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“Ennum Eluththum kanEnath thakum” and “Enneluththikalel” Avvaiyar emphasises the importance of "Numbers" and "Writing" to humans in this prologue ("Moothurai"). Scholars have remarked that writing is the best of many arts in the world and that learning other arts is difficult without knowing "numbers," which is what keeps people's lives running smoothly. (Kanak, pg-1). Only a few Tamil mathematical works were written by Tamil poets between the 18th and 20th centuries. Among those math books "KanithaDeepikai, Balakanitham, Ensuvadi, Asthana Kolakalam, Ensuvadi, EnVilakkam (Numerology)" and others. According to Tamil historians, before this, there were seven books titled "Erambam, Kilaralapam, Adisharam, Kalambagam, ThiribhuvanathThilakam, Kanikarathinam, and Siruganakku" that were burnt. (Kanak, pg-2) One party also claims that after the 18th century, particularly in the last century, Tamil experts showed little interest in developing scientific Tamil. "Kanakkathikaram" specifies six categories of accounts. Historians believe the "Kanakkathikaram" belongs to the 15th century. Quantities and mathematical riddles employed in a book from 500 to 600 years ago are still utilized even today. For example, the word saree (Pudavai) is a measurement, but now we consider saree (Pudavai) as an object. A saree (Pudavai) is 18 cubits ("Muzham") in length. Did the saree (Pudavai) receive its name because of its size? Alternatively, it is debatable if the size of the saree (Pudavai) was included in the math because of the length of the saree. Likewise the fabric is known as "Vetti" in a similar way. There is a custom in our villages of "putting crore of cloth (Kodithunipoduthal)" on funeral rituals; I assumed a "crore of cloth" meant one crore of cloth, but a "crore of cloth (kodithuni)" is also a measurement of cloth. One "crore of cloth" equals 20 "vetti" or 640 "saree (Pudavai)". Similarly, my mother still measures grains and beans using "padi and koththu" even now. This article is a research consisting of the investigation into these metrics as well as the mathematical puzzles listed in the "Kanakkathikaram."

The development of script-using ancient civilizations and their achievements in sciences such as astronomy and mathematics have been well researched. Illiterate cultures were able to attain an adequate level of knowledge and transferred this knowledge to succeeding generations via oral tradition and mnemonic artefacts. The constructions of the square ditched enclosures (Viereckschanzen) of the last phase of the La Tène period (LT D, 150-1 BC) are documentations of the mathematical knowledge of this culture and served as a kind of mnemonic artefact. The constructions comprise the application of Pythagorean theorem (in terms of Pythagorean triangles) and square-root approximation triangles. Via these types of triangles, convex quadrangles are constructed (i.e., trapezium, parallelogram, rectangle, kite, lozenge, and square). The constructions were laid out in the terrain on the basis of a consistent 'Babylonian/Egyptian' metrology. [ABSTRACT FROM AUTHOR]

This paper aims to provide an updated synthesis on the works of Archimedes and the fundamental impact these have had on subsequent mathematical practice. The influence his mathematical processes have had on modern mathematics and how these have helped develop the field is discussed in historical perspective. Some of the recent investigations into the Archimedes Palimpsest are discussed and synthesized, namely, how they alter our understanding of some of his earlier works, and how Archimedean principles are seen to have laid the foundations of possible new branches of mathematics. [ABSTRACT FROM AUTHOR]

We posit that, for many ancient thinkers, mathematical allegory was a fundamental theoretical construct in their understanding of the universe. The procedure is to establish a noteworthy mathematical structure which suggests certain physical or social applications (interpretations). We illustrate the method in regard to the Oedipus myth. Thus, we use the practice of measuring with stretched ropes to elaborate, via Pythagorean number theory, an alternative classification of triangles to that of Euclid and show how to make a perspicuous geometric representation of triangles so classified. The representation obtained is seen to be a primitive astral map, whose salient features, especially when considered in light of the macrocosm/microcosm analogy, suggest to the poetic imagination details of the Oedipus myth. Oedipus’ destiny would have been determined by casting lots on the astral map. Apparently his lot fell where “three roads meet,” which can then be interpreted as showing that the Oedipus myth (as well as Sophocles play Oedipus Rex) was understood to be a version of the Green Child myth.

The Digital Euclid project aims to publish an open, digital edition of every extant witness to the text and diagrams of Euclid’s Elements. This paper discusses the required groundwork and is divided in two parts. It first covers a survey of the surviving manuscript and print sources for the Elements that intends to identify the extent of these materials, how many of these works have already been digitally imaged, and what challenges they pose to current data extraction methods. The latter part of the paper discusses the methods used to produce machine-actionable texts and diagrams and focuses especially on the development of tools for the identification and extraction of diagrammatic data.

In this paper I discuss different approaches to past mathematical texts. The question I address is: should we stress the continuity of past mathematics with the mathematics practiced today, or should we emphasize its difference, namely what makes it a product of a distant mathematical culture? [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Ancient mathematical formulae can be directly applied to the optimization of the algebraic computation. A new algorithm used to compute decimals of the inverse based on such ancient mathematics is reported in this paper. Sahayaks (auxiliary fraction) sutra has been used for the hardware implementation of the decimals of the inverse. On account of the ancient formulae, reciprocal approximation of numbers can generate "on the fly" either the first exact n decimal of inverse, n being either arbitrary large or at least > 6 in almost all cases. The reported algorithm has been implemented, and functionality has been checked in T-Spice. Performance parameters, like propagation delay and dynamic switching power consumptions, are calculated through spice-spectre of 90 nm CMOS technology. The propagation delay of the resulting 4-digit reciprocal approximation algorithm was only ~ 1.8 uS and consumed ~ 24.7 mW power. The implementation methodology offered substantial reduction of propagation delay and dynamic switching power consumption from its counterpart (NR) based implementation. [ABSTRACT FROM AUTHOR]

Andrej I. Schetnikov (ΣΙΓΜΑ: The Centre for Educational Projects, Novosibirsk), on the basis of testimonies by Nicomachus of Gerasa, Theon of Smyrna and other later authors, reconstructs an algorithm of developing all the numerals (provisionally called the ‘algorithm of Nicomachus’) and demonstrates how this algorithm could be found in the background of the so-called Plato’s ideal numbers. Besides, he suggests that the Platonicus by Erathosthenes, preserved in a fragmentary form by Theon, could be used as a supplementary source for reconstructing the content of the famous lecture of Plato On the Good.

Andrej Schetnikov, a historian of mathematics from Novosibirsk, discusses the problem of origins of the concepts of definitions and axioms in Ancient mathematics prior Euclid. Having noticed the fact that axioms and postulates are entirely absent from geometry in the time of Plato, the author turns to the problem of definitions in Plato and Aristotle instead and reads relevant passages from their works against the background of the “Elements” of Euclid. His working hypothesis is that the greater part of proofs from the first book of the “Elements” should be traced back to the time of Aristotle‘s school rather then Plato’s Academy despite the degree of sophistication of the mathematical technique in the time of the latter.

These classroom notes offer methods of solving the quadrature of lunes, that is, the area of croissant-shaped plane figures bounded by two intersecting non-congruent circular arcs, using Hippocrates of Chios’ area conservation and similarity arguments. I also offer a method of using history in the classroom with students via dynamic geometry snapshots presented in a manner that complements the analytic and the visual approaches. [ABSTRACT FROM PUBLISHER]

In this article, I try to show how the operation of compounding of ratios as performed with the method of Proposition VIII.4 of Euclid's Elements, has possibly taken shape from experiments on the monochord. This fact, together with the systematic and exclusive use of Proposition VIII.4 in the harmonic field, strengthens the thesis concerning the connection between harmonic theory and Book VIII. A misinterpretation by Zarlino, the most important music theorist of the Renaissance, will help us explain how the musical model may have been partially responsible for the disconnection between compounding and multiplication present in the Elements in spite of the link established by Proposition VIII.5. [ABSTRACT FROM AUTHOR]

Analysis of the symbols engraved on prehistoric unknown use terracotae, the so called frying pan vessels (Teganoschema), reveal a symbolic writing that depicts astronomical phenomena, that are complex calendars based on the Sun and the Moon and all then known planets. The frying pan vessels are mainly found in Cyclades, and around the Aegean, Crete, Attica, and Thessaly. They are artifacts of the Cycladic civilization of the Early Helladic Era. They have been found mainly in graves and settlements. The first findings came to light during the late nineteenth century in the islands of Cyclades and their possible use is still causing strong scientific interest, as it is unknown. Until now, archaeologists could not determine their use and the meaning of their representations. It was believed that these vessels were used in funeral rituals, therefore depicted patterns like the sun and the sea may be associated with beliefs about the afterlife. We have studied the morphology and the representations of the Cycladic frying pan vessels that are found in museums and in literature dating from the middle of the 4th millennium. In this paper we argue that the Cycladic frying pan contain calendars that are not only based on the periodicities of the Sun and Moon, but in a very advanced knowledge of the movements of the planets, their periodicities in relation to the Earth and the phases of Venus, which is used as calendar, as well as pregnancy and birth predictor calculator.. [ABSTRACT FROM AUTHOR]

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1-32, , and Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, ). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, ). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the 'conjugate rule' or a 'completion of the square,' but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore , Ch. 11 and App. 1). [ABSTRACT FROM AUTHOR]

This paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite different intuitive model (or frame), without infinite space, infinite lines and surfaces. This ancient intuitive model of Euclidean geometry is reconstructed in connection with Plato's unwritten doctrine. The model creates a kind of "hermeneutical horizon" determining the explicit content and mathematical methods used. In the first section of the paper, it is argued that there are no actually infinite concepts in Euclid's Elements. In the second section, it is argued that ancient mathematics is based on Plato's highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled. [ABSTRACT FROM AUTHOR]

Historians believe that the discovery of the irrationality of √2, or, as it would have been described in classical Greek mathematics, the incommensurability of the side and diagonal of a square, was made in the period 550–410 bc. None of the early sources, however, includes a proof of the theorem, so historians have found it a happy hunting ground for conjectures. This article is drawn from the monographs of Knorr and Fowler, and was put together to show how much history of early mathematics is from the minds of historians. [ABSTRACT FROM AUTHOR]

The article presents a biography of Dutch professor and mathematical historian Evert Marie Bruins. Among other topics addressed are his research specialty, namely ancient mathematics, his education, and his career. Among other specific topics of research discussed are works of Egyptian mathematician Aqatun and of Greek mathematician Euclid and Babylonian mathematics.

This article describes the editorial principles behind the transcriptions in both the digital and print components of the Archimedes Palimpsest publication. Care was taken in these transcriptions to record features of the text as written on the pages of the manuscript, such as scribal abbreviations, page layout, and diagrams. This approach was adopted with the understanding that the details about how a text is committed to physical form influence how it is read and interpreted over time. In this way it becomes possible for readers and scholars to access one further dimension of ancient mathematical thought and expression. The article is based on a talk given at a conference on ‘Editing Historical Mathematics’ in December 2011 in Oxford. [ABSTRACT FROM PUBLISHER]

Abstract: Wang Xiaotongʼs Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to cubic equations which are to be solved numerically by the Chinese variant of Hornerʼs method. The problems in solid geometry give the volume of a solid and certain constraints on its dimensions, and the dimensions are required; we translate and analyze four of these. Three are solved using dissections, while one is solved using reasoning about calculations with very little recourse to geometrical considerations. The problems in Wang Xiaotongʼs text cannot be seen as practical problems in themselves, but they introduce mathematical methods which would have been useful to administrators in organizing labor forces for public works. [Copyright &y& Elsevier]

Wang Xiaotong’s 王孝通 Jigu suanjing 緝古算經 is primarily concerned with problems in solid and plane geometry leading to polynomial equations which are to be solved numerically using a procedure similar to Horner’s Method. We translate and analyze here six problems in plane geometry. In each case the solution is derived using a dissection of a 3-dimensional object. We suggest an interpretation of one fragmentary comment which at first sight appears to refer to a dissection of a 4-dimensional object. [ABSTRACT FROM AUTHOR]

In this text, I present some notes about the reception of mathematical cuneiform texts in the historiography of mathematics, since the first systematic publications in the first half of the 20th century until the present. The first section introduces the subject, by analysing fields of study in which mathematical tablets can be taken into account; here I problematise the subject of Mesopotamian "mathematics" itself. The second section summarises the very beginnings of the publication of mathematical tablets and of the preparation for a discipline of the history of Mesopotamian mathematics. The next section briefly distinguishes three moments of the historiography of the field, by showing the manner through which the historiographic reception of cuneiform mathematics was conducted: interpreted by algebraic symbolism; afterwards, with a geometric reading; eventually, a period when researchers try to link mathematical texts to texts of other genres. It will be possible to see that these different strategies provide different interpretations for cuneiform mathematics. An appendix brings, as a complement, technical help with the aim of explaining better the distinction between the algebraic and geometrical historiographic approaches. [ABSTRACT FROM AUTHOR]

The author discusses the earliest extant Chinese mathematical treatises, Jiu zhang suan shu ... (Computational procedures of nine categories), compiled no later than the first century AD, and three recently unearthed mathematical texts, Suan shu shu ... (Scripture on computations [to be performed] with counting rods), Shu ... (Numerical [procedures (?)]), and Suan shu ... ([Computational] Procedures [to be performed with] Counting Rods), all dated of the late first millennium BC. The main question addressed in the review is that of the origin and early history of the Jiu zhang suan shu. Some modern historians suggested that this treatise was considered a "canonical" text by its commentator of the third century, Liu Hui ... (fl. AD 263). The author suggests that this claim is based solely on one phrase found in the commentator's preface. Instead, the recent findings suggest that a number of mathematical texts were produced, copied and studied in China from the 3rd century BC onwards, and none of them had the status of a "canonical scripture"; these texts, the author suggests, most likely were used for educational purposes. [ABSTRACT FROM AUTHOR]

The article presents a professional biography of mathematician and historian of science Otto Neugebauer, focusing on his early work on the history of mathematics. Neugebauer's work on ancient Egyptian, Babylonian, and Greek mathematics is highlighted, and special attention is paid to his focus on the history of astronomy. Other topics include his rejection of hyper-specialization in the study of history, the scholarly reception of his work, and historians of the early 20th century.

Recently a belief has spread that the set of five Platonic Solids has been known since prehistoric times, in the form of carved stone balls from Scotland, dating from the Neolithic period. A photograph of a group of these objects has even been claimed to show mathematical understanding of the regular solids, a millennium or so before Plato. I argue that this is not so. The archaeological and statistical evidence do not support this idea, and it has been shown that there are problems with the photograph. The high symmetry of many of these objects can readily be explained without supposing any particular mathematical understanding on the part of the creators, and there seems to be no reason to doubt that the discovery of the set of five regular solids is contemporary with Plato. [ABSTRACT FROM AUTHOR]

An essay is presented on the role of mathematician Otto Neugebauer in the exportation of the Göttingen approach to the history of mathematics, as well as his ties with fellow mathematician Richard Courant. It says that Neugebauer is among the mathematicians who displayed deep interest in ancient mathematics and astronomy. The author mentions that Courant perceived that similar fields of intellectual endeavor are closely associated with each other.

This article outlines the general aims of my current research on numeracy in ancient Greece and Rome. I survey the available evidence, which is both textual and archaeological, and briefly discuss some of the main questions which will guide my exploration of this wide field. I conclude by presenting the results of some preliminary examination of two case-studies: account inscriptions from fifth- and fourth-century BC Athens, and accounts in the late Roman Republic, through the eyes of Cicero. The article is a slightly modified version of a talk presented at Gresham College May 2011 as part of the BSHM day on Early Mathematics. [ABSTRACT FROM AUTHOR]

India has had a long tradition of more than 3000 years of pursuit of Mathematical ideas, starting from the Vedic age. The Sulvasutras (which included Pythagoras theorem before Pythagoras), the Jain works, the base 10 representation (along with the use of 0), names given to powers of 10 up to 10, the works of medieval mathematicians motivated by astronomical studies, and finally the contributions of the Kerala school that came strikingly close to modern mathematics, represent the various levels of intellectual attainment. [ABSTRACT FROM AUTHOR]

Abstract: ASIC design of a high speed low power circuit for factorial calculation of a number is reported in this paper. The factorial of a number can be calculated using iterative multiplication by incrementing or decrementing process and iterative multiplication can be computed through parallel implementation methodology. Parallel implementation along with Vedic multiplication methodology for calculation of factorial of a number ensures significant reduction in propagation delay and switching power consumption due to reduction of stages in multiplication process, in comparison with the conventionally used Vedic multiplication methodologies like ‘Urdhva-tiryakbyham’ (UT) and ‘Nikhilam Navatascaramam Dasatah’ (NND) based implementation methodology. Transistor level implementation was carried out using spice specter with standard 90nm CMOS technology and the results were compared with the above mentioned conventional methodologies. The propagation delay for the calculation of 4-bit factorial of a number was only ∼42.13ns while the power consumption of the same was ∼58.82mW for a layout area of ∼6mm2. Improvement in speed was found to be ∼33% and ∼24% while corresponding reduction of power consumption in ∼34.48% and ∼24% for the factorial calculation circuitry in comparison with UT and NND based implementations, respectively. [Copyright &y& Elsevier]

A bibliography of 2011 writings is presented on a variety of topics related to the history of science including prehistoric and ancient astronomy, cosmology, and mathematics which includes the books "Chasing Shadows: Mathematics, Astronomy, and the Early History of Eclipse Reckoning," by Clemency Montelle, "In the Path of the Moon: Babylonian Celestial Divination and Its Legacy," by Francesca Rochberg, number 6 in the book series "Studies in Ancient Magic and Divination," and "Eureka Man: The Life and Legacy of Archimedes," by Alan Hirshfeld.

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