Symposium "History of Numerical Tables : Second Meeting on History of Exact Sciences along the Silk Road", XXIII International Congress of History of Science and Technology, *Ideas and Instruments in Social Context*, 28 July - 2 August 2009, Budapest, Hungary. Organizers : Dominique Tournès (France), Qu Anjing (P. R. China).

### First session : Tables in Antiquity

### Chair : JI Zhigang

### July 28, 11.30-13.30

**1. From square tables to calculation of surfaces in Mesopotamia**

**Christine PROUST**

*Abstract.* 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 archaic texts, 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.

**2. Fractional tables and water clocks in Egypt**

**Micah ROSS**

*Abstract.* 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.

**3. From lists to a table to manage grains : The evidence from the oldest extant Chinese mathematical books**

**Karine CHEMLA & MA Biao**

*Abstract.* 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* 算數書), a manuscript excavated from a tomb sealed ca 186 B.C.E. and being so far the earliest 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 彭浩, 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 倉律,” a text copied during the Qin dynasty (221 B.C.E.—206 B.C.E.) and discovered among Qin legal documents at Shuihudi 睡虎地. 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.

### Second session : Tables and Astronomy

### Chair : Alexander JONES

### July 28, 15.00-17.00

**4. The function of numerical tables in the mathematical structure of Ptolemy’s Almagest**

**Nathan SIDOLI**

*Abstract.* 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.

**5. The numerical model of Chinese planetary theory**

**QU Anjing**

*Abstract.* 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.

**6. The precision of the planetary calculation in the Song dynasty**

**TANG Quan**

*Abstract.* 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.

**7. Mercure et le second équatoire de Jean de Lignières**

**Matthieu HUSSON**

*Abstract.* 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 table is build 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 *Tabule magne* 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 allows us to wonder if the table were computed with the use of the equatory

### Third session : Tables and Arithmetic

### Chair : Karine CHEMLA

### July 28, 17.30-19.30

**8. Comments on the numerical tables and algorithms in Fibonacci’s Liber Abaci **

**JI Zhigang**

*Abstract.* 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 a useful calculating methods for the *Maestri d’abbaco*. It is well known that the gelosia method has its roots in Hindu, unit fraction 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.

**9. Cardano’s rule of proportional position in Ars Magna **

**ZHAO Jiwei**

*Abstract.* 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 $px^2+q+\sqrt*px^2+q*=n$. 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 $ay^2\pm by+c+\sqrt*ay^2\pm by+c*=n$. 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 $ay^2\pm by+c$ into $px^2+q$. 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.

**10. Earliest Factor Tables**

**Maarten BULLYNCK**

*Abstract.* 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 mathematical interest, 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.

**11. Une étude empirique de Georg Cantor**

**Anne-Marie DÉCAILLOT**

*Abstract.* L’intervention de Georg Cantor au congrès de Caen (1894) de l’Association française pour l’avancement des sciences est constituée d’un tableau de vérification empirique de la conjecture de Goldbach. Cette conjecture de théorie des nombres prévoit 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.

### Fourth session : Tables and Engineering in the 19th century

### Chair : David AUBIN

### July 29, 9.00-11.00

**12. Euler-Otto’s ballistic tables**

**Dominique TOURNÈS**

*Abstract.* 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, calculated and published in 1842 by captain Otto, of the Prussian army, will then remain in use until the late 19th century. 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.

**13. Instruments versus tables dans le calcul des déblais et remblais dans la France des années 1830-1860**

**Konstantinos CHATZIS**

*Abstract.* 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 de son 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 comme des « 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, les logiques qui ont présidé à leur élaboration…), les caractéristiques du processus de fabrication (les auteurs des tables et des instruments, qu’il soient concepteurs ou exécutants, l’organisation du travail, les divers « moyens » de production employés pour la fabrication de ces objets…), les modalités de diffusion et les pratiques d’usage des tables et des instruments relatifs au calcul des déblais et des remblais, enfin.

**14. Mathematics, analysis and mechanisation in Great-Britain (1834-1934)**

**Marie-José DURAND-RICHARD**

*Abstract.* 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.

### Fifth session : Tables and Engineering in the 20th century

### Chair : Dominique TOURNÈS

### July 29, 11.30-13.30

**15. The relationship between numerical and graphical methods in the first half of the 20th century**

**Renate TOBIES**

*Abstract.* 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 new findings 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.

**16. The Design of numerical tables for statistical quality control in industry (1920-1950)**

**Denis BAYART**

*Abstract.* 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 exemple, 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

**17. "Why might a mathematician want to add pulse circuitry to pencil and paper ?" Mathematical tables in the era of digital computing**

**Liesbeth DE MOL**

*Abstract.* 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 possibility of *automation* have nonetheless made available a new “universe of discourse” — to put it in Lehmer’s words — 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.