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Mathematics of the Vedas

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By Krishna Maheshwari


The Veda Samhitâ is composed of hymns to various deities and also hymns praising all forms of knowledge. They don't make distinctions between secular and sacred knowledge (as we define them today) because secular knowledge was thought to be a tool to be used to discover sacred knowledge. This framework, in which knowledge is seen as one whole continuum, offers the basis for these statements to be interpreted in multiple ways in multiple contexts (astronomical, spiritual, terrestrial etc). The knowledge of mathematics and geometry were all deemed important and worthy of formulations into mantra. Some of the hymns, which deal with cosmology, imply that the rishis were very familiar with geometry and the planning needed to construct complex objects.

Since the Vedas are not texts on mathematics but mention a lot of mathematical concepts, it could be construed that mathematics as a science also existed. It is unlikely that stray statements on mathematical concepts like progressions [1], concept of infinity and zero existed without mathematics as a study. Whether we can trace out texts is secondary, since Veda came down as an oral tradition for very long. So Vedic evidence is primarily indirect, and is more of an indicator of the kind of concepts that existed than a definition/explanation of those.

Contents

Rig Veda

The Rig Veda is the oldest of the Vedas and contains a wealth of knowledge on a variety of subjects. Some of them relate to the broader field of mathematics, of which a subset is described here.

Geometry

Geometry is used throughout the Rig Veda. However, unlike modern geometry, the emphasis was not on proofs but on usage. Furthermore, the rishis had knowledge of the concept of precision, and this is evident from various descriptions of applications and use of geometry. Further, the depth of precision was based on the requirements of the task or study being described. For example, descriptions of constructions were limited in precision whereas descriptions of astronomical phenomena were described in greater precision.

The following verse in the Rig Veda deals with the formation of the universe.

Who was the measurer prama? What was the model pratimâ? What were the building materials for things offered nidânam ãjyam? What is the circumference (of this universe) paridhih? What are the meters or harmonies behind the Universe chandah? What is the triangle (yoke) praugam [which connects this universe to the source of driving force, the engine]?[2]

All these Sanskrit words (ie: prama) are geometrical terms which also occur later in the Sulbasutra (where they are also defined).

Chariots are described in great detail in many different verses in the Rig Veda and Yajur Veda. Dr. Kulkarni writes:

The proficiency in chariot building presupposes a good deal of knowledge of geometry... The fixing of spokes of odd or even numbers require knowledge of dividing the area of the circle into the desired numbers of small parts of equal area, by drawing diameters. This also presupposes the knowledge of dividing a given angle into equal parts.[3]

The Rig Veda is full of references to words in rituals whose definitions we find in subsequent Brahmanas and in the Sulba Sutra to be pointing to geometrical figures. For example, three types of fire altars, garhapatya, ahavaniya and dakshina are mentioned in the Rig Veda but defined in the Shatapatha Brâhmana as being square, circular and semi circular, respectively, and also having the same area. Considering that ritualistic fire altar designs were not changed over a period of time and that the shastras were recited for several millennia before being written down, the rishis must have had a method to calculate the square root of the number two and the value of pi, without which they would not be able to determine whether the three altars had the same area or not.

Error Correction & Detection Codes

The Rig Veda was recited using a special method of recitation which is akin to modern error correction and detection codes. This feature has allowed it to be passed on from generation to generation for many millennia without the introduction of errors [4]

The rishis had focused on developing methods of chanting which can detect any errors in chanting of a mantra, such as omitting a syllable or replacing one syllable by another. For each mantra, there are several different methods of chanting, each method capable of detecting one type of error. For illustration, consider one half of the famous gayatri mantra of the seer Vishvamitra[5]. The standard method of recitation involving conjunction is called Samhita patha (given below).

Samhita Patha

tatsaviturvarenyambhargodevasyadhimahi

Separate all compound words into their constituents and number the words:

tatsavituhvarenyambhargahdevasyadhimahi
1 2 3 4 5 6

In the kramapatha chant, use a text obtained by combining two neighboring words according the rules of sandhi, resulting in six words.

1+22+33+44+55+66+6

Krama Patha

tatsavituhsaviturvarenyamvarenyambhargahbhargodevasyadevasyadhimahidhimahiti dhimahi

A Krama patha expert chants the krama-version of all the verses.

To understand its error detecting capability, divide the chant into syllables so that the syllable ends with an vowel a, i, u etc. Both the third syllable and sixth syllables are same namely vi. Suppose we commit an error and chant the third syllable as va. According to the krama chanting, the sixth syllable should be same as the third syllable. He would pronounce it as vi, since we are assuming he will make only one error. Then he notices that an error has taken place since va is different from vi. An error has obviously occurred, but he does not know which is correct, va or vi? There are other methods which detect these errors and also methods that show how to correct them.

The various forms of chanting are called as vikratis and there are eight of them [6].

First usage of Pi

In the Rigveda, a formula to find the area of a circle is mentioned showing that the Rishis knew of pi, approximating it to be equal to 22/7[7]. It was used in the formula for the area of a circle

area of a circle = pi*(AB/2)^2 where AB is the diameter of a circle.

Yajur Veda

In the Yajurveda, 1x3=3, 3x5=15, 5x7=35, etc is seen[8].

Big Numbers

In Yajurveda, numbers starting from four and with a difference of four forming an arithmetic series is discussed[9]. The Yajurveda also mentions the counting of numbers upto 10^18, the highest being named parardha[10].

In the Taittiriya Upanishad, there is a anuvaka (section), that extols the "Beatific Calculus" or a quasi-mathematical relationship between bliss of a young man, who has everything in the world to the bliss of the Brahman, or "realization". Translated roughly as follows, fear is all-pervasive. It continues by assuming that a young, good man who is fit, healthy and strong, and has all the wealth in the world, is one unit of human bliss. The anuvaka provides a precise calculation of a series of multiplications by 100 to give number 10010 units of human bliss that can be had when one attains Brahman. The previous anuvaka exhorts the aspirants to be fearless and strong, as only such a person may realize the absolute within.

Concept of Infinity

The concept of infinity was also known during Vedic times. They were aware of the basic mathematical properties of infinity and had several words for the concept-chief being ananta, purnam, aditi, and asamkhyata. Asamkhyata is mentioned in the Yajur Veda[11], and the Brihadaranyaka Upanishad as describing the number of mysteries of Indra as ananta[12]. These two statements are elaborated in the opening lines of the Isha Upanishad (Shukla Yajur Veda). This sholka is as much metaphysical as it is mathematical.

pûrnamadah pûrnamidam pûrnât pûrnamudacyate
pûrnâsya pûrnamadaya pûrnamevâvasishyate

From infinity is born infinity.
When infinity is taken out of infinity,
only infinity is left over.

Atharva Veda

The concept of one (and its mathematic properties), arithmetic progression, and arithmetric series are also seen in the Vedas as well. In Atharva Veda, the fact that 1x1 and 1/1 = 1 is stated[13].

Concept of Infinity

The Atharva Veda states that

Infinity can come out of infinity only and infinity is left over from infinity after operations on it[14].

Concept of Shunya (Zero)

The concept of Shunya, or zero void, was originally conceived as the symbol of Brahman, expressing the sum of all distinct forms. The symbol of zero and the decimal system of notation is described in the Atharvaveda[15]. it describes how the number increases by 10 by writing zero in front of it. While there is no explicit mention of zero, it must have been common knowledge based on how it is used.

In fact, the concept of shunya was not just mathematical or scientific, but is deeply rooted in all branches of thought - especially metaphysics and cosmology. Shunya is the transition point between oposites, it symboliss the real balance between divergent tendencies. Most ancient mathematicians defined zero as the sum of two equal and opposite quantities. Zero produces all figures, but is itself not limited to a certain value. Zero is the primary or final reservoir of all single numbers. The symbol of zero and the decimal system of notation is described in the Atharvaveda[16]. It describes how the number increases by 10 by writing zero in front of it.


References

  1. Krishna Yajurveda 4.7.11
  2. Rig Veda, 10.130.3
  3. Kulkarni, R.P. 1983, Geometry according to the Sulba Sutras, Pune, India; Vaidic Samshodhan Mandala, (ed) Sontakke
  4. Mathematics in India of the Vedic age, by Dr. R. L. Kashyap
  5. Rig Veda, 3.62.10
  6. Correct Chanting
  7. Rig Veda, 1.105.17
  8. Yajur Veda, 18.24
  9. Yajur Veda, 18.25
  10. Yajur Veda, 17.2
  11. Yajur Veda, 16.54
  12. Brihadaranyaka Upanishad, 2.5.10
  13. Atharva Veda, 14.3.12
  14. Atharva Veda, 10.8.24
  15. Atharva Veda 5.15, 1-11
  16. Atharva Veda, 14.1.1
  • "Ancient Indian Insights and Modern Science," by Dr. Kalpana Paranjape, Bhandarkar Oriental Research Institute, 1996
  • B. V. Subbarayappa. "India's Contributions to the History of Science" in Lokesh Chandra, et al., eds. India's Contribution to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp47-66, 1970.