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The Ganita-Sāra-Sangraha of Mahāvīrācārya with English Translations and Notes. Sanskrit text and English translation
... 33 30 The false-position technique employing such substitutions is treated also in, e.g., the Gaṇitasārasaṅgraha of Mahāvīra (ca. 850 CE) ( Raṅgācārya 1912: 40, 62; #107108) and the Līlāvatī of Bhāskara (1149/1150) (Colebrooke 1817: 23; #50-51). We thank Hayashi (2017b) for these references and for his guidance concerning the functional equivalence of śūnyam ekayutaṃ kṛtvā ("having added unity to zero," Bakhshālī sūtra N8) and śūnyasthāne rūpaṃ dattvā ("having put unity in the place of zero," Bakhshālī sūtra N12) in this context. ...
Popular attention has recently been captured by the results of the Bodleian Library's 2017 project of radiocarbon datingportions of the birch-bark fragments constituting what is known as the Bakhshālī Manuscript. In this paper, we disagree with the interpretation of the findings announced by the Bodleian team. In particular, we argue that the earliest dated folio of this manuscript is unlikely to be the date of the whole text. Rather, the latest dateable folio is logically the date of the scribal activity. This fits well with past estimates of the date of the Bakhshālī Manuscript based on historical, philological and palaeographic arguments.. And we argue that the Bakhshālī Manuscript does include written zeros that function as arithmetical operators, i.e., as numbers in their own right, and not merely as place-holders, as asserted by the Bodleian team. Finally, we express regret that the Bodleian Library chose to announce scientific results without peer-review and through a press release to newspapers and a YouTube video.
... This finally brings us to the point where we may approach Mahāvīra's 9thcentury Ganita-sāra-saṅgraha [ed., trans. Raṅgācārya 1912]. ...
In various publications [Høy95, Høy96, Høy01] I have argued for the existence in (what Western Europe sees as) the Near East of a long-lived community of practical geometers - first of all surveyors - which was not or only marginally linked to the scribe school traditions, and which (with branchings) carried a stock of methods and problems from the late third millennium BCE at least into the early second millennium CE. The arguments for this conclusion constitute an intricate web, and I shall only repeat those of them which are of immediate importance for my present concern: the links between the geometrical section of Mahāvīra’s Gaṇita-sāra-saṅgraha and the practical mathematics of the Mediterranean region in the classical ages.
... Whatever is there in all the three worlds, which are possessed of moving and non-moving beings all that indeed cannot exist without mathematics." [10] Mahavira (c817-c875). ...
The existing studies on the origin and history of mathematical economics are euro-centric and cover only the past two centuries. It is intended to show that 1) mathematical economics has an ancient origin. Kautilya wrote The Arthashastra during the fourth century BCE and used discrete marginal analysis and combinatory rules to sharpen economic analysis. 2) It is indicated that in the West, image of mathematics has changed directly and that of economics indirectly as the the-ology/philosophy of the church changed. 3) It is claimed that in India the developments in both economics and mathematics have always been simple, secular and user-friendly to solve problems related to agriculture, construction, navigation and trade.
... The straight line that is bisected is AB, the added line is BD. Raṅgācārya 1912]. Other surviving Indian medieval sources of importance are more closely linked to astronomy, but their contents is so much broader than astronomical computation that it points to the existence of a structure similar to what we find in Mahāvīra. ...
The current paper explores the potential interlink between names of individuals in a society and its collective social consciousness, particularly with reference to the pervasive occurrence of the ‘mathematical names’ in the current Hindu society spanning the Indian subcontinent and beyond. Initially, an attempt is made to put things into mathematical perspective by drawing a quick sketch of some of the stellar achievements of the Indian mathematicians. Under the six broad categories of geometry, trigonometry, numeration, arithmetic, algebra, and mathematics in the Vedic tradition, a concise simple description of these subdivisions is presented, underlining selected mathematical concepts and terms, sometimes by producing the textual references. We hypothesize that such terms permeate as names in the current Hindu society, reflecting the impressions of the tremendously rich mathematical heritage left by the stalwart Hindu mathematicians. Accordingly, an attempt is made to juxtapose these terms with the names current in the Indian Hindu society. By employing an extensive dataset of university student names in India and the directories of Facebook and LinkedIn, we produce both qualitative and quantitative evidence of the presence of such names in the Indian subcontinent. Our hypothesis has also been examined by taking surveys of people bearing these mathematical names, as well as by documenting the ‘conscious procedures’ that go behind the naming of a Hindu Indian child. In trying to investigate if such a phenomenon is unique to the Indian tradition, a stark contrast with the ‘names in mathematics’ as prevalent in the post-renaissance Europe is presented, as cultural roots of mathematics are explored. Evidently, the large magnitude and the span of such names substantiates the presence of these names as the extant remains of the colossal impact of multifarious mathematical traditions existing in India. Interestingly, the present research also brings to the fore, certain unseen facets of the Indian Hindu society as regards the education of mathematics to women – through an indirect exploration of their names. We also show that the pervasive occurrence of these names is not merely the result of semantic chance events, but must denote the richness of the Indian mathematical legacy. By presenting some cross-cultural comparisons, we bring about the specific uniqueness of Indian mathematical and scientific traditions that led to the pervasiveness of ‘mathematical names’ in India vis-à-vis other cultures. Finally, an attempt is made to clarify some subtle points on the associations between mathematics and religion in India and other cultures of the world. It is sincerely hoped that the present study may shed light on the cultural roots of mathematics and may furnish a new dimension in the study of mathematics, culture, and civilizations across the world.
The current paper explores the potential interlink between names of individuals in a society and its collective social consciousness, particularly with reference to the pervasive occurrence of the ‘mathematical names’ in the current Hindu society spanning the Indian subcontinent and beyond. Initially, an attempt is made to put things into mathematical perspective by drawing a quick sketch of some of the stellar achievements of the Indian mathematicians. Under the six broad categories of geometry, trigonometry, numeration, arithmetic, algebra, and mathematics in the Vedic tradition, a concise simple description of these subdivisions is presented, underlining selected mathematical concepts and terms, sometimes by producing the textual references. We hypothesize that such terms permeate as names in the current Hindu society, reflecting the impressions of the tremendously rich mathematical heritage left by the stalwart Hindu mathematicians. Accordingly, an attempt is made to juxtapose these terms with the names current in the Indian Hindu society. By employing an extensive dataset of university student names in India and the directories of Facebook and LinkedIn, we produce both qualitative and quantitative evidence of the presence of such names in the Indian subcontinent. Our hypothesis has also been examined by taking surveys of people bearing these mathematical names, as well as by documenting the ‘conscious procedures’ that go behind the naming of a Hindu Indian child. In trying to investigate if such a phenomenon is unique to the Indian tradition, a stark contrast with the ‘names in mathematics’ as prevalent in the post-renaissance Europe is presented, as cultural roots of mathematics are explored. Evidently, the large magnitude and the span of such names substantiates the presence of these names as the extant remains of the colossal impact of multifarious mathematical traditions existing in India. Interestingly, the present research also brings to the fore, certain unseen facets of the Indian Hindu society as regards the education of mathematics to women – through an indirect exploration of their names. We also show that the pervasive occurrence of these names is not merely the result of semantic chance events, but must denote the richness of the Indian mathematical legacy. By presenting some cross-cultural comparisons, we bring about the specific uniqueness of Indian mathematical and scientific traditions that led to the pervasiveness of ‘mathematical names’ in India vis-à-vis other cultures. Finally, an attempt is made to clarify some subtle points on the associations between mathematics and religion in India and other cultures of the world. It is sincerely hoped that the present study may shed light on the cultural roots of mathematics and may furnish a new dimension in the study of mathematics, culture, and civilizations across the world.
The current paper explores the potential interlink between names of individuals in a society and its collective social consciousness, particularly with reference to the pervasive occurrence of the ‘mathematical names’ in the current Hindu society spanning the Indian subcontinent and beyond. Initially, an attempt is made to put things into mathematical perspective by drawing a quick sketch of some of the stellar achievements of the Indian mathematicians. Under the six broad categories of geometry, trigonometry, numeration, arithmetic, algebra, and mathematics in the Vedic tradition, a concise simple description of these subdivisions is presented, underlining selected mathematical concepts and terms, sometimes by producing the textual references. We hypothesize that such terms permeate as names in the current Hindu society, reflecting the impressions of the tremendously rich mathematical heritage left by the stalwart Hindu mathematicians. Accordingly, an attempt is made to juxtapose these terms with the names current in the Indian Hindu society. By employing an extensive dataset of university student names in India and the directories of Facebook and LinkedIn, we produce both qualitative and quantitative evidence of the presence of such names in the Indian subcontinent. Our hypothesis has also been examined by taking surveys of people bearing these mathematical names, as well as by documenting the ‘conscious procedures’ that go behind the naming of a Hindu Indian child. In trying to investigate if such a phenomenon is unique to the Indian tradition, a stark contrast with the ‘names in mathematics’ as prevalent in the post-renaissance Europe is presented, as cultural roots of mathematics are explored. Evidently, the large magnitude and the span of such names substantiates the presence of these names as the extant remains of the colossal impact of multifarious mathematical traditions existing in India. Interestingly, the present research also brings to the fore, certain unseen facets of the Indian Hindu society as regards the education of mathematics to women – through an indirect exploration of their names. We also show that the pervasive occurrence of these names is not merely the result of semantic chance events, but must denote the richness of the Indian mathematical legacy. By presenting some cross-cultural comparisons, we bring about the specific uniqueness of Indian mathematical and scientific traditions that led to the pervasiveness of ‘mathematical names’ in India vis-à-vis other cultures. Finally, an attempt is made to clarify some subtle points on the associations between mathematics and religion in India and other cultures of the world. It is sincerely hoped that the present study may shed light on the cultural roots of mathematics and may furnish a new dimension in the study of mathematics, culture, and civilizations across the world.
For more than a century, there has been some discussion about whether medieval Arabic al-jabr (and hence also later European algebra) has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 bce, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and later the geometry of Elements II (where the techniques are submitted to theoretical investigation). The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European (Latin and abbaco) mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles.
The first part of the article presents, as necessary background, the development of the Near Eastern surveyors’ riddle tradition, from its pre-Old-Babylonian beginnings, over innovations that may go back to the fifth or fourth century BCE, until those that can be dated to the Seleucid-Demotic periods. It also gives a brief description of its impact on Greek theoretical as well as Pythagoreanizing and so-called practical mathematics.
Sanskrit sources from Āryabhata to Bhāskara II have a standard formulation of the rule of three. However, it is clear that mathematics must also have been spoken of and performed during this period (and before) in vernacular environments, and that the two levels must have interacted - not least because the erudite astronomer-mathematicians use commercial arithmetic as the introduction to mathematics. But we have no surviving vernacular texts.
Until some decades ago, it was customary to discuss much pre-Modern mathematics as “algebra”, without agreement between workers about what was to be understood by that word. Then this view came under heavy fire, rarely with more precision.
Now, instead, it has become customary to classify pre-Modern practical arithmetic as “algorithmic mathematics”. In so far as any computation in several steps can be claimed to follow an underlying algorithm (just as it can be explained from an “underlying theorem”, for instance from proportion theory, or from a supposedly underlying algebraic calculation), this can no doubt be justified. Traditionally, however, historians as well as the sources would speak of a rule.
The paper first goes through some of the formative appeals to the algebraic interpretation – Eisenlohr, Zeuthen, Neugebauer – as well as some of the better argued attacks on it (Rodet, Mahoney).
Next it asks for the reasons to introduce the algorithmic interpretation, and discusses the adequacy or inadequacy of some uses. Finally, it investigates in which sense various pre-modern mathematical cultures can be characterized globally as “algorithmic”, concluding that this characterization fits ancient Chinese and Sanskrit mathematics but neither early second-millennium Mediterranean practical arithmetic (including Fibonacci and the Italian abbacus tradition), nor the Old Babylonian corpus.
This is a preliminary survey of time units used or mentioned in ancient and medieval works written in Sanskrit and other Indian languages. The fields of the works surveyed are jyautiṣa, paurāṇika, uttara-vaidika, smārta, bauddha, and jaina literatures, includingChinese translations of bauddha works. No small portion of the data presented in the following sections has already been taken up and explained in the works mentioned at the end of this section, but I newly collected the data from the original sources and arranged them in my own way according to my own interest.
Geometry is the earliest recorded branch of Indian mathematics. Mathematics of the Vedic period consists of those geometric techniques needed for the construction of the altars and fire-places described by the priestly hereditary class for the performance of their rites. The link between geometry and ritual suggests that mathematical accuracy was considered of the utmost importance in this context. The study of rational figures in the Sanskrit work Gaṇita Sāra Saṃgraha, to which Mahāvīrācārya, a ninth-century ce Jaina mathematician, dedicates a special treatment, reveals striking parallelism with the earlier geometry developed in connection with the Vedic sacrifice. Mahāvīra makes extensive use of the uddeśaka or ‘sample problem’, and I suggest a new way of interpreting the uddeśaka as a significant device for constructing an ‘actual proof’ which validates and links a mathematical rule to its unmentioned premises and provides a system of knowledge based on deductive syllogism.
In 1307, a certain Jacopo da Firenze wrote in Montpellier a Tractatus algorismi that contains the earliest extant algebra in a European vernacular and probably, as is argued, the first algebra in vernacular Italian. Analysis of the text shows that it cannot descend from any of the algebras written in Latin, nor from any published Arabic treatise, for which reason it presents us with evidence for a so far unexplored level of Arabic algebra. Further, since it contains no Arabisms, it must build on an already existing Romance-speaking environment engaged in algebra. Comparison with other Italian algebras written during the next 40 years show that all are linked to Jacopo or to this environment (perhaps Catalan) and disconnected from Leonardo Fibonacci's Liber abbaci.ResumiNel 1307, un certo Jacopo da Firenze scrisse a Montpellier un Tractatus algorismi che contiene la prima presentazione sopravvissuta dell'algebra in un volgare europeo – probabilmente la prima presentazione in volgare italiano in assoluto. L'analisi del testo dimostra che l'algebra di Jacopo non è basata su nessuno dagli scritti algebrici latini, e neanche su un trattato arabo pubblicato; è dunque una testimonianza di un livello finora inesplorato dell'algebra araba. D'altra parte, Jacopo non utilizza un solo arabismo, e deve dunque aver preso la sua ispirazione da un ambiente di lingua romanza. Un'ispezione attenta di altri scritti algebrici italiani risalenti alla prima metà del Trecento svela che tutti sono legati a Jacopo o a questo ambiente (possibilmente catalano) e che nessuno ha legami con il Liber abbaci di Leonardo Fibonacci.
This paper explores some combinatoric problems which are treated in the Sanskrit literature of both ayurveda (medicine) and of jyothsastra (mathematics).
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