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

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

**ISSN:**0970-0307

**e-ISSN:********

**Subject:**Mathematics

**Periodicity:**Half Yearly

**Month(s) of Publication:**June & December

**Current Volume:**39 (2017)

- About the Journal
- Indexing & Abstracting
- Editors
- Current Issue
- Previous Issue(s)
- Instructions to Authors
- Ethics Policy
- Call For Papers

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.

**Sites are:**

ProQuest

EBSCO

Zentralblatt MATH,

Mathematical Review,

Genamics(JournalSeek) etc.

### Editor-In-Chief

- Shrikrishna G. Dani, Department of Mathematics, Indian Institute of Technology Mumbai, Powai, Mumbai-400078, Email: ganitabharati@gmail.com

### International Advisory Board

- Virender Dalal, Ramjas College, University of Delhi, Delhi - 110007

### Associate Editor

- S.L. Singh, 21, Govind Nagar nagar, Rishikesh - 249201

### Assistant Editor

- V.M. Mallayya, MELTRA-A23, T.C.25/1974(2), Deshabhimani Road, Trivandrum-695001

### Members

- S.M.S.Ansari
- Mohammad Bagheri
- S. C. Bhatnagar
- Umberto Bottazzini
- Karine Chemla
- J.W.Dauben
- Nachum Dershowitz
- Enrico Guisti
- R.C.Gupta
- Takao Hayashi
- Jan P.Hogendijik
- Subhash Kak
- Victor J.Katz
- Wenlin Li
- J.C.Martzloff
- Jean-Paul Pier
- Kim Plofker
- F.Jamil Ragep
- S.R.Sharma
- Chikara Sasaki
- M.S.Sriram
- A.M.Vadiya
- Ioannis M Vandoulakis
- D.E.Zitarelli

### 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 *

**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 *

**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 *

**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 *

**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
**

**News**
* *

### Volume 39 (2017)

#### 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 *

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

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

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

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

**Some Recent Publications in History of Mathematics**
* *

**News **
* *

**Obituary**
* *

### 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 *

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

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

**Roshdi Rashed, Historian of Greek and Arabic Mathematics
**

*by Athanase Papadopoulos*

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

**Annual Conference of ISHM - 2016 : A Report**
* *

#### 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*

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

*by G.I. Sinkevich*

**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)**

**Some Recent Publications in History of Mathematics **
* *

**Annual Conference of ISHM - 2015 : A Report **
* *

**Announcement 91-92**
* *

**CONTENTS**
* *

### 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 *

**Some Aspects of Patadhikara in Siddhantasiromani
**

*by Venketeswara Pai R., M. S. Sriram and Sita Sundar Ram*

**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*

**True Positions of Planets according to KaranakutÃ…Â«hala
**

*by Vanaja V., Shailaja M. and S. Balachandra Rao*

**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 *

**BhÃ„ÂskarÃ„ÂcÃ„ÂryaÃ¢â‚¬â„¢s Treatment of the Concept of Infinity
**

*by Avinash Sathaye*

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

**Indian Records of Historical Eclipses and their Significance
**

*by Aditya Kolachana and K. Ramasubramanian*

**Medieval Eclipse Prediction A Parallel Bias In Indian And Chinese Astronomy
**

*by Jayant Shah*

**PREFACE**
* *

### Volume 36 (2014)

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

July - December

**Hyperbolic Geometry in the Work of J. H. Lambert **
* *

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

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

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

**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**

**Some Recent Publications in History of Mathematics**
* *

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

**News**
* *

**Contents**
* *

#### 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 *

**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 *

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

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

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

**Obituriy Note**
* *

### 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 *

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

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

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

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

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

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

*by S. Balachandra Rao*

**Some Recent Publications in History of Mathematics**
* *

### Volume 34 (2012)

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

January-December

* by Jens Hoyrup *

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

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

**Eccentric Model of the Solar Orbit in China**
* by Yukio Ohashi *

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

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

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

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

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

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

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

* by Athanase Papadopoulos *

### 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 *

**Textual Analysis of Ancient Indian Mathematics**
* by Satyanad Kichenassamy *

* by Konstantinos Nikolantonakis *

**Varasankalita of Narayana Pandita**
* by Shriram M. Chauthaiwale *

**Varasankalita of Narayana Pandita**
* by Shriram M. Chauthaiwale *

* by Christopher Minkowski and Toke Knudsen *

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

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

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

**Current Bibliography of Radha Charan Gupta**
* by Takao Hayashi *

**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 *

### Volume 32 (2010)

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

January-December

**Preface**
* *

**Foreword**
* *

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

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

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

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

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

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

**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 *

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

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

### 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 *

**Eugene Jacquet and his Pioneering Study of Indian Numerical Notations**
* by JÃ©rÃ´me Petit *

**On the Figurate Numbers from the Bhagavati Sutra**
* by Dipak Jadhav *

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

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

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

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

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

**Hermann Minkowski - in Remembrance**
* *

**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 *

**Some Recent Publications in History of Mathematics**
* *

**Obituary Note**
* *

### 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*

**Serbian Mathematical School And Karamata's Theory : Historical Background
**

*by Dragan Djurcic, Ljubisa D.R. Kocinac, Malisa R. Zizovic*

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

**Vedic Mathematics And Work of Aryabhata**
* by Dr.Parmeshwar Jha *

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

**History of Mathematics For Primary School Teachers Training
**

*by Konstantinos Nikolantonakis*

**Analysis of Madhava's Series Method For Determination of Desired Rsines
**

*by V.Madhukar Mallayya*

**Letter To Editor**
* *

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

January-June

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

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

** RENAISSANCE On Mathematical Immortality**
* by Satish C. Bhatnagar *

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

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

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

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

**Contribution to History of n-Groups**
* by Malisa Zizovi *

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

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

### 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*

**Pride - Worthy Indian Contribution To Morley's Miracle**

*by V.G. Tikekar*

**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*

**Mathematics of Anuyogadwara - Sutra**

*by R.S. Shah*

**Formulation of Chinese Classical Mathematical Astronomy **

*by Yukio Ohashi*

**Algebraic Models In Paitamaha Siddhanta **

*by G.S. Pandey*

**Letters To The Editor**

**News Cafe**

**Book Review**

The History of Mathematics And Mathematician of India

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

### Volume 28 (2006)

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

January - December

**David Pingree And Indian Mathematics**

*by Kim Plofker*

**Revisiting A Chapter From The Notebooks of Ramanujan On Hypergeometric Series **

*by G. Vanden Berghe, K . Srinivasa Rao*

** On The Algorithmic Spirit of The Ancient Chinese And Indian Mathematics **

*by Wenlin Li*

** Kerala Mathematical Tradition - Some Landmarks **

*by V. Madhukar Mallayya*

** The History of The Chinese Written Zeroes Revisited **

*by Jean-Claude Martzloff*

** The Riemann Hypothesis Reviewed In A Historical Nutshell **

*by B.S. Yadav*

** Ivan Bernoulli Series Universalissima**

*by A.K. Kwasniewski*

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

*by N. Schlomiuk*

**A Concise View of The History of Mathematics In Latin America **

*by Ubiratan D Ambrosio*

**Istankapancavimsatika of Tejasimha **

*by Takao Hayashi*

**Ancient And Medieval Astronomers And Mathematicians of India **

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

**An Ancient Egyptian Problem And Its Innovative Arithmetic Solution **

*by Milo Gardner*

**Class Room Notes**

The Interplay Between Numbers And Geometry In The History of Mathematics

*by A.M. Vaidya*

** Renaissance**

Calculus Defines Civilization

*by S.C. Bhatnagar*

**News Cafe **

Meetings

The 2nd International Congress of History of Science

**Obituary **

Irving Kaplansky

*by B.S. Yadav*

**Paul Halmos - Expositor Par Excellence **

*by V S Sunder*

**GB Contents Pages Vol. 1-27 **

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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 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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### Introduction

Ganita Bhāratī, 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. It is our endeavor to be a global vehicle for dissemination of original research on topics in history of mathematics, as well expositions (in a form not currently available in literature) that would facilitate research and/or enhance awareness on
the topic of exposition, or of the subject on a broader scale. Ganita Bhāratī also publishes Book Reviews, Brief introductory comments on recent publications, Notes and News items on matters and current events of interest to the broad community involved with the study of history of mathematics, including scholars, students, as well as general enthusiasts of the subject.

This journal is in 38th year of publication and it is indexed/Abstracted in Zentralblatt MATH, Mathematical Review, Genomics(JournalSeek) etc.

### Topics

History of mathematics of various periods, from ancient to modern

### Subject Covered

Articles on a broad range of topics, all the way from mathematics from the ancient cultures around the world to historical aspects of modern mathematics are welcome.

### Instructions to Authors

### Submit Your Article

ganitabharati@gmail.com

### Frequency

2 issues per year.