**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 12^{th} 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 5^{th} 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

**
200**

* 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

**
200**

**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

** 200**

** **

** **

* 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

--------

**187.5
**** **

**-------- **

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 Mathematics **

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.