Studies in Indian Mathematics and Astronomy

Introduction

Late Professor Kripa Shankar Shukla (1918–2007) has been well known as a stalwart in the history of Hindu (Indian) mathematics and astronomy.

Throughout his career, he produced editions of important classical works in Indian mathematics together with translations and extensive notes. These included:

• Sūryasiddhānta (commentary by Parameśvara)
• Pāṭīgaṇita (Śrīdharācārya)
• Mahā-Bhāskarīya (Bhāskara I)
• Laghu-Bhāskarīya (Bhāskara I)
• Dhīkoṭida Karaṇa (Śrīpati)
• Bījagaṇitāvataṃsa (Nārāyaṇa Paṇḍita)
• Āryabhaṭīya (Commentaries of Bhāskara I and Someśvara)

These works laid the foundations for the development of astronomy in India in the fifth and later centuries. In addition, he also brought out editions of works of Devācārya, Lalla, Vaṭeśvara, Mañjula. This covers further developments in astronomy, except for the contributions of two of the greatest astronomers (and mathematicians), namely Brahmagupta and Bhāskara II, but there were already various published translations and commentaries on their works.

K.S. Shukla commented on the works of various astronomers and mathematicians when it came up during discussions. Having prepared a basic foundation for a critical modern study of Indian astronomy (as well as mathematics), K.S. Shukla continued to discuss specific developments in various topics of interest in astronomy and mathematics. His narrative is always detailed and easy to read. If different people had come up with similar formulas or rules, he never limited himself to just a summary of the results, but rather, carefully gave credit to all the contributors. Thus, while he summarized the main points, he always kept the original history intact.

K.S. Shukla was well known for his special skill of extracting meaningful statements even from corrupt manuscripts. He also ended up finding and correcting several incorrect emendations of manuscripts by others, and produced accurate readings from them. Indeed, this can be annoying to other researchers whose readings turned out to be in error, but K.S. Shukla cared more about honest readings of manuscripts than personal feelings.

In many places, during the exposition of the narrative in a commentary, his explanations of technical terms and justifications of the proposed narrative are very clear and sometimes more illuminating than the original.

Comments by Yukio Ohashi in his paper describing K.S. Shukla’s contributions resonated with me completely:

The topic of Indian planetary theory and its relation with Greek development has been discussed often. There are arguments about what the precise theory is and what its precise assumptions and details are. A prevalent argument suggests that the theory is based on old (pre-Ptolemaic) Greek theories. K.S. Shukla has discussed the varied opinions and established that the Indian theory is distinct from the Greek theory and that the borrowed parts are mostly related to astrology, rather than astronomy.

I would recommend looking up the details in the narrative presented by M.D. Srinivas in the section entitled “the survey of seminal contributions of K.S. Shukla to our understanding of Indian astronomy and mathematics (pages 39–69)”. The survey presents a detailed analysis of various developments in the understanding of planetary theory. Written in the style of K.S. Shukla himself, it shows that the traditional Indian theory is not borrowed and does not possess the alleged inaccuracies.

I will give a brief description and appreciation of the remaining parts of the book.

Studies in Indian mathematics: Bhāskara I to Nārāyaṇa

Bhāskara I

K.S. Shukla gives us a panoramic view of the works of Bhāskara I. Even though Āryabhaṭīya is a short work, Bhāskara I wrote two commentaries (i) laghu and (ii) mahā Bhāskarīya on it.

K.S. Shukla observes that much of the detailed work attributed to Āryabhaṭa is not extant but its impact can be deduced by references to it in later works. For this reason, he names it as Āryabhaṭa-siddhānta. The suggestion is that Bhāskara I was probably familiar with it.

The books of Bhāskara I give a systematic report of notations, terminology and classification of types of mathematics. Thus, we can understand the development of mathematics in India before Bhāskara I.

K.S. Shukla suggests that works by Bhāskara I and Brahmagupta also show that detailed algebraic and arithmetical processes were already well known and in use in India for quite a long time.

Bhāskara I goes on to discuss the kuṭṭaka (pulverizer) process. He also has constructed more than 100 exercises, based on the relatively short book Āryabhaṭīya. Thus, one may think of his work as an extensive guide and workbook for topics addressed in gaṇitapāda (mathematics section).

Bhāskara I, however, characterizes the mathematics section as “a little bit of mathematics” described by indications only. This is because important parts concerning astronomy appear only in the latter two sections: Kālakriyā (calculations with time) and Gola (Spherical calculations). Bhāskara I has constructed suitable exercises on these topics as well.

vargaprakṛti (square nature)

In the next sections, K.S. Shukla discusses a much-celebrated equation in mathematics known as vargaprakṛti, namely Nx^2+1=y^2.¹

Here, N is a positive non-square integer. The quantities x,y are required to be integers also.

This equation makes its first appearance in Brahmagupta’s Brāhmasphuṭa-siddhānta (628) and ultimately Bhāskara II (1150) is credited for its complete solution. However, Bhāskara II, humbly states that the majority of work was done in extensive treatises of Brahmagupta, Śrīdhara and Padmanābha and he has only put a short discussion of the method of solution.

It is unknown where and when Śrīdhara lived and his only surviving work is Pāṭīgaṇita. He has given a rational solution (which means x,y are permitted to be rational numbers). His arguments are described in detail by K.S. Shukla. It thus gives a solution to the modified equation Nx^2+z^2 = y^2 where z is allowed to be some integer.

While he did not finish the original problem, his techniques of modifying forms of a chosen solution are very useful.

Another obscure name connected with the equation is Jayadeva. There is no extant work, but reference to and some details of his work appear in a commentary named Sundarī on laghubhāskarīya by Udayadivākara.

K.S. Shukla gives a detailed quote from the commentary and shows that Jayadeva had made important progress towards the solution of the equation by the cakravāla (cyclic) method. This is the method that leads to the final solution by Bhāskara II.

Jayadeva shows how to construct many triples (x,y,c) where x is the lower root, (y) is the upper root and c is the interpolator so that (x,y,c) satisfy Nx^2+c = y^2.

In other words, we now ask “what are the possible values of c which lead to a solution?”. The main problem thus becomes “is c=1 possible?”

He showed that if (a,b,k) is a solution to the above equation, then there will be a new solution (x_1,y_1,k_1) where k_1 = \frac{t^2-N}{k} provided that \frac{at+b}{k} is an integer.²

We thus have a new interpolator k_1 to use. It was proved by Brahmagupta that if we get one of the interpolators \pm 1, \pm 2,\pm 4 then we are done (i.e. we have c=1 possible). While Jayadeva had the right tool to solve the problem completely, he did not find the process to get to the desired value c=1. The commentary describes this process as trying to land flies against the wind (in the right place).

It was the genius of Bhāskara II to create an unusual “minimization process”, which finally managed to land the flies in the correct place (i.e. solve the problem!). In view of this, it is suggested that Jayadeva’s name should be added as a part contributor to the final solution.

Series with fractional terms

A series is defined as a sum of n entities a_1+a_2+\cdots +a_n where n is an integer. But suppose that we wish to consider the sum of n terms where n is a fraction say for example n=3.5. Then what interpretation is possible?

K.S. Shukla describes problems from different works where such practical problems arise. He gives examples starting from Bakhshali manuscript all the way to Śrīdhara’s Pāṭīgaṇita.

The problems are designed so that a_j is an arithmetic sequence,
a_1=a, a_2=a+d, \cdots a_j=a+(j-1)d,\cdots ,a_n+(n-1)d.

Now if we take 3.5 terms, we declare the sum to be a+(a+d)+(a+2d) +(1/2)(a+3d). In other words, the 3.5-th term is (1/2)(a+3d) which is the natural fourth term multiplied by (0.5).

This can be summed up and interpreted as the desired answer. There is a geometric interpretation of the sum as an area under the line y=a+dx.

In other words, they seem to be taking the first step toward integration!

There is a discussion when a is negative and it corresponds to the usual problem of areas, namely if your curve crosses the x-axis, then the area under the curve can be different from the area between the curve and the x-axis.

Factoring integers

Given a positive integer n, the process of factoring it into prime factors was studied.

Śrīpati gave the usual answer of factoring an integer n thus:

• First take out the simplest factors 2,3,5.
• Next, if n is a complete square m^2, then it is enough to factor m and double up each factor of m.
• If the first two steps are carried out, then we assume that n is a non-square and not divisible by 2,3,5. Thus, we may let n=a^2+r where r\ne 0. If 2a+1-r is a square b^2, then we see that n = (a^2+r)+(2a+1-r)-b^2 = (a+1)^2-b^2. This gives the factorization n=(a+1+b)(a+1-b).

Nārāyaṇa gave further details of how to proceed if the above fails to find the square b^2.

Try to see if (2a+1)+(2a+3) - r = b^2. If b can be found, then we get n=((a+2)+b))((a+2)-b). In general try if, (2a+1)(2a+3)+\cdots (2a+2s+1) -r = b^2.

This will give us n=(a+s+1)^2-b^2 = (a+s+1+b)(a+s+1-b).

Comment: Neither of these rules explicitly discuss how to recognize when n is just a prime number. It is also not clarified why for some s, we must get the necessary b^2.

However, as K.S. Shukla informs us, based on Dickson’s “History of Theory of Numbers”, Fermat seems to have rediscovered the same rule and reported it in a letter to Mersenne. Fermat works out the factorization of the number 2027651281 as an example.

K.S. Shukla also reports alternate methods of Fermat to check if a number is a square or not without actually finding its square root. For example, in the letter, Fermat says that 49619 is not a square because its ending digits 19 are not the ending digits of any square!

K.S. Shukla also remarks that Fermat did not have such a factoring method before writing the letter, so the credit goes to Śrīpati!

Nārāyaṇa also gives a systematic method of calculating all the divisors of an integer if its full prime factorization is known. An alternate method by Nārāyaṇa for factoring is also described.

Magic squares

K.S. Shukla goes on to discuss another highlight of Nārāyaṇa’s work in constructions of magic squares as well as several other magic shapes. These are diagrams with cells filled with integers so that we get a common sum for all rows and columns and sometimes for other fixed type shapes inside the diagram.

Systematic techniques to construct magic squares are also given. A more extensive survey of the same topic appears in the next section.

Revised version of the manuscript of the Third Volume of Datta and Singh

The original book by B. Datta and A.N. Singh was designed as a three-volume book. Unfortunately both the authors passed away before the book was finished. K.S. Shukla inherited the manuscript from B. Datta. Instead of making a single third volume, he chose to edit and publish it as eight separate papers in the Indian Journal of History of Science. We are fortunate to find all of them republished here.

I will discuss the details of these in this section.

Hindu geometry

In this revised version of the third book, I will use the term “Hindus” instead of “Indians”, which was clearly favoured by the authors of the first two books Datta and Singh, as well as by K.S. Shukla.

Given the fact that historically “Hindu” referred to a region of residence, rather than a “religion”, this is appropriate.

Regular offerings to Gods were considered necessary for the nation to live in prosperity and free of disasters. The place where the offerings were made (vedī or altar) was required to be of special shapes, to be made from a prescribed number of bricks in every layer. Moreover, the area of the vedīs had to be commensurate with the measurements of the person who has undertaken the ritual.

K.S. Shukla observes that the early geometers had to tackle these types of problems:

• Construct a square on a given line (including the problem of drawing the line in a specific direction first).
• Construct a square with the same area as a given circle, and also construct a circle of the same area as a given square.
  More generally, transform various geometric shapes into other geometric shapes, without changing the area.

• Given a circle, double it in area.

Of course, these are well-known problems and Greeks asked and attempted to answer such questions. Specifically, they also raised a harder problem of determining if these can be solved by a ruler and compass alone.

It took over 3000 years to get a mathematically precise answer to these problems. Since the rituals were a daily duty, the Hindu geometers were more interested in getting a practical solution, though it may be only an approximate.

K.S. Shukla reports on the earliest treatises called the Śulbasūtras, which were parts of Vedic manuscripts and were written in a cryptic style. He also notes that Baudhāyana Śulbasūtra might have been the first to announce the Pythagorean Theorem several hundred years earlier than Pythagoras. He did not give any proofs but many of his geometric constructions make it apparent that he was taking it as a valid fact.

The sūtras also give some geometric “proofs” for specific problems like finding the area of a trapezium (Āpastamba). Around 300 BC, Jain religious books contained a fair amount of geometry and other mathematics. This includes ellipse, elliptic cylinders and a convenient approximation to \pi as \sqrt{10}.

But they did not take the next step of postulating elliptical orbits; it had to wait for Kepler and Newton!

K.S. Shukla also reports on the importation of Euclid’s books Rekhāgaṇita in Sanskrit by Samrāṭa Jagannātha 1718 AD.

It is noted that the formula for the volume of a sphere seems not to be worked out precisely, until Bhāskara II (1150 AD). K.S. Shukla notes that the development of geometry in India stayed dormant for several hundred years, until enough algebra and trigonometry were developed.

K.S. Shukla goes on to discuss various theorems discovered by Hindus, all the way up to the 16th century.

I cannot summarize the whole encyclopedic narrative here, but I will concentrate on the squaring of the circle which comes down to the discussion of \pi which is the circumference of a circle divided by its diameter.

In modern mathematics, the precise value of \pi has been proved to be a transcendental number (Lindemann 1882) and hence it cannot be written down as a finite decimal number.

However, approximate values were found and used by necessity. They range from the crudest 3 to the precise formula in terms of an infinite series in the 15th century by Mādhava. The most satisfactory result is an infinite series of rational numbers with alternating signs together with a correction term after adding up the first several terms. This shows a sophisticated approach of recognizing the meaning of the convergence of an infinite series, together with an end correction if it converges too slowly.

The game of finding digits of \pi is still a popular activity to show off the computing capabilities of newer and better computers, or computer algorithms.

Mean values: The modern concept of mean values of a sequence of numbers is reported to go back to Brahmagupta (seventh century). He discusses the average depth of an excavation in a region of given length and width.³

Hindu trigonometry

K.S. Shukla gives a detailed account of definitions and calculations of trigonometric functions commonly used in Hindu mathematics. Of course, comparison and contrast with Greek trigonometry are needed for a full understanding.

To begin with, the \sin function is not a ratio of two lengths (i.e. a dimensionless real number). For Hindu trigonometry, consider a circle of radius R with centre at O and an arc AB smaller than a quarter circle. Then its jyā is defined to be the length of the perpendicular dropped from A onto the segment OB. This jyā is denoted as R\sin(\theta) where \theta is the angle between OB and OA. This will be the usual \sin(\theta) if R=1.

The Greeks used the function “chord” which is simply the length of the segment AB. If we let the angle between OB and OA to be \alpha then the formula for the chord(\alpha) = 2R\sin(\alpha/2).

This slight change in the definition makes the trigonometric formula for addition of two angles rather complicated in the Greek system. The Hindu definitions are shown to be the same as the modern definitions, except R\sin(\theta) is nothing but R\cdot \sin(\theta).

Moreover, the modern definitions permit angles of all sizes, and not just smaller than a quarter circle. Hindus knew how to handle all sizes of angles and they knew very well that using bigger angles causes a change of sign in the formulas, depending on which quarter the angle lies in. This is necessary since the basic definition of jyā is a line segment (of possible zero length). The idea of changing rules for angles in the first or second quarters occurs in Greek trigonometry also.

After explaining that the terms sine, cosine are derived from the word jyā through various translation steps, K.S. Shukla reports that in Hindu trigonometry, mainly three trigonometric functions R\sin(\theta), R\cos(\theta) and utkramajyā or R{\rm versin}(\theta)=R-R\cos(\theta) (which may also be denoted as R{\rm ut}(\theta)) are defined and used.

R{\rm versin}(\theta) is also called {\rm Vers}(\theta) because its values are obtained by subtracting Rsines from R in reverse order (largest angle to smallest angle).

An extensive summary of trigonometric formulas follows, making it apparent that most of modern trigonometry was built in the format of Hindu trigonometry, except for the modern idea of an angle being just a real number.

For some reason, Hindus did not use symbols for R\tan(\theta) or R\sec(\theta), choosing to use explicit formulas, when needed.

After establishing the basic formula \sin^2(\theta) + \cos^2(\theta) =1 which translates to (R\sin(\theta))^2+(R\cos(\theta))^2=R^2, they went on to develop the formulas for the addition, subtraction and multiples of angles. This includes half-angle formulas as well.

K.S. Shukla reports on the explicit derivations of the relevant formulas throughout history.

One common term for Hindu trigonometry was jyotpatti—the development of jyās (i.e. sines). They developed tables for sine values for angles corresponding to 96 subdivisions of the circle (and later on many more or fewer subdivisions as desired). K.S. Shukla also reports on approximation techniques for such values.

Of special interest, is the development of power series expression in the Kerala mathematics school from 14th to 16th centuries and this is reported later.

They also calculated the functions of the usual angles 30^{\circ}, 45^{\circ}, 60^{\circ} etc. By using the half-angle formulas already developed, they evaluated sines of other related angles.

They also gave formulas for approximate values for \sin(\pi/n) for small values of n. Similar approximations were developed for the arcsine function (angle with a given sine)

Spherical trigonometry was not as well developed, but they did manage to solve spherical right-angled triangles and did the necessary calculations for more general triangles. In this topic, Greeks had the theorem of Menelaus and Ptolemy (AD 150) used it systematically to develop more advanced spherical trigonometry.

Use of calculus in Hindu mathematics

This topic is controversial and opinions range from “there was no basic concept of calculus in Hindu mathematics” to “calculus was fully developed in Hindu mathematics and exported from India”.

K.S. Shukla avoids both extremes and gives a factual survey.

The story begins in the middle of the 19th century (1858) when Pandit Bapudeva Sastri presented analysis by Bhāskara II which seems to have announced a differential formula \partial(\sin(\theta)) = \cos(\theta)\partial(\theta).

His claim was criticized by Spotiswoods by arguing the following points:

• Bāpudeva Śāstrī had overstated his case by claiming that Bhāskara II was fully acquainted with principles of differential calculus.
• The analysis of Bhāskara II contains no allusion to the infinitesimal nature of time and space divisions used.
• The approximative character of the result was not realized.

Having stated these objections, K.S. Shukla notes that no further serious investigation of the matter was undertaken except for a paper by P.C. Sengupta and subsequent discussion in a survey book “Mathematics in Ancient and Medieval India” by A.K. Bag.

K.S. Shukla then goes on to explain the details of Bhāskara II’s work.

The mean anomaly of a planet was known to have a formula (already from Āryabhaṭa and Brahmagupta) of the form e(\sin(w')-\sin(w)) where w' and w are close instants and e is the eccentricity of the planet’s orbit. Sufficiently precise evaluation of this difference when w,w' are small was needed for resolving problems about conjunctions of planets or occultations of stars. This was called tātkālika gati (instantaneous motion). Its evaluation requires a more precise calculation of the expression and the usual tool of tables is too crude for this purpose.

K.S. Shukla reports that Mañjula (932) was the first to claim a formula of the type\partial \sin(\theta) = \pm \cos(\theta)\partial \theta, but without further details. Bhāskara II gave an explanation by introducing a notion of instantaneous displacement \sin(w')-\sin(w) which in modern calculus is denoted as \Delta(\sin(\theta)). He goes on to explain that this instantaneous motion is the motion along a tangent line to the path at the time of interest and gives the position over a small enough interval where the motion may be assumed linear.

In modern terminology, he seems to be describing the idea of a differential d(\sin(\theta)) and he shows that d(\sin(\theta))=\cos(\theta)d\theta.

The formal differentials, indeed, give an alternate treatment of calculus, without using limits!

K.S. Shukla notes that Bhāskara II also makes two important observations (first noted by Sudhakara Dvivedi).

• At a maximum value of a variable its differential vanishes. This anticipates the idea of critical points.
• The equation of the centre vanishes at apogee and perigee and hence the differential of the equation of the centre vanishes at some point in between. This anticipates the mean value theorem.

K.S. Shukla also reports how Nīlakaṇṭha (AD 1500) made formulas for the differentials of \sin(\theta), \cos(\theta), formulas for second differentials of \sin(\theta) as well as the differential of inverse function of \sin^{-1} e \sin(\theta).

He also reports an instance of quotient rule by Acyuta (1550–1621 AD).

Thus, ideas of calculus (of differentials) were not unique to Bhāskara II.

Finally, the proofs of the formulas for the surface area and volume of a sphere obtained by Bhāskara II are presented which show the idea of integration using bigger and bigger subdivisions. Indeed, such integrations have also been used by Greeks and others. The only difference to be noted is that the Greeks used the method of exhaustion since they rejected the proofs using limits, due to Zeno’s paradoxes!

Remaining parts of the Revised version of the Manuscript of the Third Volume of Datta and Singh

In the interest of brevity, we will give only a brief outline of the remaining sections.

1. Use of Permutations and Combinations in India:
   The number of permutations or combinations of a given number of objects was started from Piṇgala (200 BC) for determining different numbers of meters formed from combinations of long and short syllables of a specific count.

   The subject continued, to calculate the number of different mixtures from chosen tastes to the number of different kinds of objects that arise from a chosen set of attributes.

   K.S. Shukla goes on to describe the development of permutations and combinations formed from a chosen set of either distinct or repeated objects. If integers were chosen, then the sum of the numbers resulting from permutations was also evaluated.

   They also developed what is now called Pascal’s triangle to determine the number of combinations of n objects taken r at a time (Halāyudha’s Meru prastāra from the 10th century.)

   Jains were not satisfied with just counting the number of arrangements, but wanted to assign a serial number to each arrangement in a systematic way (a hashing function) and developed the rules for creating such serial numbers.

   In short, they developed the basic structures of the theory of permutations and combinations.

2. Magic Squares in India:
   This section gives a more extensive survey of various Magic figures (not just squares), together with methods to construct a specific size of Magic square.

   The main contribution reported is by Nārāyaṇa. The general scheme is to start with one or more arithmetic progressions with the same common difference. Moreover, the used progressions have their initial terms also in an arithmetic progression.

   There are different techniques for squares of size 4n or 4n+2 or 2n+1.

   There are numerous special squares and other figures given, which are spread over more than 50 pages!

   K.S. Shukla also describes how many magic squares were reinvented after several centuries by others in the world.

3. Use of series in India:
   K.S. Shukla begins with discussion of the well known geometric and arithmetic series together with Mahāvīra’s arithmetico-geometric series (850 AD.)
   \sum_1^n t_m \ \mathrm{ where}\ t_1=a \ \mathrm{ and}\ t_m=rt_{m-1}+b \ \mathrm{ for}\ m\ge 2.

   Given a series \sum_1^n a_m, a natural new series is given by \sum_{k=1}^n\left(\sum_{m=1}^k a_m\right) which is a second order summation.

   Formulas derived for higher-order summations are discussed. The sums of \sum_{m=1}^n m^r were also analyzed.

   The crowning achievement from such development is the derivations of formal power series (in 14th to 16th century) by Kerala mathematicians for \arctan(x), \sin(x), \cos(x) and related functions which were not available elsewhere until much later.

   The \arctan series is often used to get an expansion of \pi and Kerala mathematicians used it for the same purpose.⁴

4. Surds in Hindu mathematics:
   A surd is named as karaṇī in Hindu mathematics and its first letter ka is used like the \sqrt{\ } symbol. In general, a surd is a number whose square root cannot be determined exactly (Śrīpati (1039)). A surd is a sum of a rational number (possibly zero) and one or more terms of the form \pm \sqrt{a} (or in traditional notation \pm kb \sqrt{a}) where a is a non-square.

   To add two such surds, say, \sqrt{a} \pm \sqrt{b}, it is recommended to find an auxiliary rational number c such that ac and bc are squares and then
   \[\begin{align*}\sqrt{a}\pm \sqrt{b} &= \sqrt{(1/c)(\sqrt{ac}\pm \sqrt{({bc})})^2}\\
   &=|(ac\pm bc)|\sqrt{1/c}.\end{align*}\]

   Various choices of c are possible/desirable and described by different mathematicians. However, in the rest of the discussion, it is still treated as an ordinary sum of surds as described above. Surds are multiplied by term by term multiplication, and division is carried out by the usual rationalization.

   The most interesting analysis is by Bhāskara II which gives rules of determining if a given (sum of) surds is a complete square of some other (sum of) surds. In fact, he complains that no other writers have given a way to test if such a square root exists or not, and proceeds to give a well-defined algorithm!

5. Approximate values of surds in Hindu mathematics:
   The operation of taking a square root of a given number was important in the calculation of sines (using half-angle formulas), but the solutions come out as surds or even iterated square roots of surds.

   To get usable answers, it is necessary to have reasonable approximations of them in terms of rational numbers.

   K.S. Shukla reports on the statement of an approximation of \sqrt{2} as given in Baudhāyana (800 BC)⁵:

   \sqrt{2} \approx 1+\frac{1}{3} +\frac{1}{3 \times 4} - \frac{1}{3 \times 4 \times 34}.

   K.S. Shukla goes on to describe various approximation techniques described by numerous other authors. To me, the most interesting method is to use the solution of the Cakravāla method as suggested by Nārāyaṇa (1356) which says that ifNx^2+1=y^2 has an integer solution (x,y), then \sqrt{N}\approx \frac{y}{x}.

   However, the other methods are also useful for a quick approximation in a desired form or accuracy.

Studies in Indian Astronomy: From Vedic period to the emergence of Siddhāntas

This section gives details of the development of astronomy in India starting from Vedic times and reports on various theoretical developments until modern times.

The discussion includes a historical survey of the treatises called Siddhāntas which give methods of calculations of planetary positions and also develop needed corrections to various astronomical parameters, as they change over the ages.

It also includes a critique of various translations of historical texts. It exemplifies the exceptional skill of K.S. Shukla in extracting meaning from manuscripts which are usually full of inherent errors of transcription.

Vedic Times

The earliest developments were between 2500 BC to 500 BC and most of the results are preserved in Vedas and their appendices called Vedā gas. The main interest was to identify the precise time for performing seasonal rituals. This required the precise calculation of the movement of the sun in its orbit and the relative positions of the moon on a daily basis. For ease of calculations, a five-year cycle (called Yuga) was developed and used. The text is by Lagadha and it has two recensions, one for Ṛgveda and one for Yajurveda. These are mostly similar. A third recension attached to Atharvaveda was developed later and it gives current names of the planets and more details of the calendar useful for astrology.

Siddhāntas

There are five main astronomical treatises which were developed from 80 AD (Paitāmaha) up to the 6th century AD. It, together with the remaining four: Saura, Vāsiṣṭa, Romaka and Pauliśa were described by Varāhamihira (died 587 AD) in his work called Pañcasiddhāntikā. None of these are extant and many have undergone revisions over the years. The last two have evidence of connections with Babylonians and Greeks.

However, K.S. Shukla observes that the refinements made by Ptolemy (2nd century AD) do not appear to be known in Indian treatises.

With the development of zero and decimal system in algebra and further developments in trigonometry, a new Renaissance started, which continued into the twelfth century. After that, the development in the North was affected by Muslim rule, but important work continued in the South (especially in Kerala).

After giving details of several contributors to the development, K.S. Shukla lays down the basic hypotheses of Indian astronomy:

• Hypothesis 1: Mean planets move in geocentric objects of various sizes.
• Hypothesis 2: True planets move in epicycles or eccentrics.
• Hypothesis 3: All planets have equal linear motion in their respective orbits.

For ease of calculations, they determined an epoch—a theoretical point in time when all the mean planets were at zero longitude. This helped in calculating positions of the mean planets on a given day (or time) after determining the number of days elapsed from the epoch, called day-count (ahargaṇa).

To determine the position of the true planet, four corrections were applied, based on local longitude, equation of centre, correction for eccentricity of the ecliptic and correction for local latitude. The last correction was mainly needed for the sun and moon, but their application for other planets may depend on the astronomer.

There were further corrections developed to account for the observations.

K.S. Shukla notes that unlike the Greeks, the Indian epicycles varied with planets and positions of the planets in their orbits.

K.S. Shukla also recalls many contributions of Āryabhaṭa including improved astronomical parameters, a theory of the rotation of the earth (criticized by many other astronomers), the introduction of modern (half) sines, value 3.1416 for \pi and integer solutions of linear integral equations.

Many of these were followed for centuries after his time. The current version of Saurasiddhānta is known to match Āryabhaṭa’s parameters and methods.

Pañcasiddhāntikā Texts

K.S. Shukla next comments on some of the editions of Pañcasiddhāntikā of Varāhamihira. Besides giving a comprehensive analysis of several topics, he gives specific corrections to editions by Thibaut, Dvivedī, Neugebaur, Pingree.

Āryabhaṭa I’s astronomy with midnight reckoning

As already asserted, the famous treatise Āryabhaṭīya is only a shortened version of Āryabhaṭa’s much bigger treatise Āryabhaṭasiddhānta which is no longer extant but there are numerous detailed references to it in later works, starting from Bhāskara I. One of the important one is by Brahmagupta, who after criticizing Āryabhaṭa’s work, later wrote a treatise khaṇḍakhādyaka. To give easier methods of calculations for the theory in Āryabhaṭasiddhānta. This was found to be useful and was transmitted to Arabs afterwards.

One of the main features of Āryabhaṭasiddhānta is that it starts a new day at midnight at Laṇkā. Naturally, this led to a shift in many formulas and perhaps made the calculation complicated.⁶

After giving some details of this work, K.S. Shukla goes on to describe several astronomical instruments discussed by Āryabhaṭa, as reported by a commentator, Rāmakṛṣṇa Ārādhya, of Saurasiddhānta. Various commentators have also described concrete methods of using the instruments for observing astronomical phenomena.

Development of Siddhāntic Astronomy: Some Highlights

This section describes noteworthy developments in Hindu astronomy.

Early Hindu methods in Spherical Astronomy

After a brief review of plane trigonometry, K.S. Shukla discusses the problem of solving a spherical triangle, i.e. determining all six quantities (the sides and angles), given at least two quantities (which are not both angles).

While a solution for right-angled spherical triangles was known and used systematically to get precise or approximate solutions, the same was not well developed to oblique spherical triangles in India before the twelfth century.

K.S. Shukla describes a fundamental formula for right-angled spherical triangles developed by Nīlakaṇṭha (1500 AD) in Kerala, which can be used to solve all spherical triangles in a systematic way. Apparently, Nīlakaṇṭha’s work did not travel to North India and astronomers in North India continued to work on the problem till the seventeenth century.

K.S. Shukla also notes that the theorem of Menelaus and its use by Ptolemy among the Greeks (before second century AD) for the solution of oblique spherical triangles appears to be unknown in India.

The first one to set up the procedure for right-angled spherical triangles was Bhāskara I, based on teachings of Āryabhaṭa. He did not describe a systematic method, but gave a series of twelve illustrative problems and described their solution techniques.

However, both Bhāskara I and Brahmagupta did not have precise solutions if the spherical triangle was not right-angled. Brahmagupta, using his expertise in solutions of quadratic equations, gave formulas using surds, which needed approximations for derivations of concrete answers.

Calculation of the centre controversy

One of the long-standing controversies was the formula for the equation of the centre for a planet. The method of Āryabhaṭa I did not use the calculation of the hypotenuse and this was considered as producing only an approximate answer by modern reviewers of Indian astronomy (and also including Pṛthūdakasvāmi). K.S. Shukla shows how the original method was in fact, accurate and all other commentators concur with this assessment.

Vaṭeśvara and his work

Vaṭeśvara worked in the ninth century and there are two works Vaṭeśvarasiddhānta and Karaṇasāra attributed to him. K.S. Shukla describes known history and quotations of Vaṭeśvara. His extant works are not complete, but much can be learned about them from commentaries.

Having edited Vaṭeśvara’s extant work, K.S. Shukla goes on to note that he is credited with the discovery of two corrections to the moon’s orbit, namely the evection (change in eccentricity) and the deficit of the equation of the centre.

This has a long history. Ptolemy had given two corrections to the moon’s orbit but in practice, they turned out to be not as accurate as desirable.

The modern theory of the moon’s orbit has several corrections (about 70 terms) but they depend on the use of modern instruments of observation. K.S. Shukla lists four major terms of modern lunar theory and shows how they relate to the Hindu formulas.

K.S. Shukla goes on to discuss details of the calculations and points out how the eccentric and epicyclic theories are related to the corrections.

He also observes that the Hindu methods of graphic representation explain both corrections and predate their discovery in Europe by several centuries.

Moon phases and conjunctions of stars and planets

For religious purposes exact Tithi, i.e. lunar day number of the month is important. Lunar day number needs to be determined for a particular time (when the religious activities are planned).

Moreover for astrology, it is necessary to determine if a certain time is auspicious or inauspicious and this determination changes from person to person based on their exact birthdate.

These considerations make it necessary to know the exact phase and position of the moon relative to the sun.

Further, when planets are close to specific stars or each other (in terms of the line of sight as viewed from a specific place), there are astrological consequences of such an event (called conjunction). One of the most striking events occurs when the sun and moon have a conjunction and it becomes a solar or lunar eclipse depending on whether the earth is between the sun and the moon or outside.

K.S. Shukla gives an extensive survey of procedures or formulas, given by different astronomers, to help with such problems.

The list of the topics is:

1. Moon’s phases, including the shape and width of the illuminated part of the moon.
2. The position of the horns of the moon.
3. Visibility of the true (corrected) moon at any given time.
4. Altitudes of the sun and the moon at a given position and time. K.S. Shukla gives a detailed explanation from traditional as well as modern spherical trigonometry.
5. After projecting the positions of the sun and the moon in the plane of the observer’s meridian we get a right-angled triangle with vertices S (sun’s projection), M (moon’s projection ) and A is the point on the base so that \triangle MAS is right-angled at A. The sides SA and AM are respectively called the base and the height.
6. Determination of heliacal rising and setting of any planet or star are determined by the closeness of the planet or the star to the sun in longitude.
7. Determination of the time difference between rising and setting of a planet (especially the moon).
8. Determination of a conjunction between a planet and a star.

Reviews and Responses

This last section contains reviews by K.S. Shukla of works beside his own. It also contains reviews of some of the works of K.S. Shukla by others, usually with responses by K.S. Shukla and sometimes with additional notes by him.

1. Vedic Mathematics review: In addition to the details of the book by Bhāratīkṛṣṇa Tīrtha, K.S. Shukla goes on to argue that the title of “Vedic” is misleading and the book is not actually related to any known vedic traditions.
2. Review of Rājamṛgā\dot{\mathbf{n}}ka of Bhoja, with full text based on new additional sources by David Pingree: The review consists of some small corrections and appreciation for a new complete edition, even though many additional manuscripts still exist and deserve to be analyzed.

   There is also an additional small note on some of the verses from the original (incomplete) edition of K. Madhava Krishna Sarma and translation of the same verses by S.B. Dixit.

3. Review of Karaṇaratna by Raymond P. Mercier: Karaṇaratna is by Devācārya and edited by K.S. Shukla. Mercier rejects some of the emendations by K.S. Shukla and proposes new corrected formulas.

   This is followed by a critique of the review by K.S. Shukla refuting some of the claims by Mercier.

4. Review of edition by David Pingree of the Yavanajātaka: The review restricts itself to the 79th chapter and especially to the different aspects of the yuga in the text.

   K.S. Shukla points out several mistakes in the translations and emendations and offers corrected formulas.

5. Review of Vaṭeśvarasiddhānta and Gola by David Pingree:
   The two volume critical edition with translation is reviewed by David Pingree.

   He expresses disagreement with the emendations by K.S. Shukla and claims that his editorial and translation policy does not conform to the general practice.

   The review has a long Appendix giving Pingree’s amendments to the Sanskrit text. He has not commented on the translations.

   I would have loved to see the response by K.S. Shukla but perhaps it does not exist.

Final Comments

These collected essays of K.S. Shukla outline the history of astronomy (and mathematics) in India through the ages. It is certainly a valuable source.

We recall that the original vedic astronomy was mostly concerned with maintaining the precision of time, mainly for religious reasons. Without good tools of observation and advanced techniques of calculations, this proved to be challenging.

This gave rise to siddhāntic works, which developed formulas for calculations once fundamental astronomical units are fixed (and amended over time when needed). Its development has been explained by K.S. Shukla.

I wish there were a comparable work to explain modern calendar making and the history of how the various calendar systems in use in India have evolved.

Here are the details of calendar making that I would like to see elaborated.

• The main controversy seems to be the precise determination of exact dates (tithis) and exact months. The original vedic work of Lagadha seems to rely on precisely determining the solstices and using these as markers. This determines the solar year and it is then divided into months by using a lunisolar system. Eventually, this evolved into an observation of the positions of the sun and the moon relative to stars (presumed fixed). This gave rise to the construction of the zodiac, a division of the ecliptic into 12 equal parts. To identify the parts, certain prominent stars were identified in each part.

  This produced the calendar system termed “nirayana”. It relies on the assumption that the resulting parts should be eternal, since stars are fixed!⁷ How and when did this nirayana system become fully established?

• The nirayana system, if strictly observed, loses the connection between the climate and months, but this fact was not observed until much later and a new definition for the months had to be devised, the so-called “sāyana” system. The main reason for the discrepancy is caused by the precession (called ayana) of the earth, which shifts the starting position of solstices continuously. With this adjustment, the connection between the climate and months can be restored.

  Originally, a fixed formula for the precession was devised, which nevertheless becomes inaccurate over time.

  This takes us back to the original problem of precisely determining the solstice points.

• These days, the problem of finding the day number is not so important, since several member nations of International Astronomical Union share precise observations of different planets (including the sun) and this lets us use the end of the last year as an epoch, thus minimizing the effect of errors in the calculation formulas.

  As I understand, most calendars in India use this data published by the Indian Government as a starting point and then apply whichever calculation system that they are used to.

• It would be interesting to find out what traditional astronomical formulas, if any, are used by various calendar makers and what causes the variations in different calendars. Details of the formulas used by online sites (using computer calculations) are also of interest.

  The history is outlined in many places, but the details of the methods are left out in the narrative. Hence the above wish list is created.\blacksquare

Footnotes

1. Pell’s equation: This equation acquired the name Pell’s equation in the 17th century, although Pell seems to have only revised an English translation of a book by Rahn, which had a proof using continued fractions developed by Brouncker and Wallis. In the western world, the equation was proposed as a challenge to fellow mathematicians by Fermat. Fermat was not aware that the solution to his challenge was already published by Bhāskara II some 500 years earlier. Several important developments sprang from work on the equation. The first was the concept of continued fractions developed by Brouncker and Wallis. Lagrange finally gave a proof later. It finally grew into the theory of binary quadratic forms (developed by Gauss). ↩
2. Indeed, we may take x_1=\frac{at+b}{k} and y_1=\frac{Na+bt}{k}. ↩
3. Editor’s note: For a detailed exposition, see Amartya Kumar Dutta. “Weighted Arithmetic Mean in Ancient India” Bhāvanā. Oct 2017.1(4). ↩
4. This history is fascinating and some people claim that the subject of power series might have been transmitted from India to Europe. ↩
5. Personally, I find it curious why such an approximation is given in the Śulbasūtras since it does not seem to have a practical application similar to the rest of the material. ↩
6. I was looking for a reason to use midnight as the starting point of the day in the discussion. Unfortunately, I did not find the same. It is interesting that both the midnight reckoning and sunrise reckoning are practised even in modern times!. ↩
7. Let us ignore the fact that motions of the stars are not noticeable within a short time of only a few hundred years. ↩