History of Indian mathematics(Pre-History)

3: Early Indian culture - Indus civilisation

The first appearance of evidence of the use of mathematics in the Indian subcontinent was in the Indus valley (see Figure 2.4) and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilisation was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilised ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding.
The discoveries of systems of uniform and decimal weights, over a vast area, are of considerable interest. G Joseph states:

...Such standardisation and durability is a strong indication of a numerate culture. [GJ, P 222]

Also, many of the weights uncovered have been produced in definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few) which present knowledge of basic geometry, including the circle.

This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles, leading S Sinha to state:

...The civilisation and culture of the inhabitants of the Indus valley...were of a very advanced nature. [SS1, P 71]

S Srinivasan further comments:

...There are many unique features in the construction patterns, which suggest an independent origin of ideas in ancient Indian civilisation.[SSr1, P17]

Further to the use of circles in 'decorative' design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. This clearly points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of p.

Also of great interest is a remarkably accurate decimal ruler known as the Mohenjodaro ruler. Subdivisions on the ruler have a maximum error of just 0.005 inches and, at a length of 1.32 inches, have been named the Indus inch. Furthermore, a correspondence has been noted between the Indus scale and brick size. Bricks (found in various locations) were found to have dimensions that were integral multiples of the graduations of their respective scales, which suggests advanced mathematical thinking.

Figure 3.1: Ruler found at Lothal. [SSr1, P17]

Above all else there are also brief references to an early decimal system of numeration. The seeds of what were to become the single greatest contribution of the Indian sub-continent to the world (not just of mathematics) had already been sown. My evidence comes from S Sinha who states:

...Writers on these civilisations briefly refer to the decimal system of numeration found in these excavations. [SS1, P 71]

This quote supports the theory that the Brahmi numerals, which were to go on to develop into the numerals we use today, originated in the Indus valley around 2000 BC, however this theory has been rejected by several scholars including Ifrah and Joseph. This quote could be considered a piece of overzealous reporting by the author however, on further investigation I can support the comment with some confidence.

Not only are the markings on all the excavated measuring devices decimal in nature, but there is also research currently being conducted, which is attempting, with success, to show a connection between the Brahmi and Indus scripts. This lends indirect support to suggestions of the existence of early decimal numeral forms. As I will discuss briefly later, the Brahmi numerals undoubtedly developed into the numeral forms we use today.

Although this early mathematics is generally included in histories of mathematics it is often in nothing more than a brief mention, and there is a most curious quote by J Katz who claims:

...There is no direct evidence of its (Harappan civilisation) mathematics. [JK, P4]

It is possible that he makes this comment with regards to the fact that the Indus script as yet remains undeciphered (GJ, P218). 

However R Gupta more 'sensibly' states:

...In fact the level of mathematical knowledge implied in various geometrical designs, accurate layout of streets and drains and various building constructions etc was quite high (from a practical point of view). [RG1, P131]

While Childe claims:

...India confronts Egypt and Babylonia by the 3^rd millennium with a thoroughly individual and independent civilisation of her own.[EFR/JJO'C2, P 1]

Some confusion exists as to what caused the decline of this Harrapan culture, there are several theories, the most probable of which in my opinion was the drying up of the Sarasvati River. This view is supported by S Kak and also S Kalyanaraman who has written an extensive paper on the topic and comments:

... The drying-up of the Sarasvati River led to migrations of people eastwards.

The most commonly held view by historians is that Aryan peoples from the North invaded and destroyed the Harappan culture, this view however is considered increasingly contentious. In addition to the significance the fledgling decimal system would ultimately have, the most important legacy of this early civilisation is the influence its brick technology may have had on the altar building required by the Vedic religion that followed. A theory of the 'interlinkage' of the Harappan and Vedic cultures has recently arisen from a variety of studies, and it may come to light that there was a greater interaction between the two civilisations than currently thought.

4: Mathematics in the service of religion: I. Vedas and Vedangas

The Vedic religion was followed by the Indo-Aryan peoples, who originated from the north of the sub-continent. It is through the works of Vedic religion that we gain the first literary evidence of Indian culture and hence mathematics. Written in Vedic Sanskrit the Vedic works, Vedas and Vedangas (and later Sulbasutras) are primarily religious in content, but embody a large amount of astronomical knowledge and hence a significant knowledge of mathematics. The requirement for mathematics was (at least at first) twofold, as R Gupta discusses:

...The need to determine the correct times for Vedic ceremonies and the accurate construction of altars led to the development of astronomy and geometry. [RG2, P 131]

Some chronological confusion exists with regards to the appearance of the Vedic religion. S Kak states in a very recent work that the time period for the Vedic religion stretches back potentially as far as 8000BC and definitely 4000BC. Whereas G Joseph states 1500 BC as the forming of the Hindu civilisation and the recording of Vedas and Vedangas, and later Sulbasutras. However it seems most likely that significant knowledge of astronomy and mathematics first appears in Vedic works around the 2^nd millennium BC. The Rg-Veda (fire altar) the earliest extant Vedic work dates from around 1900 BC. R Gupta in his paper on the problem of ancient Indian chronology shows that dates from 26000-200 BC have been suggested for the Vedic 'period'. Having consulted many sources I am confident at placing the period of the Vedas (and Vedangas) at around 1900-1000 BC.

Further mathematical work is found in the Sulbasutras of the later Vedic period, the earliest of which is thought to have been written around 800 BC and the last around 200 BC. I will now move on from this slightly clouded chronological discussion. It is however worth noting that there are serious underlying problems with the chronology of early Indian mathematics which require significant attention.

Although the requirement of mathematics at this time was clearly not for its own sake, but for the purposes of religion and astronomy, it is important not to ignore the secular use of the texts, i.e. by the craftsmen who were building the altars. Similarly with the earlier Harappan peoples it seems likely that (at least) basic mathematics will have grown to become used by large numbers of the population. Regardless of the fact that at this time mathematics remained for practical uses, some significant work in the fields of geometry and arithmetic were developed during the Vedic period and as L Gurjar states:

...The Hindu had made enormous strides in the field of mathematics. [LG, P 2]

It is also worthwhile briefly noting the astronomy of the Vedic period which, given very basic measuring devices (in many cases just the naked eye), gave surprisingly accurate values for various astronomical quantities. These include the relative size of the planets the distance of the earth from the sun, the length of the day, and the length of the year. For further information, S Kak is an authority on the astronomical content of Vedic works.

Much of the mathematics contained within the Vedas is found in works called Vedangas of which there are six. Of the six Vedangas those of particular significance are the Vedangas Jyotis and Kalpa (the fifth and sixth Vedangas). Jyotis was (at the time) the name for astronomy, while Kalpa contained the rules for the rituals and ceremonies. The Vedangas are best described as an auxiliary to the Vedas.

N Dwary claims, with reference to the Vedanga-Jyotis, that:

...Hindus of the period were fully conversant with fundamental operations of arithmetic. [ND1, P 39]

S Kak suggests a date of around 1350 BC for the Vedanga-Jyotis. I include this as a reminder of the time period being discussed.

Along with the Vedangas there are several further works that contain mathematics, including:

Taittiriya Samhita
Satapatha Brahmana and 
Yajur and Atharva-Veda 
Rg-Veda (of which it is thought there are three 'versions') plus additional
Samhitas

Of these the Taittiriya Samhita and Rg-Veda are considered the oldest and contain rules for the construction of great fire altars.

As a result of the mathematics required for the construction of these altars, many rules and developments of geometry are found in Vedic works. These include:

Use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
Equivalence through numbers and area. 
Equivalence led to the problem of:
Squaring the circle and visa-versa.
Early forms of Pythagoras theorem.
Estimations for p.

S Kak gives three values for p from the Satapatha Brahmana. It seems most probable that they arose from transformations of squares into circles and circles to squares. The values are:
p₁ = ²⁵/₈ (3.125)
p₂ = ⁹⁰⁰/₂₈₉ (3.11418685...)
p₃ = ¹¹⁵⁶/₃₆₁ (3.202216...)

Astronomical calculations also leads to a further Vedic approximation:
p₄ = ³³⁹/₁₀₈ (3.1389) 
This is correct (when rounded) to 2 decimal places.

Also found in Vedic works are:

All four arithmetical operators (addition, subtraction, multiplication and division).
A definite system for denoting any number up to 1055 and existence of zero.
Prime numbers.

The Arab scholar Al-Biruni (973-1084 AD) discovered that only the Indians had a number system that was capable of going beyond the thousands in naming the orders in decimal counting.

Evidence of the use of this advanced numerical concept leads S Sinha to comment:

...It is fair to agree that a nation with such an advanced and cultured civilisation and which was using the numerical system (decimal place value) knew also how to handle the associated arithmetic. [SS1, P 73]

It is in Vedic works that we also first find the term "ganita" which literally means "the science of calculation". It is basically the Indian equivalent of the word mathematics and the term occurs throughout Vedic texts and in all later Indian literature with mathematical content.

Among the other works I have mentioned, mathematical material of considerable interest is found:

Arithmetical sequences, the decreasing sequence 99, 88, ... , 11 is found in the Atharva-Veda.
Pythagoras's theorem, geometric, constructional, algebraic and computational aspects known. A rule found in the Satapatha Brahmana gives a rule, which implies knowledge of the Pythagorean theorem, and similar implications are found in the Taittiriya Samhita.
Fractions, found in one (or more) of the Samhitas.
Equations, 972x² = 972 + m for example, found in one of the Samhitas. 
The 'rule of three'.

4 II. Sulba Sutras

The later Sulba-sutras represent the 'traditional' material along with further related elaboration of Vedic mathematics. The Sulba-sutras have been dated from around 800-200 BC, and further to the expansion of topics in the Vedangas, contain a number of significant developments. 
These include first 'use' of irrational numbers, quadratic equations of the form a x² = c and ax² + bx = c, unarguable evidence of the use of Pythagoras theorem and Pythagorean triples, predating Pythagoras (c 572 - 497 BC), and evidence of a number of geometrical proofs. This is of great interest as proof is a concept thought to be completely lacking in Indian mathematics.

Example 4.2.1: Pythagoras theorem and Pythagorean triples, as found in the Sulba Sutras.

The rope stretched along the length of the diagonal of a rectangle makes an area which the, vertical and horizontal sides make together.
In other words:

a² = b² + c²

Examples of Pythagorean triples given as the sides of right angled triangles:

5, 12, 13
8, 15, 17
12, 16, 20 
12, 35, 37

Of the Sulvas so far 'uncovered' the four major and most mathematically significant are those composed by Baudhayana, Manava, Apastamba and Katyayana (perhaps least 'important' of the Sutras, by the time it was composed the Vedic religion was becoming less predominant). However in a paper written 20 years ago S Sinha claims that there are a further three Sutras, 'composed' by Maitrayana, Varaha and Vadhula (SS1, P 76). I have yet to come across any other references to these three 'extra' sutras. These men were not mathematicians in the modern sense but they are significant none the less in that they were the first mentioned 'individual' composers. E Robertson and J O'Connor have suggested that they were Vedic priests (and skilled craftsmen).

It is thought that the Sulvas were intended to supplement the Kalpa (the sixth Vedanga), and their primary content remained instructions for the construction of sacrificial altars. The name Sulvasutra means 'rule of chords' which is another name for geometry.

N Dwary states:

...They offer a wealth of geometrical as well as arithmetical results. [ND1, P 40]

R Gupta similarly claims:

...The Sulba-sutras are (quite) rich in mathematical contents. [RG2, P 133]

With reference to the possible appearance of proof is a quote from A Michaels:

...Vedic geometry, though non-axiomatic in character, is provable and indeed proof is implicit in several constructions prescribed in the Sulba-sutras. [RG2, P 133]

This is not particularly compelling evidence but does suggest that the composers of the sulba-sutras may have had a greater depth of knowledge than is generally thought.

Many suggestions for the value of p are found within the sutras. They cover a surprisingly wide range of values, from 2.99 to 3.2022.

Pythagoras's theorem and Pythagorean triples arose as the result of geometric rules. It is first found in the Baudhayana sutra - so was hence known from around 800 BC. It is also implied in the later work of Apastamba, and Pythagorean triples are found in his rules for altar construction. Altar construction also led to the discovery of irrational numbers, a remarkable estimation of 2 in found in three of the sutras. The method for approximating the value of 2 gives the following result:

2 = 1 + ¹/₃ + ¹/_3.4 - ¹/_3.4.34

This is equal to 1.412156..., which is correct to 5 decimal places.

It has been argued by scholars seemingly attempting to deprive Indian mathematics of due credit, that Indians believed that 2 = 1 + ¹/₃ + ¹/_3.4 - ¹/_3.4.34 exactly, which would not indicate knowledge of the concept of irrationality. Elsewhere in Indian works however it is stated that various square root values cannot be exactly determined, which strongly suggests an initial knowledge of irrationality.

Indeed an early method for calculating square roots can be found in some Sutras, the method involves repeated application of the formula: A = (a² + r) = a + r/2a,r being small.

Example 4.2.2: Application of formula for calculating square roots.

If A=10, take a = 9 and r = 1.
Thus 10 = (32 + 1) = a + r/2a = 3 + 1/6 = 3.16667 in (modern) decimal notation. 
10 = 3.162278 to six decimal places when calculated on a calculator. Thus, after only one application of the formula, a moderately accurate value has been calculated.

C Srinivasiengar thus states:

...The credit of using irrational numbers for the first time must go to the Indians. [CS, P 15]

Many of the Vedic contributions to mathematics have been neglected or worse. When it first became apparent that there was geometry contained within works that was not of Greek origin, historians and mathematical commentators went to great lengths to try and claim that this geometry was Greek influenced (to a greater or lesser extent).

It is undeniable that none of the methods of Greek geometry are discernible in Vedic geometry, but this merely serves to support arguments that it is independently developed and not in some way borrowed from Greek sources.

In light of recent evidence and more accurate dating it has been even more strongly claimed by A Seidenberg (in S Kak) that:

...Indian geometry and mathematics pre-dates Babylonian and Greek mathematics. [SK1, P 338]

This may be a somewhat extreme standpoint, and it seems likely that there was traffic of ideas in all directions of the Ancient world, but there is little doubt that the vast majority of Indian work is original to its writers. It may lack the cold logic and truly abstract character of modern mathematics but this observation further helps to identify it as uniquely Indian. Of all the mathematics contained in the Vedangas it is the definite appearance of decimal symbols for numerals and a place value system that should perhaps be considered the most phenomenal.

Before the period of the Sulbasutras was at an end the Brahmi numerals had definitely begun to appear (c. 300BC) and the similarity with modern day numerals is clear to see (see Figures 7.1 and 7.3). More importantly even still was the development of the concept of decimal place value. M Pandit in a recent paper (discussed in RG2) has shown certain rules given by the famous Indian grammarian Panini (c. 500 BC) imply the concept of the mathematical zero. Further to this there is a small amount of evidence of the use of symbols for numbers even earlier in the Harrapan culture. My evidence comes primarily from a paper by S Kak, which analyses some of Panini's work, and there is further support from a paper by S Sinha. B Datta and A Singh also give evidence of an early emergence of numerical forms and the decimal place value system.