The ancient Hindu books directly dealing with Mathematics are the so called Sulba Sūtras dated little later than the books mentioned above; there are seven different books. I will mention them only briefly since these books have been discussed elsewhere. My aim is to focus on the books mentioned above.

Among these the Rig Veda Samhita is the oldest dating prior to 3400 B.C.E. It has 1019 hymns and has more than ten thousand verses mainly couplets.

Highlights

1. The error-free methods of chanting the Rig Vedic Samhita closely related to the modern error correcting and detection methods in computers and communication theory.
2. The decimal system for integers, fractions, division and multiplication.
3. The so called Pythagorean triples, right angle triangles with sides in integers, the approximation for pi, the ratio of circumference to diameter and the square root of two.
4. Various geometric problems dealing with rectangles and trapezoid.

Mathematics in Veda Samhitās

As mentioned earlier, the books are the four (or five) Veda Samhitās. Then follow the Brāhmaņās books and the Sulba Sūtra books in chronological order. Somehow many writers jump to the relatively late Sulba Sūtras while discussing ancient Indian mathematics. There is considerable mathematical information, both explicit and implicit in the Veda Samhitās. Among them, the earliest is the Rig Veda Samhita.

All the Veda Samhitās are hymns to the various deities. However these hymns praise all forms of knowledge. There was no rigid distinction of the secular and sacred knowledge. The knowledge of mathematics, the knowledge of geometry, especially as related to the construction of houses and cities were all deemed important and worthy of mantrās in the hymns. Some of these hymns dealing with the series of integers are recited even today on sacred occasions. Some of the hymns, which deal with cosmology, imply that these poets were very familiar with geometry and the planning needed to construct complex objects.

Consider for example the following verse in RV (Mandala10, Sūkta130, Verse 3). The words in Italics are the words in the Sanskrit original. It deals with the creation or formation of the universe.
 

• Who was the measurer prama?
• What was the model pratimā?
• What were the building materials for things offered nidānam ājyam?
• What is the circumference (of this universe) paridhiĥ?
• What are the meters or harmonies behind the Universe chhandaĥ?
• What is the triangle (yoke) praugam [which connects this universe to the source of driving force, the engine]?

All the words in Sanskrit, prama etc., are geometrical terms which also occur in the later Sulba Sūtras with the indicated meaning. Hence it is safe to assume that these poets were aware of the construction of buildings and other artifacts.

In Atharva Veda (10.2.31) the town of gods called Ayodhya is described. It is circular in plan with eight rampart walls and nine doors. Even if the poem is interpreted metaphorically, the use of a metaphor implies that poets had the experience of real things, i.e. a real physical city. A.V. 19.58.4 declares that the town should be made unconquerable using the thing called ayasa. Whether we translate it as strength or as some metal or as iron which is the word's meaning in later times, either way it indicates that the poets were aware of complex planning of these geometrical entities.

The simplistic notion circulated by the indologists of the nineteenth century that these poets were nomads with a relatively low level of culture has absolutely no support from the hymns.

Next consider the Rig Veda Samhita which has more than 10,000 metrical couplets. Each verse has a distinct metre which imposes a structure on the verse like the total number of vowels in it and the number of vowels in each subgroup of the verse. Moreover the medium of preservation of the text was also recitation. The Vedic sages attached the greatest importance to the preservation of the text of the Rig Veda along with accent marks and developed special methods of recitation which remind us of the modern error correction and detection codes in modern communication and computer systems, i.e. what the codes do for correcting the linear printed text is done by these special methods of recitation for oral text.

Chariots are mentioned copiously in Rig Veda (1.102.3, 1.53.9, 1.55.7, 1.141.8, 2.12.8, 4.46.2...).  Chariots could also be triangular having three seats and three wheels (RV. 1.118.2, RV 1.34.2).

A spoked wheel is mentioned in many places in Rig Veda. Specifically the five spoked wheel (1.164.13) and the 360-spoked wheel (1.164.48) are mentioned. The spoked wheel has four parts, hubs nābih, fellies pradhaya, spokes shanka, āre or rim. By the time of Yajurveda (Y.V. 16.27) the number and varieties of the manufactured chariots had increased so much that a separate guild of chariot makers was developed. Dr. Kulakarni (1983) 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”.

Finally we come to the role of rituals. The Rig Veda is full of references to the words which come up in rituals, even though it does not mention any ritual in detail. The details of the rituals, especially the design of the fire altars and methods of constructing them using specially shaped bricks are given in the subsequent Brāhmaņa books and also, with more details, in the Sulba Sūtra books, the mathematical texts of the late Vedic period. Whenever any religious rituals are codified in a oral or written, the implication is that they must have been in existence for much longer time. For instance consider the three types of fire altars namely Garhapatya, Ahavaniya and Dakshina. All three are mentioned in Rig Veda. However the Shatapatha Brāhmaņa declares that the three fire altars are square, circular and semi circular in shape and, more importantly having the same area. All scholars such as Burk (1901), Seidenberg (1962, 1978), Dutta (1932) agree that this constraint of equal area must have been there even in the early Vedic age before the codification of Rig Vedic Hymns. To construct a figure of a specific area, we need to have at least an approximate method of finding the square root of the number two. To construct a circle of area equal to that of a square, one needs to have at least an approximate value of the number pi, the ratio of circumference of a circle to the diameter. Again the books like the Shatapatha Brāhmaņa or the Sulba Sūtras codify the methods existing for a long time in addition to developing new methods of drawing the geometrical figures and the associated theory. The minimum knowledge needed for finding the square root of two or for drawing isosceles trapezium is that of Pythagorean triples like 3² + 4² = 5². All this evidence implies that the Vedic sages in the early Vedic age knew some simplified versions of the Pythagorean theorem. It was there, probably that the origin of mathematics took place as argued by Seidenberg (1978), Rajaram and Frawley (1995).

Method of chanting based on error correcting codes

Rig Veda is a book of about 10,000 verses composed several thousand of years ago. Still all the available manuscripts and the authorised audio versions recorded all over India differ from each other in only one syllable. Such a feat is possible because Rig Vedic sages had developed methods of chanting reminiscent of the modern methods of error detecting codes, an advanced mathematical concept and associated technology developed only in the last fifty years of this century.

With the ubiquity of the transmission of strings of symbols over wires and wireless, it is easily realised that the string of symbols received by an user over wireless, is quite different from that sent by the sender because the symbol string received by the user has been corrupted by unwanted symbols, labeled noise. The system detects and corrects all the inserted errors; It should be obvious that if only the string of message symbols were transmitted, the received message would be a undecipherable because of the inevitable errors inserted the transmitting medium. Error detection and correction is possible only because, in addition to the symbols constituting the message, additional symbols, the so called redundant symbols, have to be added to the string to be transmitted. The procedure by which a new string is constructed from the given string by adding additional symbols is called a code in the literature on the Mathematical theory of communication. These coding methods were developed only in fifties of this century.

The users of Rig-Veda were confronted with a similar problem and came up with a similar answer six or seven thousand years earlier. The Rig Vedic sages were sure that their poems would be read in much later ages(Mandala 3, Sukta 33, Verse 8). Speech was the medium of their poetry as well as that of their preservation. In Sanskrit language, words with apparently trivial errors of transposition of syllables could have a vastly differing meaning. There is the story of a titan who wanted to ask a boon that there shall be no gods. The Sanskrit phrase is nih + devatvam = nirdevatvam. The titan, being careless, made a transposition error and asked for nidravatvam, which means to be enveloped by sleep. Each RV verse is written in a particular metre. For instance the most popular metre is Gayatri, each verse having 24 syllables, divided into three parts of eight syllable each, the so called three “feet”. Each foot is a linguistic unit (or sentence). The errors that are to be detected are:
 

1. Deletion or addition of a syllable into a word.
2. Deletion or insertion of a word into the sentence.
3. Preservation of the order of words.
4. Avoiding long jumps, i.e. Suppose there are two words a and b which are phonetically close. Let a and b occur in verses numbered x and y. Let c and d be the words next to a and b in the corresponding verses. There is a tendency while chanting to jump from a word a in verse x to word d in verse y; similarly to jump from word b in verse y to word c in verse x. This error is serious.
    

The sages developed several systematic methods of chanting (or codes) so that the errors would not only show up, but also the correction also becomes clear. Hence for each verse there is the standard method of recitation as well as these special methods recitation called here as codes. The code is formed by adding additional words, several times the number in the original. The codes have suggestive name, like krama (succession), mālā (garland), jaţa (the matted hair), danda (stick) and ghana, the hard one, the last one being the most comprehensive one. I will mention here only two such methods or codes named mālā (garland) and the double wheeled chariot, the latter deciphered for correcting the error of type 4, the so called long jumps.

MĀLĀ (Garland)

The first step is to break up the verse with euphonic combinations into individual words. Consider one half of the verse RV (10.97.22). It has six words divided into 2 parts or feet. Label the words a₁, …, a₆

Line 1:         a₁        a₂         a₃         a₄         a₅         a₆  

Rearrange the above line as six lines as shown below

a₁       a₂
a₂       a₃
a₃       a₄
a₄       a₅
a₅       a₆
a₆       a₆

Make a copy of the matrix patter, flip it top to bottom and right to left keep it next to the original as indicated below so that the bottom lines are lined up.

a₁       a₂                                           a₆                   a₆
a₂       a₃                                           a₆                   a₅
a₃       a₄                                           a₅                   a₄
a₄       a₅                                           a₄                   a₃
a₅       a₆                                           a₃                   a₂
a₆       a₆                                           a₂                   a₁

The above diagram with 6 lines looks like a garland, as the name indicates. Now chant the six lines above together as a single verse, one by one beginning with the top, left to right. The verse has 32 words since each is repeated four times.

Call this verse i_1, i_2, …, i_32, each i_k being a distinct word. This verse, Vikrati is repeated below, after bending it in the middle and reversing it

a1   a2   a6   a6   a2   a3   a6   a5   a3   a4   a5   a4		The highest number known to Greek is 104
a1   a2   a6   a6   a2   a3   a6   a5   a3   a4   a5   a4

Notice that each of the 16 columns has only one entry.

When the verse of 32 words is recited, the reciter may make unconscious errors; let the verse heard by another person be indicated below, each O_i being a word. Again bend the string in the middle and reverse it

O1           O2       O3   …….    O16
O32         O31      O30   …….  O17

If the verse recitation were perfect, there would be only one entry in each column, i.e O₁ = O₃₂, O₂ = O₃₁. If for example O₂ is different from O₃₁, then there is an indication of error.

Now note that every word like a₂ occurs at least four times in the chanted Vikrati verse (*). After imposing appropriate assumptions on the pattern of errors, and assuming no error in the first word a₁, one can prove [11] that the correct verse can be recovered namely:

a₂ = MAJ { O₂, O₃₁, O₅, O₂₈ }

a₃ = MAJ { O₆, O₉, O₂₇, O₂₄ }

where MAJ {., ., ., .} means the word among the four which occurs more than others. The details of the mathematical arguments are in [11].

Two wheeled chariot (Àvichakra ratha)

This is a code that can handle two verses which end with different words which are phonetically close, i.e. error of type 4.

8: ratnadhātamam          g: ratnadhātamaĥ

The chanting procedure ensures that word 8 is chained to 7 and the word g is chained to f and no jump can occur from 7 to g or f to 8.

Arithmetic: Numbers and decimal system

The names for the numbers one to nine found in Rig Veda are eka, dvi, tri, chatur, pancha, shat, sapta, asta, nava. The names for ten, twenty, ....., ninety occur in RV (2.18.5-6). The intermediate numbers have appropriate names. For instance ninety-four is termed four plus ninety. Nineteen is expressed one less than twenty etc. The RV (3.9.9) has a number 3339 spelled as three thousand, three hundred and thirty nine. The RV (2.14.16) uses the word hundred thousand, the modern lakh. Many lakhs are described as hundreds of thousands in RV (1.14.7). Rig Veda has more than a hundred references to numbers.

The Shukla Yajur Veda  (17.2)  mentions ayuta (10⁴) niyuta the series of 10 upto 10¹² in steps of powers of 10 namely sahasra (10⁴) niyuta (10⁵), prayuta (10⁶) arbuda (10⁷), hyabuda (10⁸), samudra (10⁹), madhya (10¹⁰), anta (10⁴), parardha (10¹²) etc. A similar list is in Taithiriya Samhita 4.40.11.4 and 7.2.20.1; Maitrayani Samhita 2.8.14; Kathaka Samhita 17.10 etc.

The atharva veda Samhita (6.25.1 thru 6.25.3, 7.4.1) specially emphasises the common relationship between one and ten, three and thirty, five and fifty, nine and ninety, clearly indicating that these persons had a good grasp of the basics of decimal system for positive integers.

The Yajur Veda (Y.V. 18.24 thru 26) mentions the series of odd numbers 1,3...33 and the series 4,8,....48. The Taittiriya Samhita (7.2.11 through 20) has in addition, the series 10,20...100, 100,200...1000; 10,100,1000...upto 10¹² and the multiplication

4 x 25=100=5 x 20=10 x 10=20 x 5.

MULTIPLICATION AND DIVISION

Shatapatha Brāhmaņa gives many instances of multiplication . For example (2.3.4.19-20) gives 360 x 2=720, 720 x 80 = 57,600.

Again the same book in the section (10.24.2 1-20) gives the result of dividing 720 by all the integers from 2 to 23 which do not give any residue. For instance it considers 720/2, 720/3, 720/4, 720/5, 720/6, 720/8, 720/9, 720/10, 720/12, 720/15, 720/16, 720/18, 720/20, 720/24.

Fractions:

Rig Veda (10.90) mentions the fractions ¼, ½, ¾ and also the fact ¼ + ¾ = 1. Shatapatha Brāhmaņa mentions these and similar results, In addition it mentions in (4.6.7.3) that 1/3 + 2/3 = 1 Y.V. (18.26) mentions the series 1/2 1 1/2, 2, 2 1/2,3, 3 1/2 and 4.

Zero:

The standard question is was there the knowledge of the arithmetic symbol zero? Even though the Vedic sages know the similarity of the relationship between 1 and 10, 2 and 20 etc, there is no explicit mention of the place value system or of the numeral zero. We want to caution that the Vedic poets used extensive symbolism. whether they expressed the numbers in a code language like the code of Aryabhatta or the code Ka¿apayādi remains to be investigated.

The concept of infinity:

The Vedic Indians were aware of the fundamental difference between a large number and infinity. They were aware that an infinite number couldn’t be produced by several finite numbers with finite number of operations.

These are many words for infinity namely ananta, purnam and aditi. The word innumerable asamkhyata occurs in Y.V. 16.54. Bŗhad Āraņyaka Upanishad (2.5.10) – (the Upanishad associated with Shatapata Brāhmaņa and Shukla Yajur Veda) in describing the count of the mysteries of Indra declares it is ananta literally meaning that which has no end anta. They stated two clear definitions. The Atharva Veda 10.8.24 states that "infinity can come out of only infinity" and "infinity is left over from infinity after operations on it". These two statements are made more precise in the invocatory verse of Isha Upanishad (chap.40, Shukla Yajur Veda).

From infinity is born infinity.

When infinity is taken out of infinity,

Only infinity is left over.

The pūrņa is not limited to the mathematical infinity. The author of the hymns is trying to define the concept of all ‘perfect’ perfects.’ Its projection to the realm of mathematics is the mathematical infinity denoted by the symbol infinity later.

Shatapatha Brāmaņa text

(Right angled triangles, Pythagorean triples, square root ...)

All the problems solved in the Sulba Sūtras are mentioned or partially solved in the Shatapatha Brāhmaņa (SB) As Kulkarni (1981) notes:

1. The different types of chits fire altars described in the Sulba Sūtrās are in SB (6.7.2.8)
2. Two types of uttaravedish fire-altars (out of six) are described in SB (7.31.27)
3. The numbers of bricks need to construct different types of Dhishnya fire altars mentioned in S.3 (9.43. 6-8)

It discusses the problem of scaling i.e. given a square shaped altar of area 7½ square purushas, how to increase its dimensions so that area becomes 101.5 square purusha. The solution is given Kathyayana Sulba Sūtras. But the original solution is in SB (10.2.37) and in other different places. However the solutions given in different places of SB are different indicating that the field of algebraic geometry was still developing and was in a state of flux.

A popular question is did the Vedic sages know the so called Pythagorean theorem, This theorem in its simpler version for integers, i.e. the knowledge of triples like (3,4,5) obeying 3² + 4² =5² was known in the Smhita period. The answer is a categorically yes. The writers of the Shatapatha Brāhmaņa knew this knowledge. The theorem and its converse were stated precisely by Baudhayana in his Sulba Sūtras. We have to realise that the authors of the Sūtras incorporated into their books all that was known earlier in addition to their own findings and Baudhayana’s Sulba Sūtra the earliest known book on mathematics is no exception.

The word akşhņayā occurring in several places of the Taittiriya Samhita, Krishna Yajur Veda, 5.2.10, 5.2.7, 5.3.5, 6.2.8, 6.3.10 etc., is the hypotenuse of a right angled triangle or the diagonal of a square or rectangle or trapezium. Shatapatha Brāhmaņa (3.5.1 to 6) gives a method for constructing a isosceles trapezium shaped fire-altar with parallel sides being 24 and 30 and the height or distance between the parallel sides being 36. Of course, the description is given in linear prose without resort to figures.

AL=LB=12, CM=MD=15, LM=36

How did they ensure that the sides AB and CD are parallel? There are only 2 possibilities: either they had access to a device for drawing right angle triangles, for which there is no evidence. Or they knew the side CB=AD=45 using the knowledge of the Pythagorean triple 3² + 4² = 5²

CB² = 36² + (12+12+3)² = 36² + 27² = 9² (4² + 3²) = 9² 5² = (45)²

So points B and A were located so that CB=AD=45 and located the midpoint L. Either way, the knowledge of the triple for integers was and its relation to right angle triangle must have been known.

There is another evidence to support the idea that the authors of Shatapatha Brāhmaņa knew the Pythagorean theorem for integers. As Dutta[5] has noted, the verse 13.8.1.5 suggests that a particular type of altar named paitŗki vedi whose corners point to the four directions must be half the area of the regular square vedi whose four sides point to the four directions. Clearly the Shatapatha Brāhmaņa must have known about the solution to the problem they noted. The Paitŗki vedi square is constructed from the regular vedi squares by joining the mid points of adjacent sides. The fact that this altar has half the area of the regular square is a clear indication of their knowledge of the Pythagorean triple. The construction of the Paitŗki vedi from the regular vedi is indicated in Baudhayana Sulba Sūtras 3.11.

Mathenmatics of Sulba Sūtrās

We give only a brief overview of the four Sulba Sūtra books associated with the names of Baudhayana, Apastamba, Katyayana and Manava. The word sulva is derived for the root sulv, to measure. Since the cord or rope, rajju was used for measuring, in course of time sulva became a synonym for rope. The date of composition of the earliest these books must be much before 1800 B.C.E when the Sarasvati river dried up and the Vedic civilisation was on the decline.

The following geometrical theorems are either explicitly mentioned or clearly implied in the construction of the altars of the prescribed shapes and sizes [Ramachandra Rao, 1997]

1. The diagonals of a rectangle divide the rectangle in four parts, two and two (Vertical, opposite) which are identical ( Ban (iii, 168,169,178)
2. Diagonals of a rhombus bisect each other at right angles
3. An isosceles triangle is divided into two identical halves by the line joining up the vertex to the middle point of base (Bau, iii, 256)
4. A quadrilateral formed by the lines joining middle points of a rectangle is a rhombus whose area is half of that of the rectangle.
5. A parallelogram and a rectangle on the same base and within the same parallels have the same area

Baudhayana theorem (earlier to 2000 B.C.E) (called as pythogoras theorem)

The diagonal of a rectangle produces both areas which the length and breadth produce separately This theorem is usually attributed to the Pythogoras (6^th B.C.E)

The Baudhayana work even states its converse.
 

Value of pi (ratio of circumference to diameter)

There are eight different approximations to pi in the different Sulba Sūtra, Baudhayana (1.61) gives the best approximation among them namely 3.088. The closest value to modern value is given by later Sūtra work Manava Sulba Sūtra (1.27) namely:

4/ (1 1/8)²  =3.16049

Square root of 2 and other surds

The earlier known value of the square root of two is obtained from one of the cuneiform tablets for the Babylonian times (1600 B.C.E) [Neugebeauer, 1952], given in sexagesimal notation Baudhayana gives the following approximation

Square root of 2 =1 +1/3 + 1/(3 x 4) –1/(3 x 4 x 34) This is a better approximation than the earlier one. For details see R.P. Kulkarni’s book(1983)

Mathematics in the Indus Seals

In 1875, in the archeological excavation near the town of Montgomery (now in Pakistan), more than three thousand inscriptions were found, all in an undeciphered script. There have been many attempts to decipher the script, with very little success. Still because of the prejudices of the early indologists (especially their Indian followers), we find frequently the claim that these indus seals have no connection with the Rig Vedic texts. If these seals have not been deciphered, how can one make such definitive statements. As Sri Aurobindo points out, in early indological literature, a mere conjecture after repeated cross references by different scholars acquires the status of “truth”. All the seals are dated 2000-3000 B.C.E by radio carbon dating methods.

Recently N.K. Jha has discovered definitive clues in deciphering the script. It is related to the old Brahmi, not the Ashokan Brahmi. It has only three symbols for vowels and 31 symbols for the consonants of the Sanskrit. Different symbols are compounded as in Sanskrit writing. It is written both for left to right and right to left. Jha has shown that 100 of the seals contain the one hundred words of the glossory on the Vedas developed by the earliest known lexicographer of India, yaska. The deciphering technique is fairly definitive.

The seals give the symbols for the numerals one through nine, ten, twenty, thirty, hundred, thousand and hundred thousand. The symbols for one through nine are close to the corresponding Roman symbols, except that alternatives are there for many of them.

Next there is one seal having three characters which have been deciphered as p k 10. k has been known to be the symbol for karni, the square root, in many places including the Bakshali manuscript of 200 AD. According to Jha, p is the abbreviated form for ‘paridhi vyas anupat’ (the circumference diameter ratio). Thus the above seal gives the square root of ten as an approximation for the pi. This approximation is 3.16. This statement also offers a solution to the long-standing problem. Why was the ratio named pi in Greek. This approximation for pi has been mentioned in the later Jain mathematical literature (500 B.C.E)

Conclusion

We have given in some detail of the mathematics found in the Rig Veda and other Veda Samhitās, and the Shatapatha Brāhmaņa and other Brāhmaņa books. These books indicate the Indians of the early Vedic age knew the decimal systems for integers including multiplication and division, the problem of increasing the square of a certain area to another of predetermined area, problems associated with isosceles trapezium. It is evident that these early Indians knew the so-called Pythagorean triples for integers, approximation to the square root 2 and 3, approximate to pi etc.

References

• Kulkarni, R.P. 1983, Geometry according to the Sulba Sūtras, Pune, India; Vaidic Samshodhan Mandala, (ed) Sontakke.
• Neugebeauer, 1951, The exact sciences in antiquity, Coperhagen.
• Sontakke N.S. and Kashikar C.G., “Rig Veda Samhita”, Vaidic Samshodan Mandala
• Burk, Albert 1901, Das Apastamba Sulba Sūtra
• Zeitschrift D.M.G vol 55, 543-91
• Datta, B.B., 1932, Science of Sulbas, reprinted by Cosmo, New  Delhi
• Shatapatha Brāhmaņa, Pub: Research Institute for ancient Scientific studies
• Bose, D.N., etal, 1971, A concise history of Science in India, Indian National Science Academy, New Delhi
• N.K. Jha, 1996, Vedic glossary on Indus Seals seah, Ganga Kaveri Pub. House, D. 35/77, Janganwadimath, Varaņasi, 221001, India
• Rajaram N.S. and Frawley D. 1997 “Vedic Aryans and the origins of civilisation”, Voice of India, New Delhi
• Balachandra Rao S. 1994, Indian Mathematics and Astronomy, Some Landmarks, National College, Basavanagudi, Bangalore 560004.
• R. L. Kashyap and M. R. Bell, “Error-correcting Code-like chanting procedures in ancient India,” available from the first author.

Back to Top