Ganita Bharati : Bulletin of The Indian Society for History of Mathematics

ISSN: 0970-0307

e-ISSN: ******

Subject: Mathematics

Periodicity: Half Yearly

DOI: 10.32381/GB.

Month(s) of Publication: June & December

Current Volume: 43 (2021)

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Ganita Bharati, the Bulletin of the Indian Society for History of Mathematics is devoted to publication of significant original articles in history of Mathematics and related areas. Although English is the official language of the journal, an article of exceptional merit written in French, German, Sanskrit or Hindi will also be considered only as a special case.

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Volume 43 (2021) Issue 1 , (January-2021 to June-2021)

January-June

Peeping into Fibonacci’s Study Room
by Jens Hoyrup

Page: 1-70

Abstract: The following collects observations I made during the reading of Fibonacci’s Liber abbaci in connection with a larger project, “abbacus mathematics analyzed and situated historically between Fibonacci and Stifel”. It shows how attention to the details allow us to learn much about Fibonacci’s way to work. In many respects, it depends crucially upon the critical edition of the Liber abbaci prepared by Enrico Giusti and upon his separate edition of an earlier version of its chapter 12 – not least on the critical apparatus of both. This, and more than three decades of esteem and friendship, explain the dedication.

DOI-https://doi.org/10.32381/GB.2021.43.1.1

Treatment of ‘Very large number’ in Cyrillic Numeration by Dionisy I. Pronin

Page: 71-86

Abstract: The paper is dedicated to signs meaning ‘very large number’, that is, for 10,000 and higher, in Cyrillic. We discuss the manuscript ‘Arithmetics’ (Saint- Petersburg, Russian National Library, Titov. 2414) of XVII c. which contains a previously unknown term ‘kony’ and its sign. The data on ‘very large number’ in this and some other manuscripts probably represent the development of new terms and extension of the counting limit. Another manuscript ‘Arithmetics’ (Saint-Petersburg, Russian National Library, Q.IX.46) of mid. XVII c. contains description of three alternative systems of values of number called small number, middle number and great number. The last one, the great number had three different variants of values for terms. We distinguish among concepts of numeral-sign and numeral-term, and discuss differences between numeral-signs and signs with meaning ‘very large number; indeterminately large number’.

DOI-https://doi.org/10.32381/GB.2021.43.1.2

Some Recent Publications in History of Mathematics

Page: 87-92

Volume 42 (2020)

Volume 42 (2020) Issue 1-2 , (January-2020 to December-2020)

January-December

The Central Role of Incommensurability in Pre-Euclidean Greek Mathematics and Philosophy by Stelios Negrepontis, Vassiliki Farmaki and Marina Brokou

Page: 1-34

Abstract: In this paper we outline the tremendous impact that the Pythagorean discovery of incommensurability had on pre-Euclidean Greek Mathematics and Philosophy. This will be a consequence of our findings that the Pythagorean method of proof of incommensurability is anthyphairetic, namely depends on Proposition X.2 of the Elements, according to which if the anthyphairesis of two line segments is infinite, then they are incommensurable.

Our fundamental finding is that the main entity of Plato’s philosophy, the intelligible Being, is a philosophical analogue/imitation of a dyad in periodic anthyphairesis.

One byproduct of our deeper and mathematical understanding of Plato’s philosophy is that we can next show (a) that Plato’s intelligible Beings coincide with the earlier Zeno’s true Beings, and (b) that the purpose of Zeno’s arguments and most exciting paradoxes is not to deny motion or multiplicity, as usually thought, but to separate the true Beings from the sensible entities of opinion.

Although Plato’s early/middle work is greatly influenced by the Pythagoreans and Zeno, in his late work he employed via philosophical imitation, the stunning discovery of the great Athenian mathematician Theaetetus, namely the palindromic periodicity theorem for quadratic incommensurabilities (established in modern era by Lagrange and Euler).

The study of incommensurability via periodic anthyphairesis produced great Mathematics and great Philosophy; however this approach could only deal with quadratic, and did not extend to solid incommensurabilities. Archytas and Eudoxus marked the beginning of a new, non-anthyphairetic era for incommensurability.

In one way or another, the Greek Mathematics (Pythagoreans, Theodorus, Theaetetus, Archytas, Eudoxus) and Philosophy (Pythagoreans, Zeno, Plato) of the pre-Euclidean era were dominated by the Pythagorean discovery of incommensurability.

DOI-https://doi.org/10.32381/GB.2020.42.1-2.1

Some Magic and Latin Squares and the Bhuvanesvari and Other Bimagic Squares by R. C. Gupta

Page: 35-54

Abstract: The nine Indian planetary magic squares of order 3 are attributed to Garga, who is said to belong to the hoary past. Formation of magic squares of order 9 from those of order 3, as bimagic squares, is found both in India and China. BhuvaneoevarÁ yantra is a bimagic square of order 8. Its full Sanskrit text, along with translation and the method of construction is described in the present paper. Construction of bimagic squares of orders 8 and 9 from simple orthogonal Latin squares is dealt with in detail. Recalling that Euler’s conjecture on Latin squares has been disproved, a counter-example of order 10 is described.

DOI-https://doi.org/10.32381/GB.2020.42.1-2.2

Fifteenth-century Italian symbolic algebraic calculation with four and five unknowns by Jens Hoyrup

Page: 55-86

Abstract: The present article continues an earlier analysis of occurrences of two algebraic unknowns in the writings of Fibonacci, Antonio de’ Mazzinghi, an anonymous Florentine abbacus writer from around 1400, Benedetto da Firenze and another anonymous Florentine writing some five years before Benedetto, and Luca Pacioli. Here I investigate how in 1463 Benedetto explores the use of four or five algebraic unknowns in symbolic calculations, describing it afterwards in rhetorical algebra; in this way he thus provides a complete parallel to what was so far only known (but rarely noticed) from Michael Stifel’s Arithmetica integra (1544) and Johannes Buteo’s Logistica (1559). It also discusses why Benedetto may have seen his innovation as a merely marginal improvement compared to techniques known from Fibonacci’s Liber abbaci, therefore failing to make explicit that he has created something new.

DOI-https://doi.org/10.32381/GB.2020.42.1-2.3

Clairaut, Euler and the Figure of the Earth by Athanase Papadopoulos

Page: 87-127

Abstract: The sphericity of the form of the Earth was questioned around the year 1687, primarily, by Isaac Newton who deduced from his theory of universal gravitation that the Earth has the form of a spheroid flattened at the poles and elongated at the equator. In France, some preeminent geographers were not convinced by Newton’s arguments, and about the same period, based on empirical measurements, they emitted another theory, claiming that on the contrary, the Earth has the form of a spheroid flattened at the equator and elongated at the poles. To find the real figure of the Earth became one of the major questions that were investigated by geographers, astronomers, mathematicians and other scientists in the 18th century, and the work done around this question had an impact on the development of all these fields.

In this paper, we review the work of the 18th-century French mathematician, astronomer and geographer Alexis-Claude Clairaut related to the question of the figure of the Earth. We report on the relation between this work and that of Leonhard Euler. At the same time, we comment on the impact of the question of the figure of the Earth on mathematics, astronomy and hydrostatics. Finally, we review some later mathematical developments that are due to various authors that were motivated by this question. It is interesting to see how a question on geography had such an impact on the theoretical sciences.

DOI-https://doi.org/10.32381/GB.2020.42.1-2.4

Apodictic discourse and the Cauchy-Bunyakovsky-Schwarz inequality by Satyanad Kichenassamy

Page: 129-147

Abstract: Bunyakovsky’s integral inequality (1859) is one of the familiar tools of modern analysis. We try and understand what Bunyakovsky did, why he did it, why others did not follow the same path, and explore some of the mathematical (re)interpretations of his inequalities. This is achieved by treating the texts as discourses that provide motivation and proofs by their very discursive structure, in addition to what meets the eye at first reading. Bunyakovsky paper is an outgrowth of the mathematical theory of mean values in Cauchy’s work (1821), but viewed from the point of view of Probability and Statistics. Liouville (1836) gave a result that implies Bunyakovsky’s inequality, but did not identify it as significant because his interests lay elsewhere. Grassmann (1862) stated the inequality in abstract form but did not prove it for reasons that can be identified. Finally, by relating the result to quadratic binary forms, Schwarz (1885) opened the way to a geometric interpretation of the inequality that became important in the theory of integral equations. His argument is the source of one of the proofs most commonly taught nowadays. At about the same time, the Rogers- Hölder inequality suggested generalizations of Cauchy’s and Bunyakovsky’s results in an entirely different direction. Later extensions and reinterpretations show that no single result, even now, subsumes all known generalizations.

AMS classification (MSC 2010): 01A65, 01A55, 01A65, 26D15, 46C05, 62-03

DOI-https://doi.org/10.32381/GB.2020.42.1-2.5

Department of Mathematics at Banaras Hindu University: A history, circa 1916-1950 by Ritesh Gupta

Page: 149-173

Abstract: A historical study of Science Colleges and their constituting departments and disciplines, viz. Mathematics, Physics, Chemistry, Zoology, Botany, et cetera established in early universities could bring to light new facts and values to the history of science in modern India. However, not much scholarship has catered to the institutional histories of Science Colleges established in the late nineteenth and early twentieth centuries. Survey and scrutiny of institutionalization of modern sciences and mathematics in Indian universities have remained rather neglected. Therefore, the present paper explores the early history of the Banaras Hindu University’s (B.H.U.) Mathematics Department. It lists out the first-generation mathematicians of the university, their education and training, the national and international collaborations, research, and scientific publications.

DOI-https://doi.org/10.32381/GB.2020.42.1-2.6

Book Reviews
Ganitagannadi, (Mirror of Mathematics) – An astronomy text of 1604 CE in Kannada by Sankaranarayana Joisaru of Srngeri by B.S. Shylaja and Seetharama Javagal

Reviewed by M.S. Sriram

Page: 175-181

A Primer to Bharatiya Ganitam; Bharatiya-Ganita-Pravesa by M.D. Srinivas (Editor), and Authors: V. Ramakalyani, M.V. Mohana, R.S. Venkatakrishna and N. Kartika

Reviewed by Avinash Sathaye

Page: 182-186

Some recent publications in History of Mathematics

Page: 187-196

Volume 41 (2019)

Volume 41 (2019) Issue 1-2 , (January-2019 to December-2019)

January-December

Brahmagupta’s Apodictic Discourse by Satyanad Kichenassamy

Page: 1-21

Abstract: We continue our analysis of Brahmagupta’s Brahmasphutasiddhanta (India, 628), that had shown that each of his sequences of propositions should be read as an apodictic discourse: a connected discourse that develops the natural consequences of explicitly stated assumptions, within a particular conceptual framework. As a consequence, we established that Brahmagupta did provide a derivation of his results on the cyclic quadrilateral. We analyze here, on the basis of the same principles, further problematic passages in Brahmagupta’s magnum opus, regarding number theory and algebra. They make no sense as sets of rules. They become clear as soon as one reads them as an apodictic discourse, so carefully composed that they leave little room for interpretation. In particular, we show that (i) Brahmagupta indicated the principle of the derivation of the solution of linear congruences (the kuttaka) at the end of chapter 12 and (ii) his algebra in several variables is the result of the extension of operations on numbers to new types of quantities – negative numbers, surds and “non-manifest” variables.

AMS classification (MSC 2010): 01A32, 01A35, 11-03, 11A05, 51-03.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.1

Reinventing or Borrowing Hot Water? Early Latin and Tuscan Algebraic Operations with Two Unknowns by Jens Hoyrup

Page: 23-67

Abstract: In mature symbolic algebra, from Viète onward, the handling of several algebraic unknowns was routine. Before Luca Pacioli, on the other hand, the simultaneous manipulation of three algebraic unknowns was absent from European algebra and the use of two unknowns so infrequent that it has rarely been observed and never analyzed.

The present paper analyzes the five occurrences of two algebraic unknowns in Fibonacci’s writings; the gradual unfolding of the idea in Antonio de’ Mazzinghi’s Fioretti; the distorted use in an anonymous Florentine algebra from ca 1400; the regular appearance in the treatises of Benedetto da Firenze; and finally what little we find in Pacioli’s Perugia manuscript and in his Summa. It asks which of these appearances of the technique can be counted as independent rediscoveries of an idea present since long in Sanskrit and Arabic mathematics – metaphorically, to which extent they represent reinvention of the hot water already available on the cooker in the neighbour’s kitchen; and it raises the question why the technique once it had been discovered was not cultivated – pointing to the line diagrams used by Fibonacci as a technique that was as efficient as rhetorical algebra handling two unknowns and much less cumbersome, at least until symbolic algebra developed, and as long as the most demanding problems with which algebra was confronted remained the traditional recreational challenges.

AMS classification (MSC 2010): 01A32, 01A35, 11-03, 11A05, 51-03.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.2

Nearest-Integer Continued Fractions in Drkkaraoa
by Venketeswara Pai R. and M. S. Sriram

Page: 69-89

Abstract: The Karaõa texts of Indian astronomy give simplified expressions for the mean rates of motion of planets. The Kerala text Karaõapaddhati (c. 1532-1566 CE) expresses these rates which involve ratios of large numerators or multipliers (guõakras) and large demominators or divisors (hÀrakas), as ratios of smaller numbers using essentially the method of simple continued fraction expansion. A modified version of this method is described in a slightly later Malayalam text named DÃkkaraõa (c. 1608 CE), also. A very interesting feature of the DÃkkaraõa algorithm is that a nearest-integer continued fraction expansion with the minimal length is implicit in it. We discuss this algorithm in this paper.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.3

Mathematics and Map Drawing in the Eighteenth Century by Athanase Papadopoulos

Page: 91-126

Abstract: We consider the mathematical theory of geographical maps, with an emphasis on the eighteenth century works of Euler, Lagrange and Delisle. This period is characterized by the frequent use of maps that are no more obtained by the stereographic projection or its variations, but by much more general maps from the sphere to the plane. More especially, the characteristics of the desired geographical maps were formulated in terms of an appropriate choice of the images of the parallels and meridians, and the mathematical properties required by the map concern the distortion of the maps restricted to these lines. The paper also contains some notes on the general use of mathematical methods in cartography in Greek Antiquity, and on the mutual influence of the two fields, mathematics and geography.

AMS Mathematics Subject Classiflcation: 01A55, 30C20, 53A05, 53A30, 91D20.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.4

On the Contribution of Anders Johan Lexell in Spherical Geometry by A. Zhukova

Page: 127-149

Abstract: In this paper, we discuss results in spherical geometry that were obtained by a remarkable mathematician of the XVIIIth century, Anders Johan Lexell. We also present a short note on the place of these results in the history of this field as well as a short biography of Lexell.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.5

Magic Squares and Other Numerical Diagrams on the Chittagong Plaster Replicas in the David Eugene Smith Collection by Takao Hayashi

Page: 151-180

Abstract: The Rare Book and Manuscript Library of Columbia University has a set of 20 plaster replicas that D. E. Smith brought from Chittagong in 1907 CE. They are twin replicas of 10 stone slabs. Most of the replicas show one or a few numerical diagrams including magic squares. In this paper I analyze them and discuss their construction methods.

DOI -https://doi.org/10.32381/GB.2019.41.1-2.6

Book Reviews
Scholars and Scholarship in Late Babylonian Uruk by Christine Proust and John M. Steel Reviewed by Jens Hoyrup

Vedic Mathematics and Science in the Vedas by S Balachandra Rao Reviewed by Shailaja D. Sharma

Page: 181-196

Volume 40 (2018)

Volume 40 (2018) Issue 2 , (July-2018 to December-2018)

Jul-Dec. 2018

Katyayana Sulvasutra : Some Observations by S. G. Dani

Page: 101-114

Abstract: The Katyayana SulvasÂtra has been much less studied or discussed from a modern perspective, even though the first English translation of two adhyayas (chapters) from it, by Thibaut, appeared as far back as 1882. Part of the reason for this seems to be that the general approach to the Sulvasatra studies has been focussed on “the mathematical knowledge found in them (as a totality)”; as the other earlier Sulvasatras, especially of Baudhayana and Apastamba substantially cover the ground in this respect, the other two Sulvasatras, Manava and Katyayana, received much less attention, the latter especially so. On the other hand the broader purpose of historical mathematical studies extends far beyond cataloguing what was known in various cultures, rather to understand the ethos of the respective times from a mathematical point of view, in their own setting, in order to evolve a more complete picture of the mathematical developments, ups as well as downs, over history. Viewed from this angle, a closer look at Katyayana Sulvasatra assumes significance. Coming at the tail-end of the Sulvasatras period, after which the Sulvasatras tradition died down due to various historical reasons that are really only partly understood, makes it special in certain ways. What it omits to mention from the body of knowledge found in the earlier Sulvasatras would also be of relevance to analyse in this context, as much as what it chooses to record. Other aspects such as the difference in language, style, would also reflect on the context. It is the purpose here to explore this direction of inquiry.

DOI-https://doi.org/10.32381/GB.2018.40.02.1

Essay Review: On the Interpretations of the History of Diophantine Analysis: A Comparative Study of Alternate Perspectives
by Ioannis Vandoulakis

Page: 115-152

Abstract: This is a review of the following two books, in particular comparing them with relevant works of I.G.Bashmakova on the topic. Les Arithmétiques de Diophante : Lecture historique et mathématique, par Roshdi Rashed en collaboration avec Christian Houzel, Berlin, New York : Walter de Gruyter, 2013, IX-629 p. Histoire de l’analyse diophantienne classique : D’Abu Kamil à Fermat, par Roshdi Rashed, Berlin, New York : Walter de Gruyter, 2013, X-349 p.

DOI-https://doi.org/10.32381/GB.2018.40.02.2

Nasir al-Din al-Tusi’s Treatise on the Quadrilateral: The Art of Being Exhaustive by Athanase Papadopoulos

Page: 153-180

Abstract: We comment on some combinatorial aspects of Nasir al-Din al-Tusi’s Treatise on the Quadrilateral, a 13th century work on spherical trigonometry. AMS classification: 01A20, 01A30, 01A35.

DOI-https://doi.org/10.32381/GB.2018.40.02.3

Book Reviews
The Mathematics of India : Concepts, Methods, Connections by P. P. Divakaran
Reviewed by Satyanad Kichenassamy
Karaõapaddhati of Putumana Somayaji with translation and explanatory notes by Venketeswara Pai, K. Ramasubramanian, M.S. Sriram and M.D. Srinivas
Reviewed by S.G. Dani and Clemency Montelle

Page: 181-198

Volume 40 (2018) Issue 1 , (January-2018 to June-2018)

January-June

T. A. Sarasvati Amma: A Centennial Tribute by P. P. Divakaran

Page: 1-16

Abstract: Sarasvati Amma published very few research papers. All her insights into the Indian mathematical (specifically, geometric) tradition are to be found in her book “Geometry in Ancient and Medieval India”, published in 1979 but prepared as her thesis in the University of Madras 20 years earlier. The present article is, consequently, an evaluation of the mathematics described in the book and of the historiographic significance of its interpretation by her. The book pays specific attention to certain themes: e.g., the key ideas of the geometry of the Vedic period, cyclic quadrilaterals, geometric algebra etc. and, especially, the infinitesimal trigonometry of MÀdhava, all in a style designed to bring out the continuity in their evolution. The case is made in this article that Sarasvati Amma’s work, along with the earlier book of B. Datta and A. N. Singh, marks the founding of an autonomous discipline of scholarship into India’s mathematical past.

DOI-https://doi.org/10.32381/GB.2018.40.01.1

The Seminal Contribution of K. S. Shukla to our Understanding of Indian Astronomy and Mathematics by M. D. Srinivas

Page: 17-51

Abstract: In this article we shall highlight some of the important contributions to the study of Indian astronomy and mathematics made by Prof. Kripa Shankar Shukla (1918 - 2007), on the occasion of his birth centenary. Shukla was a student of Prof. A. N. Singh (1905 - 1954) at Lucknow University and was also fortunate to have come in close contact with Prof. Singh’s renowned collaborator Bibhutibhusan Datta (1888-1958). Dr. Shukla became the worthy successor of Prof. Singh to lead the research programme on Indian astronomy and mathematics at Lucknow University.

Prof. Shukla brought out landmark editions of twelve important source-works of Indian astronomy and mathematics. A remarkable feature of many of these editions is that they also include lucid English translations and detailed explanatory notes. This is indeed one of the greatest contributions of Prof. Shukla since, till the 1960s, there had been very few editions of the classical source-works of Indian astronomy which also included a translation as well as explanatory notes. The editions of Shukla have become standard textbooks for the study of development of Indian astronomy during the classical SiddhÀntic period from °ryabhaÇa to OErÁpati.

Prof. Shukla also wrote over forty important research articles, which have ushered in an entirely new perspective on the historiography of Indian astronomy and mathematics. We shall discuss in some detail the following seminal contributions of Prof. Shukla:

(i) Correct explanation of the manda-saÉskÀra (equation of centre) in Indian astronomy, including the computation of the avioeiÈÇa-mandakarõa (iterated manda-hypotenuse) and its significance.

(ii) Correction of the faulty readings and translations of some of the crucial verses giving the number of civil days and other parameters of a yuga, as presented in the 1978 edition of YavanajÀtaka by David Pingree.

(iii) Publication of a revised and updated version of Part III of the ‘History of Hindu Mathematics’ by B. B. Datta and A. N. Singh.

DOI-https://doi.org/10.32381/GB.2018.40.01.2

On Old Babylonian Mathematical Terminology and its Transformations in the Mathematics of Later Periods by Jens HØyrup

Page: 53-99

Abstract: Third-millennium (BCE) Mesopotamian mathematics seems to have possessed a very restricted technical terminology. However, with the sudden flourishing of supra-utilitarian mathematics during the Old Babylonian period, in particular its second half (1800–1600 BCE) a rich terminology unfolds. This mostly concerns terms for operations and for definition of a problem format, but names for mathematical objects, for tools, and for methods or tricks can also be identified. In particular the terms for operations and the way to structure problems turn out to allow distinction between single localities or even schools. After the end of the Old Babylonian period, the richness of the terminology is strongly reduced, as is the number of known mathematical texts, but it presents us with survival as well as innovations.

Apart from analyzing the terminology synchronically and diachronically, the article looks at two long-lived non-linguistic mathematical practices that can be identified through the varying ways they are spoken about: the use of some kind of calculating board, and a way to construct the perimeter of a circle without calculating it – the former at least in use from the 26th to the 5th century BCE, the later from no later than Old Babylonian times and surviving until the European 15th century CE.

DOI-https://doi.org/10.32381/GB.2018.40.01.3

Volume 39 (2017)

Volume 39 (2017) Issue 2 , (July-2017 to December-2017)

Jul-Dec

An Indian Version of al-Kashi’s Method of Iterative Approximation of sin 1° by Kim Plofker

Page: 95-106

Abstract: The well-known “feedback loop” of trigonometry of sines, from its origin in Indian astronomy to the Islamic world in the first millennium CE and back to India in the mid-second, includes many interesting and under-studied developments. This paper examines a Sanskrit adaptation and refinement of a medieval method foRsine approximation, apparently from the court of Jai Singh in the early 18th century.

Nilakantha Critique on Aryabhata’s Verses on Squaring and Square-roots 107-124 by N. K. Sundareswaran

Page: 107-124

Abstract: Nilakantha commentary on Aryabhata's is well known for clarity and simplicity of language and for its expository nature. He goes on clarifying all the possible doubts. The way in which he formulates and interconnects ideas is simply beautiful. At times his commentary on a particular point runs into pages. But it would be a pleasure to read it, for, the language and style of argument are the same as in a polemical text of philosophy.

This paper makes a close study of the commentary on the fourth verse of Ganitapada, wherein Nilakantha explains the method for finding the square root of a number, focusing on the development of ideas and the thought process. Here Nilakantha deals, at length, with many of the rationales and the concepts involved.

The explanation given by Nilakantha for Baudhayana’s approximation of is unique. It is a fine specimen of geometrical demonstration of arithmetical ideas, a significant trend of medieval school of Kerala mathematics.

Sign and Reference in Greek Mathematics by Ioannis M. Vadoulakis

Page: 125-145

Abstract: In this paper, we will examine some modes of reference to mathematical entities used in Greek mathematical texts. In particular, we examine mathematical texts from the Early Greek period, the Euclidean, Neo-Pythagorean, and Diophantine traditions.

On the History of Analysis -The Formation of Concepts by G. Sinkevich

Page: 147-162

Abstract: Mathematical analysis was conceived in XVII century in the works of Newton and Leibniz. The issue of logical rigor in definitions was however first considered by Arnauld and Nicole in ‘’Logique ou l’art de penser’’. They were the first to distinguish between the bulk of the concept and its structure. They created a tradition which was strong in mathematics till XIX century, especially in France. The definitions were in binomial nomenclature mostly, but another type of definition appears in Cantor theory – it was the descriptive definition. As it used to be in humanities, first the object had only one characteristic, then as research continued it got enriched with new characteristics leading to a fledged concept. In this way mathematics acquired its own creativity. In 1915 Luzin laid down a new principle of the descriptive theory: a structural characteristic is done, the analytical form had to be found. New schools of descriptive set theory appeared in Moscow in the first half of the 20th century.

Book Reviews :
Mathematics in Ancient Egypt: A Contextual History by Annette Imhausen Reviewed by Jens Hoyrup
The Siddhhantasundara of Jnanaraja by Toke Lindegaard Knudsen Reviewed by Surabhi Saccidananda

Page: 163-193

News

Page: 195-197

Volume 39 (2017) Issue 1 , (January-2017 to June-2017)

Jan-Jun

Archimedes – Knowledge and Lore from Latin Antiquity to the Outgoing European Renaissance by Jens Hoyrup

Page: 1-21

On the History of Nested Intervals: From Archimedes to Cantor by G. I. Sinkevich

Page: 23-45

Explanation of the Vakyasodhana procedure for the Candravakyas by M. S. Sriram

Page: 47-53

Madhyahnakalalagna in Karanapaddhati of Putumana Somayaji by Venketeswara Pai R. & M.S. Sriram

Page: 55-74

Vedic Mathematics and Science in the Vedas (in Kannada) by S. Balachandra rao,
Reviewed by Surabhi Saccidananda

Page: 75-77

Some Recent Publications in History of Mathematics

Page: 79-90

News

Page: 91-92

Obituary

Page: 93-94

Volume 38 (2016)

Volume 38 (2016) Issue 2 , (July-2016 to December-2016)

July-December

Siddhanta-karana conversion: Some algorithms in the Grahaganitadhyaya of Bhaskara’s Siddhantasiromani and in his Karanakutuhala by Kim Plofker

Page: 93-110

The Candravakyas of Madhava by R. Venketeswara Pai, K. Ramasubramanian, M. D. Srinivas and M. S. Sriram

Page: 111-139

The poetic features in the goladhyaya of Nityananda’s Sarvasiddhantaraja by Anuj Misra, Clemency Montelle and K. Ramasubramanian

Page: 141-156

Roshdi Rashed, Historian of Greek and Arabic Mathematics
by Athanase Papadopoulos

Page: 157-182

A tribute to Syamadas Mukhopadhyaya; On the occasion of his 150th birth anniversary by S. G. Dani

Page: 183-195

Annual Conference of ISHM - 2016 : A Report

Page: 195-197

Volume 38 (2016) Issue 1 , (January-2016 to June-2016)

June

Embedding: Multi-purpose Device for Understanding Mathematics and its Development, or Empty Generalization?
by Jens Hoyrup

Page: 1-29

Rolle’s Theorem and Bolzano-Cauchy Theorem : a View from the End of the 17th century until K. Weierstrass’ Epoch
by G.I. Sinkevich

Page: 31-53

Ganitananda: Selected works of Radha Charan Gupta on History of Mathematics, Ed. by K. Ramasubramanian Reviewed by Kim Plofker (Page 55-60)

Aryabhatiyam, translation of the complete text with commentary, in Kannada, by Dr. S. Balachandra Rao Reviewed by B. S. Shylaja (Page 61-63)

Chintamani Ragoonatha Charry and Contemporary Indian Astronomy, by B.S. Shylaja
Reviewed by Surabhi Saccidananda (Page 64-67)

Pangeometry by Nikolai Lobachevsky Edited and Translated, by Athanase Papadopoulos Reviewed by Tali Pinsky (Page 67-72)

Page: 55-72

Some Recent Publications in History of Mathematics

Page: 73-84

Annual Conference of ISHM - 2015 : A Report

Page: 85-89

Announcement 91-92

Page: 91-92

CONTENTS

Page:

Volume 37 (2015)

Volume 37 (2015) Issue 1-2 , (January-2015 to December-2015)

January to December

BhaskaracaryaÃ¢â‚¬â„¢s Mathematics and Astronomy: An Overview by M. S. Sriram

Page: 1-38

Some Aspects of Patadhikara in Siddhantasiromani
by Venketeswara Pai R., M. S. Sriram and Sita Sundar Ram

Page: 39-68

The Phenomena of Retrograde Motion and Visibility of Interior Planets in BhÃ„ÂskaraÃ¢â‚¬â„¢s works
by Shailaja M., Vanaja V. and S. Balachandra Rao

Page: 69-82

True Positions of Planets according to KaranakutÃ…Â«hala
by Vanaja V., Shailaja M. and S. Balachandra Rao

Page: 83-96

The inÃ¯Â¬â€šuence of BhÃ„ÂskarÃ„ÂcÃ„ÂryaÃ¢â‚¬â„¢s Works in Ã¢â‚¬Å“WesternizedÃ¢â‚¬Â Sanskrit Mathematical Traditions by Kim Plofker

Page: 97-109

BhÃ„ÂskarÃ„ÂcÃ„ÂryaÃ¢â‚¬â„¢s Treatment of the Concept of Infinity
by Avinash Sathaye

Page: 111-123

Issues in Indian Metrology, from Harappa to BhÃ„ÂskarÃ„ÂchÃ„Ârya by Michel Danino

Page: 125-143

Indian Records of Historical Eclipses and their Significance
by Aditya Kolachana and K. Ramasubramanian

Page: 145-162

Medieval Eclipse Prediction A Parallel Bias In Indian And Chinese Astronomy
by Jayant Shah

Page: 163-178

PREFACE

Page: i

Volume 36 (2014)

Volume 36 (2014) Issue 2 , (July-2014 to December-2014)

July - December

Hyperbolic Geometry in the Work of J. H. Lambert

Page: 129-155

On the Legacy of Ibn Al-Haytham: An Exposition Based on the Work of Roshdi Rashed by Athanase Papadopoulos

Page: 157-177

Otto HÃƒÂ¶lder : A Multifaceted Mathematician by R. Sridharan

Page: 179-191

Some Identities and Series Involving Arithmetic and Geometric Progressions in PÃƒâ‚¬Ãƒâ€¡ÃƒÂgaÃƒÂµitam, GaÃƒÂµitasÃƒâ‚¬rasaÃƒÂ´grahaÃƒÂ and GaÃƒÂµitakaumudÃƒÂ by Shriram M. Chauthaiwale

Page: 193-204

Book Reviews:

Indian Astronomy - Concepts and Procedures, by S. Balachandra Rao, Reviewed by Jayant Narlikar

Bhaskaracharya virachita LÃƒÂlavatÃƒÂ (in Kannada), by S. Balachandra Rao, Reviewed by C. S. Aravinda

La ThÃƒÂ©orie des Lignes ParallÃƒÂ¨les de Johann Heinrich Lambert (in French), by A. Papadopoulos and G. ThÃƒÂ©ret, Reviewed by Ravi Raghunathan

Page: 205-218

Some Recent Publications in History of Mathematics

Page: 219-226

Report on the Annual Conference of ISHM - 2014 Dedicated to Bhaskaracarya

Page: 227-230

News

Page: 231-233

Contents

Page:

Volume 36 (2014) Issue 1 , (January-2014 to June-2014)

Jan-Dec

Mathematical Models and Data in the Brahmpaksa School of Indian Astronomy by Kim Plofker

Page: 1-11

From Verses in Text to Numerical Table - The Treatment of Solar Declination and Lunar Latitude in Bhaskara II's Karanakutuhala and the Related Tabular Work, the Brahmatulyasarani by Clemency Montelle

Page: 13-35

Intantaneous Motion (taktalikagati) and the "Derivative" of the Sine Function in Bhaskara - II's Siddhantasiromani by M. S. Sriram

Page: 37-52

On the Works of Euler and His Followers on Spherical Geometry by Athanase Papadopoulos

Page: 53-108

Sawai Jai Singh's Efforts to Revive Astronomy by Virendra N. Sharma

Page: 109-125

Obituriy Note

Page: 127

Volume 35 (2013)

Volume 35 (2013) Issue 1-2 Comb. , (January-2013 to December-2013)

January-December

A History of Pingala's Combintorics by Jayant Shah

Page: 1-54

Authenticity of the Verses in the Printed Edition of the Ganitatilaka by Takao Hayashi

Page: 55-74

Some Interesting Addenda to a Manuscript of the Kautukallavati of Ramacandra by Takao Hayashi

Page: 75-98

An Analysis of the MandaphalaTables of Makaranda and Revision of Parameters by Balachandra Rao S, Rupa K and Padmaja Venugopal

Page: 99-118

Proof-Events in History of Mathematics by Ioannis M.Vandoulakis and Petros Stefaneas

Page: 119-157

Pingala's Fountain by R. Sainudiin, R. Sridharan, M.D. Srinivas and K.Yogeeswaran

Page: 159-165

Book Review:
Sri Rajaditya's Vyavaharaganita, Ed. & Tr. by Padmavathamma, Krishnaveni and K.G. Prakash by S. Balachandra Rao

Page: 167-171

Some Recent Publications in History of Mathematics

Page: 173-181

Volume 34 (2012)

Volume 34 (2012) Issue 1-2 Comb. , (January-2012 to December-2012)

January-December

by Jens Hoyrup

Page: 1-28

A hypothetical history of Old Babylonian mathematics: places, passages, stages, development by Jens HÃ¸yrup

Page: 29-51

Mesopotamian zig-zag function of day length from Indian point of view by Yukio Ohashi

Page: 53-64

Eccentric Model of the Solar Orbit in China by Yukio Ohashi

Page: 65-75

Mathematical conceptual changes in Renaissance music by Abdounur, Oscar JoÃ£o

Page: 77-83

Treatises and tables, algorithms and approximations: the role of computation in early modern Sanskrit astronomy by Kim Plofker

Page: 85-105

Fundamentals of arithmetic according to ?a?kara V?riyar (Kerala, 1500-1560) by Pierre-Sylvain Filliozat

Page: 105-122

From Jyasam.varga-nyaya to Jyanayana: A recursive formula for tabular Rsines by Vanishri Bhat and K. Ramasubramanian

Page: 123-136

Use of Continued Fractions in Karanapaddhati by M.S.Sriram and Venketeswara Pai R

Page: 137-160

Muh.ammad Barakat's Commentary on Euclid: An Unintended Mathematics Textbook by Gregg De Young

Page: 161-191

Open issues in the new historiography of European early modern mathematics by NiccolÃ² Guicciardini

Page: 193-201

by Athanase Papadopoulos

Page: 203-224

Volume 33 (2011)

Volume 33 (2011) Issue 1-2 Comb. , (January-2011 to December-2011)

January-December

Absence of Geometric Models in Medieval Chinese Astronomy by Jayant Shah

Page: 1-14

Textual Analysis of Ancient Indian Mathematics by Satyanad Kichenassamy

Page: 15-28

by Konstantinos Nikolantonakis

Page: 29-43

Varasankalita of Narayana Pandita by Shriram M. Chauthaiwale

Page: 45-54

Varasankalita of Narayana Pandita by Shriram M. Chauthaiwale

Page: 45-54

by Christopher Minkowski and Toke Knudsen

Page: 55-87

Seasonal Poetry as Science: The Rtuvarnana in Some Astronomy Treatises by Christopher Minkowski

Page: 89-113

A Partial History of the Indian Statistical Institute by Somesh Chandra Bagchi

Page: 115-129

Book Review Tantrasangraha of Nilakantha Somayaji by K. Ramasubramanian and M. S. Sriram Reviewed by S. R. Sarma

Page: 131-135

Current Bibliography of Radha Charan Gupta by Takao Hayashi

Page: 137-172

Some Recent Publications in History of Mathematics by K. A. C. H. A. Kothalawala, Sanjay Kumar, Y. P. Singh, Mahesh Chander and H.R. Meena

Page: 173-178

Volume 32 (2010)

Volume 32 (2010) Issue 1-2 Comb. , (January-2010 to December-2010)

January-December

Preface

Page: i-iii

Foreword

Page: v-vii

Old Babylonian Ã¢â‚¬Å“AlgebraÃ¢â‚¬Â, and What It Teaches Us about Possible Kinds of Mathematics by Jens Hoyrup

Page: 1-24

The Origin of Decimal Counting: Analysis of Number Names in the R gveda by Bhagyashree Bavare

Page: 25-35

Modes of Creation of a Technical Vocabulary: the Case of Sanskrit Mathematics by Pierre-Sylvain Filliozat

Page: 37-53

A Study of the Operation Called Samkramana and Related Operations by Takanori Kusuba

Page: 55-71

The Resolution of Diophantine Equations According to Bhaskara and a Justification of the Cakravala by Krsnadaivajna by FranÃ§ois Patte

Page: 73-105

Indian Elements in Kushyar's Mathematics and Astronomy by Mohammad Bagheri

Page: 107-113

The Discovery of Madhava Series by Whish: An Episode in Historiography of Science by U. K. V. Sarma, Vanishri Bhat, Venketeswara Pai and K. Ramasubramanian

Page: 115-126

The "Ignominious Fate" of Spherical Trigonometry? by Kim Plofker

Page: 127-143

Translating the Elements into Sanskrit: Jaganna-thaÃ¢â‚¬â„¢sRekha-gan.ita. A Preliminary Assessment by Clemency Montelle

Page: 145-166

Volume 31 (2009)

Volume 31 (2009) Issue 1-2 Comb. , (January-2009 to December-2009)

January-December

On the Rationale of the Maxim Ankanam Vamato Gatih by Sreeramula Rajeswara Sarma

Page: 1-22

Page: 23-33

On the Figurate Numbers from the Bhagavati Sutra by Dipak Jadhav

Page: 35-55

Effects of Moon's Parallax: Lambana and Nati in Indian Astronomy - A Study by S. Balachandra Rao and Padmaja Venugopal

Page: 57-81

The Vakya Method of Finding the Moon's Longitude by Venketeswara Pai, Dinesh Mohan Joshi and K. Ramasubramanian

Page: 83-108

India and the World of Mathematics by M.S. Raghunathan

Page: 109-114

Professor R.C. Gupta Receives the Kenneth O. May Prize by Kim Plofker

Page: 115-118

A Centennial Tribute to Professor L.V. Gurjar by Medha S. Limaye

Page: 119-122

Hermann Minkowski - in Remembrance

Page: 123-124

Review of "Ganitasarakaumudi: The Moonlight of the Essence of Mathematics, by Thakkura Pheru, Edited with Introduction, Translation, and Mathematical Commentary by SaKHYa" by Surabhi Saccidananda

Page: 125-129

Some Recent Publications in History of Mathematics

Page: 131-135

Obituary Note

Page: 137

Volume 30 (2008)

Volume 30 (2008) Issue 2 , (July-2008 to December-2008)

July - December

Some Remarks On The History of Solitary Waves
by Wang Lixia

Page: 123-130

Serbian Mathematical School And Karamata's Theory : Historical Background
by Dragan Djurcic, Ljubisa D.R. Kocinac, Malisa R. Zizovic

Page: 131-137

On Raising A Number To Its Own Power In Ancient India by Dipak Jadhav

Page: 139-149

Vedic Mathematics And Work of Aryabhata by Dr.Parmeshwar Jha

Page: 151-158

Indian Perspective On The Ontology of A Theory by K.Mahesh , K.Ramasubramaniam

Page: 159-180

History of Mathematics For Primary School Teachers Training
by Konstantinos Nikolantonakis

Page: 181-194

Analysis of Madhava's Series Method For Determination of Desired Rsines
by V.Madhukar Mallayya

Page: 195-209

Letter To Editor

Page: 211-213

Volume 30 (2008) Issue 1 , (January-2008 to June-2008)

January-June

Mathematical Ideas in Bhagavati Sutra(BS) by R. S. Shah

Page: 1-25

Chinese Mathematical Astronomy from ca. 4th Century to ca. 6th Century by Yukio Ohashi

Page: 27-39

RENAISSANCE On Mathematical Immortality by Satish C. Bhatnagar

Page: 41-44

Vojislav G. Avakumovic (1910 - 1990)A Passionate Man of Mathematics by Vojislav Maricâ€š and Aleksandar M. Nikolic

Page: 45-60

Vatesvara's Trigonometric Tables and the Method by V. Madhukar Mallayya

Page: 61-79

ADDENDUM TO "A Concise View of the History of Mathematics in Latin America" Gaita Bh rati, 28(2006) 111-128 by Ubiratan D Ambrosio

Page: 81-83

Was India Mathematically Illiterate Until the Fifth Century A.D.? by V. Lakshmikantham

Page: 85-90

Contribution to History of n-Groups by Malisa Zizovi

Page: 91-97

A Complex-Domain Analysis of Actions of A Conscious Human Being by Radhey Shyam Kaushal

Page: 99-113

Ramanujan's First Publication: A Solution To A Problem in The Educational Times by James J. Tattersall and Shawnee L. Mcmurran

Page: 115-121

Volume 29 (2007)

Volume 29 (2007) Issue 1-2 , (January-2007 to December-2007)

January - December

A Tacit Appropriation of Hindu Algebra In Renaissance Practical Aritmetic

by Albrecht Heeffer

Page: 1 - 60

Pride - Worthy Indian Contribution To Morley's Miracle

by V.G. Tikekar

Page: 61-67

A Note On Submissions From India To The Mathematics Selections of The Educational Times And The Journal of The Indian Mathematical Society

by James J. Tattersall

Page: 69-79

Mathematics of Anuyogadwara - Sutra

by R.S. Shah

Page: 81-100

Formulation of Chinese Classical Mathematical Astronomy

by Yukio Ohashi

Page: 101-115

Algebraic Models In Paitamaha Siddhanta

by G.S. Pandey

Page: 117-133

Letters To The Editor

Page: 135-136

News Cafe

Page: 137-138

Book Review

The History of Mathematics And Mathematician of India

Reviewed By N.L. Maiti, Venugopal D. Heroor

Page: 139-140

Volume 28 (2006)

Volume 28 (2006) Issue 1-2 , (January-2006 to December-2006)

January - December

David Pingree And Indian Mathematics

by Kim Plofker

Page: 1 - 6

Revisiting A Chapter From The Notebooks of Ramanujan On Hypergeometric Series

by G. Vanden Berghe, K . Srinivasa Rao

Page: 7 - 38

On The Algorithmic Spirit of The Ancient Chinese And Indian Mathematics

by Wenlin Li

Page: 39-49

Kerala Mathematical Tradition - Some Landmarks

by V. Madhukar Mallayya

Page: 51-65

The History of The Chinese Written Zeroes Revisited

by Jean-Claude Martzloff

Page: 67-83

The Riemann Hypothesis Reviewed In A Historical Nutshell

by B.S. Yadav

Page: 85-92

Ivan Bernoulli Series Universalissima

by A.K. Kwasniewski

Page: 93-101

The Great Mathematical Research Centres of The 20th Century And The "Miracle" of Lwow I

by N. Schlomiuk

Page: 103-109

A Concise View of The History of Mathematics In Latin America

by Ubiratan D Ambrosio

Page: 111-128

Istankapancavimsatika of Tejasimha

by Takao Hayashi

Page: 129-145

Ancient And Medieval Astronomers And Mathematicians of India

by V. Lakshmikantham, J. Vasundhara Devi, S. Leela

Page: 147-156

An Ancient Egyptian Problem And Its Innovative Arithmetic Solution

by Milo Gardner

Page: 157-173

Class Room Notes

The Interplay Between Numbers And Geometry In The History of Mathematics
by A.M. Vaidya

Page: 175-181

Renaissance

Calculus Defines Civilization
by S.C. Bhatnagar

Page: 183-184

News Cafe

Meetings

The 2nd International Congress of History of Science

Page: 189-190

Obituary

Irving Kaplansky
by B.S. Yadav

Page: 191

Paul Halmos - Expositor Par Excellence

by V S Sunder

Page: 193-199

GB Contents Pages Vol. 1-27

Page: 201-229

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R.C. Gupta. Sino-Indian interaction and the great Chinese Buddhist astronomer-mathematician I-Hsing. Ganita Bhāratī, 11: 38-49, 1989. G.H. Hardy. A Mathematician's Apology . Cambridge Univ. Press: Cambridge, 1988. (Reprinted) E. von Collani. History, State of the Art and Future of the Science of Stochastics. In: Ivor Grattan-Guinness and B.S. Yadav ed. History of The Mathematical Sciences, 171-194. Hindustan Book Agency: New Delhi, 2002.
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Ganita Bharati : Bulletin of The Indian Society for History of Mathematics

Editor-In-Chief

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Volume 43 (2021) Issue 1 , (January-2021 to June-2021)

Volume 42 (2020)

Volume 42 (2020) Issue 1-2 , (January-2020 to December-2020)

Volume 41 (2019)

Volume 41 (2019) Issue 1-2 , (January-2019 to December-2019)

Volume 40 (2018)

Volume 40 (2018) Issue 2 , (July-2018 to December-2018)

Volume 40 (2018) Issue 1 , (January-2018 to June-2018)

Volume 39 (2017)

Volume 39 (2017) Issue 2 , (July-2017 to December-2017)

Volume 39 (2017) Issue 1 , (January-2017 to June-2017)

Volume 38 (2016)

Volume 38 (2016) Issue 2 , (July-2016 to December-2016)

Volume 38 (2016) Issue 1 , (January-2016 to June-2016)

Volume 37 (2015)

Volume 37 (2015) Issue 1-2 , (January-2015 to December-2015)

Volume 36 (2014)

Volume 36 (2014) Issue 2 , (July-2014 to December-2014)

Volume 36 (2014) Issue 1 , (January-2014 to June-2014)

Volume 35 (2013)

Volume 35 (2013) Issue 1-2 Comb. , (January-2013 to December-2013)

Volume 34 (2012)

Volume 34 (2012) Issue 1-2 Comb. , (January-2012 to December-2012)

Volume 33 (2011)

Volume 33 (2011) Issue 1-2 Comb. , (January-2011 to December-2011)

Volume 32 (2010)

Volume 32 (2010) Issue 1-2 Comb. , (January-2010 to December-2010)

Volume 31 (2009)

Volume 31 (2009) Issue 1-2 Comb. , (January-2009 to December-2009)

Volume 30 (2008)

Volume 30 (2008) Issue 2 , (July-2008 to December-2008)

Volume 30 (2008) Issue 1 , (January-2008 to June-2008)

Volume 29 (2007)

Volume 29 (2007) Issue 1-2 , (January-2007 to December-2007)

Volume 28 (2006)

Volume 28 (2006) Issue 1-2 , (January-2006 to December-2006)

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