# Indian Mathematics - Redressing the balance

### Ian G Pearce

### 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:

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:

S Srinivasan further comments:

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:

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:

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:

While Childe claims:

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 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

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[EFR/JJO'C2, P 1]^{rd}millennium with a thoroughly individual and independent civilisation of her own.

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.