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Über dieses Buch

This book includes 58 selected articles that highlight the major contributions of Professor Radha Charan Gupta—a doyen of history of mathematics—written on a variety of important topics pertaining to mathematics and astronomy in India. It is divided into ten parts. Part I presents three articles offering an overview of Professor Gupta’s oeuvre. The four articles in Part II convey the importance of studies in the history of mathematics. Parts III–VII constituting 33 articles, feature a number of articles on a variety of topics, such as geometry, trigonometry, algebra, combinatorics and spherical trigonometry, which not only reveal the breadth and depth of Professor Gupta’s work, but also highlight his deep commitment to the promotion of studies in the history of mathematics. The ten articles of part VIII, present interesting bibliographical sketches of a few veteran historians of mathematics and astronomy in India. Part IX examines the dissemination of mathematical knowledge across different civilisations. The last part presents an up-to-date bibliography of Gupta’s work. It also includes a tribute to him in Sanskrit composed in eight verses.

Inhaltsverzeichnis

Frontmatter

The Oeuvre of Radha Charan Gupta

Frontmatter

A Portrait of the Life of R. C. Gupta

Though I have known Prof. R. C. Gupta (RCG) for more than two decades through his writings, till recently I did not have an occasion to interact with him closely.

K. Ramasubramanian

A Birthday Tribute to R. C. Gupta

The internationally renowned historian of mathematics, Radha Charan Gupta, celebrated his 60th birthday on October 26, 1995.

Christoph J. Scriba

Professor R. C. Gupta Receives the Kenneth O. May Prize

Professor Radha Charan Gupta, who founded and nurtured Gaṇita Bhāratī as editor for over a quarter century, was awarded the Kenneth 0. May prize of 2009, jointly with Prof. Ivor Grattan—Guinness of UK—by the International Commission for the History of Mathematics (ICHM).

Kim Plofker

On Studies in History of Mathematics

Frontmatter

On the Date of Śrīdhara

H. T. Colebrooke seems to be the first modern scholar who studied the work of Śrīdhara. He had an incomplete copy of Pāṭīgaṇitasāra from which he often quoted parallel passages in his translation (1817) of the Līlāvatī.

K. Ramasubramanian

In the Name of Vedic Mathematics

In this category comes the genuine Vedic Mathematics as found in the Vedas or Vedic literature in general. The four Vedas consisting of various Vedic Saṃhitās, the several Brāhmaṇas, Āraṇyakas and Upaniṣads are the basic literary sources for this type of Vedic Mathematics.

K. Ramasubramanian

Foreign Reviews and Evaluation of Indian Works on History of Science

Although studies and research in the field of history of exact sciences in India have been going on for the last two centuries, no comprehensive and authentic chronological history of Indian mathematics has been written so far.

K. Ramasubramanian

The Study of History of Mathematical Sciences in India

Part I (1935) and Part II (1938) of the famous History of Hindu Mathematics by B. B. Datta and A. N. Singh were published from Lahore (now in Pakistan), while the promised Part III was never published in its authors’ lifetime. In 1962 a single volume edition (actually a mere reprint of Parts I and II) was brought out by the Asia Publishing House, Mumbai. This was reviewed by G. J. Toomer of Oxford (Math. Reviews, Vol. 26, 1963, p. 1142). He charged the authors of being ignorant of “historical matters” and of having “prejudice against admitting that there was any influence on Indian civilization from outside”. Their theories have been said to be often based on the “grossest errors of fact”.

K. Ramasubramanian

Historical Notes: Kali Chronograms of Nārāyaṇa Bhaṭṭatiri

Nārāyaṇa Bhaṭṭatiri (sixteenth–seventeenth century ad), son of Māṭrdatta, is one of the greatest scholar-poets of Kerala. He composed many works on diverse subjects both literary as well as technical in Sanskrit. He was a Nambutiri Brāhmin hailing from the family of Melpattur situated not far from the bank of the river Bharatappuzha. According to his grammatical work (see below), he learned Mīmāṃsā from his father, Vedas from Mādhava, logic from Dāmodara and grammar from Acyuta who was a great authority in the subject of Vyākaraṇa-śāstra.

K. Ramasubramanian

Mathematics in Ancient Period

Frontmatter

On Some Mathematical Rules from the Āryabhaṭīya

The paper deals with the controversies which arise due to different interpretations of certain mathematical rules as found in the Āryabhaṭīya of Āryabhaṭa I (born 476 ad).

K. Ramasubramanian

Decimal Denominational Terms in Ancient and Medieval India

Although the choice of ten as a base for numeration is not the best, God favoured it by giving us ten fingers. In India, ten has been the basis for counting since very early days. Later on it served the base for the place-value system of numerals which was invented in India about two thousand years ago.

K. Ramasubramanian

New Indian Values of from the Mānava-śulba-sūtra

India’s oldest written works are the Vedas. To assist their proper study, there are six ancillary texts, called Vedāṅgas (“limbs of the veda”), namely Śikṣā (Phonetics), Kalpa (Ritualistics), Vyākaraṇa (Grammar), Nirukta (Etymology), Chandas (Prosody) and Jyotiṣa (Astronomy and Astrology).

K. Ramasubramanian

The Lakṣa Scale of the Vālmīki Rāmāyaṇa and Rāmā’s Army

C. N. Srinivasiengar was perhaps the first historian of mathematics to give a modern exposition of the Lakṣa Scale as found in the Yuddhakāṇḍa (the Sixth Book) of the Vālmīki Rāmāyaṇa, the national epic of India.

K. Ramasubramanian

The Chronic Problem of Ancient Indian Chronology

Chronology is the backbone of history, and its knowledge is essential for a historian dealing with any period, culture area or subject. There cannot be a coherent history without a chronological order. Proper historical writing is not possible unless there is a sound chronology.

K. Ramasubramanian

A Problem on Interest in the Nārada-purāṇa

The Sanskrit text of the Nārada-purāṇa (abbreviated NP hereafter) was, perhaps first, published by the Venkateshwara Press, Bombay in Śaka 1845 (or ad 1923). Chapter 54 of the Pūrvabhāga of NP is devoted to mathematics and astronomy (gaṇita-jyotiṣa). The next two chapters are on astrology (phalita-jyotiṣa).

K. Ramasubramanian

Who Invented the Zero?

Obviously the answer depends on the meaning of ‘zero’. That is whether we mean the word zero or some concept of zero, the number zero or some symbol for zero, the mathematical zero or some philosophical zeroism. As a word for literal description, zero means a person or thing with no importance or independent existence.

K. Ramasubramanian

World’s Longest Lists of Decuple Terms

Perhaps the practice of using fingers for counting was responsible for the choice of ten as a base for numeration. In India, ten has been the basis for counting since the very early days.

K. Ramasubramanian

Jaina Mathematics and Astronomy

Frontmatter

Circumference of the Jambūdvīpa in Jaina Cosmography

In Jain cosmography, the periphery of the Jambu Island is taken to be a circle of diameter 100,000 yojanas. The circumference of a circle of this size, as stated in Jain canonical and geographical works like the Anuyogadvāra-sūtra and Triloka-sāra etc. is equal to 316227 yojanas, 3 krośas, 128 daṇḍas and $$13\frac{1}{2}$$ aṅgulas nearly.

K. Ramasubramanian

Mādhavacandra’s and Other Octagonal Derivations of the Jaina Value

$$\sqrt{10}$$ was one of the approximate values of $$\pi $$ used in ancient and medieval times especially in Jaina works. K. Hunrath derived it from a dodecagon a century ago, and G. Chakravarti from an octagon about fifty years ago. An ancient derivation given by Mādhavacandra (c. 1000 ad) in his Sanskrit commentary on Tiloya-sāra of Nemicandra. (c. 975 ad) has been examined in detail especially in the light of expositions given by Chakravarti and Āryikā Viśuddhamatī recently.

K. Ramasubramanian

Chords and Areas of Jambūdvīpa Regions in Jaina Cosmography

In Jaina works, the Jambūdvīpa (“Jambu Island”) is circular and of diameter $$D=100000$$ yojanas (Tiloyapaṇṇattī $$=$$ TP, IV. 11; Vol. II, p. 4; Kota, 1986). It is divided into 13 main regions by boundary lines which are all parallel to the east–west direction.

K. Ramasubramanian

The First Unenumerable Number in Jaina Mathematics

The definition of asaṃkhyāta (“unenumerable”) numbers in the ancient Indian Jaina Schools is linked to their cosmography according to which the Jambūdvīpa (“Jambū Island”) is circular in shape and has a diameter $$D_0$$ equal to one lakh yojanas. It is surrounded by a series of concentric rings (or annuli) of sea and land alternately (see Fig. 1).

K. Ramasubramanian

गोलपृष्ठ के लिये महावीर-फेरू सूत्र और विदेशों में उनकी झलक

इस लेख में जिस व्यावहारिक सूत्र की चर्चा की गई है वह है

K. Ramasubramanian

Geometry

Frontmatter

Brahmagupta’s Formulas for the Area and Diagonals of a Cyclic Quadrilateral

Let ABCD be a plane (convex) quadrilateral with sides AB, BC, CD and DA equal to a, b, c and d, respectively. Let the figure be drawn in such a manner that we may consider, according to the traditional terminology, the side BC to be the base (bhū), the side AD to be the face (mukha), and the sides AB and DC to be the flank, sides (bhujas or arms) of the quadrilateral.

K. Ramasubramanian

On the Volume of a Sphere in Ancient India

Seidenberg’s paper “On the Volume of a Sphere” appeared in the Archive for the History of Exact Sciences, Vol. 39 (1988), pp. 97–119. He had conveyed it in January 1988 but could not see

K. Ramasubramanian

Kamalākara’s Mathematics and Construction of Kuṇḍas

Kamalākara was a great astronomer and mathematician of India and was a senior contemporary of the famous Newton in Europe. He belonged to a family of jyotiṣīs and was the second son of Nṛsimha who wrote a commentary called Saurabhāṣya on the Sūrya-siddhānta in ad 1611.

K. Ramasubramanian

Area of a Bow-Figure in India

In Fig. 1, PNQP is segment of a circle (i.e. circular disc) whose centre is at O and whose radius is $$ OP=OQ=r $$. Due to the figure’s resemblance to an archer’s bow, the arc PNQ (= s in length) was called cāpa (‘bow’), the chord $$ PQ (=c) $$ was called jyā or jīvā (‘bow-string’) and the segment’s height $$ MN (=h) $$ was called bāṅa or śara (‘arrow’) in ancient India. The cāpakṣetra (‘bow-figure’) or segment of a circle had great importance in Indian cosmography and geography, especially in the Jaina school. The Bharata-kṣetra (=Bhārata-varṣa or ‘land of India’) of those times was in the shape of a bow-figure which formed the southernmost part of the central continent or Jambūdvīpa (‘Jambū Island’) which is stated to be circular and of diameter one lac (100,000) yojanas. This cartographic description may be taken to represent the oldest map of India as part of Asia. The maximum north–south breadth of the country was 526 $$ \frac{6}{19} $$ yojanas.

K. Ramasubramanian

Yantras or Mystic Diagrams: A Wide Area for Study in Ancient and Medieval Indian Mathematics

As an appliance, yantra may be an astronomical or surgical instrument, or a machine or mechanical device.

K. Ramasubramanian

Interpolations and Combinatorics

Frontmatter

Second-Order Interpolation in Indian Mathematics up to the Fifteenth Century

The computational abilities of ancient Indian mathematicians are well known. The paper deals with the second-order interpolation schemes found in a few astronomical works of India.

K. Ramasubramanian

Munīśvara’s Modification of Brahmagupta’s Rule for Second-Order Interpolation

When the values of a function are tabulated for some discrete values of the argument, the functional values corresponding to intermediary argumental values are obtained ordinarily by linear interpolation. For greater accuracy, higher order technique is necessary. It is known that the famous Indian mathematician Brahmagupta (seventh century ad) gave a rule for second-order interpolation.

K. Ramasubramanian

Varāhamihira’s Calculation of and the Discovery of Pascal’s Triangle

In ancient time, the Jaina School of Indian mathematics took great interest in the subject of permutations and combinations as is clear from their canonical and other literature. The Bhagavatī-sūtra (dated about 300 bc) is said to have mentioned combinations of n objects taken one at a time (eka-saṃyoga), two at a time (dvika-saṃyoga), three at a time (trika-saṃyoga), or more at a time.

K. Ramasubramanian

The Last Combinatorial Problem in Bhāskara’s Līlāvatī

There is no doubt that the most prominent name in the history of ancient and medieval Indian mathematics is that of Bhāskarācārya (ad twelfth century).

K. Ramasubramanian

Early Pandiagonal Magic Squares in India

In ancient India, arts and sciences were hand-maiden of religions. In fact religion has been the dominant feature of Indian culture through the ages. Almost all sciences have been attributed a divine origin. For instance, the exposition of the 54th chapter (on astronomy and mathematics) of the Nārada-purāṇa commences with the line.

K. Ramasubramanian

Trigonometry and Spherical Trigonometry

Frontmatter

Bhāskara I’s Approximation to Sine

Accuracy of the rule is discussed and comparison with the actual values of sine is made and also depicted in a diagram. In addition to the two proofs given earlier by M. G. Inamdar (The Mathematics Student, Vol. XVIII, 1950, p. 10) and K. S. Shukla, three more derivations are included by the present author.

K. Ramasubramanian

Fractional Parts of Āryabhaṭa’s Sines and Certain Rules Found in Govindasvāmin’s Bhāṣya on the Mahābhāskarīya

The commentary of Govindasvāmin (circa $$ {\textsc{ad}} $$ 800-850) on the Mahābhāskarīya contains the sexagesimal fractional parts of the 24 tabular Sine-differences given by Āryabhaṭa I (born $$ {\textsc{ad}} $$ 476). These lead to a more accurate table of Sines for the interval of 225 min. Thus the last tabular Sine becomes.

K. Ramasubramanian

Early Indians on Second-Order Sine-Differences

The well-known property that the second order differences of sines are proportional to the sines themselves was known even to Āryabhaṭa I (born ad 476) whose Āryabhaṭīya is the earliest extant historical work (of the dated type) containing a sine table.

K. Ramasubramanian

An Indian Form of Third-Order Taylor Series Approximation of the Sine

The paper describes an approximation formula for sine $$(x + h)$$ that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation.

K. Ramasubramanian

Solution of the Astronomical Triangle as Found in the Tantrasaṅgraha (AD 1500)

The spherical triangle formed on the celestial sphere by the positions of the Sun, north pole and the zenith on it is called an astronomical triangle.

K. Ramasubramanian

Addition and Subtraction Theorems for the Sine and the Cosine in Medieval India

The paper deals with the rules of finding the sines and the cosines of the sum and difference of two angles when those of the two angles are known separately. The rules, as found in the important medieval Indian works, are equivalent to the correct modern mathematical results. Indians of the said period also knew several proofs of the formulas. These proofs are based on simple algebraic and geometrical reasoning, including proportionality of sides of similar triangles and the Ptolemy’s theorem.

K. Ramasubramanian

Parameśvara’s Rule for the Circum-radius of a Cyclic Quadrilateral

The expression for the circum-radius of a cyclic quadrilateral in terms of its sides, usually attributed to L’Huilier in 1782, was known in India to Parameśvara (circa 1430). The present paper contains the original Sanskrit text of the rule, its English translation, and a discussion of its derivation as given by Saṅkara Vāriar in his Kriyākramakarī (sixteenth century) along with relevant historical remarks.

K. Ramasubramanian

Indian Values of the Sinus Totus

The predecessor of the modern trigonometric function known as the sine of an angle was born, apparently, in India. The Greek trigonometry had been based on the functional relationship between the chords of a circle and the central angles they subtend. The Indians, on the other hand, used half of a chord of a circle as their basic trigonometric function. The Indian (or Hindu) Sine (usually written with a capital letter to distinguish it from the modern Sine) of an arc in a circle is defined as half the length of the chord of double the arc. Thus the (Indian) Sine of an arc $$\alpha $$ is equal to R $$\sin \theta $$ where R is the radius of the circle of reference and $$\sin \theta $$ is the modern sine of the angle $$\theta $$ subtended at the centre by the arc $$\alpha $$.

K. Ramasubramanian

South Indian Achievements in Medieval Mathematics

The development of Hindu mathematics did not come to a standstill after the famous Bhāskarācārya or Bhāskara II (circa 1150 ad) although many scholars believed and still believe that.

K. Ramasubramanian

Munīśvara’s Traditional Sine Table

Munīśvara’s another name was Viśvarūpa. He was born in 1603 ad and his father was Raṅganātha (a commentator of Sūryasiddhānta). He resided at Varanasi and wrote the following works (in Sanskrit) related to Indian astronomy and mathematics (Census of Exact Sciences in Sanskrit, Vol. 4 of Series A, pp. 436–441; Philadelphia, 1981).

K. Ramasubramanian

Bio-bibliographical Sketches of Some Historians of Mathematics

Frontmatter

Prabodh Chandra Sengupta (1876–1962): Historian of Indian Astronomy and Mathematics

Prabodh Chandra Sengupta, the younger son of Ram Chandra Sengupta, was born in a village near Tangail in Mymensingh district (now in Bangladesh) on 21 June 1876. He had his early education in the Santosh Jahnavi H. E. School and passed the Entrance (Matric) examination with sufficient merit to obtain a scholarship.

K. Ramasubramanian

Bibhutibhusan Datta (1888–1958): Historian of Indian Mathematics

Born to a poor Bengali family, Bibhutibhusan Datta (1888–1958) was indifferent to worldly pleasures and gains. He never married. His doctoral thesis was on hydrodynamics, but he is best known for his work on the history of mathematics.

K. Ramasubramanian

Review of Pingree’s Census of Exact Sciences in Sanskrit (1992)

Series A, Volume 4, by David Pingree; American Philosophical Society, Philadelphia, 1981 (Memoirs of the A.P.S., Vol. 146); Pages 447, List Price U.S. $ 30.00.

K. Ramasubramanian

Homage to Professor Abraham Seidenberg (1916–1988)

Professor Abraham Seidenberg died on May 3, 1988. He belonged to the Department of Mathematics, University of California, Berkeley, USA, and was a member of the Editorial Board of the Gaṇita Bhāratī. He was a great scholar and was actively involved in research work on history of ancient mathematics. A few months before his death, he had sent his manuscript “On the Volume of a Sphere” for publication in the Archive for History of Exact Sciences (= AHES). He could not correct the proofs of the article which was published posthumously in December 1988. Perhaps this may be his last paper.

K. Ramasubramanian

Sudhākara Dvivedī (1855–1910): Historian of Indian Astronomy and Mathematics

Once a king wanted to know as to why the Moon (candā) is addressed (candāmāmā) or “Moon, the Maternal Uncle” in India.

K. Ramasubramanian

Clas-Olof Selenius (1922–1991): An Expert in Indian Cyclic Method

A great scholar as he was, Clas-Olof Selenius could successfully illustrate the use of modern mathematics in highlighting the significance of the ancient Indian masterpiece called Cakravāla or cyclic method in solving certain indeterminate equations.

K. Ramasubramanian

Kripa Shankar Shukla (1918–2007): Veteran Historian of Hindu Astronomy and Mathematics

Kripa Shankar Shukla’s birth took place at Lucknow on July 10, 1918. From the very early years, he was a brilliant student of Mathematics and Sanskrit. He passed the High School Examination of U.P. Board in 1934 in First Division with Distinction in Mathematics and Sanskrit and the Intermediate Examination of that Board again in First Division with Distinction in Mathematics.

K. Ramasubramanian

Obituary—T. A. Sarasvati Amma

Dr. T. A. Sarasvati Amma was born as the second daughter of her mother Kuttimalu Amma and father Marath Achutha Menon. The year of her birth was apparently 1094 of the Kollam (Kolamba) era which is prevalent in Kerala and which corresponds to ad 1918–1919.

K. Ramasubramanian

The India-Born First President of the London Mathematical Society and His Discovery of Ramachandra

Augustus De Morgan (1806–1871), the founder President of the London Mathematical Society was born in Madurai of Madras Presidency (South India) on Friday, the 27th June, 1806. His father was associated with the East India Company. His great-grand father James Dodson (died 1757) was the author of Anti-Logarithmic Canon (1742) and Mathematical Repository (1755).

K. Ramasubramanian

M. Rangacharya and His Century Old Translation of the Gaṇita-sāra-saṅgraha

Lots of Achievements in ancient Indian Mathematics as reflected in the works of Āryabhaṭa I (born 476 ad), Brahmagupta (seventh century ad) and Bhāskara II (twelfth century) were freshly made known in modern form to the Western world during the nineteenth century. Leading role in this regard was displayed by Western scholars such as R. Barrow, H. T. Colebrooke, S. Davies, C. Hutton, H. Kern, L. Rodet, E. Strachey and John Taylor.

K. Ramasubramanian

Transmission of Mathematics and Astronomy Between India and Other Civilizations

Frontmatter

Indian Astronomy and Mathematics in the Eleventh-Century Spain

According to Ibn al-Ādamī (c. 950 ad) as quoted by Qāḍī Ṣā’id al Andalūsī (d. 1070), Caliph al-Manṣūr of Baghdad (755–775 ad) ordered a Sanskrit astronomical work to be translated into Arabic.

K. Ramasubramanian

Indian Astronomy in West Asia

It is difficult to gather trustworthy knowledge of astronomy in Iran before the reign of Ardashīr I (ad 226–240) and Shāpūr I (241–272) who encouraged the spread of Indian Science in the region. The ninth century Pahlavi (Middle Persian) Dēnkart informs us authoritatively that the two kings had Indian and Greek works translated into Pahlavi and that they were revised under Khusro Anūshirwān (sixth century) (Pingree, 1964–66, p. 119).

K. Ramasubramanian

Spread and Triumph of Indian Numerals

According to Menninger, it is quite probable that due to active commercial relations with India, the first Indian numerals became known in Alexandria sometime in the fifth century ad and from there they might have penetrated farther westward.

K. Ramasubramanian

Indian Mathematical Sciences Abroad During Pre-modern Times

Views regarding the origin of mathematics have been changing fast during the last 50 years. The publication of old Babylonian texts in the thirties has not only upset the theory of the Greek origin of mathematics but rather gave rise to the view that the Greek mathematics itself was a derivative of the Babylonian mathematics. According to a tradition, Thales visited Egypt and Pythagoras is said to have even visited India.

K. Ramasubramanian

Sino-Indian Interaction and the Great Chinese Buddhist Astronomer-Mathematician I-Hsing (ad 683–727)

The rock edicts of king Aśoka (third century bc) show that he had already paved the way for the expansion of Buddhism outside India. Subsequently Buddhist missionaries took Buddhism to Central Asia, China, Korea, Japan and Tibet in the North, and to Burma, Ceylon, Thailand, Cambodia and other countries in the South. This helped in spreading Indian culture to these countries.

K. Ramasubramanian

Indian Mathematical Sciences in Ancient and Medieval China

The great Indian emperor Aśoka (usually called Devānāṃpriya, ‘Beloved of Gods’) ruled from about 272–232 bc His missionary activities are recorded in Rock Edict No. XIII as follows (Sircar, p. 54, and P. Thomas, p. 15).

K. Ramasubramanian

Indian Influence on Early Arabic and Persian Writers of Mathematical Sciences

India has given to the world outstanding gifts in sacred spiritual as well as in secular scientific fields. In spiritualism, India gave the great Buddhism through which all the Asian countries during the first millennium of our era “formed one fountainhead”.

K. Ramasubramanian

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