ancient mathematics and natural science

Hypatia of Alexandria was remembered as an inspiring teacher of philosophy, a devoted pagan, and an important mathematician. While her violent death in 415 CE has brought her the most subsequent fame, she also lived an extraordinary life. This volume presents eleven original essays as well as new translations of all the primary sources on Hypatia.

Scientific Life in Ancient Egyptian Civilization
"Historical and Analytical Study of Geometry, Mathematics, Astrology and Astronomy"
Historical Manifestations of Geometry and Mathematics in Ancient Egypt; Manifestations of Astrological Life in Ancient Egypt; Manifestations of Astronomical Life in Ancient Egypt; Linguistic Identification of Units of Time in Ancient Egyptian Language.
This book presents outlines of History of Sciences in Ancient Egyptian Civilization. The late Egyptian and Greek sources praised the ancient Egyptian cultural centers as a source for knowledge and science, thus ancient Egypt was a source of inspiration for the legislator 'Solon', as well as 'Thales of Miletus' who was a mathematician and astronomer, where He learned and practiced the sciences of ancient Egypt, then taking this knowledge to the Greeks. The First Chapter provides 'The Historical Manifestations of Geometry and Mathematics in Ancient Egypt'. The Second Chapter provides 'The Historical Manifestations of Astrological Life in Ancient Egypt'. The Third Chapter provides 'The Historical Manifestations of Astronomical Life in Ancient Egypt'. The Addendum provides 'The Historical and Linguistic Identification and Treatment of Obsolete Units and Measures of Time in Ancient Egyptian Language and Synonyms'. All of these were divided to several sections and topics to serve the scientific methodologies in ancient Egyptian civilization which was absolutely precious.

The Rhind mathematical papyrus incorporates a small group of problems focusing on pyramids and demonstrating how to calculate their seked side slopes and heights. Two other problems, pRhind 60 and pMoscow 14, have been discussed extensively in conjunction with the former problems due to the interpretive challenges they pose. The terminology they use and the context of the exercises mean that archaeology and philology can potentially provide information aiding understanding of the buildings described. Architecture uncovered in excavations may represent structures that inspired the problems outlined in the Middle Kingdom mathematical papyri, and the different classes of evidence are compared here.

Nowadays, Eratosthenes of Cyrene (276-194 BC) is mainly known for measuring the circumference of the Earth, quoted in the ancient sources as either 250000 or 252000 stades. Whereas the former figure is related to the astronomical observation reported by Cleomedes, the latter could have been derived from the geographical information available to Eratosthenes. We demonstrate that the greater of the two figures is interrelated with the number of other figures, including the equinoctial gnomon ratio in Alexandria and the width of the tropic zone, which have been attributed to Eratosthenes. In addition, it is shown how the arc of the tropic zone could be expressed as a fraction of a full circle in pre-trigonometrical times. Furthermore , we discuss a recent attempt by CARMAN and EVANS to explain the figure of 252000 stades as a result of taking into account the finite distance to the Sun. We argue that this contains some historical inconsistencies and astronomical problems.

نهضت ترجمه: سرگذشت ترجمه در فرهنگ اسلامی، از سده نخست هجری تا دوره معاصر/ نوشته سیداحمد هاشمی و دیگران، تهران: نشر کتاب مرجع، ۱۳۸۹ ش

موضوع کتاب حاضر بررسی سرگذشت ترجمه در فرهنگ اسلامی از سده‌ی نخست هجری تا دوره‌ی معاصر است. کتاب در دو بخش اصلی «نهضت ترجمه» و «ترجمه‌های جدید» تدوین شده و هر بخش مشتمل بر چند فصل است. این اثر با ارائه تاریخچه‌ای از نهضت ترجمه و نحوه‌ی انتقال آثار تمدن‌های غیرعربی به عالم اسلام شروع شده است. سپس به معرفی مترجم‌ها و نوع ترجمه آنها پرداخته و پیامدهای ترجمه‌ی آثار را بررسی کرده و شرحی بر ترجمه به عربی، ترجمه به ترکی و ترجمه به فارسی در دوره‌ی معاصر ارائه داده است

International Ancient Philosophy and Science Workshop for Women*, Non-Binary, and LGBTQIA+ Graduate Students and Early Career Researchers.

7–8 July 2022, Humboldt-Universität zu Berlin.

Organizing committee: Elena Bellini (she/her), Sinem Kılıç (she/her), Benjamin Wilck (he/him)

This is a specimen of a new comprehensive volume on Plato's Timaeus. The volume includes: a substantial introductory essay by Franco Ferrari; a new critical edition, based on seven primary witnesses, a complete review of ad litteram quotations of the dialogue and a new assessment of ancient corrections of "A", a running commentary with technical notes and tables.

This paper focuses on How there are different and dissonant values in measuring dimensions in ancient Egypt? The ancient Egyptians relied on a natural method to measure dimensions like the arm that was used as a measure of length, approximately equal to the length of a forearm. Traditionally, it was the length from the bent elbow to the tips of the fingers. Typically, almost 18 inches or 44 cm, however there was a long cubit of about 21 inches or 52 cm. The second natural method was the width of the palm of the hand. As well as, the human fingers used as digits for measuring width, where the four digits equal the sign of one palm and seven palms equals one cubit. In present-day trigonometry, cotangent requires same units for both horizontal run and vertical rise, however ancient sources like Rhind Papyrus uses palms for the run and cubits for the rise, resulting in these different, yet characteristic mathematics. In ancient Egypt there were seven palms in a cubit, in addition to the seqed that was seven times the cotangent. There are some questions are as follows: Is it true that the Egyptian seqed is the ratio of the run to the rise of a slope of a cotangent? How to measure the inclination in ancient Egypt accurately? Is the value of the seqed or the angle θ correct by applying the rules of modern trigonometry? The question arises as to whether the problems of the seqed are actually mirrored in the ancient Egyptian architecture methods and construction techniques? How are there different and dissonant values in measuring dimensions in ancient Egypt?

Contrary to popular belief, Pythagoras' investment in mathematics was very small or non-existent. Similarly, the Pythagoreans were far from uniform in their discourse in this field, and the vast majority paid no attention to it. While there are Pythagorean mathematical works in the literal sense, other speculations on numbers have led to the construction of philosophical-religious discourses on numbers. This arithmological tendency has long been considered late, but there is a large number of fragments that testify to what these speculations may have been between the 5th and 4th centuries BC. A reworking of the record shows the Pythagorean attempt to relate abstract concepts to numbers and deities in order to formulate ethical, moral or political propositions. Numbers or mathematical reasoning thus became a support for philosophical demonstration, without being a principle.

Contrairement à une idée reçue, l’investissement de Pythagore en mathématiques est resté très réduit ou inexistant. De la même manière, les pythagoriciens sont loin d’avoir eu un discours uniforme dans ce domaine, voire n’y ont prêté aucune attention pour la grande majorité. S’il existe bien toutefois des travaux mathématiques pythagoriciens au sens propre, d’autres spéculations sur les nombres ont amené à construire des discours de nature philosophico-religieuse sur les nombres. Cette tendance arithmologique a longtemps été considérée comme tardive, mais il existe un grand nombre de fragments qui témoignent de ce qu’ont pu être ces spéculations entre les Ve et IVe siècles avant notre ère. Une reprise du dossier montre la tentative pythagoricienne de relier des concepts abstraits aux nombres et aux divinités, afin de formuler des propositions éthiques, morales ou politiques. Le nombre ou le raisonnement mathématique devient donc un support de démonstration philosophique, sans être pour autant un principe.

The article explores the concept of probability in ancient Greece from a nonscientific perspective and shows how ancient decision-makers used historical data to make calculated decisions and speculate about the future. First, the paper considers how quantitative data was used by ancient Greek communities to make economic projections. It then shows how ancient Greek generals used the same conceptual tools to determine their odds of victory by tallying up and comparing the number and composition of armies and resources available to them and their enemy. In the third section, the paper examines how qualitative probability was articulated through the language of hope and likelihood to formulate chances of success in moments of crisis. Finally, the paper shows that ancient decision-makers implemented "power laws" to adapt to changing circumstances and the flow of new information, as they sought to improve their odds of success relative to their rivals.

The experience of time is among the most fundamental features of human existence. The present thereby serves as a basis by means of which we can make sense of both past and future; thus our experience of the present, which we capture in notions like “the now”, “the instant of time”, or “the present moment”, is of special concern. We are made aware of the moment of time through motion and changes, and since the present moment seems to be when we experience these changes, our conception of the “now” is strongly connected with the notion of change. But experiencing a change means experiencing that something turns from being F to not being F, or from F to not-F. If the instants when something is F and not-F are the same instant, however, then the thing seems to be both F and not-F at the same time, and we seem to end up with a contradiction. This threatening inconsistency prompted several influential answers in ancient times: for example, Heraclitus may seem to endorse this inconsistency, while Parmenides seems to have concluded that time and change are thus unreal. Plato develops the notion of exaiphnês to suggest that the turning from F to not-F occurs outside of time. And Aristotle develops both Parmenidean and Platonic intuitions to argue that change is continuous and the segmentation of time into ‘nows’ occurs in thought.

International Graduate Student Conference.
26th–28th Nov, 2020.
Humboldt-Universität zu Berlin. Online.
Co-organized with Paul Hasselkuß (Düsseldorf), Tiago Hirth (Lisbon), Deborah Kant (Konstanz), Deniz Sarikaya (Hamburg), Tobias Schütz (Mainz), Anna Kiel Steensen (Zurich), and Benjamin Wilck (Berlin)

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disciplines that since the fourth century B.C.E. had been called mathēmata or mathēmatikai epistēmai. Among the mathēmata, which in antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics, and optics, the essay also includes geography. Using a data set based on The Encyclopaedia of Ancient Natural Scientists, it considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for the purposes of this essay), from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal, and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities—ancient scientists and philosophers—allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major sites—Athens, Alexandria, Rhodes, Rome, and Byzantium/Constantinople—that appeared, in succession, as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society.

https://www.springer.com/gp/book/9783030359508

This book presents a novel methodology to study economic texts. The author investigates discrepancies in these writings by focusing on errors, mistakes, and rounding numbers. In particular, he looks at the acquisition, use, and development of practical mathematics in an ancient society: The Old Babylonian kingdom of Larsa (beginning of the second millennium BCE Southern Iraq). In so doing, coverage bridges a gap between the sciences and humanities.

Through this work, the reader will gain insight into discrepancies encountered in economic texts in general and rounding numbers in particular. They will learn a new framework to explain error as a form of economic practice. Researchers and students will also become aware of the numerical and metrological basis for calculation in these writings and how the scribes themselves conceptualized value.

This work fills a void in Assyriological studies. It provides a methodology to explore, understand, and exploit statistical data. The anlaysis also fills a void in the history of mathematics by presenting historians of mathematics a method to study practical texts. In addition, the author shows the importance mathematics has as a tool for ancient practitioners to cope with complex economic processes. This serves as a useful case study for modern policy makers into the importance of education in any economy.

This paper aims to discuss the conceptions of the stereometric body in the Platonist tradition of the Imperial age and Late Antiquity. I shall first suggest that the Middle Platonists formulated an arithmo-stereometric theory, which they combined with the traditional Euclidean stereometry. This move was meant to solve a specific philosophical problem, namely the need to describe in arithmo-stereometric terms the structure of the cosmic soul in such a way as to guarantee, on the one hand, its dissimilarity with respect to the corporeality, whose "solid" structure can be described in Euclidean terms, and on the other hand, the effectiveness of the action of the soul within the physical and three-dimensional cosmos. Precisely because of the disappearance of these requirements and for an overall change of philosophical approach, starting from Plotinus, Neoplatonism will instead be able to renounce the arithmostereometry and return to a consistently Euclidean definition of the notion of stereometric body.

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between (i) substances and non-substantial attributes of substances, (ii) different kinds of substance, and (iii) different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object.

Open access: https://philpapers.org/rec/WILEKA

Based on the analysis of various letters of dedication by Archimedes of Syracuse and Apollonios of Perge, as well as the prefaces of letters by Seneca and Diogenes of Oenoanda, the essay illustrates the strategies used by authors of ancient scientific and philosophical literature to strengthen their relationship with the recipient or to portray themselves as authors.

The oldest members of the Pythagorean communities in Southern Italy are, in general, poorly known. Nevertheless, it is possible to draw a portrait of the members of the hetairia in some cities of Magna Graecia. The example of Poseidonia shows that the Pythagoreans were probably present in the city at the end of the archaic period. Literary sources are mobilised, but archaeological and epigraphic finds also tend to show very probable Pythagorean influences in Poseidonia.

Les membres les plus anciens des communautés pythagoriciennes en Italie du Sud sont, en général, mal connus. Il est néanmoins possible de brosser un portrait des membres de l’hétairie dans certaines cités de Grande Grèce. L’exemple de Poseidonia permet de montrer que les pythagoriciens sont probablement bien présents dans la cité à la fin de l’époque archaïque. Les sources littéraires sont mobilisées, mais les découvertes archéologiques et épigraphiques tendent également à montrer de très probables influences pythagoriciennes à Poseidonia.

Relying on evidence from fifteen epigraphic collections and sixty-odd ancient sources as well as discussing a literature of over five hundred titles, the essay’s highly unorthodox conclusions are a case in point of the micrological ideal of achieving novelty on any given subject by way of transcribing and studying first-hand all relevant materials – edited and unedited alike. The paper’s ambition was to shed new light on one of the most intriguing analogies of the whole Aristotelian corpus, namely the comparison between words and pebbles. A review of all material evidence and virtually all extant sources made short work of two related, albeit mutually exclusive, misconception about ancient reckoning boards and their workings: (1) the idea that – for all practical purposes – the abacus’ arrangement mirrored the decimal system, its columns and rows conveniently matching units, tens, hundreds, thousands, etc. (2) the idea that the inscriptions on several surviving counting boards were a nuisance to the extent that, being inconsistent to a fault with the decimal system itself, they made actual calculations harder than they already were (as opposed to making them easier, as one would expect). The paper demonstrated that the first assumption – the « decimal » bias – is simply mistaken and betrays little or no awareness of the epigraphic and archaeological evidence. The paper also demonstrated that the second assumption – the « abacus riddled with complications » bias – simply defeats the purpose of resorting to the abacus in the first place and betrays a poor understanding of its practical vocation which, most assuredly, was not to add to the very problem it was meant to solve. After bulldozing its way through both misunderstandings, the essay focused on the abacus most distinctive features, that is positionality (i.e. the abacus being a positional system through and through) and hybridity (i.e. the abacus’ place value system being monetary in nature and purpose, as opposed to it being abstract and homogeneous). The final result is an interpretation that moves away from the received views by showing that the prologue of the Sophistici elenchi offers no support to the notion that, when Aristotle referred to counters, he was leaning on a kinship of sorts – or any kinship, for that matter – between abstract calculation and speech. Once we turns the pebble’s simile on its head and set it back upon its feet (Aristotle’s pebble analogy is about pebbles, what else?), it becomes clear that it presupposed numeracy all right, but it was not about numeracy itself. Moreover its goal was not to explain why computational and linguistic symbolisms succeed, but to explain how they fail – failure being the whole point; in this particular instance, failure to detect and prevent abusive value shifts affecting words and counters alike.

Der Beitrag möchte auf die Bedeutung des Unterschiedes von Sein und Werden für den sich bei Platon und Aristoteles ergebenden Zusammenhang von Naturbetrachtung auf der einen Seite sowie Ethik, Ökonomie und Politik auf der anderen Seite aufmerksam machen. Im Zentrum steht die Passage 27D 5–28A 1 aus dem platonischen Dialog 'Timaios', anhand derer dieser Unterschied in seiner Bedeutung einerseits für die aristotelische Fassung des Menschen als eines 'zôon politikón' und 'zôon lógon échon', andererseits für den aristotelischen Begriff der 'entelécheia/enérgeia' angezeigt wird. Der Geist dieser platonisch-aristotelischen Unterscheidung erweist sich dabei zum einen als von der Naturphilosophie eines Philistion von Lokroi inspiriert, zum anderen als von der athenischen Welt- und Selbstsicht getragen, wie sie sich in der thukydideischen Gefallenenrede des Perikles zwar andeutet, deren unvollendete Form aber schon der platonische 'Menexenos' kritisiert.

Attempts to tackle environmental problems against hard-to-degrade plastic materials have been widely practiced. One of them was by using Effective Microorganism 4 (EM4) bacteria. Preparations of polypropylene / chitosan (PP / chitosan) plastic with glycerol plastisiser and polyglycerol acetate (PGA) have been done by using reflux method. Plastic morphology analysis was done by Scanning Electron Microscopy (SEM) test and biodegradation characteristic was analyzed by mixed microbial batch method. Analysis of surface plastic morphology of PP / chitosan, PP / chitosan-glycerol, and PP / chitosan-PGA showed a homogeneous surface. Biodegradation test was using EM4 bacteria resulted in an optimum percentage of degradation of 71.18% obtained from PP / chitosan plastic with 20% chitosan filler and PGA plastisiser 2.5%.

A research has been done on the production of cellulose acetate films from oil palm logs (Elaeis guinensis Jack) with chloroform as solvent and triacetin as plasticizer. Cellulose from oil palm sawdust obtained by alkalization process with addition of 2% NaOH. The bleaching process was carried out with addition 2% of NaOCl, 8 drops of glacial CH3COOH and heated at 60-70°C while stirring for 1 hour. The obtained residue was added with 0.05 N HNO3 and heated at 70°C for 1 hour. The obtained cellulose is synthesized to cellulose acetate by the addition of glacial acetic acid as an activator to allow the esterification reaction to proceed smoothly. The acetylation process was carried out by addition of catalyst acetic anhydride acid and H2SO4. The obtained cellulose acetate was formed into a cellulose acetate film with a chloroform solvent and a variation amount of plasticizer triacetin 0 mL, 0.5 mL, 1 mL, and 1.5 mL. Functional group analysis was determined by FT-IR and showed the presence groups of-OH, C-H, CO for cellulose and groups of-OH, C-H, CO , C=O for cellulose acetate and cellulose acetate film. The mechanical analysis was determined by tensile test and showed cellulose acetate added with triacetin (1.8:1.5 (w / v)) having an optimum yield of 156,701 MPa, strain 0.0269 and Young's Modulus 3725.4 MPa. Thermal degradation analysis with TGA ie good heat stability occurred at temperature 294,25 o C and residual mass 1,9%. Morphological analysis with SEM is at 1000 times enlargement, the distribution of pore film spread evenly, has a smooth surface, pores are small and homogeneous.

The article explores the concept of probability in ancient Greece from a non-scientific perspective and shows how ancient decision-makers used historical data to make calculated decisions and speculate about the future. First, the paper considers how quantitative data was used by ancient Greek communities to make economic projections. It then shows how ancient Greek generals used the same conceptual tools to determine their odds of victory by tallying up and comparing the number and composition of armies and resources available to them and their enemy. In the third section, the paper examines how qualitative probability was articulated through the language of hope and likelihood to formulate chances of success in moments of crisis. Finally, the paper shows that ancient decision-makers implemented “power laws” to adapt to changing circumstances and the flow of new information, as they sought to improve their odds of success relative to their rivals.

Biton’s Construction of Machines of War and Catapults describes six machines by five engineers or inventors; the fourth machine is a rolling elevatable scaling ladder, named sambukē, designed by one Damis of Kolophōn. The first sambukē was invented by Herakleides of Taras, in 214 BCE, for the Roman siege of Syracuse. Biton is often dismissed as incomprehensible or preposterous. I here argue that the account of Damis’ device is largely coherent and shows that Biton understood that Damis had built a machine that embodied Archimedean principles. The machine embodies three such principles: (1) the proportionate balancing of the torques on a lever (from Plane Equilibria, an early work); (2) the concept of specific gravity or density (from Floating Bodies, a late work); and (3) the κοχλίας, i.e., a worm drive (invented ca 240 BCE), with the toothed wheel functioning as the horizontal axis of rotation of the elevated ladder. Moreover, the stone-thrower of Isidoros of Abydos (the second machine in Biton) also embodies the κοχλίας.

This paper studies the fractional differential problem of fractional functions, regarding the modified Riemann-Liouvellie (R-L) fractional derivatives. A new multiplication and the fractional power series method are used to obtain any order fractional derivatives of some elementary fractional functions.

Symposium at the Joint BSHM-CSHPM/SCHPM conference "People, Places, Practices".
July 2021.
University of St. Andrews, Scotland (online).
Co-organized with Deborah Kant (Konstanz).
Funded by the International Commission on the History of Mathematics (ICHM).

Number symbolism and arithmology in ancient Christian texts are here discussed: among the many examples of both Patristic and gnostic ‘serious’ numerological exegesis, the present paper aims at showing how Hermia’s treatment of Pythagorean numbers (Irrisio gentilium philosophorum, chapter 16) stands out as a curious case of ‘mathematical irrisio’.

The frame work is composed of three chapters. The first one provides "Mathematics, Geometry and Features of scientific life in ancient Egypt". The second chapter includes "Astronomy and Features of astronomical life in ancient Egypt". The third chapter provides "Astrology and Features of astrological life in ancient Egypt". All of these three chapters divided to several sections which serves the scientific methodology of the framework of the three chapters.

International workshop at Humboldt University Berlin, Germany.
14-18 April, 2012.
Co-organized by Colin G. King (Providence College) and Benjamin Wilck (Humboldt University).

https://www.philosophie.hu-berlin.de/de/lehrbereiche/antike-wissen/veranstaltungen/aristotles%20posterior%20analytics

Isolation & characterization of phenolic compound from Saputangan leaves (Maniltoa grandiflora (A. Gray) Scheff) and antioxidant activity have been carried out. The isolation process of phenolic compound was started through maceration process using methanol solvent, the partition using ethyl acetate and n-hexane, then the isolation using column chromatography. Characterization using UV-Vis, FT-IR, NMR and ESI-HRMS instruments. Antioxidant activity was carried out using DPPH method for phenolic compound. Based on the screening test, it can be stated that the Saputangan leaves contain phenolic compounds, flavonoids and terpenoids. Phenolic compound obtained has a weight of 32, 3 mg. The results of interpretation of spectroscopic data show that isolated phenolic compound is methyl gallate with the structure formula C8H8O5 and molecular weight of 184. Phenolic compound antioxidant as methyl gallate has IC50 values of 16,136 mg/L which can be categorized as very strong antioxidant.

We are fortunate to possess at least portions of a continuous tradition of Greek (and Roman) science, in contrast to MESOPOTAMIAN SCIENCE and EGYPTIAN SCIENCE, whose development was erased when their respective written cultures fell silent, the Egyptian DEMOTIC by the end of the fifth century CE, and the Mesopotamian CUNEIFORM somewhat earlier. (Sanskrit and Chinese science also provide portions of a continuous tradition.)

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disciplines that since the fourth century b.c.e. had been called mathēmata or mathēmatikai epistēmai. Among the mathēmata, which in antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics, and optics, the essay also includes geography. Using a data set based on The Encyclopaedia of Ancient Natural Scientists, it considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for the purposes of this essay), from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal, and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities—ancient scientists and philosophers—allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major sites—Athens, Alexandria, Rhodes, Rome, and Byzantium/Constantinople—that appeared, in succession, as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society.