Mesopotamian mathematics

This short article is the continuation of the “A Sumerian Numerical Palimpsest” where it is tried to find if the nomenclature of Sumerian numerals appears to be the result of the use of hand(s) for counting. That Sumerian counting in multiples of sixty has to begin with the “making” the “first” sixty is obvious, but is seldom stated. For the people who count in concrete and in any mode, a natural model - a counting device, like hands – has to be recognized so that a visual one-to-one correspondence between the individual elements of the device and the counted items became obvious. And that should apply to sixty for which I am suggesting a healthy robust spike of the six rowed barley as a natural model of a morphology that extremely closely reflects the Sumerian 60 based numerals. An idealized six rowed barley spike, še ḡeš, an intellectual achievement, which preliterate Sumerians were quite capable of, may had, or rather, could been the reckoning device on which the counting in multiples of sixty was established and prospered.

Uruk, Auruk or Warka was an ancient city of Sumer (and later of Babylonia) situated east of the present bed of the Euphrates River on the dried-up ancient channel of the Euphrates 30 km (19 mi) east of modern Samawah, Al-Muthannā, Iraq.

Uruk is the type site for the Uruk period. Uruk played a leading role in the early urbanization of Sumer in the mid-4th millennium BC. By the final phase of the Uruk period around 3100 BCE, the city may have had 40,000 residents, with 80,000-90,000 people living in its environs, making it the largest urban area in the world at the time. The legendary king Gilgamesh, according to the chronology presented in the Sumerian king list, ruled Uruk in the 27th century BC. The city lost its prime importance around 2000 BC in the context of the struggle of Babylonia against Elam, but it remained inhabited throughout the Seleucid (312–63 BC) and Parthian (227 BC to 224 AD) periods until it was finally abandoned shortly before or after the Islamic conquest of 633–638.

William Kennett Loftus visited the site of Uruk in 1849, identifying it as "Erech", known as "the second city of Nimrod", and led the first excavations from 1850 to 1854.

The Arabic name of Babylonia, which eventually became the name of the present-day country, al-ʿIrāq, is thought to derive from the name Uruk via Aramaic (Erech) and possibly via Middle Persian (Erāq) transmission.
In Sumerian, the word uru could mean "city, town, village, district, circular area"

We consider several aspects of the role and evaluation of the four-sexagesimal-place approximation to ͙2 on the well-known Old Babylonian tablet YBC 7289. By referring to what is known about OB school texts, we show that this text is most probably a school exercise by a trainee scribe who got the approximation from a coefficient list. These coefficient lists are briefly described, with their use in geometrical problems. We consider other texts involving square roots and derive an algorithm for evaluating them, which complies with all known OB examples, from a simple geometrical construction of the type that seems to underlie many other OB procedures.

Background. Over the last century, scholars have deciphered cuneiform texts on mathematics and astronomy (Neugebauer & Sachs 1955) associated with Mesopotamia's religious architecture, calendars and iconography (Curatorla 2007). However, no systematic investigation of possible astronomical orientation has been made of Mesopotamia's many ziggurats and temples. The availability of geo-referenced/sub-meter resolution satellite imagery now makes such analysis feasible; despite the region's ongoing military conflicts preventing Ground Truth field survey.

A geometric study is proposed that forms the basis of the principle of halfsums and half-differences as used by the Babylonians. Diagrams pertaining to this study are developed, and applied in the demonstration of algebraic procedures of given problems from various recovered Babylonian tablets. The geometric study of half-sums and half-differences is further developed for third degree equations, extending to the geometric construction as hypothesized by S.J. Lurje (in the well-known study of quadratic sums). In conclusion, various didactic proposals on the subject are suggested.

This short draft, the first of two, is used to explore possible use of a hand as a counting device of the inhabitants of the prehistoric alluvial Mesopotamia, and how reckoning by five digits of a hand and possible use of two hands, shaped the ancient numerals nomenclature. This is not easily discernible from the way we, commonly, think of the multiple of sixty reckoning, for it, like on a palimpsest, obscures the ancient manners of Sumerian counting by hand(s).

The focus of the second part of the essay is the thought that another commonly occurring object has been used as a model by the inhabitants of the prehistoric alluvial Mesopotamia, at first as an approximate grain measure which was sometimes later abstracted into a reckoning “device” containing exactly 60 elementary parts, še, arranged in the sixty ← ten ← one manner and conform to the ×6, ×10, ×1 rule.

A Mesopotâmia é um território situado entre os rios gémeos, Tigre e Eufrates, no Próximo Oriente, ocupado por diversos povos na antiguidade, dos quais sobressaem os sumérios, os assírios e os babilónios, tendo um carácter heterogéneo, não unificado enquanto país. O legado cultural destes povos começa a ganhar proporção com os sumérios, através da invenção da escrita cuneiforme, e consequente transição da Pré-História para a História. O seu legado alcançou uma forte projecção nas civilizações que lhe sucederam e, inclusivamente, nas sociedades actuais; noções generalizadas sobre as quais reflecte o presente texto.

The VAT 12593 laconic text, impressed on both sides of the tablet, presents the areas of sixteen square fields, obtained from multiplication of the given “widths” and “lengths.” While there is no indication of how the displayed results were attained, it was quite possible to discern an, unexpectedly, efficient algorithm by which a mental/oral abstract computing process, employing oriented and dimensionless magnitudes, would quickly yield the shown flawless results.

Ce livre présente une collection de tablettes mathématiques d’époque paléo-babylonienne (début du deuxième millénaire avant notre ère) qui ont été exhumées à la fin du XIX e siècle par une mission archéologique américaine sur le site de Nippur (Mésopotamie centrale). Ces tablettes sont aujourd’hui conservées dans les musées archéologiques d’Istanbul, de Philadelphie et de Iéna. Le lot d’Istanbul est entièrement édité dans cet ouvrage et dans le CD qui l’accompagne (photos, copies, transcriptions). Les tablettes mathématiques de Nippur sont principalement des brouillons d’écoliers. Sans doute considérées comme trop élémentaires, elles avaient jusqu’à une date récente peu attiré l’attention des épigraphistes et des historiens, et elles étaient restées ignorées dans les réserves des musées. Pourtant, les tablettes scolaires apportent de précieux témoignages sur la vie intellectuelle qui s’est épanouie à Nippur, la grande capitale culturelle de la Mésopotamie, et notamment sur la place qu’y occupaient la langue sumérienne et les mathématiques, dans leurs raffinements les plus abstraits. L’étude des textes scolaires mathématiques, en prolongeant celles qui ont été menées sur les textes scolaires lexicaux et littéraires sumériens, permet une reconstitution remarquablement détaillée du cursus de formation des scribes. Précisément parce qu’ils sont des textes d’apprentissage, ces modestes brouillons d’écoliers donnent accès aux conceptions originales en matière de métrologie, de numération et de calcul qui étaient inculquées aux jeunes scribes et qui donc contribuaient au fond culturel des milieux érudits. Par ailleurs, trois textes mathématiques savants, dont un texte inédit conservé à Istanbul, ont été retrouvés à Nippur. Leur contenu est particulièrement intéressant, car il concerne différents aspects du calcul des volumes et des racines cubiques. Si on les aborde selon les conceptions élaborées par les scribes eux-mêmes, telles qu’elles leur ont été enseignées, et non au moyen de nos outils algébriques et arithmétiques actuels, ces textes livrent toute la singularité et la finesse des mathématiques qui se sont développées à cette époque.

This is my second book that intends to discover the history of the Flower of Life symbol. Now my main focus is on the interpretation of the symbol and on meanings attached to it throughout the history, starting from the Near East Bronze Age until our days.

The reader is expected to be familiar with basics of the Flower of Life geometry. Elementary knowledge of the ancient Near East history and history of mathematics is helpful. Knowledge of visual arts, geometry, and comparative religion studies are also useful.

I hope that my independent research provides valuable resources for people to further investigate the topic.

In this essay I will present continuity of historical artifacts of the geometrical symbol that is known as the "Flower of Life" in modern day. Origins of the symbol dates at least to the beginning of the 2nd millenium BC in Mesopotamia. Intermediate knowledge of the ancient Near East and western history of mathematics is required from the reader as well as elementary knowledge of art, geometry and religions. Reader is also expected to know basics of the Flower of Life geometry. I hope document will provide key sources for investigators willing to do further research with the topic.

The Riemann zeta function is one of the most Leonhard  Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.

In her contribution, Victoria Altmann-Wendling presents on the one hand the practical application of the lunar phases for the calendar, the regulation of religious festivals and the priests’ services. These indirect pieces of evidence will be further elaborated by Demotic and Greek papyri that contain computations about the moon’s movement and phases. Furthermore, the role of the moon in astrology is elucidated. On the other hand, she shows how the moon and the astronomical phenomena related to it were integrated into the religious imagery and symbolism of Ancient Egypt. Since the sources derive mainly from the Graeco-Roman period in which a multicultural society was living in Egypt, reflections on possible exchange, transfer or adoption of knowledge are possible.


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[1] TM.75.G.1565 = MEE 3 67 -This small rounded tablet, published among the "esercizi scolastici" in PETTINATO 1981:261, is a "word list with personal names" for ARCHI 1992:14 (and see also CIVIL 1984:81 n. 10). For Pettinato "il testo si compone di due parti nettamente distinte tra loro: la prima parte (recto) contiene un elenco di parole sumeriche ordinate acrograficamente in base all'elemento iniziale nì-; la seconda invece (verso) abbraccia una serie di nomi di persona che contengono tutti l'elemento lugal-in prima posizione. Mentre la serie che inizia con nì-ricorda da vicino l'omonima serie dei testi lessical šè-bar-unken, la seconda va rapportata al testo [MEE 3] n. 59 r. II:1-7". However, the comparison with TM.75.G.2014 = MEE 3 59 (see below) suggests that also in TM.75.G.1565 = MEE 3 67 the six personal names were instead written before the seven common nouns of the acrographic section ninda (and note not only "r. ? " and "v. ? " in the edition in PETTINATO 1981:261, but also the inversion of the obverse and the reverse in the photographs of the tablet in MEE 3, Tav. XXXVIIa-b).

The written mathematics of late neolithic Mesopotamians emerged from a cultural impetus to control the flow of surplus economic goods in their settled societies: grains and grain products, sheep and other herded animals, jugs of dairy fats and beer, rope, textiles, etc. It then progressed, as all human inventions do, through a graded sequence of innovations, each expanding upon and rendering more efficient the ability to communicate and record quantitative information. Success relied on the use of visual signs whose meaning was accepted by convention, and whose widening repertoire was standardized through scribal schools associated with the temple economy of the early city-state (initially Uruk c.3500 BCE).

Mathematically, quantity was governed by a comprehensive set of metrological systems each with its own factor relations, sums were made which required grouping and replacement operations to ensure all quantities were written in canonical form, and fractions and scaling were used as a matter of course to document production inputs. By the end of the fourth millennium BCE, economic control through writing and mathematics was a standard part of how city-states were run, touching off a 1000-year period in which Sumerian city-states would joust for dominance (Early Dynastic period).

It is argued that iterative computations which are attested in Mesopotamian and other ancient sources can be productively analyzed and interpreted in a simulation-based framework. Ancient Mesopotamia presents us with a rich body of textual evidence for computational practices over a period of more than three millennia. This paper is concerned with Mesopotamian iterative computations of empirical phenomena, where each iteration updates the values of certain quantities from one state to the next state. It will be argued that these computations can be fruitfully interpreted in the so-called simulation-based framework, which was recently developed by philosophers of science in order to better account for the role of simulations in modern science. This is exemplified on the basis of a text from the Ur III period (2100-2000 BCE) about the growth of a cow herd. Other Mesopotamian sources with iteratively computed sequences, in particular various types of mathematical tables, are ignored here, because they do not directly correspond to any phenomena. Section 1 briefly addresses some developments in the philosophy and historiography of science in order to introduce the simulation-based framework. Section 2 discusses the textual example. Section 3 contains the conclusions.

Печатается по решению редакционно-издательского совета государственного эрмитажа научный редактор: н. в. Козлова труды государственного эрмитажа : [т.] 95 : материалы международной конференции, посвященной 100-летию со дня рождения игоря михайловича дьяконова (1915) / государственный эрмитаж. -сПб. : изд-во гос. эрми тажа, 2018 настоящий сборник включает материалы международной конференции, посвященной 100-летию со дня рождения игоря михайловича дьяконова , которая состоялась в государственном эрмитаже и институте восточных рукописей ран 13-15 января 2015 года. состав сборника отражает широту научных интересов и. м. дьяконова. статьи посвящены различным областям ассириологии, переднеазиатской археологии и семитологии. в них затрагиваются проблемы политической и социально-экономической истории месопотамии и сопредельных стран, месопотамской религии и литературы, лингвистики и истории письменности.

How far back in time can we trace mathematical understanding and mathematical practice?  When did we acquire the neurological circuitry for the sense perceptions and cognitive and linguistic capabilities on which mathematics depends?  Over the past 50 years (and especially the past 30), there have been an array of advances that have fundamentally changed what we know about our past and about the biological capacity for and cultural impulses behind cognitive precision (language, number sense, cultural transmission).  These questions will take us on a journey across multiple disciplines including archaeology, Assyriology, artifact analysis (close reading theory), anthropology, genomics, linguistics, neurobiology, and animal cognition.

In this article, we will proceed backwards in time, starting from c.3000 BCE just after the the birth of written mathematics in Sumeria (modern S.Iraq).  From here we will move to the prehistoric evidence of practitioner geometry in the cultures of the late Neolithic as evidenced in the layout of permanent houses, granaries and temples (c.5000 BCE in Ubaid culture), and in geometric pottery designs (c.6000 BCE, Halaf and Samarran cultures).  Further back, we see the appearance of plain tokens from c.7000-8000 BCE, the same plain tokens that we know were used for counting by herders and which were instrumental in the invention of writing for book-keeping purposes within the temple institutions running centralized economic control in the urban city-states.  Looking beyond the Near East, in Paeleolithic/Neolithic Europe and Britain there is evidence for monolithic monuments c.4000-7000 BCE oriented toward midwinter and midsummer solstice that suggest an awareness of the periodicity of the solar and lunar cycles, and the relation of the solar cycle to the seasons. 

Before 10,000 BCE, before the retreat of the last ice age, the density of artifactual evidence is insufficient to draw firm conclusions: there are fewer than 5 isolated finds of artifacts between 70,000-18,000 BCE (one find per 10,000 years), with contested interpretations.  So we switch to indirect evidence (genetic, anthropologic, and linguistic) to establish the capability of symbolic thought in anatomically modern humans (H. Sapiens) from c.200,000 BCE onwards.  Here culture becomes a critical factor, under-scored by the fascinating example of the modern Piraha tribe in Brazil that have broken previous assumptions about the inevitability of symbolic thought in anatomically modern humans.

From anatomically modern humans c.200,000, we jump backwards to c.2.4 million years ago and consider the capacity for conceptual thinking implicit in the tool-making capability of early hominids.  We look at C.S. Peirce's of conceptual and linguistic development, and conclude that bladed tool-making (Lokalalei site) required at least stage 2 or stage 3 conceptual development.  Having gone back as far as we can with the capabilities of humans and hominids, we consider the origin of number sense in humans, animals, birds, and reptiles, and trace back the neurological circuitry supporting an analogue number sense to a latest common ancestor (LCA), a stem reptile that would have existed some 260 million years ago.

Sídla, ktoré spĺňajú všetky atribúty v mnohých prípadoch aj súčasného mesta vrátane jeho správy a byrokracie, sa jednoznačne a plnohodnotne vyformovali aj počas existencie sumerskej civilizácie v dolnom povodí riek Eufrat a Tigris. V tých časoch začali byť neoddeliteľnou súčasťou riadenia a existencie mesta aj písomné, úradné záznamy.

The working group at Hirschbach met to address a set of comparative historical questions about how ancient labor (often, a rubric for mass or forced labor) was compelled or procured in ancient economies. This paper will interrogate some of the premises for those questions themselves, by way of an econometric exercise. What are the grounds for assuming that coercion or inequality (political, social, or economic) should be the determining theoretical problems for research ? Are either social and legal degrees of "unfreedom" or political and economic inequality the crucial explanatory forms to seek out and examine ? In what other contexts of political economy could ancient mass labor be understood? What tools can we use to think about the largest possible formal representations of the ancient economy, given the massive metho dological and documentary challenges presented by the sources? * This essay was helped immeasurably by advice and input over the years from

All contributions, manuscripts and books for review should be addressed to director of editing SUMER (recently) Jacob Jawdat State Board of Antiquities and Heritage, Baghdad. The preferred method of submission is an MS Word attachment with accompanying PDF file in an email to the editor as contact information below. Articles may be written in Arabic, English, French or German. The maximum length of the manuscript is 10,000 words, excluding the figures and bibliography, font size 12 for text and 10 for footnote (Times New Roman). The article should be consist on abstract, introduction and conclusions in addition to keywords. As well the figures should be suitable for length of article with resolution at least (300 dpi). Concerning in the cuneiform studies, all fonts used should be included. For the transliteration of cuneiform characters, please use the system established in Rykle Borger, Assyrisch-babylonische Zeichenliste. Sumerian words in English context should be set in capitalized and Akkadian should be italicized wherever it appears, depending on the font used. Footnotes should be numbered consecutively and should appear at the end of the page. Each author will receive a pdf of his or her article. The responsibility for the views expressed rests with the individual author(s). Abbreviations Standard abbreviations should follow the usage of AHw, CAD or CDLI; any other abbreviations should be explained in the first footnote. When citing a specific text in an edition, abbreviated journal, book, or series titles (e.g., MAD, MSL, OIP) should be italicized; catalog numbers (e.g., Msk, RS, TM) are set in roman. Full citations must be provided, even for well-known works. Authors may use author-date citations (e.g., Smith 2009: 83), in which case a reference list (including publisher and city) must be appended to the article. Alternatively. References The bibliography at the end of the article should include all the publications and websites referred to in the article. The Social Sciences format may be used for footnotes and bibliography thus: For footnotes, Smith 2015: 75, and then full details in bibliography.

The Author has explored, identified, and concluded the origin of our civilization and presented his stunning findings, largely original and unprecedented, by means of extensive multi-discipline studies of archaic Sumerians and neolithic Chinese. The written language, numerology, and mathematics, metrology, calendar, astronomy and astrology, wheeling, a stratified society with specialized professions and centralized government, etc. first emerged and matured in a special neolithic settlement at the northern latitude of 36 degrees where millet was cultivated, a crop requiring hot temperatures however with a brief planting season. From this river valley plain, its neolithic civilization spread to the Euphrates and Tigris Rivers valley, to the Nile River valley, and the Indus River valley. This newly identified neolithic civilization (5400BC) is the earliest and predates the Mesopotamia Sumerians (ca3500BC) by two thousand years. And it is the neolithic Yellow River downstream of China where the Mesopotamia civilization originates including the written language.