was the sturdiest edifice in the kingdom of knowledge. For a taste of the Euclidean method, consider Euclid’s proof of proposition 32 in book 1: that the sum of the angles of any triangle is equal to two right angles—or, as we would say, 180 degrees. Euclid, at this point, has already proven that when a straight line falls on two parallel lines, it creates the same angles with one parallel line as with the other (book 1, proposition 29). He makes good use of this theorem here: Proposition 32: In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. Proof: Let ABC be a triangle and let one side of it be produced to D. I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle, ABC, BCA, CAB, are equal to two right angles. Figure 2.1. The sum of the angles in a triangle. 92 This was Salviati’s (Galileo’s) theory of matter, and as he himself admitted, it was a difficult one. “What a sea we are slipping into without knowing it!” Salviati exclaims at one point. “With vacua, and infinities, and indivisibles … shall we ever be able, even by means of a thousand discussions, to reach dry land?” Indeed, can a finite amount of material be composed of an infinite number of atoms and an infinite number of empty spaces? To prove his point that it could, he turned to mathematics. -
BADO E MART AUCTIONS
BADO e MART AUCTIONS Rare Books, Manuscripts and Prints Tuesday - June 28, 2016 Rare Books, Manuscripts and Prints 1: Collection of 12 Elzevirs EUR 2,000 - 3,000 Collection of 12 travel and history books, first editions or rare editions.16mo; Contemporary full vellum or calf bindings; Titlepages engraved on copper.1 - AA.VV. Respublica, sive status regni Scotiae et Hiberniae. Leiden, Ex officina Elzeviriana, 1627 [at the end: 1630]2 - CUNAEUS, Petrus. De Republica Hebraerum. Lugd. Batavor. [Leida], Ex Officina Elzeviriana, 16323 - GILLES, Pierre. De Bosporo. Lugduni Batauorum, Apud Elzevirios, 1632. Elzeviriana first edition first issue.4 - LAET, Joannes De. De Imperio Magni Mogolis, sive, India vera. Leida, Ex officina Elzeviriana, 1631. Elzeviriana first edition, second issue.5 - LAET, Joannes De. Gallia, sive de Francorum regis dominus et opibus commentarius. Leida, Elzevier, 1629. Contemporary binding in brown calf. Elzeviriana rare edition, released the same year of the first.6 - LAET, Joannes De. Hispania, sive De regis hispaniae regnis et opibus commentarius Ex Officina Elzeviriana, Lugduni Batavorum [Leida] 1629. Second edition published the same year of the first. Exemplary missing of 4 leaves.7 - MONTALBANO, Giovanni Battista. Turcici imperii status. Seu discursus varij de rebus Turcarum. Leida, Ex officina Elzeviriana, 1630. First Elzeviriana edition.8 - SIMLER, Josias. Helvetiorum Respublica. Lugd[uni] Bat[avorum], [Leiden], Ex Officina Elzeviriana, 1627. Original edition, first edition.9 - SIONITA, GABRIEL - AA.VV. Arabia, seu Arabum vicinarumque gentium Orientalium leges, ritus, sacri et profani mores, instituta et historia. Amsterdam, J. Jansson, 1633.Edizione originale10 - SMYTH, Thomas. De Republica Anglorum. Leida, Officina Elzeviriana 1641. In first issue.11 - SPRECHER, Fortunat. -
The Royal House of Hanover
The Royal House of Hanover George I (George Louis), 1660-1727, Duke of Brunswick and Lüneburg, from 1698 Elector of Brunswick and Lüneburg (“The Electorate of Hanover”), from 1714 King of Great Britain and Ireland. George II (George Augustus), 1683-1760, Elector of Hanover and King of Great Britain and Ireland. George III (George William Frederick), 1738-1820, Elector of Hanover and King of Great Britain and Ireland. George IV (George Augustus Frederick), 1762-1830, King of Hanover and King of the United Kingdom of Great Britain and Ireland. William IV (William Henry), 1765-1837, King of Hanover and King of the United Kingdom of Great Britain and Ireland. End of the Union of Crowns. Ernest Augustus, 1771-1851, Duke of Cumberland, from 1837 King of Hanover. George V, 1819-1878, King of Hanover. 1866: End of the Kingdom of Hanover. 2 Welcome! Dear Readers, Dear Visitors following in the footsteps of the Guelphs, For centuries, the history of the former Kingdom of Hanover and of the present-day State of Lower Saxony was closely intertwined with the story of the Electors and Kings of Hanover. When George Louis, Elector of Hanover, ascended the British throne as King George I in 1714, this marked the beginning of the 123-year reign of the House of Guelph over Great Britain, where it was known as the House of Hanover. As a result, the Guelphs, the oldest royal dynasty in Europe, left a decisive imprint on European, American and world history. Numerous testimonies to the days when the region was ruled by the House of Guelph are still to be found in and around Hanover today. -
Baroque Music Guide
Arcangelo Corelli Cardinal Pietro (1653-1713) Ottoboni (1667-1740) a founding influence librettist, important in baroque music St Peter’s, Rome music patron, Rome Antonio Vivaldi (1678-1741), major Concert in a typical Venetian Ospedale, Francesco Geminiani baroque composer or Orphanage, where Vivaldi was employed (1687-1762) composer based in Venice as tutor, composer and conductor pupil of Corelli George Friderick Henry Purcell HANDEL (1659-1695) (1685-1755) German- Concert in the major English born, English-adopted Bach-Haus, Eisenach composer A baroque cantata rehearsal Clavichord Baroque German Hunting Lodge Silbermann Organ, High baroque Burgk Castle Church interior Silbermann Organ J.S Bach, portrait Interior of above Pedal-harpsichord Freiberg Cathedral in old age, late 1740s CONTENTS WHAT IS BAROQUE? 1 THE GRAND TOUR OF EUROPE 6 MUSIC PUBLISHING IN EUROPE 14 Antonio STRADIVARI - Violin maker - Cremona 17 Tomaso ALBINONI - Italy 18 Arcangelo CORELLI - Rome 20 Francois COUPERIN ("the Great") - France 24 Johann Joseph FUX - Austria 25 Francesco GEMINIANI - Italy, England 26 George Frideric HANDEL - Germany, Engla28 24 Henry PURCELL - England 34 Jean Philippe RAMEAU - France 36 Alessandro SCARLATTI (father) - Italy 38 Domenico SCARLATTI (son) - Italy, Spain 41 John STANLEY - England 43 Georg Philipp TELEMANN - Germany 45 Antonio VIVALDI - Venice 49 Unico Willem van WASSENAER - Holland 53 Jan Dismas ZELENKA - Prague, Dresden 56 Johann Sebastian BACH: EISENACH - MÜHLHAUSEN 58 WEIMAR (second term) 64 CÖTHEN 66 LEIPZIG 1 - Cantor and Director of Music 70 LEIPZIG 2 - The Collegium Musicum 77 LEIPZIG 3 - The Introspective Years 82 BAROQUE KEYBOARD INSTRUMENTS 84 GOTTFRIED SILBERMANN: Master organ builder 88 BAROQUE MUSIC PERFORMANCE: 92 Produced by the editors at www.BaroqueMusic.org where further detail on baroque music can be found.