# ICHM Co-sponsored Symposia at the 23rd Congress of History of Science and Technology Budapest, July 28 – August 2, 2009

Five of the symposia at ICHST09 were co-sponsored by the ICHM. These symposia are listed below with the names of their organizers, descriptions of their purpose, as well as the names of speakers, with titles of their papers and abstracts.

## Symposium S06 "Transmission and Transformation of Mathematics and Mathematical Instruments in their Social Contexts, East and West" Organizers, Joe Dauben (USA) and Liu Dun (China)

Description:
This symposium will focus on the history of Asian mathematics and instruments related thereto, including the use of gnomons, the abacus, and other instruments meant to record of facilitate computations.  Emphasis will be given to indigenous developments, transmission between cultures, and how mathematics takes on local characteristics when transplanted from one part of the world to another.

Speakers:

QIN JIUSHAO’S DIVINING METHOD AND IN WHICH THE MATHEMATICAL THEORY CONTAINED
Gang HOU

Department of Mathematics, Tianjin Normal University, Tianjin, P. R. China
Qin Jiushao was a mathematician in Southern Song China. In his mathematical work, Shushu Jiuzhang (Mathematical Treatise in Nine Sections), he advanced a method for divination by counting yarrow stalks that was distinct from the divining method derived from Zhouyi (Book of Changes).Qin’s method had its own origin, which was involved in the first problem titled Shigua Fawei of the first chapter of his work. Qin obviously took shigua fawei as a mode lbrought to light the pith of Dayan zongshu method that, in nowadays mathematics terms, is a systematic solving process for simultaneous linear congruence equations. He not only clarified the algorithm in accordance with the theory of congruence but also revealed the connection between the algorithm and the method for divination by counting yarrow stalks.

THE INTRODUCTION OF NAPIER’S RODS IN CHINA
Cervera, Jose A.

Tecnologico de Monterrey, Mexico
The Scottish John Napier (1550-1617), besides his logarithms, also developed other instruments devoted to facilitate computations in his book Rabdology (1617). One of these methods consists of a set of small rods (Napier’s rods or bones), that enable fast multiplications, divisions and square and cubic roots. This method was popular for several decades in Europe, especially in Scotland, but it was soon replaced with other methods, such as logarithms.
It is really interesting that the Rabdology was one of the first methods for arithmetic calculations introduced in China by Jesuits. In 1628, Giacomo Rho (Luo Yagu, 1592-1638) wrote his Chou Suan (Calculus with rods). The Chou Suan was included in the Xiyang Xinfa Lishu (Calendar compendium according to the Western new methods), an encyclopedic work on mathematics and astronomy reedited by Adam Schall von Bell (Tang Ruowang, 1592-1666) in 1645 after the Chongzhen Lishu (Calendar compendium of the Chongzhen era), which contained many European mathematical and astronomical treatises translated into Chinese by Rho and Schall between 1630 and 1635.
Probably, Rho’s Chou Suan would not have been very important for Chinese mathematics if Mei Wending (1633-1721) had not written his own Chou Suan. Mei Wending is considered one of the Chinese mathematicians more influential of his time. His Chou Suan has several differences from Rho’sone. For example, rods are not vertical ,but horizontal ones.
In this paper, I will give a general survey on Rho’s and Mei’s Chou Suan, and I will compare these books with Napier’s Rabdology as a typical example of adaptation of a European computation device in China.

KAREL SLAVÍÈEK AND YAN JIALE METHOD
Liu Dun

Institute for the History of Natural Science, Beijing, China
It is well-known that the Jesuits had played important role to the transmission of some knowledge of Western science to China during the 17 and 18 centuries. Among them, Karel Slavíèek (1678—1735), with the Chinese name Yan Jia-le, was a versatile scholar but unfortunately ignored by most researchers in the field concerned. Slavíèek was born in Moravia in 1678, arrived in Macao in 1716 and died in Beijing in 1735.
During the 19 years of his stay in China, in the immediate aftermath of the Chinese Rites Controversy, like his contemporarycolleagues,Slavíèekexperiencedtremendouslimitationtoactivitiesofevangelization;nevertheless,thanks to his profound knowledge of astronomy, mathematics, machinery and musicology, Slavíèek was high in the favor of Emperor Kangxi (1654—1722), the sole monarch in Chinese history who favored mathematics and natural sciences.
There is evidence suggesting that Slavíèek was involved in writing many literature on astronomy, part of which was compiled into Lixiang Kaocheng Houbian (later edition of the established system of calendric astronomy, 1743). While in China, he carried out a number of scientific activities, including and not limited to the drawing of a map of Beijing, measuring the altitude at which the Northern Star rises above the horizon, observing the motion and location of the moon, drawing a map of the lunar surface, analysis of Chinese record of solar eclipse and research in Chinese calendar, musical tones and chronology.
The works mentioned above could be gleaned from his letters to personages in and outside the European churches. In addition, in Chishui Yizhen (precious relic along the red river, before 1744) by Chinese mathematician Mei Juecheng (1681-1763), the approach to determining geographical latitude through the height of a star, and its hour angle is recorded, and this is called “Method of Western Scholar Yan Jia-le”.

ON THE HISTORY OF THE COMPASS
Galina A. Zverkina

Moscow State University of Railway Engineering, Russia
The compass is one of the most ancient tools that mankind has used since the Stone Age. Apparently, one of the most ancient images of a compass is from ancient China, where a stone relief from a Han dynasty tomb depicts the mythical figure, Fu Xi,with a gnomon and his consort,NuWa, holding a compass. These tools symbolize divine knowledge. It is possible to divide all compasses into four basic groups:
1. Cord compasses: cords with pegs at the extremities. Such compasses are still applied in various engineering operations (for example, by roofers). 2. Bow compasses: compasses with two legs connected by a hinge. One modification of the bow compass is the “perfect” or “elliptic” compass, created in antiquity for drawing hyperbolas, parabolas, and ellipses (whole-and-half compasses), and reconstructed by Arabic scientists in the tenth century. 3. Beam compasses: compasses ith two short legs that move along a ruled rod (bar), today used basically for measurements. 4. Proportional compasses: These are used for reducing or enlarging drawings, and have legs crossing so as to present a pair on each side of a common pivot. By means of a slit in the legs, and the movable pivot, the relative distances between the points at the respective ends may be adjusted to any required proportion. The second type of proportional compass has legs in the form of two wide bars on which various scales are plotted; with their help one can easily solve complicated geometrical and algebraic problems.
In ancient times, the compass and carpenter’s square or gnomon were symbols of knowledge, and we find traces of the widespread use of geometrical algebra. The mathematics of all ancient civilizations has surpassed this stage of development. For a long time geometrical calculations were the basic tool of science and engineering. In the Middle Ages the art of using compasses was widely appreciated, and the compass was regarded as a divine creation. The compass was adopted as the emblem of craft communities, a secret symbol of masons, and a figure that appeared on the arms of noblemen.
The complexity (and sometimes the impossibility) of using compasses in certain problems led to new designs for drawing tools used for the resolution of practical problems. New compasses based on various adaptations (usually having a basis in established, classical designs) were also created.
Later, when numerical systems were more highly developed, geometrical methods receded into the background. However, the art of geometrical constructions was highly regarded and was an important basis of mathematical training. The geometry of ruler and compass was fundamental to the development of new mathematics in Europe. And in practical designing and measuring, there is nothing than can replace the compass.
Now there are new models of compasses, including electronic digital compasses, but these nevertheless are based on designs as ancient as humanity.

CIRCLES AND SQUARES, CUBES AND SPHERES, EAST AND WEST
Joseph W. Dauben

Herbert H. Lehman College, The City University of New York, USA
Among the oldest extant mathematical records are methods for finding the side of a square with a given area, or the volume of a cube or sphere. Were the methods for approaching and solving such mathematical problems similar or different in Egyptian, Babylonian, Greek, Indian, Islamic, and Chinese contexts? By approaching this question from a comparative point of view, the differences in methods are as striking as their similarities. Comparisons within a given culture are also instructive, for instance the methods found in the earliest yet-known Chinese mathematical text, the Suan Shu Shu (A Book on Numbers and Computations, ca. 186 BCE) and the classic texts of ancient Chinese mathematics, especially the Jiuzhangsuanshu (Nine Chapters on the Art of Mathematics; the edition with commentary by Liu Hui dates to 263 CE).

LEIBNIZ’S VIEW OF THE I CHING
Maria Sol de Mora
UPV/EHU, Spain

The I Ching or Book of Changes is one of the oldest Chinese classic texts. Although it is a set of oracular statements, it is traditionally said that Confucius wrote a set of philosophical commentaries on the I Ching known as the Shi Yi. And there are statements in this text that are very near to the philosophy of Leibniz, for example; the dynamic balance of opposites, the evolution of events as a process, or the acceptance of the inevitability of change.
Leibniz was in contact with the French Jesuit R.P. Bouvet who informed him about the I Ching, and this came as great news to Leibniz and reinforced his idea of the universality of knowledge.
1. Explication de l’Arithmétique binaire, Mémoires de l’Académie des Sciences de Paris, 1703. Dutens, III, 390-4.
GM, VII, 223. 5p.
2. De inventione Arithmeticae Binariae, excerpt. ex Vita Leibnitii a D. Jaucourt scripta, Dutens, III, 345-8.
3. Epistolae duae ad Schulenburgium De Arithmetica Dyadica, 1724, Dutens III, 349-54.
4. Erklärung der Aritmeticae binariae, Journal des Sçavans, 81-112. Etc.
We will comment in particular on the binary system as a very important concept in the theory of combinatorics.

THE CALCULATING PROGRAM OF THE LUNAR MOTION IN YUZHI LIXIANG KAOCHENG (1725)
Lu Dalong

Institute for the History of Natural Sciences, Chinese Academy of Sciences, P. R. China
In the early period of the Qing Dynasty (1644-1911), the four calendars had been put into use, which are Xiyang Xinfa Lishu (Treatise on Mathematics (Astronomy and Calendrical Science) according to the Western Method, this encyclopedia was issued in the Ming (1635) as Chongzhen reign-period Treatise on (Astronomy and) Calendrical Science, first form of the Jesuit astronomical encyclopedia, reissued as the former by Johann Adam Schall von Bell (1591-1666)in1645,1628-1827,andtheyearof1628wasselectedastheepochoftheCalendar),KangxiYongnianLifa (The Eternal Calendar of Kangxi Emperor, compiled by Ferdinand Verbiest (1623-1688) in 1669, 1828-3827), Yuzhi Lixiang Kaocheng (Complete Studies on Astronomy and Calendar, 1684-1983) and Yuzhi Lixiang Kaocheng Houbian (The Supplement to Complete Studies on Astronomy and Calendar, 1723-2022).
The theory of the lunar motion in Yuzhi Lixiang Kaocheng was derived from the model of epicycle- oblique circlesub-epicycle- sub-oblique circle (Benlun- Junlun- Cilun- Cijunlun). The diameters of the epicycle, oblique circle, sub-epicycle and sub-oblique circle were 1,160,000, 580,000, 434,000 and 235,000 respectively when the distance between the Earth and the Moon was supposed as 10,000,000. The calculating program for the lunar motion in Yuzhi Lixiang Kaocheng has been realised and the correspondingly preliminary conclusions as followed.
1. The characters of leap years in Xiyang Xinfa Lishu (1645), Kangxi Yongnian Lifa (1669) and Yuzhi Lixiang Kaocheng (1725) had a common continuity.
2. The Ersan Junshu Biao in Yuzhi Lixiang Kaocheng, and Ersan Junshu Zongshu Jiajian Biao in Xiyang Xinfa Lishu
as well, could be derived by the above mentioned model as the result of the eight calculating formulae.
3. TheMoon’s greatest distance from the Earth, the parallaxes of the Moon and the apparent diameters of the Moon in
Yuzhi Lixiang Kaocheng were different from the values given by Claudius Ptolemaeus (c.90-168), Nicolaus Copernicus (1473-1543), Tycho Brahe (1546-1601), and Johannes Kepler (1571-1630).
4. The ratio of the diameter of Earth and the diameter of the Moon in Yuzhi Lixiang Kaocheng is 3.72 and 1, which was different from the value of 3.5 and 1 given by Nicolaus Copernicus and appeared in Xiyang Xinfa Lishu.
5. The calculating formulae for semi-diameter of the Sun (Ri Banjing), semi-diameter of the Moon (Yue Banjing), and shadow’s semi-diameter of the Earth (Ying Banjing), have been outlined, which are different from that in Copernicus’ Revolutions (1543).
6. The theory of the lunar motion in Isaac Newton’s 1702 Theory of the Moon’s Motion and the second edition of
Principia (1713) had not introduced in Yuzhi Lixiang Kaocheng, which was slightly revised and incorporated in Yuzhi Lixiang Kaocheng Houbian.
The author gratefully acknowledges the supports of K. C. Wong Foundation, Hong Kong, National Science Foundation of China (NSFC), and the China-Portugal Center for the History of Sciences (CPCHS).

“WASAN” MATHEMATICIANS, TECHNOCRATS AND SAMURAI DURING THE EDO PERIOD IN JAPAN –SEKI TAKAKAZU’S RESIDENCE
AND THE SOCIAL STATUS OF MATHEMATICIANS
Shigeru JOCHI

National Kaohsiung First University of Science and Technology, TAIWAN
Seki Takakazu (1642?-1708) lived in the age of Samurai warriors, or Samurai technocrats. In the middle of the 17th century, there was nothing for Samurai warriors to do, but Samurai technocrats controlled the Shogun government. Samurai technocrats surveyed their own land. Seki was the second son of Uchiyama Nagaakira (?-1646?/1662?), an unemployedSamurai;asthesecondson,itwasquitedifficultforSekitobecomeaSamuraiaswell. However,Sekistudied Wasan, traditional Japanese mathematics, and was adopted as a foster son by Seki Gorozaemon (?-1665£©. Seki Takakazu became an examiner of accounts for the lord of -han, Tokugawa Tsunashige. Consequently, we can conclude that the Wasan was a means for success in life for Samurai in this era. Thus we could call this era the “Kanjo-gata Wasan era”since the appearanceof Seki’s HatsubiSampo (Seki,1674). After Seki, there were Takebe Katahiro(1664-1739) and Yamaji Nushizumi (1704-1773) who also studied Wasan and lived the successful life of the Samurai.
We must consider where Seki lived, because we can determine the social position of Kanjo-gata from their residential area. In the Kofu-sama Goninshu Bugencho (1695), it is said that Seki lived in Tenryuji. The problem is, in which Tenryu-ji did Seki live? That is to say, Yotsuya’s Teryu-ji or Ushigome’s Tenryu-ji, because Tenryu-ji was in Ushigome before the big fire of Tenwa (1682). We can examine the maps of Gofunai Enkaku Zusho (1808-1861), where Uchiyama Nagasada’s residence was in the former Tenryu-ji, or where the Ushigome police station is now. Moreover, the residence of Seki Gorozaemon, Seki’s father, was in the former Tenryu-ji; also, the Jorin-ji temple of both Seki’s family and Uchiyama’s family is in Ushigome. Therefore we can conclude that Seki lived in Ushigome. Kanjo-gatatodayliveinShinjuku,that is to say,the“newtown.” We can know something of the lifestyle of Kanjo-gata through Seki’s former residence.

"TRIGONOMETRIC TABLES, THEIR UTILITY, AND MAKING IN LATE IMPERIAL CHINA
Jiang-Ping Jeff Chen
St. Cloud State University, Minnesota, USA

Integration of Jesuit and Chinese methods in astronomy was one of the phases Xu Guangqi (1579-1659) prescribed for theCalendarReforminlateMing.Oneoftheobstaclesinintegrationofthetwomethodswastheincommensurabilityof trigonometric tables the Jesuits introduced and the measuring unit for arcs in the traditional Chinese system. As an important computing “instrument,” trigonometric tables in China as well as its utility and making warrant close examination.
The Jesuits utilized trigonometric tables to simplify the computations in astronomy while Chinese astronomers, before the arrival of the Jesuits, employed the method of interpolation to serve the same computational needs. Although the Jesuits provided the basic principles of making trigonometric tables, there were technical details left unexplained, which made the reconstruction of a complete trigonometric table with impossible. Some Chinese scholars in the 18th century tried to remedy this situation by changing the measuring unit of arcs from 360 degree for the full circle to some other units. Such changes rarely had any following. In the 19th century, after the publication of Geyuan milü jiefa (Quick Methods for Circle-Division and Determining the Precise ratio of the circle) by Ming Antu (1692?-1765?), a trigonometric treatise which discussed the power series approach of finding the length of an arc from its sine value and related properties provided an easy and fast way to complete the construction of trigonometric tables. In this paper, I compare the Chinese indigenous computation methods, trigonometric tables introduced by Jesuits, and trigonometric tablesconstructedbyChinesescholarsinthe17th-19th century utilizing various computation methods to investigate how Chinese scholars at different time viewed and received trigonometric tables. Moreover, I examine the how scholars’ views on Western learning influenced their approach to making trigonometric tables.
Key Words: Seki Takakazu (Kowa)’s Residence, Ushigome Jorin-ji, Kofu-han, Gofunai Enkaku Zusho

## Symposium S03 “Status in Mathematics: In Particular the Role of Applications in the First World War" Organizers, Jeremy Gray (UK) and Reinhard Siegmund-Schultze (Norway)

Description:
The aim of this symposium on the history of mathematics is to take the theme of ideas in social context and focus it on how status is acquired in mathematics. We do this by considering aeronautics, an emerging topic in applied mathematics, and on complex function theory, a major, and related, topic in pure mathematics, concentrating on the period around the First World War. Particular attention is paid to the way these two strands were involved in the development of institutions (universities, research establishments, and journals) and the biographies of the protagonists and their social context. Aeronautics as a mathematical topic will also be approached as part of a presentation on the impact of the First World War. to, and promotion of, complex analysis"

Speakers:

THE RISE OF COMPLEX ANALYSIS IN FRANCE AND GERMANY
Jeremy J Gray

Open University, Milton Keynes, United Kingdom j.j.gray@open.ac.uk
Complex analysis was created as a researchsubject in different ways andfor different reasons by Cauchy, Riemann, and Weierstrass. By and large it was left to their followers to bring it into the university syllabuses and to give it a central place in the teaching curriculum. This talk examines the different ways thiswas done in France and Germany by looking at the textbooks that were written up to 1914.

CAUCHY'S AND WEIERSTRASS'S 'SCHOOLS' ON COMPLEX FUNCTION THEORY: A COMPARISON
Umberto Bottazzini

Universita di Milano, Dipartimento di Matematica, Italy umberto.bottazzini@unimi.it
In the second half of the 19th-century the leading role in complex function theory (and more generally, in mathematics) shifted from France to Germany, more precisely from Paris to Berlin. Cauchy had obtained a number of fundamental results,andhiswork–combinedwithcontemporarystudiesbyLiouville,Laurent,Puiseuxandothers–hadcontributed to the establishment of a ‘French school’ in complex analysis. However, despite being tremendously influential in France, Cauchy’s scientific heritage prevented the French mathematicians from appreciating new developments taking place abroad, in the Prussian capital in particular. Led by Weierstrass, a ‘school’ grew up there that promoted a theoretical and abstract approach to analysis. Perhaps because of his very different training, Weierstrass did not share with the ingénieur-savant Cauchy his broad interest in applied mathematics. Instead, he made a point of establishing complex analysis (and the whole of analysis in general) on rigorous, arithmetical foundations. Eventually, his arithmetical approach became dominant, and the German term /Funktionenlehre/ became synonymous with analytic function theory according to Weierstrass’s principles.

FUNCTION THEORY FOR WAR: BALLISTICS AND FLUID MECHANICS IN FRANCE, 1915-1930
David Aubin
Université Pierre et Marie Curie, France

Starting in 1915, mathematicians played an especially prominent part in French military efforts in the field of ballistics and fluid mechanics. In both cases, nontrivial aspects of function theory were the tools of choice mobilized by mathematicians. This paper will show how an urgent need for new kinds of ballistic tables gave rise to new computing techniques that used function theory in order to provide better assessments of errors. Once computing procedures were established, the main problem shifted to physics and meteorology. Many in particular felt the need to improve the resistance law of a body in motion in a fluid (which was also useful for aviation). French mathematicians therefore endeavored to developed sophisticated mathematical techniques that ultimately gave rise to important new development in function theory. Mathematicians involved in this story include Jules Haag, Henri Villat and Jean Leray. This paper will discuss the various ways in which mathematicians argued for their social usefulness in the aftermath of World War I.

THE BRITISH USE OF MATHEMATICS IN THE FIRST WORLD WAR
June E. Barrow-Green

Open University, Milton Keynes, United Kingdom J.E.Barrow-Green@open.ac.uk
‘This is a Mathematical War’ declared a veteran British mathematician to his colleagues in January 1915. Two years later his words were echoed at the front by a young British soldier who found himself fighting in a ‘war of guns and mathematics’. Were these accurate descriptions or isolated observations? What effect did the war have on British mathematicians and on their subject? To answer these questions, I shall consider the extent to which British mathematicians were encouraged to contribute to the war effort and the nature of their contributions.

DEVELOPING A THEORY OF BALLISTICS FROM EXPERIMENTATION AND MATHEMATICS: OSWALD VEBLEN, FOREST RAY MOULTON,
AND THE ABERDEEN PROVING GROUND PROJECT
Deborah Kent

Hillsdale College, Science Division, Mathematics, Hillsdale, USA dkent0@gmail.com
The introduction of long-range artillery, high-altitude fire, and anti-aircraft guns in the First World War rendered Siacci theory largely outmoded. At the Army Ordnance office in Washington, F.R. Moulton assumed the role of developing a new, more effective theory of ballistics. This effort incorporated data collection and computation overseen by Veblen at Aberdeen together with new applications of mathematics to ballistics. This paper considers mathematical work at the Aberdeen Proving Ground in context of the new relations it created between the mathematical community and both the United States government and military.

THE RISE OF EXTERIOR BALLISTICS IN AMERICA, 1880-1929
Alan Gluchoff Villanova University, U.S.A

Exterior ballistics – the study of the flight of a projectile after its firing from a gun – evolved in several stages in the United States. The first–from the founding of the country until about 1880–involved somewhat cursory adaptations of studies in Europe, notably those o f Leonhard Euler, Benjamin Robins,and Francis Bashforth. These appeared primarily in textbooks issued from the United States Military Academy at West Point, founded in 1802. The second began in the early 1880’s with the first textbook in America devoted to exterior ballistics – James Ingalls’ « Exterior Ballistics in the Plane of Fire ». This was also an adaptation, this time of Francesco Siacci’s widely adopted solution of the differential equations of motion by means of approximations. The second stage – the period from 1880 to the beginning of World War I – was dominated by the elaboration and use of this theory for American guns, an effort initiated by military officers at the Coast Artillery School at Fort Monroe, Virginia. This school, established in 1824, had evolved into an institution for advanced officer training, providing a followup to a West Point education. The third stage came as a result of a call to the mathematical community at large for assistance in solving new problems arising as a result of challenges of the World War. This era, dating roughly from 1917 through 1929, saw a rise in the sophistication of the mathematics brought to bear on the problems and the entrance of this material into university settings as well as newer specialized military schools. Many of the contributions of this era were for the first time made by professional mathematicians, not military personnel, and were independent of developments in Europe.
This paper examines the second and third eras and reviews the increasing status of exterior ballistics as a topic in military and civilian educational institutions during these times. New institutional courses devoted to the material will be mentioned, as will new military journals whose articles were often devoted to aspects of it. The lives of some of the men who brought about the changes are examined, and their roles in the rise of exterior ballistics explained. The establishment of the Aberdeen Proving Grounds in 1917 as a model of scientific treatment of ballistic problems will be discussed. Several mathematical topics were a part of the new American treatment of the subject in the third era, and each experienced an increase in prominence, albeit short-lived, as a result of their use ; details on this phenomenon will be given. Mathematicians wrote some new books on exterior ballistics as well as articles appearing in mathematical journals ; their work will be described. A long-term project involving general ballistic tables was undertaken in 1919 using the new methods ; we will see how this project developed. We relate the context in which these changes occurred and the fate of exterior ballistics, in both university and military settings, in the late 1920’s.

MATHEMATICS IN A DEAD END: ARNOLD SOMMERFELD AND THE TURBULENCE PROBLEM OF THE EARLY 20th CENTURY
Michael Eckert Deutsches Museum, Germany

During the last decades of the 19th century, natural philosophers, mathematicians, physicists and engineers (like Lord Kelvin, Lord Rayleigh, Osborn Reynolds, Joseph Boussinesq, Hendrik A. Lorentz) tried in vain to bridge the gap between theoretical and practical hydrodynamics - a gap which was mainly rooted in the different modes of laminar and turbulent flow. Only the former was in agreement with mathematical analysis. The latter, most often met in practice and addressed as ‘hydraulic’, appeared intractable.
In 1908, Arnold Sommerfeld presented at the Fourth International Mathematical Congress in Rome a mathematical approach for deriving the transition to turbulence for a plane laminar flow with a linear velocity profile (plane Couette flow). By linearization of the differential equations for such a flow he obtained a transcendental equation which offered the opportunity to derive criteria for the onset of turbulence. One year before and unknown to Sommerfeld, the Irish mathematician William McFadden Orr had arrived at similar results. This approach (later named after Orr and Sommerfeld) was regarded as a promising avenue to explain the transition from laminar to turbulent flow. Although Sommerfeld pursued the conceptual paths paved by the British protagonists, the context of his approach was different. Sommerfeld’s mathematical roots had grown in Göttingen (as Felix Klein’s assistant) and Aachen (as professor of mechanics), where he became used to apply sophisticated mathematical techniques to a variety of physical and technical problems. Prior to his paper for the Rome Congress, Sommerfeld was involved both in discussions about the principles of hydrodynamics (with Lorentz and Hilbert) and in technical applications (for example, he had published in 1904 a theory of lubrication).
After the Rome Congress,“the turbulence problem”, as the challenge to describe the transition from lamina rto turbulent flow was labelled subsequently, became a fashionable topic for further analysis. However, despite the efforts of Sommerfeld himself and a number of his disciples (among them Otto Blumenthal, Ludwig Hopf, Fritz Noether and Werner Heisenberg), the Orr-Sommerfeld approach did not live up to its expectations. Even worse, it provided mathematical evidence that certain flows could never perform the transition to turbulence – in obvious contrast to experiments. Applied mathematicians outside Sommerfeld’s school, most prominently Richard von Mises, arrived at the same result. Although based on solid theoretical ground and sparked by the prospect of technical applications, the Orr-Sommerfeld approach was unable to bridge the gap between theoretical hydrodynamics and practical hydraulics. By 1920, the turbulence problem was widely recognized as an outstanding challenge for applied mathematicians. Despite (or because?) its paradoxical results, it contributed to the consolidation of applied mathematics as a specialty of its own right.

RICHARD VON MISES: A PIONEER OF PRACTICAL AND THEORETICAL AERODYNAMICS
Reinhard Siegmund-Schultze University of Agder, Kristiansand, Norway

Richard von Mises (1883-1953), who studied mechanical engineering at the Technical University in Vienna (1901-1905), was a pioneer of teaching (Strasbourg 1912) and research (wing theory 1917/20, based on complex functiontheory)inaerodynamics.DuringWorldWarI,asanofficerintheAustrian-HungarianFlyingCorpsinVienna, he built a ‚huge airplane’ (Grossflugzeug) which, however, never went into service.
The talk considers the interplay of the practical and theoretical dimensions in von Mises’ work in aerodynamics and compares it briefly with other, in the end more successful approaches (von Kármán, Prandtl).
The talk reflects also on the status of function theory acquired by important applications such as founding wing theory.

AVIATION AND AERODYNAMICS ALONGSIDE MATHEMATICS: THE CASE OF THE UNIVERSITY OF LEIPZIG
Schlote, Karl-Heinz
Sächsische Akademie der Wissenschaften, Leipzig Germany

As many others, academic institutions around Leipzig and Halle too were influenced by the rapid and impressive development of aviation in the first decades of the 20th century. The talk will answer the question how scientists at Leipzig University took part in this process. It seems at first glance that only physicists tackled problems of aerodynamics. But mathematicians contributed to the theoretical investigations, too. With reference to Harry Schmidt’s treatise on aerodynamics it will be shown in which way the work of men like Leipzig’s well known mathematician L. Lichtenstein attacked aerodynamical problems.

## Symposium S04 "Mathematical Analysis from the Eighteenth to the Nineteenth Centuries" Organizers, Craig Fraser (Canada) and Michiyo Nakane (Japan)

Description:

“Mathematics underwent, in the nineteenth century, a transformation so profound that it is not too much to call it a second birth of the subject - it’s first having occurred among the ancient Greeks...” - Howard Stein (1988)

The purpose of the symposium is to explore changes in the nature and understanding of analysis from the eighteenth to the nineteenth centuries. Much historical work over the past few decades has been devoted to uncovering the precepts and philosophical outlook of analysis in the eighteenth century. There have also been many studies of fundamental nineteenth-century developments in foundations, infinite series, functions of a complex variable, and other subject areas within analysis. Important themes  have included the status of the numerical continuum, conceptual versus algorithmic modes of thought, algebraic analysis versus geometric approaches, and differing conceptions of an analytic function of a complex variable.
The period of primary concern is the period from about 1750 to about 1850, beginning with Euler’s Introductio in Analysin Infinitorum of 1748 and ending with Riemann’s Grundlagen für eine allgemeine Theorie der Funktionen of 1851. These  temporal endpoints should not be interpreted too rigidly, and relevant developments somewhat earlier or later could also be considered.
Until the end of the nineteenth century the term analysis was sometimes understood fairly broadly to refer to any part of advanced mathematics that employed symbolic methods and algorithms. As late as 1869 the English mathematician James Joseph Sylvester could write that work on algebraic invariance had  “led to a complete revolution in the whole aspect of modern analysis.” Nevertheless, the sense of analysis as something concerned primarily with continuous processes had by 1850 gained substantial currency, a shift that reflected the radical overhaul of calculus-related parts of mathematics initiated by Cauchy. A goal of the symposium will be to identify the various changes in meaning attached to the concept of analysis during the period from 1750 to 1850.
In keeping with the general theme of the Congress (“Ideas and Instruments in Social Context”), the symposium will include discussion of the idea of analysis in the wider societal context of physics, engineering and education. One may identify external factors that contributed to changes in how analysis was understood. For example, national education systems became a factor in both the transmission and internal development of mathematical theories. The symposium will address the question of how the positioning of research within technical schools and universities affected the way in which analysis was understood and practised.

Speakers:

ELABORATION OF EULER‘S IDEAS ON SERIES IN THE EARLY 19th CENTURY
Hans Niels Jahnke
Universitä Duisburg-Essen

In a paper of 1760 titled “De seriebus divergentibus” L. Euler justified the 18th century use of divergent series by the definition „the sum of any series is a closed expression out of whose development that series has been formed“.
Surely, this was a rather informal definition, but in the early 19th century there were some mathematicians who tried to elaborate Euler’s idea by distinguishing systematically between a notion of “formal equality” between power series (close to the modern idea of formal power series) and numerical equality. They felt a need for more rigour but tried to be more faithful to 18th century practices than Cauchy’s radical ban of divergent series. One of them was the Berlin mathematician Martin Ohm (1792-1872). He published his approach in 1822 one year after Cauchy’s famous Analyse algébrique. In 1823 Ohm was able to correct one of the most discussed incorrect results of the time by consciously using divergent series.

EULER AND FUNCTIONS OF A COMPLEX VARIABLE
Craig Fraser

In the last ten years of his life Euler wrote several papers on functions of a complex variable. Published posthumously, these researches contributed to the initial identification of complex analysis as an autonomous part of analysis. The paper examines Eulers work, focussing on some ideas that would become important in the later development of the subject. A general question of interest is how differential forms and integration were understood in different parts of analysis during the period. Particular attention will be paid to Eulers De integrationibus maxime memorabilibus ex calculo imaginariorum oriundis, a paper submitted to the St. Petersburg Academy in 1777 and published in 1793.

TWO WAYS OF UNDERSTANDING THE NATURE OF TRANSCENDENTAL FUNCTIONS IN MATHEMATICAL ANALYSIS
Carlos Alvarez
National University of Mexico (UNAM) Mexico

It is a remarkable fact that the development in power series of the transcendental function ex or the trigonometric functions sinx, cosx can already be found in some of Newton’s and Leibniz’s texts, but new proofs to support these developments are provided throughout the 18th century, by Jacques and Jean Bernoulli and Euler among others, and it is still a major discussion point for Cauchy in his Cours d’Analyse of 1821.
Our aim in this talk is to give an interpretation of why the study of these transcendental functions and their development in power series became such an important matter in the period treated in our symposium. Whether they appear in relation to a geometric problem (for instance the section of angles) for Bernoulli or in the wider scope of Euler’s infinitesimal analysis, or in Cauchy’s real and complex analysis, it turns out that these functions provide an important support to the “analysis” that each of them developed and worked on.
After a brief introduction related to the geometric origin of the problem, we will focus on the way in which Euler and Cauchy understood these transcendental functions and the role they played in their respective treatises on analysis.

FROM LAGRANGE TO FREGE: IS A FUNCTION AN EXPRESSION?
Marco Panza
CNRS, REHSEIS (UMR 7596), France

Frege’s Grundgesetze (1893) opens with the assertion that one should not confound a function with an expression that designates it. This claim was apparently intended to exclude the definition of a function admitted by many 18th-century mathematicians, according to which a function is just an expression. Lagrange was one of these mathematicians, more particularly one who tried to draw more radical conclusions from this definition and related ideas about functions and quantities. Still, in spite of this opposition between Frege’s claim and Lagrange’s definition, it is possible to identify important analogies in their respective views on the subject.
The aim of my talk is to describe and discuss these analogies and to explore their underlying rationality. I argue that this comparison sheds some light both on Lagrange’s and Frege’s foundational program.

DA CUNHA, STOCKLER AND MATHEMATICAL ANALYSIS IN PORTUGAL IN THE PERIOD 1770-1820
Luis Saraiva
CMAF/University of Lisbon, Portugal

From about the middle of the 16th century until the middle of the 18th century there was little significant mathematics done in Portugal, and the country lost touch with the more advanced European countries.
AchangearrivedwiththecomingtopowerofKingD.Joséin1750.UnderthedirectionoftheMarquisofPombal,the King’s Prime Minister, a policy of reforms was instituted, the more significant one being the 1772 reform of the University, then located in Coimbra, the first reform of the Portuguese University in 160 years. With this reform came the establishment of the first faculty of mathematics in Portugal. Two important Portuguese mathematicians were called to lecture in mathematics, José Monteiro da Rocha (1734-1819) the mastermind behind the University’s reform, who mostly worked in applied mathematics and astronomy, and José Anastácio da Cunha (1744-1787), the most distinguished Portuguese mathematician of the 18thcentury, who made important contributions to mathematical analysis.
However, the King died in 1777, Pombal was dismissed, and there was a purge of the intelligentsia, which in particular led to da Cunha’s expulsion from the University, cutting short (but not eliminating) his influence on the development of mathematics in Portugal.
The most important scientific institution during the 19th century was the Lisbon Academy of Sciences, founded in 1779. Up to the second half of the 19th century, the Academy’s Memoirs was the only journal where mathematical papers could be published. An important figure, at one time the Academy’s secretary, was the mathematician Francisco de Borja Garção Stockler (1759-1829), who also specialized in mathematical analysis.
In the first half of the 19th century publications in mathematics stagnated. To a significant degree this was due to the successive wars that ravaged the country, starting with a short war with Spain (1801), then the Napoleonic invasions (1807-1811), which caused the court to move to Rio de Janeiro. Brazil became from 1807 to 1822, the year of its independence, the center of the Portuguese empire. There followed a long period of civil unrest that culminated in a civil war (1832-1834). In the 19th century the military was the dominant group among mathematicians, and so any civil unrest would affect their research directly. In our talk we analyze the contributions of da Cunha and Stockler, in the context of Portuguese mathematics of the time, with a particular emphasis on the 1772 reform of the University and on the work of the Lisbon Academy of Sciences in its first decades.

LE STATUT DE L’ANALYSE MATHÉMATIQUE: DE L’ENCYCLOPÉDIE
AU COURS DE L’ECOLE POLYTECHNIQUE
Christian Gilain Université de Paris 6, France

Nous nous proposons d’étudier le phénomène d’institutionnalisation de l’analyse comme science mathématique spécifique, entre le milieu du XVIIIe siècle et le milieu du XIXe siècle. On étudiera notamment les rapports complexes entre le statut de l’analyse et celui de l’algèbre dans cette période. Pour cela, nous regarderons particulièrement comment la situation se présente dans les grandes encyclopédies et les rubriques académiques, puis dans le cadre de l’Ecole polytechnique de Paris laquelle, dès sa fondation, institue un cours d’analyse au rôle important.
Dans l’Encyclopédie, éditée par d’Alembert et Diderot, le statut de l’analyse en mathématiques est ambivalent : il désignetantôtuneméthode,tantôtunediscipline.Danscederniercas,laprésentationdesonrapportàl’algèbreestaussi variable. Il est utile de comparer cette situation à celle que l’on trouve dans la Cyclopædia de Chambers, dont l’Encyclopédie devait être, au départ, une simple traduction.
Une évolution importante apparaît avec l’Encyclopédie méthodique, nouveau grand projet éditorial qui reprend les matériaux de l’Encyclopédie en les actualisant et en les organisant par « ordre de matières » au lieu de l’ordre alphabétique. La partie Mathématiques, éditée (sauf l’astronomie) par Bossut, Condorcet et Charles, paraît de 1784 à 1789. Elle comprend une « Table de lecture » destinée à permettre une utilisation du dictionnaire comme un traité. Cette table présente une classification des sciences mathématiques en dix rubriques dont l’originalité est l’introduction explicite d’une partie « Analyse », distincte en particulier de la partie « Algèbre ». Pour l’essentiel, la rubrique analyse comprend des articles relatifs à des concepts ou des théories faisant intervenir des quantités infinies, au lieu de quantités finies pour les articles d’algèbre. Beaucoup d’articles classés en analyse concernent le calcul intégral et reflétent les travaux effectués sur ce sujet dans la période récente. On peut penser que c’est l’expansion rapide alors du domaine du calculintégralquiconduitcestroismembresdel’AcadémiedessciencesdeParisàintroduirel’analysecommenouvelle science mathématique, distincte à la fois de la géométrie et de l’algèbre.
Cependant, cette nouvelle classification des mathématiques est loin d’être encore stabilisée. La création de l’Ecole polytechnique, au moment de la Révolution française marque aussi l’institutionnalisation de l’analyse comme une discipline, faisant l’objet d’un cours considéré comme fondamental. Mais le programme de ce cours ne correspond pas, au départ, au contenu de la partie analyse de l’Encyclopédie méthodique ; il recouvre, plus largement, l’ensemble des mathématiques pures à l’exception de la géométrie. La structure du programme officiel du cours d’analyse de Polytechnique va évoluer, sous l’effet convergent de positions pourtant profondément différentes : celle de Cauchy, préoccupé par l’établissement de nouveaux fondements du domaine, et celle de la direction de l’Ecole, soucieuse d’orienter davantage l’enseignement vers les applications. A partir de la fin des années 1820, et pour longtemps, le contenu du cours d’analyse va s’identifier au calcul différentiel et intégral.

NUMBERS, LIMITS AND CONTINUITY IN GAUSS. SOME OBSERVATIONS ON THE FOUNDATIONS OF MATHEMATICS AROUND 1800
Giovanni Ferraro University of Molise, Italy

In his analytical writings, Gauss introduced many novelties in the fabric of eighteenth-century analysis. He, indeed, rejected the formal methodology and the traditional notions of complex number, function, integral and of the sum of series. However, the investigation of some of his theorems shows he conceived the continuum in a way substantially different from the modern one. For instance, in the definition of the arithmetico-geometric mean, he assumes the following hidden lemma: if an increasing (decreasing) sequence k has an upper (lower) bound, then there exists a real number that is the limit k for k            .
This and other hidden lemmas are due to the lacking of an adequate construction of real numbers and show that Gauss’s mathematics was based upon a revised version of the traditional concept of continuous quantity, which he had inherited from eighteenth-century mathematicians. The traditional continuum did not consist of points but was given as a whole. It was an intensional idea characterized by the relation between the whole and its possible parts, unlike in Dedekind-Cantor theory, which is based on extensional set theory. This conception have remarkable consequences in the calculus: it is sufficient to think that an interval was always thought of as including its endpoints (in modern terms, it was always a closed interval). In my opinion, the main difference between Gauss and the traditional 18th-century continuum is that the latter thought to a geometrical continuum, immediately referred to geometric quantities, while the former to a numerical continuum, which, in principle, embodies the flowing of time.
Moreover, Gauss’s concept of the continuum has consequences on his notion of continuous functions and on some procedures (for instance, the interchange of limits), which are similar to those later used by Cauchy in his Coursd’analyse.
References
Breger, H. Les Continu chez Leibniz. In Salanskis, J.-M. and Sinaceur, H. (Eds.), Le Labyrinthe du Continu, Paris: Springer-Verlag France, 1992, 75-84.
Ferraro, G., Analytical symbols and geometrical figures. Eighteenth Century Analysis as Nonfigural Geometry, Studies In History and Philosophy of Science Part A, 32 (2001), 535-555.
Ferraro, G., Differentials and differential coefficients in the Eulerian foundations of the calculus, Historia Mathematica, 31 (2004) 34-61.
Ferraro, G., The foundational aspects of Gauss’s work on the hypergeometric, factorial and digamma functions, The foundational aspects of Gauss’s work on the hypergeometric, factorial and digamma functions, Archive for History of Exact Sciences 61 (2007), 457-518.
Gauss, C.F., Disquisitiones generales circa seriem infinitam                                                                             +etc.
Pars Prior, Commentationes societatis regiae scientiarum Gottingensis recentiores, 2 (1813). In Werke, 3:125-162. Gauss, C.F. Determinatio seriei nostrae per aequationem differentialem secondi ordinis, in Werke, 3: 207-229. Gauss, C.F., Zur Lehre von den Reihen in Werke,, 10, Abt.1, 382-428. Gauss, C.F., Zur Metaphysik der Mathematik, in Werke, 12: 57-61.
Gauss, C.F., De origine proprietatibusque generales numerorum mediorum arithm. Geometricorum,in Werke,, 3: 361-374.

“THROWING SOME LIGHT ON THE VAST DARKNESS THAT IS ANALYSIS”:
NIELS HENRIK ABEL’S CONTRIBUTIONS TO THE REORIENTATION OF ANALYSIS IN THE 1820s
Henrik Kragh Sørensen. Department of Science Studies, University of Aarhus, Denmark hks@ivs.au.dk

In the 1820s, the young Norwegian mathematician Niels Henrik Abel (1802-1829) became one of the earliest adherents to and advocates of Augustin-Louis Cauchy’s (1789-1857) new approach to analysis. Ivor Grattan-Guinness has even characterised Abel as “more Cauchyian than Cauchy himself”. In Abel’s notebooks, a single scribble has been found mentioning the Czech Bernard Bolzano (1781-1848), who simultaneously with Cauchy undertook a similar reorientation of analysis. In letters from his European Tour back to his mentors in Norway, Abel vowed to help “throwing some light on the vast darkness that is Analysis”. This paper is devoted to investigating what that darkness consisted in for Abel and how he went about illuminating it.
Cauchy advanced his new analysis under the banner of rejecting any and all arguments conducted “by the generality of algebra”. By that slogan, Cauchy insisted on an interpretation of mathematical equality as numerical equality instead of the formal equivalence central to an older Eulerian, formula-centred style. Such a radical reinterpretation of a fundamental concept in mathematical analysis meant that mathematicians – including Abel – began to look at established procedures and proofs to figure out whether they were still valid, and if so, why ungrounded reasoning could lead to correct results.
Abel’s mathematical production falls in between the formula-centred style of Euler that he followed in his work on elliptic functions and the new more concept-centred style of Cauchy’s new analysis. Abel’s contributions to the new stylerestedmainlyonthreepillars.First,heexemplifiedthenewstyleofrigourandconceptualreasoninginhisproofof the most general case of the binomial theorem, published 1826. That theorem had already been the vehicle by which Bolzano sought to reform the standards of rigour in analysis, and its central position in the analytical edifice is a good starting point for evaluating the impact of the conceptual reorientation. Second, Abel employed the new style in discussing certain criteria for convergence of series in a dispute with the otherwise largely unknown mathematician Louis Olivier. Third, albeit of more local importance only, Abel’s letters documenting the attraction to the new analytical style were – and are – of great interest in understanding the early reception of Cauchy’s work.
Thistalkwillbeginbyoutliningtheissuesinvolvedinthereorientationofanalysisfromits18thcenturyformula-centred styletoits19thcenturymoreconcept-centredstyle.Then,IwilldiscussinmoredetailthecontributionsofAbelandtheir contextualisation within the larger transformation. In particular, I will emphasise the role of ‘critical revision’ as part of connecting the new concept-centred style to the existing body of analytical knowledge.

WHAT MAKES MATHEMATICAL ANALYSIS RIGOROUS? COUNTEREXAMPLES AND PATHOLOGICAL FUNCTIONS
Michiyo Nakane
Rikkyo University, Japan

The 19th century is known in the history of mathematics as the age of rigor. Mathematicians began to discuss some important notions in analysis that could not be grasped by geometric intuition. The adoption of - inequalities was an important part of this trend. The present paper explores the relationship between the development of new theories in analysis and the increasing emphasis on rigor.
In his Cours d’analyse, published in 1821, Cauchy described the geometric idea of a limit concept in terms of inequalities Although he developed his theory in an abbreviated style by means of - methods, his analysis was not completely free of geometric notions. Another important aspect of his approach was the rejection of what he called “the generality of algebra.” He noted that a general formula for power series may not hold for particular values, a fact to which18th mathematicians were largely indifferent. His special attention to counterexamples was an important factor in his formulation of the notions of convergence and divergence.
The publication of Fourier’s theory of series in 1822 was a crucial event in the emergence of the concept of rigor. Fourier expansions provided counterexamples to results about series traditionally assumed to be valid. Mathematicians modified existing theories using - inequalities. They also sought necessary and sufficient conditions for the convergence of a series. This program of research led them to reconsider the concept of a function. Riemann arrived at his famous example of an everywhere continuous but nowhere-differentiable function in the course of his investigation of functions that were represented by Fourier series. The evidence suggests that he had this example by 1861 at the latest.
Weierstrass realized that confusion could result if one tried to grasp the properties of Riemann’s function by means of geometric intuition.Weierstrass defined the limit concept in terms of – inequalities without any reference to geometric conceptions. His primary motivation was not the clarification of the notion of uniform convergence but rather the construction of a purely algebraic theory of analysis using the method of - inequalities. He developed this new framework in his 1861 lectures, a work that is often regarded as a prototype of the new rigor in analysis.

THE PLACE OF ANALYSIS IN 19th-CENTURY BRITISH MATHEMATICS

While one could easily name 19th-century British mathematicians who made substantial contributions to algebra (e.g. Boole, Cayley, Sylvester), geometry (Salmon, Clifford) and mathematical physics (Stokes, Maxwell, Kelvin), the task of finding those who were first-rate analysts is somewhat harder. In fact, it is rare to find anyone of note in 19th-century Britain who worked on analysis purely for its own sake. This is all the more remarkable considering that analysis was probably the major forte of British pure mathematics in the first half of the 20th century. All this leads to some interesting questions: Why was the subject neglected by British mathematicians for so much of the 19th century? When did this situation change? And what were the reasons for the turnaround? In order to provide some sort of answers, this paper will look at those who comprised the British mathematical community (to the extent that such an entity existed) at that time, their mathematical training and backgrounds, and, most importantly, what was understood in 19th-century Britain by the word “analysis” itself.

MITTAG-LEFFLER AND WEIERSTRASSIAN ANALYSIS
Laura E. Turner University of Aarhus, Denmark

In recent years the Swedish mathematician Gösta Mittag-Leffer (1846-1927) has attracted the attention of historians as an important organizer of mathematics, a journal editor, and, to some extent, a mathematician. However, very little attention has been paid to his role as a teacher, and relatively little has been done to investigate his impact on the shape and development of Swedish mathematics in the late 19th century through this area of his career.
This paper has two aims: first, to describe Mittag-Leffler’s “mission” to promote specialized mathematical study at Stockholms Högskola (founded 1878), where he held the institution’s first chair of mathematics, and to establish a research ethos there. In attempting to do so Weierstrassian analysis, specifically in connection with Mittag-Leffler’s research investigations concerning the analytic representation of single-valued functions, played a central role in the early- to mid-1880s.
Second, I will analyze the impact that this mission had on two of Mittag-Leffler’s first students there, Ivar Bendixson (1861-1935) and Edvard Phragmén (1863-1937), focusing on among other things Mittag-Leffler’s roles in problem-selectionandconceptshifts,andthetransmissionofcertainvaluesregardingmathematicalpracticeandstudy.

RIGOUR VS. INTUITION: TEACHING AND RESEARCH IN ANALYSIS IN TURIN IN THE SECOND HALF OF THE NINETEENTH CENTURY
Erika Luciano
Department of Mathematics, University of Turin

Thispaperwillidentifysomeinternalandexternalfactorsthatcontributedtochangesanddevelopmentsinhowanalysis was taught and understood in Italy – and especially in Turin – in the last decades of the nineteenth      century. On one hand, the emergence of an interest in foundational studies and the goal of elevating Italian mathematics to a level comparable to the leading countries of Europe led to a complete renewal of studies in infinitesimal calculus, thanks to the contributions of E. Betti, F. Brioschi, F. Casorati and U. Dini. On the other hand, the birth of the Italian national education system stimulated the study of teaching programs for universities and technical schools and let to the writing of specialized treatises. In Piedmont, among the main events that caused a change in the research and teaching of analysis were the lectures of Augustin-Louis Cauchy during his stay in Turin (1831-33) and those of Angelo Genocchi, Felice Chiò and Francesco Faà di Bruno, the latter all being teachers of Giuseppe Peano. A lively scientific and didactic tradition in calculus was established, which was characterised by a rigorous and abstract approach, and gave rise to significant results in the theory of real and complex functions and in foundational studies on continuous functions, derivatives and series.
To document these changes we examine Peano’s lectures on analysis, held at the Turin University and at the Military Academy(1882-1901),which illustrate both progress and differences with respect to Genocchi’s lectures. Also relevant were the lively debates that surrounded Peano’s treatises Genocchi-Peano (1884), Applicazioni geometriche del Calcolo infinitesimale (1887) and Lezioni di Analisi infinitesimale (1893). The study of the correspondence between Peano, Genocchi and other contemporaries (H.A. Schwarz, C. Hermite, F. Casorati, P. Tardy, E. Cesàro …) will shed light on the scientific and institutional context of the publication of these books, which formed the basis for the subsequent encyclopaedic Formulaire de Mathématiques (1895-1908). An examination of Peano’s and Genocchi’s published and unpublished lectures will show: the extent of the influence exerted by Genocchi on his assistant; their internal interaction in choices of research themes and teaching practice; and the critical studies they completed of earlier and contemporary literature.
Finally, Peano’s autograph marginalia in his treatises allows one to follow the development of his scientific thinking, to understand the role of logical notations in his analysis, and to appreciate his preference for formal-algorithmic procedures over synthetic-geometrical approaches.
References
Roero C.S. (ed.) 1999, La Facoltà di Scienze Matematiche Fisiche Naturali di Torino, 1848-1998, 2 volls., Torino, Deputazione Subalpina di Storia Patria.
LucianoE. 2007, Il trattato Genocchi-Peano (1884) alla luce di documenti inediti, Bollettino di Storia delle Scienze Matematiche, 27, 2, p. 219-264."The Introduction of

## Symposium S36 “Modern Mathematics into Iberoamerican Countries" Organizers, Clara Sanchez (Colombia), Sergio Nobre (Brazil), and Alejandro Garciadiego (Mexico)

Description:
This session will be divided into two components---"The Introduction of Logic" and "The Introduction of Set Theory".

Speakers

THE TRANSLATION OF KURT GRELLING’S THEORY OF SETS INTO SPANISH
Universidad Nacional Autónoma de México, México

In the early 1940s three related events marked the process of professionalization of modern mathematics in Mexico: the establishment of the Mathematics Department within the Faculty of Sciences, the foundation of the Institute of Mathematics and the appearance of the Mexican Mathematical Society. At the beginning, the student population was not very significant, in terms of quantity and most of the textbooks used were kept in their original languages, mainly English and French.
But almost as soon as this decade started, a German textbook on set theory was translated into Spanish. In a previous talk, I have discussed why the translator selected this text. Now, I’ll analyzed the translator’s academic background.

THE INTRODUCTION OF SET THEORY IN COLOMBIA
Clara Helena Sánchez B., Víctor Albis G.
Departamento de Matemáticas, Universidad Nacional de Colombia Grupo PROCLO, Bogotá, Colombia

In Colombia, in the 1940 decade, we find evidence of two clear attempts to formally introduce set theory in the teaching of mathematics. One of them is the book Introducción a la teoría de conjuntos (Introduction to set theory) (1944), a compilation of lectures given in 1942 by FRANCISCO VERA, the well known Spanish mathematician and historian of sciences and Republican exile. The other one are two outstanding expository papers by WALDEMARBELLON, a German mathematician who arrived to Colombia fleeing from the Nazi regimen in 1938; these papers were published in the journalRevistadelaUniversidadNacionaldeColombia(Revistatrimestraldeculturamoderna).Oneofthepurposesof this presentation is to review these works taking into account the Colombian political and sociological situation at that time. Also, we will examine the steps taken in order to introduce set theory in higher education for the first time in the Universidad Nacional de Colombia undergraduate mathematics program created in 1951. Finally, we will present the reforms, starting in 1960, in primary and secondary education levels, containing rudiments of set theory.

LOGIC AND SET THEORY IN PRIMARY AND SECONDARY EDUCATION IN SPAIN
Elena Ausejo
University of Zaragoza, Spain

The world wave of mathematics reform of the sixties and seventies also reached the isolated Spain under Franco, then committed to a technocratic process of modernizing the economy. Thus, in 1961, just two years after the famous Royaumont Seminar (1959), the Ministry of Education promoted a meeting of Professors of Mathematics in Secondary Education in which the Professor of Projective Geometry of the Central University of Madrid, Pedro Abellanas, explainedtheneedforacomprehensivereformofthemathematicscurriculumofsecondaryeducation.Modernalgebra, set theory and to a lesser extent, mathematical logic started an entry in non-university education that would lead them to primary school with the Education Act 1970.

HILBERT’S “GRUNDLAGEN DER GEOMETRIE” TRANSLATED INTO SPANISH:
A CASE OF A FAMOUS MATHEMATICAL TEXT AND ITS CONTEXTS
Leo Corry
Tel Aviv University, Israel

Around 1940 a translation to Spanish of Book I of Euclid’s Elements was published at the Universidad Nacional Autonoma de Mexico. This was intended as a first step towards what should be a definitive, full translation of this work into Spanish. The entire project, however did not materialize. Nevertheless, the published translation of Book I is accompanied by a translation of David Hilbert’s Grundlagen der Geometrie, by David Gracia Bacca. In this talk I will explain some of the background and context to Garcia Bacca’s translation, as well as to other translations of Hilbert’s epoch-making book.

Speakers for the part on set theory will be drawn from this list: Angel Ruiz (Costa Rica), Luis Radford (Guatemala), Jose Ferreiros (Spain), Clara H. Sanchez (Colombia), Eduardo Ortiz (Argentina), Mario H. Otero (Uruguay), Sergio Nobre (Brazil), Alejandro R. Garciadiego (Mexico), and Javier de Lorenzo (Espana).

Speakers for the part on logic will be drawn from this list: Walter Carnielli (Brazil), Itala de Ottaviano (Brazil), Renato Lewin (Chile), Carlos di Prisco (Venezuela), Gonzalo Serrano (Colombia), Fernando Zalamea (Colombia), Luis Vega (Spain), and Francisco Rodriguez-Consuegra (Spain).

MISCHA COTLAR’S FIRST STUDIES ON MEASURE AND INTEGRATION THEORY (1939-1944)
Luis Carlos Arboleda Universidad del Valle, Cali, Colombia
Carlos D. Galles Universidad Nacional de Rosario, Argentina

In 1940 a series of publications began to appear in the Institute of Mathematics in the city of Rosario, under the direction of the eminent Italian mathematician Beppo Levi, who was to play a critical role in the development of mathematical studiesinArgentina.LevihadjusttakenupofficeasdirectoroftheinstituteintheNationalUniversityofLitoral.Oneof the primary contributors to the new magazine was Mischa Cotlar, a mathematician of Russian origin, who had lived in Uruguay since his childhood, mainly self-taught. Cotlar submitted a paper on non-measurable sets and a generalization of the Lebesgue integral. Cotlar’s ideas strongly attracted the attention of Levi, who realized that this was an important work and that its author possessed the creative qualities of someone destined to be an eminent mathematician.
In fact, although Cotlar would only obtain his doctorate much later on (at the University o fChicago, in 1953) by the time this publication was written he was thoroughly acquainted with the state of the art of research in Measure Theory and Integration (Lebesgue, Fréchet, de la Vallée Poussin, Banach, Kuratowski, Saks, Sierpinski, among others). His particular field of interest was the problem of determining the conditions of the measure of a set from which it I spossible to generalize the Lebesgue integral of an abstract space.
The epistolary exchanges that took place between Levi and Cotlar focused specifically on the most appropriate way to provide a formal presentation of the results of Cotlar in this matter: definition of the pseudo-measure of a set, generalization of the Lebesgue integral for non-measurable functions and demonstration of the main theorems of the classicaltheory.AnimportantissuethroughouthisresultsisCotlar´semploymentadditivefunctionsofabstractsetsthat Fréchet introduced in 1915 and in 1922 in his theory of the integration of generalized functions.Taking into account this fact and the consideration of other works of the same period, it will be shown that Cotlar was one of the first mathematicians to choose the topology of abstract spaces as the privileged scenario for his investigations.
In this communication we study the evolution of Cotlar’s ideas, from his first contributions at the end of the 30s in what he called “theory of anágenos”, right up to the mid 40s. The correspondence Levi and Cotlar exchanged during the publication of the article in the magazine directed by Levi is also studied with special care.

THE IMPACT OF ANTONIO MONTEIRO ON THE
ESTABLISHMENT OF ALGEBRAIC LOGIC IN LATIN AMERICA
Eduardo Ortiz

SUR UNE CONTRIBUTION DE PI CALLEJA AU PROBLÈME D’INDUCTION ET RÉCURSIVITÉ DANS LES AXIOMES DE PEANO POUR LES NOMBRES NATURELS
Luis Carlos Arboleda Universidad del Valle, Cali. Colombia
Un des livres les plus influents dans la diffusion de l’approche formaliste des systèmes numériques dans l’enseignement, a été “Grundlagen der Analysis” d’Edmund Landau. Initialement publié en 1930 et traduit comme “Foundations of Analysis” à Chelsea en 1951, “Grundlagen” systématise les notes des cours de Landau à Göttingen depuis 1909. Dans la préface adressée aux enseignants de l’ouvrage, Landau se réfère au docteur Grandjot, son assistant à Göttingen, qui a corrigé le manuscrit du livre en cherchant en particulier de présenter de la manière la plus rigoureuse et complète les Axiomes de Dedekind-Peano pour les naturels.
Karl Grandjot est plus connu dans l’Amérique Latine comme l’un des pionniers de l’introduction de l’enseignement moderne des mathématiques au Chili (1929-1967). L’ “objection de Grandjot”, à laquelle concerne la préface de “Grundlagen”, est liée à la confusion répandue depuis lors dans plusieurs textes, que la “Définition par récursivité” (de la somme et le produit de naturels) est permise par la possibilité de la “preuve par induction”. Landau pense avoir répondu à l’objection sans introduire axiomes supplémentaires à ceux de Peano, en utilisant une procédure suggéré par le mathématicien hongrois L. Kalmár.
En 1949, le mathématicien catalan Pedro Pi Calleja réalise une étude détaillée de la question visant à justifier la position de Grandjot. Le travail a été publié en “Mathematicae Notae”, la revue dirigée par Beppo Levi à Rosario (Argentine), et les comptes rendus de Church et Curry montrent qu’il a eu une certaine notoriété internationale. Sur la base du formalisme logique des “Grundlagen der Mathematik” de Hilbert et Bernays et en interaction étroite avec Levi, Pi Calleja montre l’ampleur réelle de la “objection de Grandjot”, en ce qui concerne la nécessité de préciser le statut logique des “Définitions récursives” d’addition et de multiplication fondées sur les axiomes de Peano, et de répondre à l’exigence de nouveaux postulats formulés dans une logique plus forte que la logique de prédicats de premier ordre.
Dans la présente communication on essaie de situer l’importance du travail de Pi Calleja par rapport à l’État d‘art de la questiond’InductionetRécursivitéavantlesannées1950,etdanslecadredeseffortsdéployésparLevietsescollaborateurs, pour étayer l’enseignement du calcul dans nos pays sur l’étude rigoureuse de l’axiomatique des systèmes numériques.

A BRASILIAN GENEALOGY OF THE MATHEMATICS FROM LUIGI FANTAPPIÈ
Plínio Zornoff Táboas
Universidade Federal do ABC – UFABC, Brazil

This work is a inicial contribution for the construction of a brasilian Mathematic genealogy. It begins with the study of the arrival of the first Foreign Mission in Brazil for the equipment of the Faculdade de Filosofia, Ciências e Letras – FFCL from Universidade de São Paulo – USP.
What is done in this study is to observe Fantappiè’s academical activities in São Paulo from 1934 until 1939 and, then, research and present the mathematic affiliation that he left in Brazil, focusing the mathematicians that worked in the Universidade de São Paulo.
The first people classified in this genealogy, from Fantappiè, were chosen first of all because of their participation in the Seminário de Matemática e Física da USP in the year of 1935, that was published in the first and unique number of the Revista de Matemática Pura e Aplicada da Universidade de São Paulo in 1936, and second because they became mathematicians.
Besides this, the central idea of this work is to classify this affiliation according to the areas of Mathematics research and try to observe the influence of cultural capital in the professional choice of each Mathematician listed in this genealogy.

THE BIRTH OF THE RESEARCH IN MATHEMATICS INSIDE THE STATE OF SÃO PAULO - BRAZIL
Sergio Nobre UNESP - Brazil

In this lecture we will talk about the creation of centers of mathematicians’ formation and of research in mathematics inside the state of São Paulo, especially in the Faculty of Philosophy, Sciences and Letters of Rio Claro’s city and in the School of Engineering of São Carlos city (belonging to the University of São Paulo). Both these institutions were founded in the fifties of the 20th century, decade when, it can be said, it began the process of professionalization of the scientific research in mathematics inside the state of São Paulo. Personalities like the Italian mathematicians, Achile Bassi, Jaurès P. Cecconi and Ubaldo Richard, that had the responsibility of creating the Department of Mathematics of the School of Engineering of São Carlos, and especially Nelson Onuchic and Mário Tourasse Teixeira, founders of the Department of Mathematics of Faculty of Philosophy of Rio Claro will be evidenced in this talk. Nelson Onuchic worked in Rio Claro until the year of 1966, when he moved to São Carlos, beginning his work in the School of Engineering,where it stayed. MárioTourasse  Teixeira was in Rio Claro until the death in 1993. Both had many students that followed them in their academic activities. Prof. Onuchic dedicated their mathematical researches in the area of the Differential Equations and Prof. Mário Tourasse in the area of the Foundations of the Mathematics. It will be done a short report about these two mathematicians and of some of their students.

INTUITION, FORMALISM AND PURITY IN BRAZILIAN MATHEMATICS
Rogério Monteiro de Siqueira
EACH, Universidade de São Paulo, Brasil

In 1957, a small report, almost a resume of ideas, entitled “A Natureza dos Juízos Matemáticos” caused some disagreements between two important figures of Brazilian mathematic community. The report presented by Newton Carneiro Affonso da Costa at the section for logic and philosophy of science of a congress of philosophy was sent by a college to Omar Catuda. In spite of this friendly gesture, Catunda explicitly disagreed from Costa’s ideas in a posterior letter.The main reason for the controversy was the distinction between intuitive and symbolic mathematics which Costa had sustained in the report, whereas Catunda had advocated the construction of mathematical concepts could not be separated from practical and real examples.
Intuitionism, formalism and logicism are old personages of the history and philosophy of mathematics very explored in the modernist movement. They can be observed, for example, in early communities like the German mathematical community, where Felix Klein, David Hilbert and Hermann Weil had acted. In the same way, Catunda and Costa revive in Brazil a very common motif for divergences. However, in the Brazilian fresh community of mathematics, established at thirties decade, such modernist face will gain other colors, depending on the disciplinary origins of the scientific actors and the available space within this new community.
The symbolic view of Costa, for example, can be identified in many episodes as a searching for an independency of his objects and themes of research. At the fifties decade, his discourse is proclaimed in an interesting way: It is an amalgam of his interest in philosophy and mathematics that found a “secure” place in congresses and journals of philosophy. By the other side, in spite of a mathematician with deep interest in structures, like sets, groups and rings, behind the mathematical ideas, Catunda was very concerned about teaching of mathematics, as many other professors at university in this period. At the following decade, Catunda participated in a very influential group responsible for the modern experience in the Brazilian secondary school. In his pedagogic view, there is not space for pure or symbolic ideas without the use of intuition and practical examples.
Therefore, in the genesis of modern mathematics in Brazil, also in the debate performed by Costa and Catunda, emerge three phenomenona: the reception of international early divergences, the beginning of the professionalism in mathematics and philosophy in Brazil, and the establishment of new Brazilian experts in the scientific community. The aim of this work is to analyse the confluence of these three phenomena with the rising of modern mathematics in Brazil.

ON THE DEVELOPMENT OF PARACONSISTENT LOGIC AND THE BRAZILIAN SCHOOL OF LOGIC
Itala M. Loffredo D’Ottaviano
Centre for Logic, Epistemology and the History of Science, Philosophy Department State University of Campinas – UNICAMP
itala@cle.unicamp.br

A theory is inconsistent if there is a formula of its language such that the formula and its negation are both theorems of the theory; otherwise, the theory is consistent. A theory is trivial if all formulas of its language are theorems. Paraconsistent logics are the logics of inconsistent but non-trivial theories. A deductive theory is paraconsistent if its underlying logic is paraconsistent. In paraconsistent theories the Aristotelian principle of (non-)contradiction is not valid in general.
The first logician to construct a formal system of paraconsistent logic, restricted to the propositional level, was Stanislaw Jaskowski in 1948. In 1958, the Brazilian Newton C.A. da Costa, independently of Jaskowski, began the general study of contradictory systems.
From 1963, da Costa has developed several systems and theories related to paraconsistency, apparently becoming the first logician to develop strong paraconsistent logical systems which could be useful for mathematics, as well as for empirical and human sciences. DaCosta and collaborators, from Brazil and several other countries, have introduced and studied many paraconsistent logics and set theories, appropriate semantics and algebras associated to the systems, decidability procedures, paraconsistent model theories, a paraconsistent differential calculus; and have studied applications to the foundational analysis of physical theories and to partial truth. Nowadays, ‘paraconsistency’ has become a field of knowledge and there have been applications of paraconsistent logic not only to the foundations of science and its philosophical analysis, but even to informatics and technology.
In this talk we shall present a general survey on the development of paraconsistent logic and da Costa’s work, emphasizing the role played by the Brazilian School of Logic.
Da Costa, N.C.A., Krause, D., Bueno, O. Paraconsistent logics and paraconsistency. In: Dale Jacquete (Ed.), Handbook of the Philosophy of Science. Philosophy of Logic. Elsevier: 2007, pp. 791-911.
D’Ottaviano, I.M.L. O n the development of paraconsistent logic and da Costa’s work. The Journal of Non-Classical Logic 7 (1/2), 1990, pp. 9-72.

SOME HISTORICAL CONSIDERATIONS ON THE PARACONSISTENT LOGIC IN BRAZIL
Carlos Roberto de Moraes
UNIARARAS, Brasil

In Brazil the paraconsistent logic approach had its beginning in the works carried out by Professor Newton da Costa in the  fifties in the twentieth century. His thesis was published in 1963 and was entitled “Inconsisten tFormal Systems”. In 1974 he published an abstract from his thesis in the Notre Dame Journal of Formal Logic entitled “On the Theory Of Inconsistent Formal Systems”. This abstract is considered the starting point for researches in paraconsistent logic in Brazil. Besides being one of the first researchers to develop works in paraconsistent logic at an international level, Professor Newton da Costa and his students have studied several applications of paraconsistent logic into philosophy problems, computing, artificial intelligence and medicine fields. We want to present an overview on the historical development of this line of research in Brazil.

Symposium S35  "History of Numerical Tables: The Second Meeting on History of the Exact Sciences along the Silk Road". Organizer: QU Anjing.

Speakers:

FROM SQUARE TABLES TO CALCULATION OF SURFACES IN MESOPOTAMIA
Christine PROUST
CNRS, France

In this paper, I will analyse links between some numeric tables and field texts. I will rely on two groups of documents. The first group include presargonic (mid third millennium B. C.) tables of surfaces and some field texts dating from the same period. In contrast to these archaict exts,I will present the Old-Babylonian (beginning of second millennium B.C.) coherent metrological system attested in school tablets and the relationship between this system and calculation of surfaces. My aim is to draw up a link between the surface problem (transformation of unidimentional magnitudes into bidimentional ones) and the apparition of place value notation in Mesopotamia.

FRACTIONAL TABLES AND WATER CLOCKS IN EGYPT
Micah Ross Upland, IN, USA

The best known and most far-reaching Egyptian contribution to astronomy division of the day and night into twenty-four hours is. Somewhat less known but more remarkable given the latitude of Egypt is the fact that the difference between the seasonal hours and the equinoctial hours was also an Egyptian observation. The Egyptian approximation of the seasonal hours is documented by two sources document: fractional tables and water-clocks.
These two sources have often been perceived as being in opposition. Several divergent methods of approximation are preserved in the fractional tables. Neither were all water clocks constructed by the same principals. In fact, in some cases, water clocks may be used to explain the meaning of the fractional tables. A coordination of elements from these two sources establishes several correspondences and eliminates several disparities.
Similar fractional tables of seasonal hours exist in cuneiform sources. A variety of proposals once related these fractional tables to the construction of Babylonian water clocks but the recent discovery of more explicit texts has established the fractional tables as shadow lengths. Even though shadow lengths cannot explain all the fractional tables in Egypt, this approach to understanding fractional tables in an Egyptian context demands consideration.
Because the use of water clocks in Egypt was described in (fantastical) detail by Macrobius, his account also merits re examination.The historical context for his account is demonstrably wrong, but several errors in his description betray a confused, second-hand account of probable Egyptian practices.

FROM LISTS TO A TABLE TO MANAGE GRAINS:
THE EVIDENCE FROM THE OLDEST EXTANT CHINESE MATHEMATICAL BOOKS
Karine Chemla & MA Biao
CNRS, France & Yamaguchi University, Japan

The oldest extant Chinese texts devoted to mathematics contain a passage related to equivalences between grains, a product, the management of which was an essential task for the imperial bureaucracy. In the Book on mathematical procedures (Suanshushu),a manuscript excavated from a tomb sealed ca186B.C.E.and being so far theearliest known mathematical text from China, the equivalence between various kinds of grain is provided in the form of several sentences. Each of them states a sequence of equivalent amounts of given types of grain, expressed with respect to measure units of weight and capacity (bamboo slips 88—90, Peng Hao, 2001: 80). The editor Peng Hao showed that the format and the content of this passage were essentially identical to what can be found in the “Regulations for granaries canglü,” a text copied during the Qin dynasty (221 B.C.E.—206 B.C.E.) and discovered among Qin legal documents at Shuihudi. By contrast, in The Nine chapters on mathematical procedures, a writing probably completed in the 1st century C.E., the corresponding passage takes the form of a table with a homogeneous pattern, in which all grains are gathered and associated with an abstract figure. There is no known legal document from the Han dynasty that can be compared to this passage from The Nine Chapters.
The main question that the talk will address is: how can we account for the difference between the two texts shaped to handle equivalences between grains? To deal with the issue, the authors will discuss the systems of measure units underlying the two passages and cast light on the nature of the data recorded in them. Moreover, they will show that the values used in both texts constitute different types of quantities, which they will relate to the distinct mathematical contexts evidenced by the two books, within which the passages were inserted. Lastly, they will suggest hypotheses linking the differences between the two passages and the differences between the practices of managing grains at the two distinct time periods. In particular, they will address the issue of the difference between the shape of the texts: sequences versus a table.
PENG Hao, 2001. Zhangjiashan hanjian «Suanshushu» zhushi (Commentary on the Book of mathematical procedures, a writing on bamboo slips dating from the Han and discovered at Zhangjiashan), Beijing: Science Press (Kexue chubanshe), 2001.
The research presented in this communication could be completed during two months that the authors spent at the Max Planck Institute for the History of Science, Berlin, in the summer 2007.

THE FUNCTION OF NUMERICAL TABLES IN THE DEDUCTIVE STRUCTURE OF PTOLEMY’S ALMAGEST
Nathan Sidoli Waseda University, Japan

The Almagest has a fairly concise deductive structure and Ptolemy uses tables in a number of interesting ways to advance the mathematical argument of the text. In the course of the argument, Ptolemy uses tables both the facilitate calculations that could be made using straightforward arithmetic or the underlying geometry, or to make possible calculations that cannot generally be solved using Greek geometry. Tables, however, also function as objects of knowledge. Tables, like theorems or problems, are presented both as the results of mathematical research and as important tools used in developing new knowledge. In this talk, I will look at a number of examples of Ptolemy’s tables performing this dual function.

THE NUMERICAL MODEL OF CHINESE PLANETARY THEORY
Qu Anjing
Northwest University, Xian, 710069, China

Astronomical table as tool played an important role in Chinese mathematical astronomy. Quite different from the geometrical system in Western tradition, the planetary theory appeared in old China took the numerical model which was always constructed with several difference tables. The function and precision of these tables will be discussed in this talk.

THE  PRECISION OF THE PLANETARY CALCULATION IN THE SONG DYNASTY
TANG Quan
Xianyang Normal University, Xianyang, 712000, China

Planetary theory is one important part of traditional Chinese mathematical astronomy. Ancient Chinese calendar-makers usually regarded the precision of planetary calculation as one standard for verifying whether one calendar was excellent or not.
According to Chapter of Calendar in the Histories of Song, the maximum error of planetary calculation that calendar-makers of the Northern Song dynasty permitted was only two degree, and in the Southern Song dynasty, the maximum error of planetary calculation that calendar-makers permitted was only one degree.
By analyzing the precision of planetary calculation of Jiyuan Li, one calendar compiled in the Northern dynasty, we point out that the computational error of the Jupiter and Saturn in Jiyuan Li could meet the requirement of precession that the calendar-makers of Northern Song dynasty expected, but the computational error of Mars, Mercury and Venus couldn’t.

MERCURE ET LE SECOND ÉQUATOIRE DE JEAN DE LIGNIÈRES
Matthieu Husson
EPHE, France

Our aim in this presentation is to analyse and compare the second equatory of John of Ligneris and the Tabule magne of the same author. Both were produced in Paris during the early fourteenth century. Both are means to compute the equation of a planet in the ptolemaic model.
We will first confirm, using an ad hoc adaptation of Beno Van Dalen parameter evaluation methods, that this set of tables is built on Alphonsine parameters. We will, on this basis, study the error pattern of the Tabule magne. Two possible families of models will be examined: the geometrical models, and the tabular models with the standard Ptolemy interpolation for the equation of planets. This study will show that the error pattern of the Tabulemagne is very specific: all the planets except Mercure appear to be closer to the geometrical model. Mercure is closer to the tabular model. A study of the equatory will demonstrate that the geometrical instrument present exactly the same error pattern. This fact may allow us to wonder if the table were computed with the use of the equatory.

COMMENTS ON THE NUMERICAL TABLES AND ALGORITHMS IN FIBONACCI’S LIBER ABACI
Ji Zhigang
Department for the History & Philosophy of Science Shanghai Jiaotong University, Shanghai,200030, China

Fibonacci’s Liber Abaci(1202) is one of the most important books on mathematics of Middle Ages. Its effect was enormous in dissemination the Hindu number system and the methods of algebra throughout Europe. The Hindu numerals with the place system are used both to make the calculation and to write down the result. So these calculating procedures and the results formed a plenty of numerical tables in Liber Abaci.
By presenting a classification of different numerical tables in Liber Abaci, this paper will demonstrate how those numerical tables were used for calculation. Some famous algorithms such as gelosia method, galley division, Egypt, unit fraction, the systematic proportion based diagram method, and the method of false position are additional to those numerical tables. Just these numerical tables and algorithms provided useful calculating methods for the Maestri d’abbaco.
It is well known that the gelosia method has its roots in Hindu unit fractions in Egypt, algebra in Arab, this paper will also point that in Liber Abaci there are some problems and algorithms which are similar to those in ancient China.

CARDANO’S RULE OF PROPORTIONAL POSITION IN ARTIS MAGNAE
ZHAO Jiwei
(Centre for History of Mathematics and Sciences, Northwest University, China)

Cardano’s Artis Magnae in 1545 is a milestone in the development of algebra. It is credited especially for the first publication of the solutions of cubic and quartic equations. Many books on the general history of mathematics tend to explain Cardano and Ferrari’s method of solving quartic equation by means of 5-term equations. However, from the rules in chapter 7, 26, 34 and 39 of this book, it seems that Cardano and Ferrari have not got the general method for the 5-term quartic equations.
In chapter 33 of Artis Magnae, Cardano intends to find two numbers such that the sum or difference of them is given, and the sum of the squares of certain parts of the two numbers added to its square root is also given. Cardano discovers the rule of proportional position by which he could solve the problem through the equation
.
Thus, by the traditional method, i.e., letting the root alone on one side of the equation and square both sides, it will lead to a solvable bi-quadratic equation.
The rule of proportional position is explained by 7 numerical examples. Cardano gives the procedure of the calculation on the two proportions of the two numbers. However, he does not explain why he needs to discover a new rule, nor does he explain why it should be operated in such way. This paper is to respond to these questions. The purpose of this rule is to avoid 5-term quartic equation. For if by simple position, the above problems will lead to equations of the form
.
If solving it by traditional method, it will lead to a 5-term quartic equation which is unsolvable to Cardano. By this rule, Cardano could transform a y2 by cintopx2 q.As for the procedure of the calculations, Cardano calculate by unknowns to find the results firstly, and then he transforms the result into procedure of calculation on the related proportions. Cardano’s reasoning is complemented.

EARLIEST FACTOR TABLES
Maarten Bullynck
Belgium

The earliest factor tables were produced as an aid for solving classic Greek number problems,viz. perfect and amicable numbers. Frans van Schooten’s table (1657) in the Exercitationes Mathematicae was, however, embedded in the more ambitious project of divulging and promoting the Cartesian method in mathematics. As a reaction to van Schooten’s table, John Pell organized the calculation of the first extensive factor table, upto 102,000 in 1668. For Pell, the factor table had not only mathematica linterest, but was to be a specimen of a more general tool, viz. a table of simple ideas could be combined to form truths. Both the cultural and mathematical contexts in which these two early factor tables were produced will be discussed, and the fabrication and usage of this tabular tool in mathematical problems will be illustrated by examples.

UNE ÉTUDE EMPIRIQUE DE GEORG CANTOR
Anne-Marie DECAILLOT
University of Paris 5

L’intervention de Georg Cantor au congrès de Caen (1894) de l’Association française pour l’avancement des sciences est constituéed’untableaudevérificationempiriquedelaconjecturedeGoldbach.Cetteconjecturedethéoriedesnombresprévoit que tout nombre pair est la somme de deux nombres premiers. Elle n’a reçu de nos jours encore aucune démonstration.
Cantor en vérifie la validité jusqu’au nombre 1000 en donnant toutes les décompositions des nombres pairs, compris entre 2 et 1000, en somme de deux nombres premiers. Il établit ainsi, dans les limites fixées, la table de la fonction empirique qui associe à un nombre pair le nombre de ses décompositions de Goldbach.
Mais l’examen de sa correspondance avec les mathématiciens Charles Hermite ou Felix Klein révèle une tout autre ambition. Cantor est à la recherche de lois vérifiées par la fonction précédente et fait à ce propos des conjectures audacieuses dont nous apprécions la valeur à la lumière de recherches récentes.

EULER-OTTO’S BALLISTIC TABLES
Dominique TOURNES University of La Reunion, FRANCE

In 1753, Euler gives a new method of numerical integration for the differential equation of the motion of a projectile in a resistant middle, and provides the computation scheme of a set of numerical tables for the use of artillery. These tables, calculatedandpublishedin1842bycaptainOtto,of the Prussian army,will then remain in use until the late 19thcentury.
We shall analyze Euler-Otto’s tables and we shall compare them with the other projects of calculation of ballistic tables conceived during the period 1750-1850 by Graevenitz, Lambert, Borda, Bezout, Legendre, Obenheim, Poncelet, and Didion. It will allow us to draw up a state of numerical and graphical methods of computation used in this time, and to study the circulation of knowledge which could exist in Europe between mathematicians and artillerymen.

INSTRUMENTS VERSUS TABLES DANS LE CALCUL DES DÉBLAIS ET REMBLAIS DANS LA FRANCE DES ANNÉES 1830-1860
Konstantinos Chatzis Université Paris-Est – France

Les années 1830-1860 constituent une période faste pour les travaux publics en France et accueillent la réalisation de multiples projets en matière de routes et de canaux et, à partir de 1842, de chemins de fer. Ces projets demandent de nombreux calculs fastidieux des surfaces de déblais et de remblais sur les profils en travers de ces différentes voies de communication. Les ingénieurs du corps des Ponts et chaussées, soumis à la pression d’un volume de travail accru, essaient alors différents procédés de calcul plus ou moins expéditifs. Plusieurs tables numériques donnant directement les surfaces en fonction d’un certain nombre de caractéristiques de la route et des on environnement, telles que la largeur de la chaussée ou l’inclinaison du terrain naturel, sont alors fabriquées. Pendant la même période, des ingénieurs du corps inventent aussi plusieurs instruments à calculer rapidement toutes sortes de surfaces sur un plan. Notre communication propose une vue panoramique sur cette production protéiforme selon une perspective qui envisage les tables et les instruments commedes «objets» qui sont produits selon un «processus de fabrication», mis sur «marché» et « consommés » (utilisés) par les praticiens. Nous allons ainsi étudier à la fois le « produit » (les caractéristiques de l’objet,leslogiquesquiontprésidéàleurélaboration…),lescaractéristiquesduprocessusdefabrication(lesauteursdes tables et des instruments, qu’il soient concepteurs ou exécutants, l’organisation du travail, les divers « moyens » de productionemployéspourlafabricationdecesobjets…),lesmodalitésdediffusionetlespratiquesd’usagedestableset des instruments relatifs au calcul des déblais et des remblais, enfin.

MATHEMATICS, ANALYSIS AND MECANISATION IN GREAT-BRITAIN (1834-1934)
Durand-Richard Marie-José
LAGA (Université Paris 8) & UMR 72 (CNRS-Paris7) Paris, France

When Charles Babbage conceived his « difference engine » and his « analytical engine » in 1834, his main goal was to mechanise the algebraic analysis, by transfering to the machine the organisational principles of the division of labor. So the machine could produce directly some numerical tables, essentially for astronomy and navigation. From the second part of the 19th century, the methods induced by the mechanisation of analysis were essentially different. The harmonic analyser (1876) of Lord Kelvin, as well as the differential analyser (1931) of Vannevar Bush, as soon realized by Douglas R. Hartree in Manchester and Cambridge, applied the reading, analysis and drawing of continous curves. Nevertheless, these machines were largely involved in the making out for firing tables during the World War II. My talk will precise what kinds of differential equations were so resoved, and how analogous and numerical methods interact during this period.

THE RELATIONSHIP BETWEEN NUMERICAL AND GRAPHICAL METHODS IN THE FIRST HALF OF THE 20th CENTURY
Renate Tobies
Technical University of Braunschweig, Germany

Numerical and graphical methods became a focal point of applied mathematics in the first half of the 20th century not only at universities but also in industry.
An international figure was Carl Runge (1856-1927) who introduced these methods not only at German and American universities but also in German industry.There are newfindings that his eldest daughter Iris Runge(1888-1966) became one of his most important followers and that a book written by the British automobile factory owner and aircraft researcher F.W.Lanchester(1868-1946),which was translated into German by Carl Runge, his wife, and Iris, promoted the enthusiasm for using and developing graphical methods.
From 1923 to 1945, Iris Runge worked as a (single) mathematical consultant to engineers in German communication industry, using a wide range of mathematics. I would like to show the relationship between numerical methods (equations and tables) and graphical representations in this context.

THE DESIGN OF NUMERICAL TABLES FOR STATISTICAL QUALITY CONTROL IN INDUSTRY (1920-1950)
Denis Bayart
Ecole polytechnique & CNRS, PREG-CRG, Paris, France

Statistical methods for the control of quality of manufactured products are used in industry when the characteristics of the products are sensitive to random variations in the production processes. Methods currently in use in many industries under various appellations (e.g. “six sigma” in electronics industry) have been devised since the years 1920s, nearly at the same period but independently, in industrialized western countries (USA, Germany, France, UK).
From the beginning, such methods required the treatment of large volumes of numerical data running through various operations : data collection, presentation, computation of statistical summaries, hypothesis testing and conclusions (ASTM, 1933). Graphical representations have been used extensively (histograms, distribution charts, control charts...) as well as lists or tabular representations of data. Both kinds of representation, graphical and numerical, complement each other in a dialectical process attempting to better catch the properties of series of numerical data.
The methods under scrutiny are most generally intended to be put to use by industrial workers not trained in higher mathematics or statistics, such as shopfloor technicians, quality inspectors, or even machine workers. Control chart methods, for example, rely on a very intuitive graphical display(Shewhart,1931),allowing ordinary workers to perform a periodic sampling of the production and draw correct conclusions. On the other side, acceptance sampling methods rely on sets of numerical tables where the numbers are digitally expressed (Dodge & Romig, 1944).
After presenting the different genres of numerical tables implied in statistical quality control, I’ll concentrate on numerical tables for acceptance sampling. Successive publications of these tables will be compared in relation with the historical context (first publication in 1928, the same as a standard published in 1944, and a different form designed for the war industries). I hypothesize that the tables are designed to become cognitive instruments fitted to specific working situations, and I try to show how such orientations shape and modulate the scientific structure at the foundation of the tables.
ASTM, 1933. American Society for Testing Materials. Manual on Presentation of Data. Published by American Society for Testing Materials
Dodge H.F. & Romig, H.G, 1944. Sampling Inspection Tables - Single and Double Sampling. New York: Wiley Shewhart, W.A., 1931. Economic Control of Quality of Manufactured Products. New York : McGraw-Hill

WHY MIGHT A MATHEMATICIAN WANT TO ADD PULSE CIRCUITRY TO PENCIL AND PAPER?’’ MATHEMATICAL TABLES IN THE ERA OF DIGITAL COMPUTING.
Liesbeth De Mol

Centre for logic and philosophy of science, Universiteit Gent, Belgium Boole Centre for Research in Informatics, University College Cork, Ireland
In his paper Computer technology applied to the theory of numbers dated 1969, the number theorist Derrick Henry Lehmer provided his answer to the question Why might a number theorist want to add pulse circuitry to pencil and paper?’’ by summing up several different usages of the computer in number theory in order of increasing machine involvement. This list of number-theoretical computer usages ranges from computing sequences of numbers to find counter-examples to conjectures to real computer-assisted proofs. Included in this list is the actual construction and inspection of mathematical tables, while the several other usages often make implicit use of tables in some way or the other. Although almost each of these usages were, theoretically,not beyond human reach before the rise of the computer (including the inspection and construction of mathematical tables) the gain in speed and memory as well as the possibilityofautomationhavenonethelessmadeavailableanew“universeofdiscourse”–toputitinLehmer’swords– that was not accessible before. Mathematical tables play a fundamental role here, since most of the applications involve automated (explicit or implicit) construction and/or inspection of tables. The aim of this talk is to come to a better understanding of the methods of construction and use of mathematical tables since the rise of the digital, electronic general-purpose computer in order to trace the impact they have (had) on mathematics. The starting point will be Lehmer’s ideas on mathematical tables in relation to computing machines. He not only made extensive use of computing machines for doing mathematics, often involving the construction and inspection of tables, but was also, on several occasions, quite explicit about how the computer might change mathematics. On the basis of our analysis of Lehmer’s work and ideas on the topic, we will consider several examples of the use of mathematical tables throughout the history of digital computing up to now, evaluating them in the light of Lehmer’s ideas on using pulse circuitry in number theory, and, more generally, mathematics.