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Three papers by Christoph J Scriba
We give below versions of three of Christoph J Scriba's papers. These come from very different parts of Scriba's career. The first, although published ten years after his unpublished doctoral thesis, nevertheless is based on that thesis.
1. Christoph J Scriba, Zur Entstehung der Royal Society, Sudhoffs Archiv 52 (3) (1968), 269-271.
The second paper, in memory of the Norwegian number theorist and historian of mathematics, Viggo Brun, was written when Scriba was in the main part of his career as head of the Institute for the History of Science at the University of Hamburg.
2. Christoph J Scriba, VIGGO BRUN IN MEMORIAM, Historia Mathematica 7 (1980), 1-6.
The third paper was written one year after Scriba retired for the 60th birthday of the historian of mathematics Radha Charan Gupta.
3. Christoph J Scriba, A Birthday Tribute to R C Gupta, Historia Mathematica 23 (2) (1996), 117-120.
Paper 1.
Zur Entstehung der Royal Society, by Christoph J Scriba, Hamburg.
On the origins of the Royal Society. The Royal Society was founded at a time when England was shaken by major political and ecclesiastical conflicts. The struggle for supremacy between the king and Parliament had led to civil war, which came to a temporary end in 1649 with the execution of Charles I and the establishment of the Commonwealth by Cromwell. Soon after Cromwell's death, who had also failed to govern in coalition with Parliament, the monarchy was finally restored in 1660 with the accession of Charles II. On 28 November of the same year a group of London citizens, who had been meeting regularly for some time to discuss scientific problems, decided to found a society, prompted by a proposal "of founding a College for the Promoting of Physico-Mathematical Experimental Learning." On 15 July 1662, this society was granted the Royal Society charter; since receiving a second, improved charter on 22 April 1663, it has borne the name The Royal Society of London for Promoting Natural Knowledge.
However, only a few years after its founding, the society found itself faced with the necessity of responding to malicious attacks and defending its own position, its work, and its aims. This task was undertaken, under the guidance of various committees, by Thomas Sprat, a young cleric who had not personally witnessed the events leading up to its founding In 1667, he published a book, "The History of the Royal Society", in which he identified the beginning of the meetings as the period after the end of the Civil War, naming Wadham College, Oxford, as their original location, where Warden John Wilkins had played a prominent role. Eleven years later, in a polemical pamphlet, John Wallis pushed the origins back several years, to around 1645, when similar meetings were already taking place in London, in which he himself, as well as Wilkins, had participated. According to Wallis, the Oxford group of the 1650s was merely an offshoot of the London group, formed after several participants had moved to Oxford since 1648. With some variations, Wallis described these events again in his autobiography, written in 1697, which, however, was not printed until 1725, long after his death, when none of the original participants were still alive.
Up to now, the contradiction between the accounts of Sprat and Wallis has usually been explained by pointing to Sprat's close relationship with Wilkins, in whose college he had studied. Therefore, Sprat, presumably without malicious intent, did not mention the older London circle. It is the thesis of a recently published book by Margery Purver [1] (already expressed in the author's unpublished Oxford doctoral dissertation and first published in a small pamphlet in 1960) that it was Sprat who described the situation correctly, not Wallis. The existence of the London group is not denied, but an attempt is made to demonstrate that it had nothing to do with the Oxford group.
This can only be done indirectly: Miss Purver therefore presents a detailed summary, enriched with many quotations, of the ideas of Lord Chancellor Bacon, dispelling some long-held misunderstandings. She then shows how the Oxford Circle, and subsequently the Royal Society that emerged from it and was founded in London following the return of several members, saw itself as the society that sought to realise Bacon's visionary, though by then diluted by popularisation, concept of a "new philosophy" in its original sense. She sees the unique character of the Royal Society and its Oxford predecessor in this distinction from the many similar practical, utilitarian, and humanitarian endeavours of the time, and in the emphasis on the "new philosophy," which was more than just a new method. Further chapters on the Royal Society's relationship to Gresham College, Boyle's "Invisible College," Comenius and Hartlib's Pansophia, and religious questions supplement her argument, which is - certainly deliberately - one-sided, but likely intended precisely to challenge the opposing viewpoint.
The reaction to this book will show whether, among other things, the political, religious, and perhaps also the social conditions must be given greater consideration than has been done here. Finally, the further question arises whether the first meetings, including those of the Oxford Circle under Wilkins, were not perhaps far less fervently committed to the concept elaborated by Miss Purver than she attempts to portray them; for it is quite conceivable that only the attacks that followed the founding and recognition prompted the Fellows to reflect more deeply and formulate more concisely the actual aim of their work.
References.
1. Margery Purver, The Royal Society: Concept and Creation. With an introduction by H R Trevor-Roper (London: Routledge and Kegan Paul 1967).
Paper 2.
VIGGO BRUN IN MEMORIAM (1885-1978), by Christoph J Scriba, University Of Hamburg, Hamburg, Federal Republic Of Germany.
On 15 August 1978, Professor Viggo Brun died at his home at Drøbak, Norway, in his 92nd year. In several ways he was an extraordinary mathematician. Before World War I, when he first began to attack Christian Goldbach's hypothesis of 1742 that every even natural number (greater than 2) can be expressed as the sum of two prime numbers, and also the question whether there exist infinitely many prime twins (such as 5,7; 11,13; 17,19) [1], most mathematicians were doubtful that such elementary methods as the now famous "Brun's sieve" could produce anything more than trivial results. Edmund Landau, the systematiser of number theory, left Brun's manuscript untouched in his drawer for six years. Even after Brun had shown in 1919 [2] by steadfastly refining his methods that the sum of the reciprocals of twin primes is finite (while the sum of the reciprocals of all primes is infinite) - so that there cannot exist "too many" prime twins - only a few experts tried to follow his line of thought.
In the late forties Berkeley Rosser developed an important refinement of Brun's sieve but by a strange coincidence his discovery remained unpublished for more than twenty years and hence could not produce an effect. A new impetus was given to sieve methods after World War II by Atle Selberg. It was taken up in the fifties by A I Vinogradov, B V Levin, M B Barban; in the sixties by N C Ankeny and H Onishi, A A Buchstab, W B Jurkat, and H-E Richert. In 1974, six decades after Brun's first work, H Halberstam and H-E Richert published their London Mathematical Society monograph "Sieve Methods" (Academic Press, London/New York). In the introduction they express hope that their exposition (to which I owe some of the information given above) of Brun's sieve demonstrates "how powerful and fruitful a method it is even today."
While it is still not known whether there exist infinitely many twin primes and whether Goldbach's conjecture is true, V Brun's early results [3] already included the following:
(i) there exist infinitely many integers such that both and have at most nine prime factors;
(ii) every sufficiently large even integer is the sum of two numbers each having at most nine prime factors.
Brun had obtained these powerful results by means of skilful improvements of a sieve method that goes substantially beyond his historic source: the sieve of Eratosthenes (3rd century B.C.).
The idea is to remove from a given set of integers all those that are divisible by a certain set of primes; the number of those integers left over must then be counted by suitable combinatorial procedures. Many improvements of (i) and (ii) based on Brun's sieve have been obtained in recent years. Though in addition the authors had to employ other arithmetical devices; this shows how fruitful Brun's ideas have been and still are since they have found access into the repertoire of methods in number theory.
Another topic which interested Brun throughout his life was also rooted in ancient Greek mathematics: the theory of higher-dimensional continued fractions. Taking the Euclidean Antaneiresis method ("taking away in turn," Euclidean Algorithm) for a start, he generalised it in 1919/1920 to three [4] and later to four and more numbers. Brun's algorithm is distinguished from a similar one introduced by Jacobi by the fact that it only employs repeated subtraction, no divisions, and that it does not suffer from the setbacks of a related algorithm by Poincaré.
Among applications, there is one of particular interest related to musical theory, studied by V Brun in several papers [5], [6], [7]. If one wants to divide the octave (ratio 2:1) by intermediate tones in such a way that each one has the same ratio to the previous one and that the simple ratios 3:2 and 4:3 are included among the possible chords, one has to determine numbers , and such that , or
Brun's algorithm yields and , which corresponds to our normal division of the octave: there are 12 half-tone steps of which the 7th is the quint, the 5th the quart (with ratios 3:2 and 4:3, approximately). If one wants to include, besides 2:1, the ratios 7:4, 6:4, and 5:4 among the tones of an octave, one arrives for instance at the solution . This is Huygens' suggestion for a temperament with 31 fifth-tone steps to the octave. When A D Fokker obtained this solution again, empirically, in 1947, he stated that mathematicians were not able to give a derivation - not knowing that Brun had studied just such an algorithm of three and four dimensions which would easily produce the desired results. Brun himself called it an irony of fate that he, who was very unmusical, should write about music.
Though not musically gifted, Viggo Brun had a strong sense for harmony and geometric symmetry. He gave some lectures about mathematics and aesthetics, and he published a number of papers which testify to his lively geometric intuition. This enabled him [8] to give a geometric representation of Leibniz's series
using tangent functions in polar coordinates: ; the resulting rosette-like picture is composed of sicle-formed parts enclosing each other and showing the symmetry of the cyclic group . In this connection Brun quoted a sentence from Leonardo da Vinci's Trattato della pittura: "If you, historians, or poets, or mathematicians had not seen things with your eyes you could not report them in writing."
A fine example of what he did see with his eyes is to be found in the closing words of another of Brun's charming little articles [9]. After having described the truncation of a cube (by cutting off the eight corners) to a cube-octahedron (one of Archimedes' semiregular polyhedrons), and that of a tetrahedron to an octahedron which in turn also reduces to a cube-octahedron when its corners are chopped off, Brun concluded: "If therefore two stones, one being a cube and the other a regular tetrahedron, should roll for some millions of years in the wells at the coast, their shape would probably be more and more equal."
It will not surprise the reader that the mathematician Viggo Brun whose work was so deeply rooted in the history of mathematics also published a number of biographical and historical studies. These include biographies of Caspar Wessel [10], Sophus Lie [11], Axel Thue [12], and Carl Størmer [13], and two books: Regnekunsten i det gamle Norge [14], and Alt er tall. Matematikkens historie fra Oldtid til Renessanse [15]. The latter is the text of a course given at Oslo University in 1961 under the Pythagorean motto "all is number." The former was provoked by a remark about Abel in which this great Norwegian mathematician was not considered the product of development but compared to the burst of a bomb. This remark caused the author to inquire into the art of reckoning in old Norway from the earliest times to about 1800. One must regret that these books are only available in the Norwegian Language, so that they could reach but a limited public. Perhaps the most impressive of Brun's historical discoveries was made in 1952 during a trip to Italy, where he rediscovered in Florence Abel's famous manuscript which had disappeared in 1826 after submission to the French Académie des Sciences! Convinced that a sound knowledge of the history of mathematics is of great importance for future teachers of mathematics, he pleaded for the creation of a professorship in the history of mathematics at every large university [16].
In 1955 I met Professor Brun at the second conference on the history of mathematics at Oberwolfach. His wide interests and his humanistic outlook, combined with his modesty, kindness, and totally un-professorial habitus made the strongest impression upon me. This, our only meeting, was also the beginning of an intermittent correspondence, in which Professor Brun's last sentences (written shortly after his 90th birthday - after he had just undergone an operation) were the following: "Upon my return, I hope to be able to resume my mathematical work. After all, I have enjoyed a happy old age so far."
Only after his death did I learn that Viggo Brun, who was born on 13 October 1885, in Lier as the youngest of ten children, had lost both his parents at a very young age. His father, Søren Martens Brun, had been a captain of the artillery, his mother was Lorentze Thaulow, née Petersen. Viggo grew up under the care of his elder sisters, then went to the University of Oslo in 1903. Owing to the regulations of the teacher examination in those days, he had to study a very broad spectrum of mathematical and scientific areas, which left no time for specialisation. In later life he always enjoyed more his own research than reading mathematical literature. So, as a mathematician, he was more a scout than a cultivator of land that had already been made arable by others.
In 1910 Viggo Brun undertook a journey to Göttingen, and it was at that time that he began to attack some of the most difficult problems in number theory with his original methods. In contrast to the analytical methods then in vogue, the sieve method was called elementary, although in its details it was anything but easy and simple. After returning from Göttingen, Brun received a research grant for some years and served in the armed forces. In 1921 he became assistant for applied mathematics in Oslo, in 1923 professor at the Technical University in Trondheim, and in 1946 professor at the University of Oslo. In 1955, at the age of 70, he retired.
Professor Brun married Laura Michelsen, who had a school for dressmaking in Oslo, in 1940. They lived in an ancient wooden house in Drøbak, which had been built in 1770 and which his father had bought in 1888. In recent years he and his wife were active in helping to preserve those old houses in Drøback. He loved to go for walks in the forest where he would collect curious pieces of wood or roots from which he produced nice handiwork.
During his long and fruitful life, in which he was also engaged in politics and worked for peace, Professor Brun received many honours for his mathematical work: the Fridtjof Nansen prize in 1939, the Norges Tekniske Högskoles "Högskolefondets prize" in 1946, the Gunnerus medal of Det Kongelige Norske Videnskabers Selskab in 1958, and an honorary doctorate from the University of Hamburg in 1966. He was a member of the scientific societies or academies of Oslo, Trondheim, Uppsala, and Finland.
His research during the last years was inspired by the study of virus structures as they had been revealed by the electron microscope. Twenty years earlier Brun had already become interested in the similarities between the structures of pollen grains and semiregular (Archimedean) polyhedra [17]. He even baked spherical breads from 32 conical pieces of dough (in a spherical cake tin) and found that normally the surface will show a structure of 12 pentagons and 20 hexagons, which - though not completely regular - resembles a truncated icosahedron. The photograph shows the Norwegian mathematician a few days before his 90th birthday, sitting at his desk under a portrait of Abel and surrounded by the objects of his last article [18]. His name will be mentioned by mathematicians as long as they remember that of Eratosthenes. Those who had the good fortune to meet him and "to see him with their eyes" will not be able to forget this extraordinary man.
References.
1. Über das Goldbachsche Gesetz und die Anzahl der Primzahl-paare, Archive for Mathematik og Naturvid. В 34 (1915), No. 8, 19 pp.
2. ... La série + ou les denominateurs sont "nombres premiers jumeaux" est convergent ou finie, Bulletin des Sciences Mathématiques (2) 43 (1919), 100-104, 124-128.
3. Le crible de Eratosthène et le théorème de Goldbach, Skrifter Norske Videnskabers-Akademie Kristiana I (1920), No. 3.
4. En generalisation av kjedbrøken, I, II, Norske Vid. Selsk. Skrifter, Math.-Natur. Klasse, Oslo (1919), No. 6; (1920), No. 6.
5. Music and ternary continued fractions, Det Kongelige Norse Videnskabers Selskabs Rorhandlinger 23 (1950), No. 10.
6. Musikk og Euklidske algoritmer, Nordisk Matematisk Tidskrift 9 (1961), 29-36.
7. Euclidean algorithms and musical theory, L'Enseignement Mathématique 10 (1964), 125-137.
8. Leibniz' formula for π deduced by a "mapping" of the circular disc, Nordisk Matematisk Tidskrift 18 (1970), 73-81.
9. An application of a "carpenter's curve" to Simpson formulas, Nordisk Matematisk Tidskrift 7 (1959), 20-24 (with "A note on the preceding paper by V Brun" by E S Selmer, pp. 25-26).
10. Caspar Wessel et l'introduction géométrique des nombres complexes, Revue d'Histoire des Sciences 12 (1959), 19-24
11. Sophus Lie. Eine biographische Skizze, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 31 (1967), 119-124.
12. Axel Thue, In Axel Thue: Selected mathematical papers, pp. XV-XXV. Oslo, 1977.
13. Carl Størmer in memoriam, Nordisk Matematisk Tidskrift 5 (1957), 169-175.
14. Regnekunsten i det gamle Norge, Bergen: Universitetsforlaget. Oslo, 1962.
15. Alt er tal. Matematikkens historie fra Oldtid til Renessanse, Bergen: Universitetsforlaget Oslo, 1964.
16. Visst behövos undervisning i matematikens historica! [Certainly one needs an instruction in the history of mathematics!] Dagens Nyheter [a Swedish newspaper], August 8, 1970.
17. Fritzsches Tegninger av visse pollenkorn sammenlignet med et Arkimedespolyeder, Blyttia 13 (1955), 85-89 (with "Noen bemerkninger om pollenkornene av Gomphrena globosa og Chrysanthemum carinatum" by S. Manum, pp. 90-95).
18. On some problems in solid geometry inspired by virology, Nordisk Mat. Tidsskrift (normat), 25/26 (1978), 113-119 (1979).
Paper 3.
A Birthday Tribute to R C Gupta, by Christoph J Scriba.
The internationally renowned historian of mathematics, Radha Charan Gupta, celebrated his 60th birthday on 26 October 1995. [We note that in fact R C Gupta was born on 14 August 1935 but his High School Certificate wrongly gave 26 October 1935 and somehow this incorrect date appeared in all official records from that time.] According to custom in his native India, this implies official retirement from his post as Professor of Mathematics at the Birla Institute of Technology (BIT) in Mesra, Ranchi. His many friends, as well as the many colleagues who have been in contact with him, realise, however, that "retirement" and "official retirement" are two different things; they look forward to benefiting from the continuing labours of this active and productive historian of mathematics towards the promotion of the discipline both in his native country and on an international level.
Born in Jhansi, Uttar Pradesh, Gupta graduated from Lucknow University in 1955. Two years later, he passed the M.Sc. examination (with a major in mathematics) in the first rank from the same university, and in 1965, he earned a diploma in mechanical engineering from the School of Careers in London. Ranchi University awarded him a Ph.D. for his research in the history of mathematics in 1971. His achievements were further acknowledged in 1986 with an honorary doctorate in the history of science from the World University (U.S.A.).
After teaching at Lucknow Christian College for a year, he joined the staff of the Birla Institute of Technology in 1958. He served there in the ranks of first assistant professor and then associate professor prior to his promotion to full professor of mathematics in 1982. Beginning in 1979, he was Professor-in-charge of the Research Centre for the History of Science at BIT.
R C Gupta has travelled widely in India and abroad and has given presentations of his research before many audiences. In 1977, he addressed the British Society for the History of Mathematics at Cambridge and attended the XVth International Congress on the History of Science in Edinburgh. Three years later, he lectured in Germany, the USA, and Canada. He also participated in the XVIIth International Congress of Mathematicians in Berkeley, California, in 1985.
In his several hundred papers - among them a series of 16 popular articles, entitled "Glimpses of Ancient Indian Mathematics" - Gupta has always striven to deepen and broaden our knowledge and understanding of the development of mathematics on the Indian subcontinent. He is thus a successor to his compatriots, B B Datta (1888-1958) and A N Singh (1901-1954). Their important book, History of Hindu Mathematics (Lahore, 1935, 1938), although now some 60 years old, remains a standard reference work for those of us who are unable to read the native Indian languages. It is to be hoped that Gupta's forthcoming book on Indian mathematics (in Hindi) will soon be translated for a wider readership. In the meantime, we can gain from his insights by reading the pages of Ganita Bhāratī, the official journal of the Indian Society for the History of Mathematics, which he founded in 1979 and which he continues to edit. As the cumulative index (in volume 13 (1991)) for volumes 1 through 12 of that journal reveals, Gupta has published many articles, notes, and reviews there under both his own name and the pseudonym "Ganitanand." One recent article, "The Chronic Problem of Ancient Indian Chronology" (Ganita Bhāratī 12 (1990), 17-26), is characteristic of his scholarship. For reasons of space, the selected bibliography below is limited to some of the more extensive papers that Gupta has published in English.
R C Gupta's scientific achievements have received acknowledgement in many ways during his fruitful and ongoing career. Most recently, he was elected Fellow of the National Academy of Sciences in India in 1991, President of the Association of Mathematics Teachers of India in 1994, and Corresponding Member of the International Academy of the History of Science in 1995. He has also represented India on the Executive Committee of the International Commission on the History of Mathematics for many years. An active sportsman, he has won numerous medals and prizes for his athletic prowess. May his health continue and enable him to pursue his researches in the history of mathematics for many years to come.
Selected Bibliography of Radha Charan Gupta by Takao Hayashi.
1. Bhāskara I's Approximation to Sine, Indian Journal of History of Science 2 (1967), 121-136.
2. Second-Order Interpolation in Indian Mathematics up to the Fifteenth Century, Indian Journal of History of Science 4 (1969), 86-98.
3. Fractional Parts of Aryabhata's Sines and Certain Rules Found in Govindasvāmin's Bhāṣya on the Mahābhāskarīya, Indian Journal of History of Science 6 (1971), 51-59.
4. Early Indians on Second-Order Sine Differences, Indian Journal of History of Science 7 (1972), 81-86.
5. An Indian Form of Third-Order Taylor Series Approximation of the Sine, Historia Mathematica 1 (1974), 287-289.
6. Solution of the Astronomical Triangle As Found in the Tantrasangraha (A.D. 1500), Indian Journal of History of Science 9 (1974), 86-99.
7. Sines and Cosines of Multiple Arcs As Given by Kamalākara, Indian Journal of History of Science 9 (1974), 143-150.
8. Addition and Subtraction Theorems for the Sine and the Cosine in Medieval India, Indian Journal of History of Science 9 (1974), 164-177.
9. Some Important Indian Mathematical Methods As Conceived in Sanskrit Language, Indological Studies (Journal of the Department of Sanskrit, University of Delhi) 3 (1974), 49-62.
10. Circumference of the Jambūdvīpa in Jaina Cosmography, Indian Journal of History of Science 10 (1975), 38-46.
11. Sine of Eighteen Degrees in India up to the Eighteenth Century, Indian Journal of History of Science 11 (1976), 1-10.
12. Parameśvara's Rule for the Circumference of a Cyclic Quadrilateral, Historia Mathematica 4 (1977), 67-74.
13. On Some Mathematical Rules from the Aryabhațīya, Indian Journal of History of Science 12 (1977), 200-206.
14. Indian Values of the Sinus Totus, Indian Journal of History of Science 13 (1978), 125-143.
15. Munīśvara's Modification of Brahmagupta's Rule for Second-Order Interpolation, Indian Journal of History of Science 14 (1979), 66-72.
16. Square Root of 164 in the Berlin Papyrus 11529, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 2 (1980), 29-31.
17. Indian Mathematics and Astronomy in the Eleventh Century Spain, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 2 (1980), 53-57.
18. Bibhutibhusan Datta (1888-1958), Historian of Indian Mathematics, Historia Mathematica 7 (1980), 126-133.
19. The Marīci Commentary on the Jyotpatti, Indian Journal of History of Science 15 (1980), 44-49.
20. The Process of Averaging in Ancient and Medieval Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 3 (1981), 32-42.
21. A Bibliography of Selected Sanskrit and Allied Works on Indian Mathematics and Mathematical Astronomy, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 3 (1981), 86-102.
22. Indian Mathematics Abroad up to the Tenth Century A.D., Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 4 (1982), 10-16.
23. Candū, an Astronomer of Medieval Rajasthan, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 4 (1982), 134-135.
24. Decimal Denominational Terms in Ancient and Medieval India, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 5 (1983), 8-15.
25. Spread and Triumph of Indian Numerals, Indian Journal of History of Science 18 (1983), 21-38.
26. On Some Ancient and Medieval Methods of Approximating Quadratic Surds, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 7 (1985), 13-22.
27. Jinabhadra Gaņi and Segment of a Circle between Two Parallel Chords, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 7 (1985), 25-26.
28. On Derivation of Bhāskara I's Formula for the Sine, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 8 (1986), 39-41.
29. Some Equalization Problems from the Bakhshālī Manuscript, Indian Journal of History of Science 21 (1986), 51-61.
30. Mādhavacandra's and Other Octagonal Derivations of the Jaina Value π = √10, Indian Journal of History of Science 21 (1986), 131-139.
31. South Indian Achievements in Medieval Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 9 (1987), 15-40.
32. On the Date of Śrīdhara, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 9 (1987), 54-56 (under the pen name Ganitanand).
33. Mādhava's Rule for Finding the Angle between the Ecliptic and the Horizon and Aryabhata's Knowledge of It, in History of Oriental Astronomy (Cambridge: Cambridge Univ. Press, 1987), 197-202.
34. Chords and Areas of Jambūdvīpa Regions in Jaina Cosmography, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 9 (1987), 51-53, and 10 (1988), 124.
35. On the Values of π from the Bible, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 10 (1988), 51-58.
36. Tombstone Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 10 (1988), 69-74.
37. Volume of a Sphere in Ancient Indian Mathematics, Journal of the Asiatic Society 30 (1988), 128-140.
38. On Some Rules from Jaina Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 11 (1989), 18-26.
39. Sino-Indian Interaction and the Great Chinese Buddhist Astronomer-Mathematician I-Hsing (A.D. 683-727), Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 11 (1989), 38-49.
40. The Laksa Scale of the Valmiki Rāmāyaṇa and Rāma's Army, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 10-16 (under the pen name of Ganitanand).
41. The Chronic Problem of Ancient Indian Chronology, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 17-26.
42. A Few Remarks concerning Certain Values of π in Ancient India, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 33-38 (under the pen name of Ganitanand).
43. The Value of π in the Mahābhārata, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 45-47.
44. Sudhākara Dvivedī (1855-1910), Historian of Indian Astronomy and Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 83-96.
45. An Ancient Approximate Rule for the Area of a Polygon, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 12 (1990), 108-112.
46. The 'Molten Sea' and the Value of π, The Jewish Bible Quarterly 19 (2) (1990-1991), 127-135.
47. On the Volume of a Sphere in Ancient India, Historia Scientiarum [Japan] 42 (1991), 33-44.
48. The First Unenumerable Number in Jaina Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 14 (1992), 11-24.
49. Varāhamihira's Calculation of , and the Discovery of Pascal's Triangle, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 14 (1992), 45-49.
50. Abu'l Wafa' and His Indian Rule about Regular Polygons, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 14 (1992), 57-61.
51. On the Remainder Term in the Madhava-Leibniz's Series, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 14 (1992), 68-71.
52. Rectification of Ellipse from Mahāvīra to Ramanujan, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 15 (1993), 14-40.
53. A Problem of Interest in the Nārada-purāņa, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 15 (1993), 67-69.
54. Sundararāja's Improvements of Vedic Circle-Square Conversions, Indian Journal of History of Science 28 (1993), 81-101.
55. Six Types of Vedic Mathematics, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 16 (1994), 5-15.
56. A Circulature Rule from the Agnipurāņa, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 16 (1994), 53-56.
57. Areas of Regular Polygons in Ancient and Medieval Times, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 16 (1994), 61-65.
58. Marx and His Mathematical Work, Ganita Bhāratī (Bulletin of the Indian Society for History of Mathematics) 16 (1994), 66-69 (under the pen name Ganitanand).
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