A. A. Krishnaswami Ayyangar (AAK), my father, was born on 1 December, 1892. He got his M.A. in Mathematics at the age of 18 and subsequently started teaching Mathematics at his own alma mater, Pachaiappas College, Chennai (Madras). He was married to Seshammal and they had four sons, A. K. Srinivasan, A. K. Ramanujan, A. K. Rajagopal, A. K. Vasudevan, and two daughters Mrs. Vedavathi Bhogishayana and Mrs. Saroja Krishnamurthi. In 1918 he joined the Mathematics Department, University of Mysore and retired from there in 1947. He passed away in June 1953. During his tenure at the University, for nearly three decades, he made many contributions to Geometry, Statistics, Astronomy, the History of Indian Mathematics, and other topics.

The subject matter of interest, presented in this CD, concerns only his works on the history of Indian Mathematics. This work involved not only translations from the Sanskrit texts but also transforming some of them in modern notation and language. The originality of AAK’s work was brought home to Rajagopal on a casual conversation with Professor R. Sridharan of the Tata Institute of Fundamental Research, Bombay in 1998. Professor Sridharan gave him an article he had written in Science in the West and India, edited by B. V. Subbarayappa and N. Mukunda, (1995, Himalaya Publishing House, Bombay). In this he points out that AAK‘s article on the Chakravala method (included in this CD) shows how it differs from the method of continued fractions. He recounted that this point was missed by A. Weil, who thought the Chakravala method was only an “experimental fact” to the Indians and attributed general proofs to Fermat and Lagrange much later! On a different occasion, Professor Subhash Kak of Louisiana State University, Baton Rouge, stressed to Rajagopal that AAK’s presentations of Indian works were unique and should be known generally in the community of mathematicians today who are interested in the History of Mathematics. These incidents prompted us to present some of AAK’s papers in a CD with brief account of each of the articles included as an aid to the reader.

Without the keen interest evinced by Professor Kak and his drive to get Indian contributions to the World of Astronomy, Mathematics, Music, and everything else, this collection would have suffered the quiet death of so many things Indian! Two new publications may be mentioned: Indian Mathematics: Redressing the balance by Ian G. Pearce (2002, http://www.history.mcs,st-andrews.ac.uk/history/Projects/Pearce/index.html) and The Crest of the Peacock, Non-European Roots of Mathematics (2000, Princeton University Press) by George Ghverghese Joseph which do not contain any citations from AAK’s works, most certainly because of their obscurity! We owe Professor Kak a debt of gratitude for urging us to bring the work of AAK into the collection of Indian contributions to Mathematics and Astronomy.

It was not easy to obtain the relevant articles from the various journals particularly because AAK’s publications date back to early 1920’s and onwards. This collection of AAK’s articles on Indian Mathematics and Astronomy, in this CD, is a partial culmination of a much larger search for all his writings. In this form, this CD is our family’s tribute to our father’s memory. It is gratifying to find these works to be of interest to people engaged in writing about the World History of Mathematics and Astronomy to day. I hope this collection is a modest contribution to this endeavor.

Rajagopal’s many conversations with Professor Kak on Indian contributions to various human endeavors over the millennia started all this. Professor M. S. Rama Rao of Mysore, a good friend of Rajagopal was approached to help in obtaining some of the papers from the University of Mysore. We are deeply indebted to Professor Rama Rao for his persistence in going through the authorities of the University of Mysore and making photocopies of two of the articles in the half-yearly Journal of the Mysore University. Dr. J. Sheshidhara Prasad, Vice-Chancellor of the University of Mysore, and Dr. S. L. N. Arora, Director of Prasararanga, University of Mysore, for helping Professor Rao in his search for these articles. Many thanks are due for their help. Our thanks are also due to the President, Mythic Society, Bangalore, for making photocopies of AAK’s articles published in the Quarterly Journal of the Mythic Society. Without the help of Mr. Saranatha Gopalan, who accompanied me to the Library of the University of Madras and other libraries, searched for the Journals and made photocopies of the articles this compilation would have taken a longer time. Many thanks to him. Last but not least, we wish to thank Mr. S. L. Narasimhan (Yessel), who transformed the jpg files into PDF files, in spite of his busy office schedule.


A. K. Srinivasan         

C-7 Hiranya

67/69 Greenways Road

R A Puram

Chennai - 600 028, INDIA

email: attipat22@yahoo.com



Titles in the ABSTRACTS and CONTENTS are linked to the articles.

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Educational Review, 1935, Vol.41

Mathematically, the five-fold specification of the Panchanga (vara, thithi, nakshatra, yoga, & karana) suggests that we are living in a five dimensional world. We require five coordinates to specify our place-time, a blended entity of four dimensions according to Einstein. A reliable panchanga gives these five coordinates as accurately as possible for the orthodox persons following religious traditions. We have, now, a conflicting set of astronomical systems handed down from ancient times and followed by different sections of people with a blind and uncritical devotion to the letter of the texts. Those who have studied European astronomy do not have any conception of Hindu astronomy. The Indian astronomers following the Sanskrit texts have not an inkling of European astronomy. To bring the two together, Universities should teach Hindu and European Astronomy while in the Sanskrit Colleges European and Hindu Astronomy should be taught.


Current Science, Vol.7, No.4, Oct. 1938

The principle in Bhaskara’s solution of the biquadratic was hidden away in a numerical example. Bhaskara’s is the earliest attempt at the solution of the biquadratic  recognizing only the real positive roots. This article generalizes his principle of his method of solving a biquadratic.


Journal of Indian Mathematical Society, 1929-30, Vol.18

Saradakanta Ganguly draws attention to a rule regarding the solution of the general case of Simultaneous Indeterminate Equations of the first degree, found in two palm leaf manuscript copies of Bhaskara’s Lilavati.  The rule for the solution of the general case of simultaneous indeterminate equations mentioned in the manuscripts is known as samclishta bahusamanya kuttaka sutram .For brevity this is called samclishta sutra. This note proposes to reopen the question of genuineness of Bhaskara’s authorship. It examines the rule in the light of other Indian solutions of the same problem and also in the light of Indian mathematical tradition. Earliest Indian attempt of indeterminate analysis takes the form of solving simultaneous congruences.

Aryabhata mentions only two congruences but can be extended to more than two cases by repeating the process. This is done by Mahaviracharya. This paper gives an example to solve a set of congruences which is a repetition of a process with no new principle involved. Thus it does not justify an explicit enunciation by Bhaskara. If he had done so, he has played a role of a commentator. Therefore, it is likely that a commentator has interpolated it in Bhaskara’s text. 

Aryabhata was the first to hint the general problem of finding a number having given residues with respect to given modulii. Mahaviracharya has given a rule different from the samclishta rule. Bhaskara suggests a novel method (Samclishta Kuttaka) different from samclishta rule for solving simultaneous equations in the particular case of the same modulus for all congruences. It furnishes novel method of solving simultaneous congruence equations, with common modulus. Bhaskara does not point out the cases of failure, which is a normal custom. This paper indicates the condition in which his rule fails, and gives numerical example.

                        5 x ≡ 7 (mod. 63)

                        30 x ≡ 14 (mod. 63)

Bhaskara’s method will give 35 x ≡ 49 (mod. 63) or 5 x ≡ 49 (mod. 9) as the representative congruence with x ≡ 5 as the least value. This does not satisfy the given congruences and have to choose the next higher value 14.


Journal of the Indian Mathematical Society, 1929-30, Vol.18

This note gives new proofs (by AAK) of Apollonius and Brahmagupta’s theorems.


Journal of the Indian Mathematical Society, 1929-30, Vol.18

More than thousand years before Euler, Indians were the first to give a general solution to the equation

x2 – N y2 = 1    (1)

It was based on the principle of composition of quadratic forms.

Euler’s theorem

If x = a, y = b satisfies α x2 + p = y2,   (2) and
x = c,  y = d satisfies α x2 + q = y2,    (3) then

x = bc ± ad, y = bd ± αac  satisfies α x2 + pq = y2                       (4)

has been dealt by Brahmagupta in his ‘Vargaprakriti’ and drawing important corollaries from it.
Special identities are also given by him for deriving integral solutions of Ny2+1=x2 and of Ny2±4=x2.

About 1150 A.D., Bhaskara gives integral solutions, designating it as ‘Chakravala’ after the ancients. Western writers attribute the cyclic method to Brahmagupta but the credit should go to Bhaskara. Bhaskara may have given credit to his predecessors for the name ‘chakravala’, as applied to all iterative operations in mathematics.

Brahmagupta’s formula for the composition of quadratic forms is also repeated by Bhaskara. Using this both of them derive the solutions of equation (1) from those of

N y2± 2 = x2      or    N y2 ± 4 = x2        (5)

Bhaskara goes one step further to show that the roots of the equations (1) and (5) can be derived, by successive reduction, from the more general equation N y2 ± k = x2, where k is an integer. The method of such reduction is the chakravala method (Cyclic Method). The first complete proof of Bhaskara’s Cyclic Method is sketched in this paper.

This paper discusses the following:

·                     Bhaskara’s Condition for Reduction

·                     Reduced Forms

·                     Properties of Bhaskara’s Forms

·                     Bhaskara and Fermat



The Mathematics Student, Vol. I, 1933

The early Hindus developed mathematics as a limb of Vedas (Vedanta). The world owes some of the basic ideas such as place value system of notation in Arithmetic, the generalizations of Algebra, the sine function of Trigonometry, and the foundations of Indeterminate Analysis. The early texts contain merely rules, results, and a number of problems but rarely a full worked out mathematical argument.

·                     Vedanga Jyotisha is the oldest work bearing astronomy. The style of the text is characterized by brevity, archaisms, and want of connection between consecutive verses. The text gives rules to calculate relative lengths of day and night. It gives positions of the moon at different parts of the year and the time of the day. The early Hindus, evidently, were well acquainted with arithmetical manipulations including fractions.

·                     Sulva Sutra (about 200 A.D.) deal with the construction of sacrificial altars. It gives construction of squares, rectangles, triangles and parallelograms with given specifications. It also contains the first arithmetical solution of indeterminate equation x2 + y2 = z2 and rational approximations to √2, (√п)/2.

·                     Bhakshali Manuscript, a work on Arithmetic, was discovered in 1881 A.D. It has a second approximation for √(a2+r). This result can also be obtained from the rule found in all the Indian arithmetical works.

·                     Geometrical problems were first analyzed as problems in arithmetic and then expressed in geometrical language. Area of a cyclic quadrilateral, Pythagorean Theorem and others were analyzed as arithmetical problems first.

·                     The next stage after Sulva Sutra is the age of Aryabhata, Varahamihira, Brahmagupta, and the unknown author of Suryasiddhanta (400-650 A.D). Aryabhata is placed by tradition at the head of the Indian mathematicians but he appears to be the last of the earlier school. His work appears to be the first to give form and an individuality to the mathematical knowledge existed before him. His treatises were translated into Latin, French, and English. He was the first to in India to hint the daily rotation of the earth on its axis. In his work one can find new alphabetic notation adopting both consonants and vowels to denote large numbers in a concise way. His admirer, Lalla abandoned it in favor of the more popular and the more ancient word numerals. Vedanga Jyotisha, Surya Siddhanta, and Varahamihira use the numeral words. Brahmagupta was the first to explicitly enunciate this system.

·                     A list of important mathematical topics on which there are contributions from the ancient and the medieval Hindus. Among the contributions of ancient Hindus to higher mathematics, the most outstanding, is on Indeterminate Analysis. Aryabhata was the first to indicate a method of getting integral solutions for mx + a = ny + b (m, n, a, b being integers). He proceeds by successive reductions to simpler equations until one reaches a solution. Brahmagupta has given a clearer exposition of the same. Bhaskara and others followed with improvements, alterations and extension. Brahmagupta was the first to enunciate the principle of composition of roots to solve the indeterminate equations of the second degree. This is similar to the modern principle of compositions of quadratic forms. It may be noted that the so called Pellian Equation x2 – Ny2 = 1 should be called the Brahmagupta Equation.

·                     Spherical trigonometry was developed earlier than Plane trigonometry to be used in Astronomy. The semi-chord and the arrow gave rise to the sine and the versed-sine functions. Hindus divided the circumference of a circle into 21600 minutes. Sine is considered as a length, and not a ratio, and expressed in terms of the arc. Hindus attributed to the arc all the properties of the angle. They established fundamental formulae of trigonometry and managed their astronomical formulae with the help of three functions: sine, the versed-sine, and the cosine. The works on astronomy from Suryasiddhanta to Siddhanta Siromoni give interpolation and extrapolation formulas for the tabulation of successive sines.

·                     Vedic hymns contain description of the motions of Sun and the Moon, the seasons, the number of days in a year, the two ayanas (Devayana and Pitriyana), the intercalary month etc. There are sufficient data in the Vedas for postulating the phenomenon of precession of the equinoxes and determining its rate. After the Vedas we have the twenty or so Siddhantas. These were the earliest efforts at crystallizing, in a scientific form, the knowledge of astronomy. Of these twenty, Suryasiddhanta and Brahma Siddhanta are the most popular and have been revised by other writers starting from Aryabhata. Siddhantas mention the number of revolutions of the sun, moon, and the planets, and their nodes. They assumed the motions of the planets have mean circular orbits during a Kalpa and all of them travel the same number of Yojanas. This constant is described as Khakaksha the circumference of the sphere to which solar rays extend. Bhaskara’s value for this constant is 187120692 x 108 while Suryasiddhanta gives value as 18712080864 x 106. Indian astronomers give the moon’s distance from the earth but erred in the distances of the sun and the planets as they were calculated from the assumption of constant velocity. There is enough evidence that the Indian astronomers actually verified the astronomical elements by personal observations of the heavenly bodies.

·                     Names, after Bhaskara, of other astronomers are given.


The Mathematics Student, Vol. VII, 1939

This paper is a re-examination and re-assessment of the Bhakshali Manuscript (BM). G. R. Kaye has written introduction and notes after the earlier work of Hoernie. G. R. Kaye’s biased view of Greek influence of anything Indian prompted others to re-examine his statements. This article examines the manuscript under the following headings:

1.          Method of presentation

2.          Peculiar terminology, abbreviations, and the cross symbol for subtraction

3.          Decimal Notation and the absence of word numerals

4.          Symbol for unknown

5.          Rule of Regula Falsi

6.          Square-root rule and the process of reconciliation, and

7.          Series and sequences.


·         Rules and examples are presented in verse in Sloka meter and the explanations are given in prose. Verification method is incorporated to prove a mathematical problem, and is common among Indian mathematical commentators as early as 9th century A.D. It seems as though the Bhakshali Manuscript was a teaching notes by a tutor.

·         The mathematical terminology adopted in BM, generally follows other Hindu mathematical works, but contains a few exceptions. There are several abbreviations like yu (yutam) for addition and gu (gunitam) for multiplication are used between, sometimes after, or often dropped. Examples of the inconsistencies are given in the paper. There is a unique symbol () is used consistently to denote Rina (negative) and placed to the right of the number which it qualifies. This is peculiar to BM and does not appear in any other Hindu mathematical works.  The dot-symbol () for the negative, from the time of Bhaskara was a fashion in India, is absent in BM. Hence this work must have preceded Bhaskara.

·         Special numerals are used and the decimal notation employed in BM but not the word numerals. Absence of word numerals, used by Varahamihira in the sixth century, is a sure index that it belongs to an earlier period. The use of the decimal notation is another indication of the early period of the Christian era.

·         The dot () symbol is used with two different kinds of significance. It was primarily a symbol of ‘emptiness’ and secondarily has become a symbol for the unknown or absent quantity. In BM the dot symbol is used simultaneously for several unknowns. The symbol for ‘zero’, ‘negative’, and for the ‘unknown’ seem to have a common ancestry and their nebulous beginnings are reflected in BM.

·         The ‘Rule of Regula Falsi”  is the same as that of analytical geometry to find the coordinates of the point of intersection of the line joining two given coordinate points with the x-axis. One can say it is a precursor of the interpolation theory. This rule should not be confused with the Indian ‘ishtakarma’ which is an operation with an assumed number, used in cases where the final result, arrived at by a series of manipulations, is proportional to the number originally assumed.  The use of Regula Falsi in BM is a fanciful deduction based on a misunderstanding of an ingenious method of generalized ishtakarma. This method is peculiar to BM and to trace this to the medieval Regula Falsi is far-fetched argument.

·         A striking feature in BM is the chapter on the square root rule, also found in the Arab work of the 12th century. The rule is merely a corollary of the general square root rule found in all Indian mathematical works from Aryabhata onwards. The earliest evidence of concrete numerical application of this rule even to a higher order is found in Sulva Sutras. The BM rule gives only the second approximation but is suggestive of a process permitting repeated application to any desired order of approximation. Brahmagupta gives a similar rule in connection with the square root of a number in sexagesimal notation.

·         The problems on series and sequences are very original, varied and interesting. In most of the problems, the sum to n terms happens to be proportional to the first term and therefore Ishtakarma can apply. This paper discusses eight types of problems using modern notation. The identity

Tn  = (1-a1 )(1-a2 ) . . . . (1-an-1 ) an

is common in Hindu mathematical works from at least the 8th century onwards. The Arithmetical Papyrus of Akhim (9th century) contains such a problem. Mr. Kaye infers from this that BM must belong to the tenth century or later.


The Bhakshali text, apart from peculiarities, is more or less a replica of other Hindu mathematical works such as Ganitasara Sangraha. It contains in common with them the following:

1.      Practical and commercial problems

2.      Problems of income and expenditure

3.      Motion-problems

4.      Profit and loss

5.      Interest

6.      Bills of exchange or hundika, and

7.      Miscellaneous problems involving literary and social references.


In BM solutions are given in general form as to be nearly algebraic character without employing adequate symbols. This is the characteristic of all Indian works. There are omissions, such as expressions for sums of squares and cubes, indeterminate equations of the first and second degree, shadow problems, permutation and combination. The omissions may indicate an early date of composition, probably prior to tenth century. The BM may be placed near the times of Sridhara, Mahavira, and Chaturveda with whose works the BM has many points in common. The systematic presentation of the working steps and methods of verification shows it was written for the benefit of the students. It has a great value for a practical teacher since Bhakshali was near to Taxila, the renowned University centers in Ancient India.


The Mathematics Student, Vol. V, 1937

This describes the rule of classifying the yearly calendars into fourteen distinct types (seven ordinary and seven leap years) which recur in a given century.


The Mathematics Student, Vol. 18, 1950

Bhaskara’s rational approximation to sin (π/n) can be written as 16(n-1) / (5n2-4n+4), which is the best rational approximation. This can also be expressed as [n2 – (n-2)2] / [n2 + (1/4) (n-2)2] which is Ganesa’s (16th century astronomer) variant.


Quarterly Journal of the Mythic Society, Vol. 16, 1926

One of the most prominent and scientific writers is Aryabhata of Kusumapura born in the year 476 A. D. There is another Aryabhata who is known by his work Mahasiddhanta. There is a confusion of recognizing between the two. Sudhakara Dwivedi gives dubious references from Bhaskara to show that Bhaskara did not know of the older Aryabhata but only the younger one. This may be due to incomplete and erroneous manuscripts of Aryabhatiyam being in circulation.

Aryabhata’s treatise on Algebra has been translated into several European languages. Many works have been attributed to him but the Aryabhatiyam is the only work which can be called his. It consists of Dasagitika Sutra, Ganita, Kalakriya, and Gola dealing respectively with astronomical tables, mathematics, the measure of time, and the spherics. He was one of the first to expound the principles of the Indian astronomical system in a condensed and technical form. His was the original statement that the daily rotation of earth was on its axis but could not assert and maintain this against the then popular geocentric theory. He goes against the prevailing orthodox notions: in his theory of the eclipses and in the subdivision of chatur-yuga into four equal parts. He was an innovator in astronomy and he attempted to reform some of the prevailing notions and doctrines. In the eyes of the orthodox teachers he was a heretic. His high originality can be seen in the mathematical works in Dasagitika and in the Ganitapada. He can be considered as the father of Indian mathematics by looking at the later writings by Brahmagupta, Bhaskara, Mahaviracharya, and Sridhara.

Aryabhata gives in his Dasagitika, a peculiar notation for expressing numbers in terms of the letters of the Sanskrit alphabet. The twenty-five varga letters of the alphabet represent the numbers one to twenty-five respectively in the square or odd places while the avarga letters represent the numbers 30, 40, ….., up to 100 occupying the even or non-square places. The nine vowels attached to any consonant indicate that the value of the consonant is multiplied by 1, 100, 1002. . . 1008 respectively. This system was used merely for mnemonic purposes and not followed in the Ganitapada. His notation and numeration indicate that the Hindus of that age were acquainted with the principle of the position system in the decimal or the centesimal scale. He defines the square and the cube of a number and gives rules for finding the square-root and the cube-root. Sridhara, Mahaviracharya, Bhaskara and Brahmagupta give more or less identical rules for the extraction of the square-root and the cube-root, while no method of extracting the cube-root is given by an early Greek writer. He has given some mensuration formulas, some of which are obviously wrong. The area of the triangle is given correctly but the volume of a solid with six edges is given to be equal to the product of half the area of the base and the height. This is because he considered the solid of six edges as the analogue of the triangle. There is a general direction for determining the area of any figure by decomposing it into trapeziums.

The value of π is given by the rule “when the diameter is 20,000, the circumference will be 62,832 approximately”. It is remarkable that Aryabhata should have given it while nothing like it occurs in the Greek works. This rule occurs before his rule for the formation of the sine-tables. This leads one to suppose that the above value of π was used only for the construction of the sine-tables at intervals of 3.75o and that the less approximate values, such as √10, were used elsewhere. He gives the rule for deriving successive sine-differences, in the modern notation, {d2(Sin x)/dx2} = - Sin x. The ‘Sine” is equivalent to the modern sine multiplied by 3438. According to the rule, each sine-difference diminished by the quotients of all the previous differences and itself by the first difference gives thee next difference. The differences given in Dasagitika are: 225, 224, 222, 219 ....... 37, 22, 7. Same results are also given in Suryasiddhanta with the same rule. Mr. Kaye holds that this rule may be an attempt at the enunciation or application of Ptolemy’s Theorem, but the trigonometry of Ptolemy does not give it. The author (AAK) gives a possible reasoning how the rule may have been developed. As there were some discrepancies in some differences it is possible corrections were made by comparing the results with the actual ones obtained by direct calculation for the common angles 30o, 45o, 60o, 75o and 90o.

Another topic is on the mathematics of Sun-dial and the Shadows. In discussing this topic, constructions are given for drawing a circle, a triangle (equilateral!) given a side and a rectangle (square!) given a diagonal. Directions are given for determining experimentally the horizontal and vertical planes by means of water and plumb-line respectively. Pythagorean rule is used for finding the radius of the gnomon-circle given the height of the gnomon and the length of the shadow. Two rules are given for determining (1) the lengths of the shadow of a gnomon of given height and (2) the height of the source of light and its distance from two equal gnomons casting known shadows. The rules are:

(1)   Shadow = (height of gnomon x the distance of the light from the gnomon) / (the difference between and the heights (the gnomon  & the light)

(2)   The distance between the end of a shadow and the base of the light, when two gnomons are considered = (the length of the shadow x the distance between the ends of the shadows) / (the difference)

It is not clear which difference is meant in (2).

Aryabhata in an Eclipse Problem a property of the circle is enunciated which is ‘that in a circle, the product of the arrows is the square of the semi-chord of the arc’. He gives a theorem, derived from this property, to use in the calculation of eclipses. The next topic in the Ganitapada is the arithmetical progression. A general formula is given for the sum of the terms beginning with the (p+1)th  term.

n = [a + ((n-1)/2 + p) d]

where a is the first term, d the increment. An alternative form is also given as ‘add the beginning and end terms and multiply by the sum by half the number of terms’. These are followed by another formula determining the number of terms. Next follows the contents of a triangular pile and a square pile as

n(n+1)(n+2)/6  and  n(n+1)(2n+1)/6

The formula for the sum of the cubes is given by ∑ n3  =  (∑ n)2.

A pair of semi-geometrical identities to solve simultaneous equations of the types x ± y = p, x y = q;

x ± y = p, x2 + y2 = q; x y = p, x2 + y2 = q. Next topic is his rule for finding the interest on the Principle (P), the interest on the amount (A) and the time (t). Solving the quadratic equation P r2 t2 + Prt = A determines the rate of interest. After giving this rule he proceeds to enunciate the principle of the rule of three and the usual rules for the division of one fraction by another and for reducing all fractions to a common denominator.

Rules for reversing the steps in a mathematical process are enunciated by Aryabhata. This principle is useful for verification purposes and for the solution of equations where one has to clear the variable from all the ramifications in which it is involved. He gives an identity. In modern notation it is

∑ [∑(Xir – Xr) / (n-1)] = ∑ Xr ,  i = 1, . . . ,n;  r = 1, . . . ,n

From the fact that two particular cases of this theorem occur in Diophantus it is argued that the problem is of Greek origin. Mahaviracharya gives many more varieties of problems not found in Diophantus or in any other ancient Greek works. Hence one cannot infer the Hindus copied from Greek works. After the Indian version of the Epanthem follows the ordinary method for the solution of a simple equation where both sides are linear functions of the variable. This is succeeded by a discussion of the relative velocity of one moving body with respect to another, when both are moving (i) in the same direction, and (ii) in opposite directions.

Aryabhata’s solution of the linear indeterminate equation is the crown of his mathematics. He writes the problem as to find a number which leaves residues n, n’ with respect to the modulii m, m’ respectively”. If n›n’, m is called (adhikaagrabhaagahaar) the divisor corresponding to the greater residue and m’ is called (oonaagrabhaaghaar) the divisor belonging to the lesser residue. While the residue for the modulus mm’ is called ‘dwichchedagra’. The rationale and genesis of this method can be explained by an example.

Let m’ = 29, n’ = 15, m = 45, n = 19

We have to find x and y to satisfy the equation:

29x + 15 = 45y + 19.

The method is to express x in terms of y as

x = y + (16y + 4) / 29.

Since (16y +4)/9 should be an integer let it be equal to z.

Then y = z + (13z – 4)/16. Again let p = (13z – 4) /16 leading to z = p + (3p + 4)/13. At this stage, since the coefficient is small, choose p = 3 so that 3p +4 is divisible by 13. To find the value of x, we retrace the steps and get z = 4, y = 7, x = 1. Aryabhata calls p as ‘mati’.

Thus Aryabhata’s rule is a method of detached coefficients for carrying out the backward process for evaluating successively z, y, and x. This process was called “Vallikakuttaakaara” by later mathematicians. Bhaskara’s method the ‘creeper’ (valli) is extended to its utmost length. Mahaviracharya gives exactly Aryabhata’s rule using Aryabhata’s nomenclature.

The Ganitapada ends with the Indeterminate Equations and the rest of his work is Astronomy. The impression left by the Ganita is that it is a collection of working rules necessary for solving practical problems such as interest problems, problems in astronomy, and problems of survey. The author’s style is business-like, lacking the richness of imagination, the zeal in problem-setting, and the extravagant poetry characteristic of other Indian authors like Bhaskara and Mahaviracharya. Later Hindu astronomer-mathematicians are indebted to Aryabhata. He was the first to give a form and individuality to the scattered bits of mathematical knowledge that existed before his time.


Quarterly Journal of the Mythic Society, Vol. 18, 4, 1928, and Vol.19, 1, 1929

There were four different kinds of numerals in use in India from early times. They are Kharoshti, Brahmi, symbolic word notation, and alphabetic notation, before the decimal came into being with the nine symbols and the zero.

The Kharoshti script was written from right to left and the numerals followed this direction. These numerals occur in the Saka inscriptions (first century B.C). The symbols used are:

(a) one, two, three vertical strokes for 1, 2, 3 respectively, (b) an inclined cross for 4, (c) a symbol for 10, (d) a cursive combination of two tens for twenty, (e) a sign resembling the Brahmi symbol with a vertical strike to its right for ‘one hundred’. In this notation not more than three repetitions are allowed of any symbol and a new symbol is always used to avoid the fourth repetition. The Kharoshti numerals with their additive and iterative principles appear to be the first stage in the growth of the Hindu notation. These are absorbed to and superseded by the Brahmi notation.

Brahmi notation is the most important of the early Hindu notations. Some fragments of these numerals occur in Asoka’s edicts (300 B.C). They also appear in the Nanaghat cave inscriptions of the second century B.C. The next evidence of these numerals is found in Nasik cave inscriptions in which the principle of right adjunction of the smaller unit, as in Kharoshti numerals, with a multiplicative significance, is evident. Brahmi numerals belong to a non-place value system and have only a limited scope since they cannot represent large numbers. (Item 30 in the Appendix of this article gives the Brahmi symbols and its parallel notation in the present numerical notation).

The early Hindus counted in the ten-scale as so many units, tens, hundreds, and so on in successive powers of ten. The Sanskrit numeration has ‘one and ten’, two and ten’ (ekadasa, dwadasa) etc. Large number such as 108 was expressed as ‘ashtottharashata’ (eight above hundred). Later it took a convenient form where the number of units, tens, hundreds etc., occurred in a number. For example: five, seven and two meant five units, seven tens and two hundreds. The Hindus adapted this notation in verses denoting the numbers according to a set of rules. For example: one was represented by anything which is unique such as earth, moon etc., two by those which occur in doubles (eyes, black & white), naught by heaven, twelve by the names of the Sun. Aryabhata thought of substituting in its place his scheme of ‘katapayadi’ notation and the decimal notation thus combining the advantages of both. This notation was difficult for an average person and it was soon forgotten. But this notation suggested a method of using the same symbol with variations to denote a multiple powers of a hundred.

When the word-numeral notation was being used, the Brahmi symbols were at hand and a notation for a new symbol had to be invented for zero. The word ‘Sunya’ or ‘aakaash’ denoting the absence of a power of ten in the word-numeration must have suggested the symbol ‘0’. An earlier or an alternative form of this symbol is the dot ().

The Hindus called the decimal notation ‘anka palli’; the word ‘anka’ means a mark or symbol. In the word-numeral notation adapted by Varahamihira (6th century A.D.) and others ‘anka’ used to represent the numeral 9, probably because nine not ten numerals were in common use. By 9th century, at least, all the ten symbols were in place. In connection with these ten numerals of the new notation (date unknown), a rule came (ankaana vaamathogathi) that is, the order of the numerals is from left to right. One does not know whether this rule refers to the order in writing or to an arrangement in some form of abacus. It is said that Hindu astrologers were using a wooden calculating board called ‘paati’, hence ‘paati ganita’ (name for Arithmetic).

This paper traces the history of the Hindu numerals. It establishes that Western writers, who had a bias towards Greek mathematicians, are wrong in saying that Hindus did not have their own system of numerals and these numerals were spread by Arab merchants to their country. From there these numerals were adapted by others. Leonardo Fibonacci spread the Hindu numerals in Europe and wrote in 1202 his thesis Liber Abaci.


Current Science, Vol.6, 1937-38

This proposes a new Half Regular Continued Fraction development of quadratic surds, distinct from suggested by Minnegerod. Such a development was implied in the method developed by Bhaskara for obtaining the integral solutions of the indeterminate equation

X2 – NY2 = 1.

This article rebuts the erroneous view that Bhaskara’s method gives rise to the simple continued fraction. The full development of this method is described in the subsequent articles.


The Half-Yearly Journal of the Mysore University, Vol. I, 1940, No.1 and Vol. I, 1941, No. 2

This paper was inspired by the remark of Sir Thomas Little Heath “Indian Cyclic Method of solving the equation x2 – Ny2 = 1 in integers due to Bhaskara in 1150 is remarkably enough, the same as that which was rediscovered and expounded by Lagrange in 1768”. This paper investigates the Indian method of half-regular continued fraction (h.r.c.f.). This new continued fraction (the nearest square continued fraction) is a natural sequel to Bhaskara’s cyclic method. This theory was developed with the help of the simplest mathematics known to Hindus about the fifth century A.D.

It starts with the definition of the quadratic surd of the form [(P + √R) / Q] as a standard form if R is a non-square positive integer, and P (≠ 0), Q, [(R – P2) / Q] are integers having no common factor. If P = 0 it is sufficient (R / Q) and Q are relatively prime integers.

Starting with some elementary results goes on to state and prove several Theorems to the development of a new continued fraction Bhaskara continued fraction (B. c. f.) or the nearest continued fraction. This paper gives a table of B.c.f.’s equal to the square-roots of non-square integers less than 100.

In conclusion, the B.c.f. has a complicated individuality of its own and further investigation into the character of the cyclic part, the transformations that convert the simple continued fraction into the continued fraction to the nearest square, and the associated quadratic forms.


The Half-Yearly Journal of the Mysore University, Vol. V, 1945, No. 2

This paper traces the ancient Hindu mathematics from 800 B.C. to 1200 A.D. It has a brief description of the mathematics and astronomy developed by Indian mathematicians of that era. The following are the topics selected:

1.      Vedanga Jyothisha (1200 B.C.)

2.      The Sulva – Sutras (between 800 and 500 B.C.)

3.      Surya Siddhanta (gone through revisions from 500 A.D. to 100 A.D.)

4.      Aryabhata (Aryabhatiyam: 499 A.D.)

5.      Varahamihira (6th century)

6.      Brahmagupta (598 A.D.)

7.      Sridhara (probably 8th century)

8.      Mahavira (period between Brahmagupta and Bhaskara)



The Tamil Encyclopedia, Vol.3, 1st Edition

This is written in Tamil language. It covers more or less the topics on mathematics covered in some of the above articles.



1.      Ancient Hindu Mathematics, Educational Review, Madras, 1921 (to be traced)

2.      The Training of Indian almanac mathematics, Ibid, 1935

3.      A misunderstood chapter on Indian mathematics, Ibid , 1940 (to be traced)

4.      The Hindu sine Table, Journal of Indian Mathematical Society, 1923-24 (to be traced)

5.      A classical Indian puzzle problem, Ibid, 1923-24 (to be traced)

6.      Bhaskara and samclishta kuttaka, Ibid, 1929-30

7.      New proofs of old theorems – Apollonius and Brahmagupta, Ibid, 1929-30

8.      New light on Bhaskara’s Chakravala or cyclic method of solving indeterminate equations of second degree in two variables, Ibid, 1929-30

9.      The Earliest Solution of the Biquadratic, Current Science, Vol. 7, 1938

10.   The New Continued Fraction, Current Science, Vol.6, 1937-38

11.  Theory of nearest square continued fraction, Journal of Mysore University, 1940

12.  Peeps into India’s mathematical past, Ibid, 1945

13.  Some glimpses of ancient Hindu mathematics, Mathematics Student, 1933

14.  Fourteen calendars, Ibid, 1937

15.  The Bhakshali Manuscript, Ibid, 1939

16.  Remarks on Bhaskara’s approximation to the sine of an angle, Ibid, 1950

17.  Astronomy – past and present, Mysore University Magazine, 1930 (to be traced)

18.  Mathematics of Aryabhata, Quarterly Journal of Mythic Society, 1926

19.  The Hindu – Arabic Numerals, Ibid, 1928


20.  History of Indian Mathematics (in Tamil), Kalai Kalanjiyam (Tamil Encyclopedia, 1st Edition)

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