Over the last century, several books have been published, detailing the history of mathematics with particular attention to India, focusing on different time periods, from the Vedic times to the modern era. Some give a general survey; some concentrate on specific areas of mathematics while some others are restricted to a critical study of specific manuscripts or mathematicians. Most of these books typically provide the details about “when, who and what” in mathematical history.

Divakaran’s book, however, is in a class of its own. In addition to the “when, who and what”, he also discusses the “how and why”, and connects historical developments in linguistics, philosophy and cultural traditions; at the same time he observes their meaningful relations with mathematical history. He addresses previous discussions and controversies, and presents his own views and opinions about the feasibility of others’ arguments. His commentaries are critical without being dogmatic.

The book provides details of mathematics developed primarily in India, bringing out its salient points while using modern notation and conventions to help foster a better understanding and appreciation of the topic on hand. One problem with some of the books on history of mathematics in India is that they do not present the original text from a manuscript during an exposition, leaving the reader in doubt about the accuracy of the translation presented. Divakaran alleviates this problem by providing the necessary text from the original manuscripts (at least in Roman IAST^{1} notation), or sufficient references in places where the text itself is missing.

The book comprises three major parts: (1) Mathematical development in ancient India, (2) Āryabhaṭan revolution (covering the period from Āryabhaṭa to Bhāskara II), and (3) The Nīla school in Kerala starting from Mādhava. It refrains from merely providing a list of dates and personalities; it gives, instead, vivid descriptions of developments, bringing in the right amount of detail.

### Mathematical Development in Ancient India

The story begins with the Indus Valley civilization. In a sense, this is a dark period since their script remains undecoded, and no manuscripts survive. However, archaeology suggests an urban culture with well developed and organized cities that had regular floor plans. Moreover, there is also evidence of uniform measures for length and weight measurements in use across different settlements. This uniformity exists despite an apparent absence of a central authority (no sign of wars either). The picture of this civilization that evolves is that of a peaceful, prosperous society with agriculture, manufacturing and commerce. In Mesopotamia, Indus seals have been found and there are references to a country called Meluha, which is conjectured to be a region on the banks of the river Indus.

One of the mathematically interesting features of this civilization is a sequence of cubic stone weights forming a regular series of weights, which are in multiples by two for small weights, evolving into complicated binary and decimal multiples for higher weights.

Another aspect of mathematical interest is the architecture that consists of well-designed rectangular structures for plots and roads. They must have known enough geometry. However, the dimensions of the bricks in Indus Valley do not seem to use any usual Pythagorean triple! Divakaran proposes an intriguing idea that rather than using the Pythagorean principle to generate right angles, they might have used the property that the line through the intersection points of two circles is perpendicular to the line joining their centres. Moreover, if these circles have equal radii, then the perpendicular passes through the midpoint of this line. This is a technique that is seen used later in Śulba sūtra geometry.

It is also suggested that they had a numerical system with base 8, while 10 played a special role. The proposed connections with Tamil language are very interesting. Unfortunately, the available script-samples from this period are few in number, and seem to be isolated sentences. Moreover, as mentioned earlier, nobody has yet succeeded in deciphering the script. The author gives numerous details about the nature of the script and its short appearances in other parts of India as well as in the Middle East. The “Indus Valley book” is thus still awaiting a comprehensible interpretation.

The Vedic era begins from roughly around 1200 bce. This period witnessed numerous interesting developments, including the composition of the Veda and Vedāñga—the sacred scriptures from India. The author goes on to describe this huge body of literature, its language and its most prominent feature, namely, the oral tradition of preserving and perpetuating the content.

Even though they are considered scriptures, the Vedas play a central role in Indian culture and progress. One discovers the beginning of astronomy and the development of the number systems in this great body of work. The fundamental notion of “zero” arises from the philosophical significance of śūnya, and notions of combinatorics spring from organization of metres.

The development of Bauddha and Jaina religions (circa 500 bce) also contributed to the mathematical traditions later on.

Divakaran next turns to the details of Vedic geometry in the Śulba sūtras which span the period from 800 bce to 400 bce. Although the Śulba sūtras are described as a manual for altar construction, they do carry certain interesting mathematical ideas. Starting from simple constructions of basic directional lines and perpendiculars, the description progresses into transforming figures, while keeping the area constant (or a fixed multiple).

The author’s commentaries are critical without being dogmatic

There is an interesting reason behind the need for such transformations. First, the unit of length is not universal, but rather determined by the body measurements of the performer of the vedic rituals. Second, the areas and the number of bricks for each layer were prescribed, and often different shapes of bricks are needed for different altar designs.

The Śulba sūtras state the theorem of the diagonal (popularly known as the Pythagorean theorem), and their geometric constructions leave no doubt that they viewed the theorem as valid in general. Divakaran gives numerous arguments for the truth of this assertion. In particular, he observes that they demonstrate a technique to generate an infinite sequence of right triangles with rational lengths.

The Vedic geometers also ran into the famous problem of transforming a circle into a square and vice-versa. They gave approximate solutions, of course, but in the process, they ran into issues of approximating square roots by rational numbers. Divakaran presents a very interesting argument about why and how they might have developed their approximations. He also observes that some of the approximation formulas were of a more theoretical nature, rather than a “convenient formula” for design.

The next chapter revisits the Indus Valley mathematics (outlining the ideas) already described above to discuss possible connections between the Indus Valley civilization and other ancient civilizations in detail. The discussion of the development of decimal numbers appears thereafter, where the author proposes that the decimals with place value were already present in the Vedic lore and gives extensive evidence for the same.

He traces the known history of the decimal system, from Āryabhaṭa onwards up to its eventual spread to the rest of the world. The place value system is described as a set of atomic symbols that, together with a base, produces higher numbers. The author brings out its similarity with the Sanskrit grammarian Bhartṛhari’s analysis of how sentences are built from words. Bhartṛhari suggests that the process is similar to the construction of numbers from a few atomic units (the digits 1,2,…, 9). In case of numbers, a string of these gives rise to a sentence called number whose meaning derives from its structure!

Divakaran also discusses both the zero and the infinity in their roles as numbers, and also the ideas of positive and negative numbers. An early appearance of combinatorics in Piñgala (about 200 bce) is also noted.

### The Āryabhaṭan Revolution

After a general historical narrative, the author begins discussing Āryabhaṭa’s work (fifth century ce) in this part that covers the period from Āryabhaṭa to Bhāskara II.

Āryabhaṭa’s work on astronomy was influential in Indian mathematics. His book on astronomy, has only a small section on mathematics but, among other things, it develops the Kuṭṭaka, the technique of solving the equation by-ax=c in integers when a,b,c are all integers. This technique is considered central to many astronomical calculations and Brahmagupta has a whole chapter in his book dedicated to this process. Divakaran remarks that the Kuṭṭaka process was essential for concordance between different astronomical systems.

The book refrains from merely providing a list of dates and personalities, and instead gives vivid descriptions of developments

Among numerous praiseworthy achievements of Āryabhaṭa, the author notes his treatment of the Earth as one of the planets with a fixed number of revolutions that rotates on its own axis. The controversy about this daily rotation is discussed in section 6.3 of the book, given in detail, together with Divakaran’s own observations and conjectures.

The section starts with Āryabhaṭa’s suggestion that the Earth rotates on its own axis daily, so the stars and planets appear to move towards west (at the equator), just as a boat moving in a stream causes fixed objects (on the shore) to be moving backwards. This was seriously objected to, ridiculed and summarily dismissed by other astronomers as well as his own followers. Some of his commentators attempted to reinterpret words to derive an opposite meaning from the same verse. The author describes this as a poor attempt to discredit the original verse by adding verse 10 which repeats the orthodox explanation that some wind called the great mover (pravaha) causes actual motion of these objects while the Earth stays fixed.

In an extensive narrative, the interactions between Greek (Hellenistic) theories and ancient Indian ones are examined by Divakaran. Indeed, two of the five astronomical systems are connected with Greek theories. Despite the sexagesimal system of numbers for time and angles (orbital positions), the Indian systems made decimal calculations. Similar to the Greeks, the Indian systems also seem to use the notions of cycles and epicycles for their theory of orbital motion. Here, the author argues that despite the borrowed concepts from the Greeks, the knowledge of algebra and geometry already developed in India was superior and led to many innovations.

This discussion is followed up by one on the historical context of Āryabhaṭa and his influence over much of southern India, including his serving as the inspiration for the Nīla school.

Divakaran then moves on to an examination of the Bakhshali manuscript, which are parts of a problem-book of mathematics. Discovered accidentally during an excavation, the date of the work has been a very controversial topic. Divakaran proposes a date between 300 to 500 ce. The recent carbon-dating evidence is discussed by the author in a separate article.^{2} The Bakhshali manuscript contains documentary evidence of the existence of decimal numbers at the time, including the zero (written as a dot). The manuscript contains descriptions of algebraic notation that uses initial syllables of the corresponding operations and discusses techniques to approximate square roots with rational numbers. Unfortunately, the fragmented manuscript stands alone and other related works appear to have been lost entirely.

### The Mathematics of Āryabhaṭa

Divakaran next gives a survey of mathematics in the astronomical treatise of Āryabhaṭa, the Gaṇitapāda. Notable among the topics are a definition of sine, which is in terms of a half chord rather than the Greek chord, an iterative formula for the values of \sin(\theta) for 96 angles in a circle, and a value of \pi. The change from using a Greek chord to a half-chord for sine may seem like a minor change but it leads to simpler addition formulas that are used in modern mathematics. The iterative formula is a general method of using double differences to build the sine function from a value of \sin(\theta) for small \theta, of the form \pi/(2m).

Āryabhaṭa declares the value of \pi=3.1416 to be sufficiently approximate without justification. It is close to but not the same as Ptolemy’s value and also differs from other approximations in vogue at the time, such as \sqrt{10}. This is how the author counters arguments that suggest that Āryabhaṭa’s estimate was simply borrowed from Ptolemy.

Another interesting discussion concerns the formula for the surface area of a sphere, which is incorrect in Āryabhaṭa’s work. Using the sine values from Āryabhaṭa, Bhāskara II (12th century ce) derives the correct formula. Divakaran’s discussion on getting a precise formula from numerical integration is especially worth a close reading.

### Brahmagupta, Bhāskara II, Nārāyaṇa

The succeeding topics span seven centuries and highlight the further development of mathematics in India.

Brahmagupta arrived about a century after Āryabhaṭa and has to his credit several mathematical gems. After giving a detailed exposition of kuṭṭaka, he goes on to develop bhāvanā, the multiplicative rule for pairs (a,b) of integers that are really the multiplication rules for surds of the form a+b\sqrt{D}, where D is a non-square positive integer. Brahmagupta used this to make significant progress in the problem of vargaprakṛti, which we know today as the problem of finding all solutions of the Brahmagupta–Pell equation, namely, Nx^2 + 1 = y^2. Brahmagupta started the analysis of this problem by generalizing the problem to Nx^2+C=y^2 and then, using his bhāvanā process, arguing that if the equation is solvable for C=-1, C=\pm 2, C=\pm 4, then there is a solution for C=1.

The author then gives an extensive history that describes progress made by Jayadeva, Bhāskara II, and Nārāyaṇa on this problem, after the original impetus from Brahmagupta. This involves rational solutions as well as a complete algorithm to generate all solutions when C=1, but there was no proof offered by Brahmagupta for the existence of integral solutions.

Bhāskara II gave a solution and discussed the cases N=67, 61 as an illustration of the use of his algorithm. After Fermat thought of the equation for N=61 and presented it as a challenge to other mathematicians, the problem received special attention.This resulted in a formal analysis of the equation using the theory of continued fractions at the hands of Euler, Brouncker, Wallis and Lagrange. The size of the smallest solution for N=61 (x=226,153,980, y=1,766,319,049) illustrates the non-trivial nature of the problem. Related problems remain a topic of further investigation in number theory.

Another spectacular achievement of Brahmagupta is his analysis of the cyclic quadrilateral, which is a quadrilateral whose corners lie on a circle. This analysis was followed up by subsequent mathematicians Mahāvīra, Bhāskara II and Nārāyaṇa. The topics developed included formulas for the diagonals and areas of cyclic quadrilaterals in terms of the lengths of their sides. Brahmagupta’s original analysis was critiqued by Bhāskara II for not making the cyclic hypothesis explicit since the diagonals of a quadrilateral are not otherwise determined by its sides. Combining this work with a modern paper (2010) by S. Kichenassamy, the author presents a beautiful and lucid summary of the various achievements and impressive theorems generated in this area over seven centuries.

I would further add that Nārāyaṇa’s work may be thought of as a significant step into the geometry of a cyclic hexagon.

### The Nīla School

Divakaran writes with great passion and authority about the crowning achievements of the mathematical descendants of the Āryabhaṭan school. These developments took place starting from about 13th century and flourished into the 17th century. Referred to as the Nīla school, the tradition came to an unexplained end, leaving a certain gap with regard to extant signal contributions from India until Rāmānujan arrived on the scene in the 20th century.

Mādhava (1350) is the most prominent name in the Nīla school and is credited with developing the fundamental power series of calculus, namely, \arctan(\theta), \sin(\theta) and \cos(\theta), which did not appear on the European scene until 300 years later. What is more impressive is that even the topic of power series itself was not yet developed. Another interesting fact is that no work of Mādhava is extant, though details of his contributions are available from narratives by other writers.

Divakaran writes with great passion and authority about the crowning achievements of the mathematical descendants of the Āryabhaṭan school

A noteworthy feature of Mādhava’s work is that, in addition to developing the series expansion, he gives techniques to obtain good error estimates and also develops other series which converge faster. Thus, he seems to have clearly anticipated the future trends in mathematical analysis.

One of the most important resources to learn about the Nīla school is the book Yuktibhāṣa by Jyeṣthadeva (16th century), which is written in Malayalam. It is the first book in India to provide detailed proofs of mathematical statements. The book was not widely known and printed copies of its Sanskrit version and English translation have appeared only in the last decade. Divakaran provides extensive details of chosen proofs from Yuktibhāṣa as well as other available manuscripts (from the Nīla school) as needed during a discussion.

Divakaran presents several topics which are often discussed and debated, such as the history of decimal numbers, what is and what is not in Piñgala, details of the Bakhshali manuscript, the Mādhava series and its connections with calculus and so on. Throughout the book, he is very careful in his analysis, pointing out merits and faults of various arguments, while making a strong case for his own informed opinions.

In the conclusions, he raises doubts against the dictum (a principle suggested by van der Waerden) that a great discovery in mathematics must be presumed to be discovered by a single, equally great individual, and therefore all subsequent instances of its discovery must be cases of transmission. In doing so, he thus raises the rhetorical question of whether mathematicians of today would really accept this dictum for the discoveries of Mādhava.

Although the book traces the history from the point of view of mathematics, the discussion excludes their motivating viewpoint of astronomy! One hopes that someday there will be a similar book replacing mathematics with astronomy in its title.

### Footnotes

- International Alphabet of Sanskrit Transliteration. ↩
- P.P. Divakaran. “The Bakhshali Manuscript and the Indian Zero”.
*Bhāvanā*. Oct 2018.**2**(4): 13–21. ↩