Ganita Bharati

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.
 

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.
 

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.
 

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.
 

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

Abstract
The aim of this article is to present at first the geometrical rules of the 10th century Indian mathematician-astronomer °ryabhaÇa II that are included in his work MahÀsiddhÀnta, composed in c.950 CE. Thereafter an attempt will be made to throw light upon some concepts of present-day geometry (including mensuration) which are implicit in this medieval work.

Author :
Dr. Sanatan Koley : Former Headmaster, Jagacha High School (H.S.), Howrah-711112. Present Address : Kadambari Housing Complex, Block-I, Flat-IA, 144 Mohiary Road, Jagacha, P.O. GIP Colony, Howrah-711112, W.B.
 

Abstract
The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. In this second part, we show that its introduction clarifies the approach to epistemic cultures. We also show that the notion of fantasy echo is relevant in Epistemology, as suggested by J.W. Scott. We then continue our earlier analysis of Brahmagupta’s Prop. 12.21-32 on the cyclic quadrilateral and identify discursive strategies that enable him to convey definitions, hypotheses and derivations encoded in the very structure of the propositions stating his new results. We also show that the statements of mathematical formulae in words also follow definite discursive patterns.

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.
 

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

Abstract
In the light of our re-interpretation of Plato’s philosophy and of our reconstruction of the proofs of quadratic incommensurabilities by the Pythagoreans, Theodorus, and Theaetetus, in terms of periodic anthyphairesis, we re-examine the Geometric Algebra hypothesis in Greek Mathematics, originally enunciated by Zeuthen and Tannery and supported by van der Waerden and Weil, but challenged by Unguru and several modern historians. Our reconstruction of these proofs employs, for the computation of the anthyphairetic quotient at every step, the solution of a Pythagorean Application of Areas, either in excess or in defect, and is thus qualified as “school algebra” in the spirit of van der Waerden. For the Application of Areas in defect in the Theaetetean Books X and XIII of the Elements, by which the alogoi lines are characterized, the periodic nature of their anthyphairesis is revealed by the Scholia in Eucliden X.135 and 185 and by our re-interpretation of the ill-understood Meno 86e-87b passage. In conclusion, the pre-Eudoxean uses of Applications of Areas fall under the description of “school algebra” solutions of quadratic equations. It is interesting that these early uses stand in sharp contrast to the later uses of more general versions of Application of Areas by Appolonius in his Conic Sections, and which, according to Zeuthen, qualify as Geometric Algebra too, but in the form of pre-Analytic Geometry.
 

Authors:
Stelios Negrepontis : Department of Mathematics, Athens University, Athens 157 84, Greece
Vasiliki Farmaki : Department of Mathematics, Athens University, Athens 157 84, Greece
Demetra Kalisperi : Department of Mathematics, Athens University, Athens 157 84, Greece
 

Abstract
Our main goal in this paper is to analyze the two rules for solving “hundred fowls” type of problems described in Mahāvīrācārya’s well-known Gaṇitasārasaṅgraha. This will be done based on two manuscripts that Prof. M. Rangacharya consulted to prepare his edition and translation of the text, in 1912, and which are still available at the Government Oriental Manuscripts Library and Research Centre – Chennai (Madras). One of the manuscripts contains a running commentary in a medieval form of Kannada that is particularly useful for clarifying the steps of the algorithms. It allows us to see how Rangacharya, in an unusual way, deviated for the first example from the solution given in the manuscripts and provided his own solution instead. It will also allow us to appreciate the uniqueness and originality of Mahāvīrācārya’s second rule. We are fortunate that four well-known Sanskrit texts propound independent rules for this type of problems and give as illustration an identical example involving the buying of four species of birds. This is a rare instance that can help us revise previous understandings regarding the meaning of technical terms such as kuṭṭaka and kuṭṭīkāra – usually considered as synonyms and translated as “pulverizers” – and suggest new perspectives.

Author:
Catherine Morice-Singh : c/o Laboratoire SPHERE, 8 Rue Albert Einstein, Bâtiment Olympe de Gouge. Université Paris Cité, F-75013 Paris, France
 

Abstract
After canvassing the solutions for indeterminate linear equations (kuṭṭaka), Indian scholars deliberated on the common solution for the two systems of similar equations under the caption “Conjunct Pulverisers (saṃśliṣṭakuṭṭaka).” Āryabhaṭa II, Mahāvīra, Śrīpatī, Bhāskara II, Nārāyaṇa Paṇḍita, Kṛṣṇa Daivajña, and Devarāja is the chain of the Indian scholars who explained similar or different methods for extracting the solutions. B. Datta discussed some of these methods, and T. Hayashi commented on Devarāja’s methods. S. K. Ganguli discovered an alternative method from the manuscript copies of Līlāvatī. This paper provides the juxtaposed mathematical formats of the methods after translating the relevant verses. Later, these methods are compared. Illustrations from the referred texts are quoted with answers.

Author:
Shriram M Chauthaiwale : Lecturer (Rt) in Mathematics, Amolakchand College, Yavatmal (M.H.)
 

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.

Author:
Jens Hoyrup
Roskilde University, Section for Philosophy and Science Studies, Denmark.
 

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’ (SaintPetersburg, 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 The paper is dedicated to signs meaning ‘very large number’, that is, for 10,000 and higher, in Cyrillic. We discuss the manuscript ‘Arithmetics’ (SaintPetersburg, 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 numbers 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’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 numbers 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’.

Author :
Dionisy I. Pronin
Independent Researcher, Russia, Yakutsk.
 

Abstract
The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. We briefly recall why this introduction was necessary. We then show that this form of discourse is widespread among scholars, even in contemporary Mathematics, in India and elsewhere. It is in India a natural outgrowth of the emphasis on non-written communication, combined with the need for freedom of thought. New results in this first part include the following: (i) ?ryabha?a proposed a geometric derivation of a basic algebraic identity; (ii) Brahmagupta proposed an original argument for the irrationality of quadratic surds on the basis of his results on the varga-prak?ti problem, thereby justifying his change in the definition of the word karani.

Abstract
In Indian mathematical texts the term savarnana “reduction to the same color” is usually found in the context of calculation for fractions. A number of explanations for the term have been offered in previous studies, but they slightly differ from each other. The Trisatibhasya is an anonymous commentary on Sridhara’s Trisati. In the present paper, I survey the meanings of savarnana in each text and the usage of it in the Trisatibhasya.

Abstract
In a commentary on example verse 112 for rule verses 97-98 in the mathematical series of the Patiganita, various solutions of a problem are described. After solving the problem according to the given rule, the commentator shows alternative methods: Aryabhatiya 2.19, linear equations, and rule verses 99-101. Also in the commentary on example verse 113 for rule verses 99-101, he again employs Aryabhatiya 2.19. The present paper has a threefold objective. First, we fully investigate the ways of solving which the commentary exhibits for the two examples. Secondly, we point out particularly where Aryabhatiya 2.19 is applied, although neither the author Aryabhatiya nor the title of his work is cited in the commentary. And thirdly, we study excerpts of rules concerning the bijaganita quoted there.

Abstract
As a Medieval Muslim polymath, al-Biruni had also been an observer of Indian astronomy. He gave some opinions on Indian theory of precession in his Tahqiq ma li-l-Hind. Al- Biruni adhered to Ptolemaic theory of the movement of the sphere of the fixed stars, which is opposite to medieval Indian theory of precession. It was such a contradiction that made al- Biruni misjudge medieval Indian theory of precession. This case reveals a particular aspect, both of the difference between pre-Ptolemaic Greco-Indian astronomy and Ptolemaic Greek one, and of the influence of Greek thought on Muslim scholars including al- Biruni.

Abstract
At the Annual conference of the Indian Society for History of Mathematics in 2020 I spoke about the scattered use of several algebraic unknowns in Italian algebra from Fibonacci to Pacioli, and in 2021 about Benedetto da Firenze’s introduction of symbolic algebraic calculations with up to five unknowns in 1463 – the latter having no impact whatsoever on future developments. Here I shall complete what was not originally planned to become a triptych, looking at the development of the technique from Pacioli onward in the writings of Rudolff, Stifel and Mennher. In the end I shall consider the likely influence on Viete’s and Descartes’ algebras, together with the reasons for their unprecedented introduction of abstract coefficients.

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.

Authors :
Stelios Negrepontis
Professor Emeritus, Mathematics Department, Athens University, Athens, Greece.

Vassiliki Farmaki
Professor Emeritus, Mathematics Department, Athens University, Athens, Greece.

Marina Brokou
Ph. D. Candidate, Mathematics Department, Athens University, Athens, Greece.

 

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. BhuvaneœvarÁ 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.

Author :
R. C. Gupta
R20, Ras Bahar Colony P.O. Sipri Bazar, Jhansi, U.P., India.
 

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.

Author :
Jens Hoyrup
Roskilde University, Section for Philosophy and Science Studies, Denmark.
 

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.

Author :
Athanase Papadopoulos
Universite de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France.

 

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.

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

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.

Author :
Ritesh Gupta
Ph.D. Research Scholar, Zakir Husain Centre for Educational Studies, Jawaharlal Nehru University, New Delhi.
 

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
Prof. K.V. Sarma Research Foundation, 42, Venkatarathnam Nagar, Adyar, Chennai, India.

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
Department of Mathematics, University of Kentucky, Lexington KY, U.S.A.
 

Abstract 
We continue our analysis of Brahmagupta’s BrÀhmasphuÇasiddhÀnta (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 kuÇÇaka) 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.
 

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.
 

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.
 

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.
 

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.
 

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.
 

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.
 

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 Siddhantic period from Aryabhata to Sripati.
 

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.
 

Abstract
The KÀtyÀyana ŒulvasÂtra has been much less studied or discussed from a modern perspective, even though the first English translation of two adhyÀyas (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 ŒulvasÂtra studies has been focussed on “the mathematical knowledge found in them (as a totality)”; as the other earlier ŒulvasÂtras, especially of BaudhÀyana and °pastamba substantially cover the ground in this respect, the other two ŒulvasÂtras, MÀnava and KÀtyÀyana, 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 KÀtyÀyana ŒulvasÂtra assumes significance. Coming at the tail-end of the ŒulvasÂtras period, after which the ŒulvasÂtras 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 ŒulvasÂtras 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.
 

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’Ab KÀmil à Fermat, par Roshdi Rashed, Berlin, New York : Walter de Gruyter, 2013, X-349 p.
 

Abstract
We comment on some combinatorial aspects of Nasir al-Din al-Tusi Treatise on the Quadrilateral, a 13th century work on spherical trigonometry.
 

The Mathematics of India : Concepts, Methods, Connections by P. P. Divakaran
Reviewed by Satyanad Kichenassamy

Karanaapaddhati 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

Abstract
With Apuleius and Augustine as the only partial exceptions, Latin Antiquity did not know Archimedes as a mathematician but only as an ingenious engineer and astronomer, serving his city and killed by fatal distraction when in the end it was taken by ruse. The Latin Middle Ages forgot even much of that, and when Archimedean mathematics was translated in the 12th and 13th centuries, almost no integration with the traditional image of the person took place. Petrarca knew the civically useful engineer and the astrologer (!); no other fourteenth-century Humanist seems to know about Archimedes in any role. In the 15th century, however, “higher artisans” with Humanist connections or education took interest in Archimedes the technician and started identifying with him. In mid-century, a new translation of most works from the Greek was made by Jacopo Cremonensis, and Regiomontanus and a few other mathematicians began resurrecting the image of the geometer, yet without emulating him in their own work. Giorgio Valla’s posthumous De expetendis et fugiendis rebus from 1501 marks a watershed. Valla drew knowledge of the person as well as his works from Proclus and Pappus, thus integrating the two. Over the century, a number of editions also appeared, the editio princeps in 1544, and mathematical work following in the footsteps of Archimedes was made by Maurolico, Commandino and others. The Northern Renaissance only discovered Archimedes in the 1530s, and for long only superficially. The first to express a (purely ideological) high appreciation was Ramus in 1569, and the first to make creative use of his mathematics was Viète in the 1590s.
 

Abstract
The idea of the principle of nested intervals, or the concept of convergent sequences which is equivalent to this idea, dates back to the ancient world. Archimedes calculated the unknown in excess and deficiency, approximating with two sets of values: ambient and nested values. J. Buridan came up with a concept of a point lying within a sequence of nested intervals. P. Fermat, D. Gregory, I. Newton, C. MacLaurin, C. Gauss, and J.-B. Fourier used to search for an unknown value with the help of approximation in excess and deficiency. In the 19th century, in the works of B. Bolzano, A.-L. Cauchy, J.P.G. Lejeune Dirichlet, K. Weierstrass, and G. Cantor, this logical construction turned into the analysis argumentation method. The concept of a real number was elaborated in the 1870s in works of Ch. Méray, Weierstrass, H.E. Heine, Cantor, and R. Dedekind. Cantor’s elaboration was based on the notion of a limiting point and principle of nested intervals. What discuss here the development of the idea starting from the ancient times.

Abstract
The CandravÀkyas of MÀdhava give the true longitude of the Moon for each day of an anomalistic cycle of 248 days. These coincide with the computed values within an error of one second. Traditionally, a vÀkyaœodhana (exculpating vÀkyas) has been prescribed to check the correctness of the numerical values given by the vÀkyas, if there is any doubt about any of them. We are not aware of any explanation for this procedure in any commentary. In this article, we provide an explanation for the vÀkyaœodhana - procedure, based on the traditional trairÀœika or the “rule of three” procedure.
 

Abstract
Madhyahnakalalagna is the time interval between the rise of the equinox and the instant when a star with a non zero latitude is on the meridian. Algorithms for finding the Madhyahnakalalagna are given in the text Karaõapaddhati of Putumana Somayaji. These have no equivalents in the other Kerala astronomical works. Only a person with a great deal of insight into the subject of spherical trigonometry could have arrived at these algorithms. In this paper, we present four algorithms to find the Madhyahnakalalagna as described in the text. We also provide the detailed derivation of two of them, adopting the method of Yuktibhasha for such problems, as it is very likely that the author would have followed such a method.
 

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.
 

Abstract
NÁlakaõÇha’s commentary on °ryabhaÇÁya 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 GaõitapÀda, wherein NÁlakaõÇha explains the method for finding the square root of a number, focusing on the development of ideas and the thought process. Here NÁlakaõÇha deals, at length, with many of the rationales and the concepts involved. The explanation given by NÁlakaõÇha for BaudhÀyana’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.
 

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.
 

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.
 

Abstract
“Embedding” as a technical concept comes from linguistics, more precisely from grammar. The present paper investigates whether it can be applied fruitfully to certain questions that have been investigated by historians (and sometimes philosophers) of mathematics:
1. The construction of numeral systems, in particular place-value and quasi place-value systems.
2. The development of algebraic symbolisms.
3. The discussion whether “scientific revolutions” ever take place in mathematics, or new conceptualizations always include what preceded them.
A final section investigates the relation between spatial and linguistic embedding and concludes that the spatio-linguistic notion of embedding can be meaningfully applied to the former two discussions, whereas the apparent embedding of older within new theories is rather an ideological mirage.
 

Abstract
We discuss the history of the famous Rolle’s theorem “If a function is continuous at [a, b], differentiable in (a, b), and f (a) = f (b), then there exists a point c in (a, b) such that f’(c) = 0”, and that of the related theorem on the root interval, “If a function is continuous on [a, b] and has different signs at the ends of the interval, then there exists a point c in (a, b) such that f (c) = 0”.
 

Abstract 
One of the great and unique achievements of Sanskrit mathematical astronomy is its wealth of ingenious approximation formulas to substitute for laborious trigonometric computations. This paper examines some intriguing and highly sophisticated examples of such approximations in the twelfth-century work of Bhaskara (II, Bhaskaracarya).
 

Abstract
Many astronomical works in India, like those in other intellectual disciplines, were composed in beautiful verses. While most studies focus on the technical contents of these verses, very few have examined the poetic features, here known as alaôkÀra, that the authors employed to add poetic charm to their treatises. We consider the use of such alaôkÀras in the Gola chapter of a seventeenth century work in Sanskrit astronomy, the SarvasiddhÀntarÀja of Nityananda, and, by highlighting several examples, we examine the ways in which specific embellishments have been woven into the text to make the medium of communication as beautiful as the content.
 

Abstract
We survey the work of Roshdi Rashed, the Egyptian-French historian of mathematics. Surveying Rashed’s work gives an overview of the most important part of Greek mathematics that was transmitted to us in Arabic, as well as of the finest pieces of Arabic mathematics that survive.
 

Abstract 
Bhaskara’s works incorporate most of the results and methods of mathematics and astronomy in India in his times, and carry them forward significantly. There are clear explanations and proofs of the assertions in the verses in the main texts in his own commentaries on them. This article provides a bird’s eye view of his works.
 

Abstract
Vyatipata and Vaidhrta occur when the magnitudes of the declinations of the Sun and the Moon are equal, and one of them is increasing, while the other is decreasing. In this paper, we discuss the calculations associated with them in the patadhikara in the Grahaganita part of Siddhantsiromani. Some of these are similar to the computations in Brahmasphutasiddhanta and Sisyaddhidatantra, but the computation of the golasandhi appears here for the first time. We also compare Bhaskara’s procedures with the ones in Tantrasangraha (c. 1500 CE).
 

Abstract
In this paper we present the interesting phenomena of the retrograde motion of taragrahas as also the visibility of Budha (Mercury) and Sukra (Venus) in the eastern and western horizons.
Bhaskaracarya in his astronomical works has dwelt at length on these phenomena and provided the relevant critical and stationary points. WE work out the details in the case of the two interior planets and compare the results with modern ones.
 

Abstract
We will be presenting briefly the procedure of determining the mean and true positions of the Sun, the Moon and the tÀrÀgrahas (planets) according to KaraõakutÂhala of BhÀskara II. We work out the true planetary positions for a contemporary date of BhÀskara’s period and compare the results with those of a couple of other traditional texts and modern procedures for validation of BhÀskara’s procedures and parameters.
 

Abstract
The well-known treatises of Bhaskara II or Bhaskaracarya (b.1114) are unanimously recognized as canonical in Sanskrit mathematics and mathematical astronomy, but the specific details of their influence on later works remain largely unexplored (partly because most of those later works themselves still await comprehensive study). This article examines a few texts from the sixteenth to eighteenth centuries whose authors were familiar with some aspects of Greco-Islamic astronomy and mathematics, and discusses their continued use of BhÀskara’s works as a model.
 

Abstract
Bhaskaracarya’s treatment of the concept of infinity in his Algebra book is strikingly different and his corresponding exercises are sometimes criticized as erroneous. We discuss his ideas and propose a new explanation in terms of an extended number system with idempotents.
 

Abstract
Numerous systems of units were developed in India for lengths, angles, areas, volumes, time or weights. They exhibit common features and a continuity sometimes running from Harappa to BhÀskarÀchÀrya, but also an evolution in time and considerable regional variations. This paper presents an overview of some issues in Indian metrology, especially with regard to units of length and weight, some of which are traceable all the way to the Indus-Sarasvati civilization. It discusses, among others, the aôgula and its multiple variations, and the value of yojana and its impact on calculations for the circumference of the Earth.
 

Abstract
Among the various techniques that are employed in determining the variation in the length of day (LOD), the recorded observations of ancient eclipses play a crucial role, particularly for estimating variations in the remote past. Scholarly investigations of these records preserved in different cultures around the world, for the above purpose, have completely ignored the Indian record of historical eclipses on the presumption that “no early records appear to be extant”. Consequently, estimates of the variations in LOD are entirely based on the records of only a few civilisations - Arabia, Babylon, China, Europe, and lately Japan and Korea. In this paper we aim to show that this presumption is ill informed, and that Indian records of historical eclipse observations are reasonably well extant. We also provide a few examples of eclipses recorded in India which maybe useful for finding the variation in LOD (?T).
 

Abstract
Since lunar and solar parallax play a crucial role in predicting solar eclipses, the focus of this paper is on the computation of parallax. A brief history of parallax computation in India and China is traced. Predictions of solar eclipses based on NÁlakaõÇha’s Tantrasaôgraha are statistically analyzed. They turn out to be remarkably accurate, but there is a pronounced bias towards predicting false positives rather than false negatives. The false positives occur more to the south of the ecliptic at northerly terrestrial latitudes and more to the north of the ecliptic at southerly latitudes. A very similar bias is found in Chinese astronomy providing another hint at possible links between Indian and Chinese astronomy. The Chinese have traditionally used different values for the eclipse limit north and south of the ecliptic, perhaps to compensate for the southward bias.
 

Abstract
While many of the innovative mathematical techniques developed by medieval Indian astronomers have been studied extensively, much less is known about how they chose to select and apply specific mathematical models to physical phenomena. This paper focuses on the paks.a or astronomical school associated with Brahmagupta (628 CE) and investigates what some of its characteristic features may tell us about the evolution of Indian mathematical astronomy.
 

Abstract
A twelfth century set of astronomical tables, the Brahmatulyasa-ran. , poses some interesting challenges for the modern historian. While these tables exhibit a range of standard issues that numerical data typically present, their circumstances and mathematical structure are further complicated by the fact that they are purported to be a recasting of another work by Bhaskara II that was originally composed in verse, the Karan.akutu- hala (epoch 1183 CE). We explore this relationship by considering the tables for solar declination and lunar latitude and comparing them to their textual counterparts.
 

Abstract
It was well known even before Bha-skara-ca -rya that the daily motion of a planet would vary from day to day. In his Siddha-nta´siroman. i, Bha-skara-II observes that the rate of motion would vary even during the course of a day, and discusses the concept of an instantaneous rate of motion, which involves the derivative of the sine function. We discuss how Bha-skara could have arrived at the correct expression for this, and also how it was applied to find the ‘apogee’ of a planet.
 

Abstract
We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler’s use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical results, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell’s Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid’s Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps.
 

Abstract
The paper reviews Sawai Jai Singh’s (1688-1743) efforts to revive astronomy in his domain. For this reviving, he erected observatories, designed instruments of masonry and stone, assembled a team of astronomers of different schools of astronomy such as the Hindu, Islamic and European, and finally sent a fact finding scientific delegation to Europe. Jai Singh did not succeed in his efforts. The paper explains that poor communications of his times and a complex interaction of intellectual stagnation, religious taboos, theological beliefs, national rivalries and simple human failings were responsible for his failure.
 

Abstract
The memoir Theorie der Parallellinien (1766)* by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author’s aim was, like many of his predecessors’, to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean axioms are kept except the parallel axiom and where the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print.
 

Abstract
We report on the work of Ibn al-Haytham, an Arabic scholar who had settled in Cairo in the eleventh century, and worked in several fields, including mathematics, physics and philosophy. We review some of his work on optics, astronomy, number theory and especially spherical geometry. Our report is mostly based on the books published by Roshdi Rashed, a specialist on Ibn alHaytham and the world expert on Arabic and Greek mathematics and their interaction. We also provide a report on the life of Ibn al-Haytham, his influence, and the general background in which he flourished. The year 2015 has been declared by the UNESCO the “International Year of Light”, and one reason is that we celebrate this year the thousandth anniversary of Ibn Al-Haytham’s fundamental work on optics, KitÀb al-ManÀzir.
 

Abstract
Otto Hölder (1859-1937) was a many sided German mathematician and he worked in diverse areas like Analysis, Group Theory, Mathematical Mechanics, Geometry, Foundational questions in Mathematics and Number theory. The aim of this article is to give a brief sketch of his life and work, paying tribute to this mathematician for his manifold contributions to various branches of mathematics, and how in spite of his remarkable achievements he had to face many obstructions in his academic life.
 

Abstract
The celebrity Indian mathematician trio ŒrÁdhara, MahÀvÁra and NÀrÀyaõa Paõçita elaborates on some identities which are either algebraic sums of numbers with the series on natural numbers or sums of series of natural numbers. Series involving arithmetic and geometric progressions are also found discussed. In this article we discuss these identities and the series, explaining the format of each one followed by their formulation. The rationale for the nontrivial results is provided. The results are amended and extended when necessary.