Prof. Makkhan Lal
Like the British historians, Marxists, too, have virtually ruled out the presence of science and technology, not only during the Vedic period but also throughout the entire period of over 7,000 years, i.e., before the arrival of the Muslims at the end of 12th century A.D. Only casually, and with great reluctance, do they mention Aryabhat, Varaha Mihira and the invention of zero. This, they must willy-nilly do because the world over the contributions of Aryabhat and Varaha Mihira and the zero are mentioned in science books even at the most basic level. Any further mention of science and technology during the ancient period in India is nothing less than an anathema. After all, how could the Vedic people, who were merely ‘pastoral nomads’, have thought of science?
They refuse to acknowledge that the Vedic people had accurate knowledge of the movements of heavenly bodies and could make a near perfect calendar. They refuse to accept that the Vedic people had advanced knowledge of mathematics with the help of which they could handle fairly complex calculations involving a large number of digits and a sophisticated system. We have seen in Chapter V how just a mention of the knowledge of zero in the Vedic period created a furor among Marxist historians. Even this invention and use of zero is never taken by colonialist and Marxist historians to be earlier than 5th century A.D.135
Anyone who talks about science, mathematics, astronomy, etc. during the Vedic period is dumped as being communal, obscurantist, woolly-headed, illiterate, a duffer, and so on, and all discussion is damned as ‘a realm of fantasy’, ‘absurd’, ‘without any proof’, entirely baseless’, etc.136
Several books dealing with the history of scientific development around the world have acknowledged in detail the contributions made by ancient Indians. One of such books which came out in 2002 is The Lost Discoveries: The Ancient Roots of Modern Science –– from the Babylonians to the Maya, authored by Dick Teresi.137 Teresi introduces the Indian contributions (indeed, referring to basically the primary sources) in the following words:
“Twenty four centuries before Isaac Newton, the Hindu Rig-Veda asserted that gravitation held the Universe together…. The Sanskrit speaking Aryans subscribed to the idea of a spherical earth in an era when the Greeks believed in a flat one. The Indians of fifth century A.D. somehow calculated the age of earth as 4.3 billion years; scientists in nineteenth century England were convinced it was 100 million years. The modern estimate is 4.6 billion years.”138
“The concept of infinite numbers was grasped by Indian thinkers in sixth century B.C. and by Alhazen in tenth century A.D. It entered Europe nearly a thousand years later, when the nineteenth-century German mathematician George Cantor refined and categorized infinite sets.”139
Regarding ancient India’s strides in mathematics, Teresi writes:
“The earliest recorded Indian mathematics was found along the banks of the Indus… The precise mathematical expertise of the Harappan culture, which lasted from 3000 to 1500 B.C., is difficult to pinpoint, as Harappan script has never been deciphered. There are physical clues, though. …Archaeologists have uncovered several scales, instruments, and other measuring devices. The Harappans employed a variety of plumb bobs that reveal a system of weights based on a decimal scale. For example, a basic Harappan plumb bob weights 27.584 grams. If we assign that a value of 1, other weights scale in at .05, .1, .2, .5, 2, 5, 10, 20, 50, 100, 200, and 500. These weights have been found in sites that span a five-hundred-year period, with little change in size.
“Archaeologists also found a “ruler” made of shell lines drawn 6.7 millimeters apart with a high degree of accuracy. Two of the lines are distinguished by circles and are separated by 33.5 millimeters, or 1.32 inches. This distance is the so-called Indus inch. …Most interesting are their [bricks] dimensions: while found in fifteen different sizes, there length, width, and thickness are always in the ratio of 4 : 2: 1.
“Bricks and religion are at the root of the Vedic period of Indian mathematics. Vedic literature, one of the largest and oldest literary collections, encompasses works of hymns and prayers, songs, magic formulas and spells, and most important to us here, sacrificial formulas. One collection of Vedic literature, called the Brahmanas, spells out the rules for conducting sacrifices. Another collection, known as the Sulbasutras, meaning “the rules of the cord”, dictates the shapes and areas of altars (vedi) and the location of the sacred fires. Square and circular altars were okay for simple household rituals, but rectangles, triangles and trapezoids were required for public occasions.
“These altars sometimes took extravagant forms, such as the falcon altar, made from four different shapes of bricks: (a) parallelograms, (b) trapeziums, (c) rectangles, and (d) triangles.
“The sulbasutras were written between 800 and 600 B.C., making them at least as old as the earliest Greek mathematics. According to George Joseph, researchers in the nineteenth century made a point of emphasizing the religious nature of the sulbasutras – and certainly they are religious – but ignored their mathematical content. Joseph sees in the sulbasutras a link between the Harappan culture and the highly literate Vedic culture, by means of the Harappan brick technology, which was put to geometrical and religious uses in Vedic sacrifices. To ignore the mathematical component of Vedic rituals is akin to characterizing the Gregorian calendar as a religious exercise rather than a mathematical and astronomical accomplishment.
“The earliest sulbasutras were composed by the priest-craftsman Baudhayana somewhere between 800 and 600 B.C. and include a general statement of the Pythagorean theorem and a procedure for obtaining the square root of 2 to five decimal places. Baudhayana’s motivations were religious and practical; he needed a mathematics that would help scale altars to the proper size depending on the sacrifice. His version of the Pythagorean theorem is: “The rope that is stretched across the diagonal of a square produces an areas double the size of the original square.” Another sulbasutra states: “The rope (stretched along the length) of the diagonal of a rectangle makes an (area) that the vertical and horizontal sides make together.
“The sulbasutras contain instructions for the building of a smasana, a cemetery altar on which soma, an intoxicating drink, was offered as a sacrifice to the gods. The smasana’s base was a complicated shape called an isosceles trapezium, which comprised, among other figures, six right triangles of different sizes. It’s obvious that the Indians of this era knew the Pythagorean rule.
“The most basic right triangle, with sides of 3, 4, and 5 units in length, might be stumbled upon by chance. Using a rope marked off with knots at 3, 4, and 5 units would allow builders to ascertain the squareness of corners, and the Egyptians, for example, did just that. Mathematicians have pointed out to me that ancient nonwhite people might by accident come up with a triangle with sides of 3, 4 and 5 and note that it always formed a right angle.
“However, the instructions given for a smasana in the Sulbasutras dictate that six right triangles be used in the construction, consisting of sides of 5 : 12 : 13, 8 : 15 : 17, 12 : 16 : 20 (a multiple of 3 : 4 : 5), 12 : 35 : 37, 15 : 20 : 25 (another multiple of 3 : 4 : 5), and 15 : 36 : 39. That’s a lot of luck. In addition, the Sulbasutras employed right triangles with sides of fractional and even irrational lengths.
“The Vedic sacrificers figured out a method of evaluating square roots. Joseph suspects the technique evolved from a need to double the size of a square altar. Say you wish to double the area of an altar with sides 1 unit long. Obviously, doubling the lengths of the sides would result in an altar four times the size. It becomes clear that one needs a square whose sides are the square root of 2, and thus one needs a technique for calculating square roots. The Sulbasutra square root of 2 is 1.414215… the modern value is 1.414213…. No one is certain how the Indians arrived at their method, but it probably involved positing two equal squares with 1-unit sides, then cutting the second square into various strips and adding those strips to the first square to make a square with twice the area, then converting the strips to fractions to construct a numerical formula. This may have been the first recorded method of evaluating square roots.
“Early Indian geometry is filled with fantastic and phantasmagorical dynamic constructions, such as the sriyantra, or “great object,” which belongs to the tantric tradition. In it nine basic isosceles triangles form forty-three others, encircled by an eight-petaled lotus, a sixteen-petaled lotus, and three circles, which in turn are surrounded by a square with four doors. The meditator concentrates on the dot, called a bindu, in the center, and moves outward, mentally embracing more and more shapes, until he reaches the boundary. Or the meditation can be done in reverse.
“The sriyantra is typical of Indian geometry, with its religious originality, mysticism, and even playfulness, qualities we rarely see in Greek geometry, which remains “uncontaminated” by religion. Various special “numbers” are integrated into the sriyantra, such as pi and another irrational number, the golden ratio, or approximately 1.6183. The golden ratio is found in the pyramids at Giza and in the later construction of the Parthenon and other classical Greek buildings.
“Is 1.61803 a better number when found in later secular Greek architecture than in earlier Indian religious patterns? Interestingly, as Vedic sacrifices declined around 500 B.C., so, too, did the practice of mathematics among Indians.
“The Ancient Indian practiced a very sophisticated form of mathematics. They had the usual arithmetic operations – addition, subtraction, multiplication, division – but also algebra, indices, logarithms, trigonometry, and a nascent form of calculus.”140
This long quotation actually refutes all the biased and strange objections and opinions, including the knowledge of the decimal system. How else, otherwise, could the Sulvasutras have known the value of the square-root of 2? The passage above sums up the knowledge of mathematics available in the Vedic period.
As mentioned earlier, a very advanced level of mathematics was required to construct the Vedic altars. Below is given the measurements of various kind of bricks that are required in the construction of a five-layered brick altar. It may be seen from the drawing that the total area of each layer of bricks remains the same but in order to avoid the joints coming over one another the sized and shape of bricks changed but the out line of the altar and the total covered area remained the same.141
1 Aindri, “for Indra” 1
2 Vibhakti, “Share” 1
3 Mandala, “Circle” 1
4 Retahsic, “Seed Discharging” 1
5-16* Skandhya, “Shoulder” 12
17-36 Apasya, “Watery” 20
37-86 Pranabhrt, “Supporting Exhalation” 50
87-98 Samyat, “Stretch” 12
99-148 Apanabhrt, “Supporting Inhalation” 50
149-153 Mukham, “Face” 5
154-158 Angam, “Limb” 5
159 Prajapatya, “for Prajapati” 1
160 Rsabha, “Bull” 1
161-200 Lokamprna, “Space Filler” 40
* 15-16 are called Samyani, “way.”
Names of Bricks in the Fourth Layer
Number Name of Bricks Number of Bricks
1 Vibhakti, “Share” 1
2-6 Skandhya, “Shoulder” 5
7-34 Aksnayastomiya, “With Diagonal Stoma” 28
35-51 Srshti, “Creation” 17
52-67 Vyushti, “Dawn” 16
68-72 Mukham, “Face” 5
73-77 Angam, “Limb” 5
78 Prajapatya, “For Prajapati” 1
79 Rshabha, “Bull” 1
80-200 Lokamprna, “Space Filler” 121
Number Name of Bricks Number of Bricks
1 Vibhakti, “Share” 1
2-3 Skandhya, “Shoulder” 7
9-13 Asapatna, “Unrivaled” 5
14-53 Viraj, “Sovereign” 40
54-84 Stomabhaga, “Chant Sharing” 31
85-89* Nakashat, “Sitting in the Sky” 5
90-94* Coda, “Protuberance” 5
95-123 Chhandas, “Meter” 29
124-130 Krttika, “Pleiads” 7
131-135 Vrshtisani, “Rain Bringing” 5
136-143 Aditya 8
144-148 Ghrta, “Clarified Butter” 5
149-153 Yashoda, “Glory Giver” 5
154-158 Bhuyaskrt, “Augmenting” 5
159-163 Apsusad, “Sitting in Waters” 5
164-168 Dravinoda, “Wealth Giver” 5
169-175 Ayushya, “Life Giver” 7
176-180 Rtunama, “Season’s Name” 5
[118 pebbles: see Table 13]
181-185 Shashthi citi, “Sixth Layer” 5
186-190 Mukham, “Face” 5
191-195 Angam, “Limb” 5
196 Prajapatya, “For Prajapati” 1
197 Rshabha, “Bull” 1
198-202 Lokamprna, “Space Filler” 5
203 Mandala, “Circle” 1
204 Retahsic, “Seed Discharging” 1
205 Vikarni, “Without Ears” 1
* The Nãkasat and Coda are twenty half-bricks, equal to ten whole bricks.
* 175 is called Pañcajanya
Area of Bricks in the First, Third, and Fifth Layer
Number Area per Bricks Area
Pancami 38 1.0 0 38.0
Sapada 02 1.25 02.5
Adhyardha 56 1.50 84.0
Panchamyardha 60 0.5 0 30.0
Adhyardhardha 44 0.75 33.0
However, since the question of the scientific achievements of ancient Indians covers many other fields as well, a detail quotation is hereby provided from the series, Cultural Heritage of India (Vol.VI) which are respected around the world for their authenticity and thoroughness.
Vedic Hindus evinced special interest in two particular branches of mathematics, viz. geometry (sulva) and astronomy (jyotisa). Sacrifice (yajna) was their prime religious avocation. Each sacrifice had to be performed on an altar of prescribed size and shape.… So the greatest care was taken to have the right shape and size of the sacrificial altar. Thus originated problems of geometry and consequently the science of geometry. The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice… In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake…
The Chandogya Upanishad (VIII. 1.2.4) mentions among other sciences the science of numbers (rasi). In the Mundaka Upanishad (I.2.4-5) knowledge is classified as superior (para) and inferior (apara). In the Mahabharata (XII. 201) we came across a reference to the science of stellar motion (naksatragati).
The term ganita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedanga Jyotisa gives it the highest place of honour amongst all the sciences which form the Vedanga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the ganita at the top of the sciences known as the Vedanga.’ (yajur vedic recension, verse 4). At that remote period ganita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.
Available sources of Vedic mathematics are very poor. Almost all the works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three recensions, namely, the Arca Jyotisa, Yajusa Jyotisa, and Atharva Jyotisa. There are six small treatises on Vedic geometry belonging to the six schools of the Veda.
There is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brahmana period when astronomy came to be regarded as a separate science called naksatra-vidya (the science of stars). An astronomer was called a naksatra-darsa (star-observer) or ganaka (calculator).
According to the Rig-Veda (I.115.1, II. 40.4, etc.), the universe comprises prthivi (earth), antariksa (sky, literally meaning ‘the’ region below the stars’), and div or dyaus (heaven). The distance of the heaven from the earth has been stated differently in various works. The Rig-Veda (I.52.11) gives it as ten times the extent of the earth, the Atharva-Veda (X. 8.18) as a thousand days’ journey for the sun-bird, the Aitareya Brahmana (II.17.8) as a thousand days’ journey for a horse… All these are evidently figurative expressions indicating that the extent of the universe is infinite.
There is speculation in the Rig-Veda (V. 85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air (IV.53.3). the Satapatha Brahmana describes it expressly as parimandala (globe or sphere). There is evidence in the Rig-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.
According to the Rig-Veda (VI.58.1), there is only one sun, which is the maker of the day and night, twilight, month, and year. It is the cause of the seasons (I.95.3). It has seven rays (I.105.9, I.152.2, etc.), which are clearly the seven colours of the sun’s rays. The sun is the cause of winds, says the Aitareya Brahmana (II.7). It states (III.44) further: ‘The sun never sets or rises. When people think the sun is setting, it is not so; for it only changes about after reaching the end of the day, making night below and day to what is on the other side. Then when people think he rises in the morning, he only shifts himself about after reaching the end of the night, and makes day below and night to what is no the other side. In fact he never does set at all.’ This theory occurs probably in the Rig-Veda (I. 115.5) also. The sun holds the earth and other heavenly bodies in the their respective places by its mysterious power.
In the Rg-Veda, Varuna is stated to have constructed a broad path for the sun (I.28.8) called the path of rta (I.41.4). This evidently refers to the zodical belt. Ludwing thinks that the Rig-Veda mentions the inclinations of the ecliptic with the equator (I.110.2) and the axis of the earth (X.86.4). the apparent annual course of the sun is divided into two halves, the uttarayana when the sun goes northwards and the daksinayana when it goes southwards. Tilak has shown that according to the Satapatha Brahmana (II.1.3.1-3) the uttarayana begins from the vernal equinox. But it is clear from the Kausitaki Brahmana (XIX.3) that those periods begin respectively from the winter and summer solstices. The ecliptic is divided into twelve parts or sign of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at the various parts of the zodiac, and thus has originated the doctrine of twelve adityas or suns.
The Rg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relation to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus ) and Manthin are mentioned by name.
The Rg-Veda mentions thirty-four ribs of the horse (I.162.18) and thirty-four lights (X.55.3). Ludwig and Zimmer think that these refer to the sun, the moon, five planets, and twenty-seven naksatras (stars). The Taittiriya Samhita (IV.4.10.1-3) and other works expressly mention twenty-seven naksatras. The Vedic Hindus observed mostly those stars which lie near about the ecliptic and consequently identified very few stars lying outside that belt….
It appears from a passage in the Taittiriya Brahmana (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Rig-Veda, a passage of which states that Atri observed a total eclipse of the sun caused by its being covered by Svarbhanu, the darkening demon (V.40.5-9). Atri could calculate the occurrence, duration, beginning, and end of the eclipse. His descendants also were particularly conversant with the calculation of eclipses. In the Atharva-Veda (XIX.9.10) the eclipse of the sun is stated to be caused by Rahu the demon. At the time of the Rig-Veda the cause of the solar eclipse was understood as the occultation of the sun by the moon. There is also mention of lunar eclipses.
In the Vedic Samhitas the seasons in a year are generally stated to be five in number, namely, Vasanta (spring), Grisma (summer), Varsa (rains), Sarat (autumn), and Hemanta-Sisira (winter). Sometimes Hemanta and Sisira are counted separately, so that the number of seasons in a year becomes six. Occasional mention of a seventh season occurs, most probably the intercalary months are termed ‘twins’. Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterism. Vasanta used to be considered the first of the seasons as well as the beginning of the year (Taittiriya Brahmana, I.1.2.67; III. 10.4.1). The Taittiriya Samhita (VI.1.5.1) and Aitareya Brahmana (I.7) speak of Aditi, the presiding deity of the Punarvasu naksatra, receiving the boon that all sacrifices would begin and end with her. This clearly refers to the position of the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mrgasira from whence, in course of time, it receded to Krttika. Thus there is clear evidence in the Samhitas and Brahmanas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.
Sulva (geometry) was used in Vedic times to solve propositions about the construction of various rectilinear figures; combination, transformation, and application of areas; mensuration of areas and volumes, squaring of the circle and vice versa etc. One theorem which was of great importance to them on account of its various applications is the theorem of the square of the diagonal. It has been enunciated by Baudhayana (c.600 B.C.) in his Sulvasutra (I.48) thus: ‘The diagonal of a rectangle produces both (areas) which its length and breadth produce separately.’ That is, the square described on the diagonal of a rectangle has an area equal to the sum of the areas of the squares described on its two sides. This theorem has been given in almost identical terms in other Vedic texts like the Apastamba Sulvasutra (I.4) and Katyayana Sulvasutra (II.11). The corresponding theorem for the square has been given by Baudhayana (I.45) separately, though it is in fact a particular case of the former: ‘The diagonal of a square produces an area twice as much.’ That is to say, the area of the square described on the diagonal of a square is double its area.
The converse theorem –– if a triangle is such that the square on one side of it is equal to the sum of the squares on the two other sides, then the angle contained by these two sides is a right angle –– is not found to have been expressly defined by any sulvakara (geometrician). But its truth has been tacitly assumed by all of them, as it has been freely employed for the construction of a right angle.
The theorem of the square of the diagonal is now generally credited to Pythagoras (c.540 B.C.), though some doubt exists in the matter. Heath asserts, for instance: ‘No really trustworthy evidence exists that it was actually discovered by him. The tradition which attributes the theorem to Pythagoras began five centuries after his demise and was based upon a vague statement which did not specify this or any other great geometrical discovery as due to him. On the other hand, Baudhayana, in whose Sulvasutra we find the general enunciation of the theorem, seems to have been anterior to Pythagoras. Instances of application of the theorem occur in the Baudhayana Srautasutra (X.19, XIX.1,XXVI) and the Satapatha Brahmana (X.2.3.7–14). There are reasons to believe it to be as old as the Taittiriya and other Samhitas. With Burk, Hankel, and Schopenhauer, we are definitely of the opinion that the early Hindus knew a geometrical proof of the theorem of the square of the diagonal.”142
I hope that from these two long extracts from Dick Teresi’s Lost Discoveries: The Ancient Roots of Modern Sciences and the Cultural Heritage of India, Vol. VI, it must be clear that during the Vedic period in India, science, astronomy, mathematics, etc. were highly advanced branches of knowledge.
1. For details see I. Habib et al. 2003, History in the New NCERT Textbooks – a Report and an Index of Errors, New Delhi; R.S. Sharma, 1999, Ancient India: A Textbook for class XI, NCERT, New Delhi, Romila Thapar, 1987, Ancient India: A Textbook for Class VI, NCERT, New Delhi.
2. The writings of an Indian Marxist historian are considered academically sound the historian having reached the age only when his writings are punctuate with such epithets listed in the text against those who do not agree with them. For this see the Writings of I. Habib, R.S. Sharma, Romila Thapar and a scores of other Marxist historians and social scientists.
3. Dick Teresi, 2002, The Lost Discoveries: The Ancient Roots of Modern Science –– from the Babylonians to the Maya, New York.
4. ibid. pp. 7-8
5. ibid. p. 22.
6. ibid. pp. 59-64.
7. Frits Stall et al., 1983, Agni: The Vedic Ritual of Fire Altar, Vols. I & II; (Indian edition published in 2001, New Delhi).
8. Cultural Heritage of India, Vol. VI, pp. 18-22.