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Ganita Bharati

Published in Association with Bulletin of The Indian Society for History of Mathematics

Current Volume: 45 (2023 )

ISSN: 0970-0307

Periodicity: Half-Yearly

Month(s) of Publication: June & December

Subject: Mathematics

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

Online Access is Free for Life Member

250

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.

The ISHM aims to Promote study, research and education in history of mathematics. It provides a forum for exchange of ideas and experiences regarding various aspects of history of mathematics. In addition to the annual conferences, ISHM aims at organizing seminars/symposia on the works of ancient, medieval and modern mathematics, and has been bringing out the bulletin Ganita Bharati. Scholars, Teachers, Students and all lovers of mathematical sciences are encouraged to join the Society.

ProQuest
EBSCO
Zentralblatt Math
Mathematical Review
Genamics(JournalSeek)

 

Editor
S.G. Dani

UM-DAE Centre for Excellence in Basic Sciences
Vidyanagari Campus of University of Mumbai
Kalina, Mumbai 400098, India


Managing Editor
Ruchika Verma

Ramjas College
University of Delhi
Delhi-110007, India


Assistant Editor
V. M. Mallayya

Meltra-A23, 'Padmasree'
T. C. 25/1974(2)
Near Gandhari Amman Kovil,
Thiruvananthapuram, Kerala,
PIN: 695001, India.


Members
S.M.S. Ansari

Muzammil Manzil Compound
Dodhpur Road
Aligarh 202002, India.


R. C. Gupta

R-20, Ras Bahar Colony
P. O. Lahar Gird,
Jhansi-284003, India


Kim Plofker

Department of Mathematics
Union College
Schenectady, NY 12308
USA


Mohammad Bagheri

Encyclopedia Islamic Foundation
PO Box 13145-1785
Tehran
Iran


Takao Hayashi

Science & Engg. Research Institute
Doshisha University
Kyotanabe Kyoto 610-0394
Japan


F. Jamil Ragep

Islamic Studies
McGill University
Morrice Hall, 3485 McTavish Street
Montreal, Quebec,
Canada H3A 1Y1


S. C. Bhatnagar

Department of Mathematics
University of Nevada
Las Vegas
USA


Jan P. Hogendljk

University of Utrecht
P.O. Box 80010
3508 TA Utrecht
The Netherlands


S. R. Sarma

Höhenstr. 28
40227 Düsseldorf
Germany


Umberto Botttazzni

Universita degli Studi di Milano
Dipartimento di Matematica
Federigo Enriques Via Saldini 50
20133, Milano 
Italy


Jens Hoyrup

Roskilde University
Section for Philosophy and Science Studies
Denmark


Karine Chemla

REHSEIS-CNRS and
University Paris7, 75019,
Paris, France


Subhash Kak

Dept. of Computer Sc.
MSCS 219
Oklahoma State University
Stillwater, OK 74078, USA


Chikara Sasaki

University of Tokyo
3-8-1 Komaba,
Meguro-Ru,
Tokyo 153-8902
Japan


J. W. Dauben

The Graduate Centre
CUNY, 33, West 42nd Street
New York, NY 10036
U.S.A.


Victor J. Katz

University of the D.C.
4200 Connecticut Ave.
N.W.Washington, D.C 20008
USA


M. S. Sriram

Prof. K.V. Sarma Research Foundation
Venkatarathnam Nagar
Adyar, Chennai - 600020

 


Nachum Dershowitz

Department of Computer Science
Tel Aviv University,
Tel Aviv
Israel


Wenlin Li

Academy of Mathematics & Systems Science
Chinese Academy of Science,
No. 55, Zhongguancun East Road,
Haidan District, Beijing, 100190,
China


Ioannis M. Vandoulakis

The Hellenic Open Unversity
School of Humanities
23, Syngrou Avenue,
GR-11743, Athens, Greece.


Nachum Dershowitz

Department of Computer Science
Tel Aviv University,
Tel Aviv
Israel


Wenlin Li

Academy of Mathematics & Systems Science
Chinese Academy of Science,
No. 55, Zhongguancun East Road,
Haidan District, Beijing, 100190,
China


Ioannis M. Vandoulakis

The Hellenic Open Unversity
School of Humanities
23, Syngrou Avenue,
GR-11743, Athens, Greece.


Enrico Giusti

Dipartimento di Matematica
Viale Morgagni, 67/A
I-50134 Firenze, Italy


Jean-Paul Pier

Société mathématique du Luxembourg
117 rue Jean-Pierre Michels
L-4243 Esch-sur-Alzette
Luxembourg


D. E. Zitarelli

Department of Mathematics
Temple University
Philadelphia, PA 19/22, USA.


Volume 45 Issue 1 , (Jan-2023 to Jan-2023)

New Perspectives on the Development of the Indian Positional System in the Light of Sanskrit, Pāli and Tamil Sources

By: Satyanad Kichenassamy

Page No : 1-21

Abstract
We show that the Indian number system, now in nearly universal use, was preceded by a system capable of expressing arbitrarily large numbers using only twelve symbols, and no zero, the last three symbols serving as place separators. It is alluded to by Brahmagupta in 628 under the name gomūtrikā that, later on, took a different meaning. It is very similar to the standard Indian way of writing equations. The turning point seems to have been the introduction of visual patterns in Sanskrit poetry. There are also traces of it in Pāli grammar as well as in Tamil mathematics. In Tamil, this system was extended to express arbitrarily small fractions. We suggest that the place separators were eventually omitted as a result of the development of methods for root extraction and of avyakatagaṇita or algebra.
 

Author
Satyanad Kichenassamy : 
Professor of Mathematics, Université de Reims Champagne-Ardenne, Laboratoire de Mathématiques de Reims (CNRS, UMR9008), B.P. 1039, F-51687 Reims Cedex 2.
 

DOI : https://doi.org/10.32381/GB.2023.45.1.1

Price: 251

From Concrete to Abstract in Indian Mathematics

By: Jaidev Dasgupta

Page No : 23-43

Abstract
Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the historical conditions under which it originated and evolved has not been studied much. The focus has been more on achievements than on how they developed. One also tends to read the ancient texts with the present concepts and methods in mind. The absence of writing over a long stretch of Indian history too gets overlooked in such readings. The purpose of this article is to explore the journey of mathematics by examining what the ancient texts on arithmetic, geometry and algebra tell us about the nature of mathematics in their times. These investigations reveal that over a period of a thousand or more years Indian mathematics transitioned from concrete and context-bound phase to context-free, abstract phase accompanied by several conceptual leaps. Invention of writing in the 3rd century BCE greatly facilitated this transition.
 

DOI : https://doi.org/10.32381/GB.2023.45.1.2

Price: 251

How did the All-purpose Parenthesis Come about in European Algebra?

By: Jens Høyrup

Page No : 45-75

Abstract
One of the characteristics that distinguishes modern algebra from al-Khwārizmī’s as well as abbacus algebra is the general “parenthesis function”. A parenthesis, it must be observed, is not a bracket but what is kept together possibly by a pair of round brackets. An algebraic parenthesis is an expression which as a whole can be operated upon or serve as argument for a function. Abbacus algebra made use of various “special-purpose” parentheses: most important of these being the numerator and the denominator of formal fractions, and composite radicands. Descartes adds a way to keep together a composite coefficient, but even this remains a special-purpose parenthesis. Oughtred, Newton and others start using a vinculum (a stroke above a composite expression), which could have served to define any kind of parenthesis, but actually only keeps together polynomials as factors. Neither their algebra nor any earlier algebra in the tradition had any need going beyond that. That need emerged in the second half of the 17th century, when algebra gave rise to infinitesimal analysis. Wallis, Leibniz and Johann Bernoulli needed and used the vinculum or the round brackets for new purposes, and thereby transformed them into general tools for parenthesis delimitation – probably without being aware that they had taken an important step. After 1750, even algebra proper adopted these new tools. A link to the emergence of the function as a mathematical category is sketched but not explored in depth.

Author
Jens Høyrup :
 Roskilde University, Section for Philosophy and Science Studies Max-Planck-Institut für Wissenschaftsgeschichte, Berlin, Germany.
 

DOI : https://doi.org/10.32381/GB.2023.45.1.3

Price: 251

Leibniz’s Contested Infinitesimals: Further Depictions

By: Mikhail G. Katz , Karl Kuhlemann

Page No : 77-112

Abstract
We contribute to the lively debate in current scholarship on the Leibnizian calculus. In a recent text, Arthur and Rabouin argue that non-Archimedean continua are incompatible with Leibniz’s concepts of number, quantity and magnitude. They allege that Leibniz viewed infinitesimals as contradictory, and claim to deduce such a conclusion from an analysis of the Leibnizian definition of quantity. However, their argument is marred by numerous errors, deliberate omissions, and misrepresentations, stemming in a number of cases from flawed analyses in their earlier publications. We defend the thesis, traceable to the classic study by Henk Bos, that Leibniz used genuine infinitesimals, which he viewed as fictional mathematical entities (and not merely shorthand for talk about more ordinary quantities) on par with negatives and imaginaries. 2020 Mathematics Subject Classification. Primary 01A45, 01A61 Secondary 01A85, 01A90, 26E35.

Authors
Mikhail G. Katz : Department of Mathematics, Bar Ilan University, Ramat Gan 5290002 Israel.
Karl Kuhlemann : Gottfried Wilhelm Leibniz University Hannover, D-30167 Hannover, Germany.
 

DOI : https://doi.org/10.32381/GB.2023.45.1.4

Price: 251

Book Review

The Buddhivilāsinī – Commentary of Gaṇeśa Daivajña on the Līlāvatī of Bhāskarācārya II : A Critical Study of Proofs in Indian Mathematics of the Sixteenth Century, By V. Ramakalyani; D.K. Printworld, New Delhi, 2024, xxix + 471 pp.

By: M.S. Sriram

Page No : 113-120


Reviewed by
M.S. SRIRAM : 
Prof. K.V. Sarma Research Foundation, Chennai, 27, CGE Housing Colony, Kuppam Beach Road, Thiruvanmiyur, Chennai, 600041, India.

Price: 251

Instruction to the Author

It is preferred that the article is created in MS Word using 12-point Times New Roman type throughout. Once an article has been accepted the final version may be submitted in TeX / LaTeX also, together with the corresponding PDF file. The title, numbered equations and tables, should be centered. Everything else must be aligned to the left without any indent. A double space above and below all headings is required. If special characters (e.g. Chinese, Cyrillic) other than Latin or Greek alphabets and common mathematical symbols are used, PDF files should be supplied to indicate their placement. In fact a PDF file showing complete article with everything embedded as it should appear in the print, must be supplied.

The main body of the article should be divided by appropriate numbered section and sub-section headings all in upper/lower bold type and aligned to the left. An Acknowledgment section may be included before the list of references. Manuscripts must generally be organized in the following manner:

(i) Title (bold face) followed by author name(s) only [centered], (ii) Abstract and Key Words, (iii) Article Text, (iv) Acknowledgments, (v) References, (vi) Appendices.

The abstract should be followed by three to seven keywords that would be useful in identifying it for reference purposes.

Please avoid using any Footnotes. All references in the text must be cited by author surname and year, like (Smith, 1993) or Smith (1985b). List all the cited references at the end of the article, in alphabetical order of the surnames (writing initials first followed by the surnames), strictly in accordance with the following examples:

J.W. Dauben. The first international connexions in history of mathematics: The case of the Encyclopadie. Historia Mathematica, 26: 343-359, 1999.

R.C. Gupta. Sino-Indian interaction and the great Chinese Buddhist astronomer-mathematician I-Hsing. Ganita Bh?rat?, 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.
As a last section, please provide brief information about each contributing author's contact details, including his/her current affiliation(s), email addresses and URL (if any). The corresponding author will receive galley proofs as a PDF file via E-mail, to enable him/her to point out any corrections to be made.

All the manuscripts submitted for the Ganita Bharati should accompany a covering letter giving an undertaking following certain principles under Ethical Policy.

The cover letter should include a written statement from the author(s) that:
1. The manuscript is an original research work and has not been published elsewhere including open access at the internet.

2. The data used in the research has not been manipulated, fabricated, or in any other way misrepresented to support the conclusions.

3. No part of the text of the manuscript has been plagiarised.

4. The manuscript is not under consideration for publication elsewhere.

5. The manuscript will not be submitted elsewhere for review while it is still under consideration for publication in the Ganita Bharati.

The cover letter should also include an ethical statement disclosing any conflict of interest that may directly or indirectly impart bias to the research work. Conflict of interest most commonly arises from the source of funding, and therefore, the name(s) of funding agency must be mentioned in the cover letter. In case of no conflict of interest, please include the statement that “the authors declare that they have no conflict of interest”.

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