Astronomy in ancient india

Astronomy in Bharat-varsha*

*the

ancient name of India

A presentation summarizing and based on
the publication
“Remarks on the astronomy of the Brahmins”
by

John Playfair (FRSE, FRS)
Transactions of the
Royal Society of Edinburgh,
1790

 Modern day interpretations of ancient Bharat-Varsha (India) are often
distorted by biases introduced by British colonizers, in line with the
colonial agenda.

colonial agenda.
 These interpretations obscure/ignore the fundamental contributions
made by Bharat-Varsha to science and technology, and the debt which
modern civilization owes to the Indian civilization.

colonial agenda.
 A more unbiased and reliable picture emerges from the records left
by early colonizers and scientists in the 18th century, by whom a careful
observation of Indian society and traditions was undertaken, both for
gathering technological know-how and for establishing political control.

colonial agenda.
 A perusal of such records is necessary for dispelling stereotypical
notions and myths regarding ancient Bharat-Varsha.

colonial agenda.
 A perusal of such records is necessary for dispelling stereotypical
notions and myths regarding ancient Bharat-Varsha.
 And hence, this discussion regarding the publication mentioned, is
meant to serve as a small step in this direction.

What this publication discusses

Three sets of astronomical tables originating from India


(1)
Brought from Siam in 1687 to Europe


(1)
(2)
Sent from Krishnapuram in about 1750 to Europe


(1)
(2)
Sent from Krishnapuram in about 1750 to Europe

(3)
Brought from Tiruvallur in 1772 from Europe

An example of one of the tables from Krishnapuram

published in the book “Traité de l’astronomie Indienne et orientale”, by
M. Bailly, Paris, 1787.

“From the materials furnished by Monsieur Le Gentil and
Monsieur Le Bailly, Mr. Playfair has gone even beyond those
authors, in establishing, by scientific proof, the originality of the
Hindoo astronomy, and its superior antiquity to any other that is
known.”
-Quentin Craufurd, ‘Sketches chiefly relating the history,
religion, learning, and manners, of the Hindoos’, Vol I, 1792.

Celestial sphere- An imaginary sphere of arbitrarily large radius centered at the
earth, useful for describing positions of heavenly objects observed from the
earth.
Celestial coordinates- Coordinates used for describing the position of a
heavenly object on the celestial sphere.
Celestial equator- Intersection of the equatorial plane of the earth with the
celestial sphere.
Ecliptic- The circle on the celestial sphere in which the sun appears to revolve
around the earth in the course of a year. This motion is due to the orbital motion
of the earth around the sun. The ecliptic plane and the celestial equatorial plane
are tilted by about 23.50. This tilt is called obliquity of the ecliptic, and gives
rise to the seasonal variations.
Points of Equinox- The points where the ecliptic and the celestial equator
appear to intersect. This happens twice a year at diametrically opposite points of
the celestial sphere: once during spring (spring/vernal equinox) and once during
autumn (autumnal equinox).

(2) Precession of the equinoxes

The direction of the earth’s axis of rotation itself precesses (rotates)
about a direction perpendicular to the ecliptic plane tracing out a cone
with semi-angle 23.5o. A full precession is completed in about 26,000 yrs.

Implications
Due to precession of the earth’s axis of rotation, every time the earth
completes one revolution around the sun, the orientation of the earth’s axis of
rotation, with respect to the sun, would have changed slightly.
This implies that the periodic change of the seasons occurs with a regularity
that is slightly different from the time taken by the earth to complete one
revolution around the sun, and this difference accumulates with time.
This leads us to two definitions of the year:
SIDEREAL YEAR and TROPICAL YEAR:

SIDEREAL YEAR:
The time taken by the earth to complete one revolution around the sun
with respect to the fixed stars.

SIDEREAL YEAR:
The time taken by the earth to complete one revolution around the sun
with respect to the fixed stars.

TROPICAL YEAR:
The time taken by the earth to return to the same cycle of seasons. It
can also be defined as the time required by the sun to traverse around
the ecliptic from, say, the vernal equinox and back again to the vernal
equinox, and thus returning back to the cycle of seasons.

 The sidereal year is longer than the tropical year by about 20 minutes. This
means that the vernal equinox shifts slightly westwards as seen from the
earth, with respect to the fixed stars, after each revolution of the sun along the
ecliptic.

 Earth's precession was historically called precession of the equinoxes
because the equinoxes move westward along the ecliptic relative to the fixed
stars, opposite to the motion of the Sun along the ecliptic.

What they contain:

Positions of the sun, moon and the planets on the celestial
sphere calculated for different times, and the times of eclipses.
Consequently, they give information on the ellipticity of the
earth’s and the moon’s motion, precession of the equinoxes,
and other details to a high level of precision.

Why they are special:
 Offer solid evidence that astronomical observations were
carried out in Bharata-Varsha at least 5000 years ago, if not
considerably earlier, and with no inputs from any of the later
civilizations.

civilizations.
 Most of the predictions are in general agreement with
contemporary observations and theoretical calculations based
on the theory of gravitation.

civilizations.
 Most of the predictions are in general agreement with
contemporary observations and theoretical calculations based
on the theory of gravitation.
 Since predicting the positions of heavenly objects is a task
involving considerable complexity, these tables offer a
tantalizing glimpse into the level of mathematical sophistication
achieved in Bharat-Varsha, and as such point to the existence of
an age of which almost every vestige has been lost.

Some terms…
(1) Epoch: The time at which actual observations are
carried out, and which serve as the reference for
calculating the positions of heavenly objects at a future
instant of time.

Some terms…
instant of time.
(2) Mean motion/position: This refers to position of a
heavenly object in the celestial sphere at a given instant in
time if its motion is subjected to no irregularities.

Some terms…
instant of time.
(2) Mean motion/position: This refers to position of a
heavenly object in the celestial sphere at a given instant in
time if its motion is subjected to no irregularities.
(3) Correction: This refers to the correction, arising due
to irregularities, which must be added/subtracted to the
mean position of a heavenly object in order to obtain its
true position at a given instant of time. The irregularities
are caused by factors such as ellipticity of the orbits,
precession of the equinoxes, or perturbing effects of other
planets.

Thus the tables contain…

The mean positions for the sun, moon and the planets at
various times, the corrections that must be added to the
mean positions at the various times, and the rules for
adding these corrections depending on the time one wants
the position of a heavenly object.

The tables from Siam
Originally meant for the location of Benares, as deduced by the French
astronomer Cassini.

astronomer Cassini.
Supposed epoch of the tables: 21st March, year 638, as deduced by
Cassini.

astronomer Cassini.
Cassini.
Sidereal year according to the tables: 365d, 6h, 12’, 36” *
Tropical year according to the tablesϯ: 365d, 5h, 50’, 41” ϯϯ

*

d = days, h = hours, ( ’ ) = minutes, ( ” )=seconds.

Thus the equinoxes precess at a rate of once every 24,000 years, which is slightly more than
the current rate of once every 26,000 years.
ϯ

The duration of the sidereal and tropical years is nearly the same in all the other tables, and
hence the rate of the precession of the equinoxes is about once in 24,000 years.
ϯϯ

astronomer Cassini.
Cassini.
Sidereal year according to the tables: 365d, 6h, 12’, 36” *
Tropical year according to the tablesϯ: 365d, 5h, 50’, 41” ϯϯ

“This determination of the length of the year is but 1’, 53” greater than
that of De La Caille, which is a degree of accuracy beyond what is to be
found in the more ancient tables of our astronomy.” **

*

d = days, h = hours, ( ’ ) = minutes, ( ” )=seconds.

Thus the equinoxes precess at a rate of once every 24,000 years, which is slightly more than
the current rate of once every 26,000 years.
ϯ

The duration of the sidereal and tropical years is nearly the same in all the other tables, and
hence the rate of the precession of the equinoxes is about once in 24,000 years.
ϯϯ

Statements within quotes, unless otherwise stated, should be understood as being directly
quoted from the article.
**

On the motion of the moon - I

According to the Siamese (and presumably other) tables, the
moon completes 235 revolutions with respect to the line joining
the earth and the sun* in a period of about 19 years. The
discovery of this so-called Metonic cycle is attributed to the
Greek astronomer Meton, even though Indian astronomy seems
to have been independently well aware of this fact.

*

i.e., 235 lunar months.

On the motion of the moon - II
According to these tables, the apogee* of the moon was in the beginning
of the moveable Zodiac 621 days after the epoch. This matches with
contemporary records to within a degree.
And from this point, the apogee of the moon is supposed to make an
entire revolution in the heavens in the space of 3232 days. This differs
from contemporary records only by 11h, 14’, 31”.

“and if it be considered that the apogee is an ideal point in the heavens
which even the eyes of an astronomer cannot directly perceive, to have
discovered its true motion, so nearly, argues no small correctness of
observation.”

The apogee is the point where the moon is farthest from the earth. The motion is
slowest at the apogee.
*

The tables from Krishnapuram
Epoch: Supposed to be 10th March, 1491, but as we shall see,
probably much older.


“The places which they assign at that time (epoch) to the sun
and moon agree very well with the calculations made from the
tables of Mayer * and De La Caille.**”

* Tobias
** Abbé

Mayer (1723-1762), German astronomer famous for his tables of the moon.

Nicolas Louis de Lacaille (1713-1762), French astronomer.


“The places which they assign at that time (epoch) to the sun
and moon agree very well with the calculations made from the
tables of Mayer * and De La Caille.**”
Equationϯ of the sun’s center: 2o, 10’, 30” ‡
Equation of the moon’s center: 5o, 2’, 47”

* Tobias
** Abbé

Mayer (1723-1762), German astronomer famous for his tables of the moon.

Nicolas Louis de Lacaille (1713-1762), French astronomer.

The equation of the center is, roughly speaking, a measure of the ellipticity of the orbit and is
measured in units of the angle.
ϯ

The angle is measured in units of degrees (o), minutes (’) and seconds (”) . One degree equals
sixty minutes and one minute equals sixty seconds.
‡

Epoch:
3102 B.C.
Commencement of the Kali Yuga

The tables of Tiruvallur

“We must, therefore, enquire, whether this epoch is real or fictitious, that
is, whether it has been determined by actual observation, or has been
calculated from the modern epochs of the other tables. For it may
naturally be supposed, that the Brahmins, having made observations in
later times, or having borrowed from the astronomical knowledge of
other nations…have only calculated what they pretend that their
ancestors observed.”


“In doing this, however, the Brahmins must have furnished us with
means, almost infallible, of detecting their imposture. It is only for
astronomy, in its most perfect state, to go back to the distance of fortysix centuries, and to ascertain the situation of the heavenly bodies at so
remote a period. The modern astronomy of Europe…could not venture on
so difficult a task, were it not assisted by the theory of gravitation, and
had not the integral calculus…been able, at last, to determine the
disturbances in our system, which arise from the action of the planets on
one another.”


“Unless the corrections for these disturbances be taken into account, any
system of astronomical tables, however accurate at the time of its
formation, and however diligently copied from the heavens, will be found
less exact for every instant, either before or after that time, and will
continually diverge more and more from the truth, both for future and
past ages.”


“…it may (therefore) be established as a maxim, that, if there be given a
system of astronomical tables, founded on observations of an unknown
date (epoch), that date may be found, by taking the time when the tables
represent the celestial motions most exactly.”
“Here, therefore, we have a criterion, by which we are to judge of the
pretensions of the Indian astronomy to so great antiquity.”


The location of Aldebaran in 3102 B.C.



Aldebaran


According to the tables from Tiruvallur

Aldebaran



40’ minutes before the vernal equinox
in 3102 B.C.

Aldebaran


According to contemporary calculations

Aldebaran


1) If the rate of precession of equinoxes
is assumed constant, it was 1o, 32’
before the vernal equinox in 3102 B.C.,
based on observations made in 1750.

Aldebaran


1) If the rate of precession of equinoxes
is assumed constant, it was 1o, 32’
before the vernal equinox in 3102 B.C.,
based on observations made in 1750.
2) But the rate of precession of the
equinoxes is itself variable, as discovered
by La Grange, and “this result is to be
Aldebaran

corrected…by the addition of 1o, 45’, 22”
to the longitude of Aldebaran, which
gives the longitude of that star 13’ from
the vernal equinox, at the time of the
Calyougham
(Kali-yugam),
agreeing,
within 53’, with the determination of the
Indian astronomy.”


“This agreement is the more remarkable, that the Brahmins, by their own
rules for computing the motion of the fixed stars, could not have
assigned this place to Aldebaran for the beginning of the Calyougham,
had they calculated it from a modern observation. For as they make the
motion of the fixed stars too great by more than 3” annually,* if they had
calculated backward from 1491, they would have placed the fixed stars
less advanced by 4o or 5o, at their ancient epoch, than they have actually
done.”

i.e., the rate of precession of equinoxes is faster than the observed rate of once every 26,000
years in Indian astronomy.
*


“This agreement is the more remarkable, that the Brahmins, by their own
rules for computing the motion of the fixed stars, could not have
assigned this place to Aldebaran for the beginning of the Calyougham,
had they calculated it from a modern observation. For as they make the
motion of the fixed stars too great by more than 3” annually,* if they had
calculated backward from 1491, they would have placed the fixed stars
less advanced by 4o or 5o, at their ancient epoch, than they have actually
done.”
“This argument carries with it a great deal of force, and even were it the
only one we had to produce, it would render it, in a high degree,
probable, that the Indian Zodiac was as old as the Calyougham.”

i.e., the rate of precession of equinoxes is faster than the observed rate of once every 26,000
years in Indian astronomy.
*


The location of the Sun in 3102 B.C.



10s, 3o, 38’, 13”
(Mean position)


1) Assuming a uniform rate of precession of the equinoxes,
it was, calculating back from later observations, at 10s, 1o,
5’, 57”.


5’, 57”.
2) After adding the correction by La Grange due to the
variation in the rate of precession, it is 10s, 2o, 51’, 19”.


5’, 57”.
2) After adding the correction by La Grange due to the
variation in the rate of precession, it is 10s, 2o, 51’, 19”.

“…not more than 47’ from the radical place in the tables of
Tirvalore. This agreement is near enough to afford a strong
proof of the reality of the ancient epoch…”


The location of the Moon in 3102 B.C.



10s, 6o, 38’
(Mean position)


1) Assuming a constant rate of motion for the moon, its
mean place, calculated by Mayer’s tables, is 10s, 0o, 51’,
16”.


16”.
2) “But, according to the same astronomer, the moon is

subject to a small, but uniform acceleration…the real mean
place of the moon, at the astronomical epoch of the
Calyougham…is therefore 10s, 6o, 37’ .”


16”.
2) “But, according to the same astronomer, the moon is

subject to a small, but uniform acceleration…the real mean
place of the moon, at the astronomical epoch of the
Calyougham…is therefore 10s, 6o, 37’ .”
“…a degree of accuracy that nothing but actual observation
could have produced.”

On the acceleration of the moon


(according to data from Mayer’s tables)


(according to data from Mayer’s tables)

“But, according to the same astronomer, the moon is subject to
a small, but uniform acceleration, such, that her angular motion,
in any one age, is 9” greater than in the preceding, which in an
interval of 4801 years, must have amounted to 5o, 45’, 44”.


(according to the theory of gravity)


(according to the theory of gravity)
“…that acceleration…is a phenomenon, which M. De La Place* has, with
great ability, deduced from the principle of universal gravitation, and
shown to be necessarily connected with the changes in the eccentricity
of the earth’s orbit, discovered by M. De La Grange; so that the
acceleration of the moon is indirectly produced by the action of the
planets, which alternately increasing and diminishing the said
eccentricity, subjects the moon to different degrees of that force by
which the sun disturbs the time of her revolution around the earth. It is
therefore a periodical inequality, by which the moon’s motion in the
course of ages, will be as much retarded as accelerated; but its changes
are so slow, that her motion has been constantly accelerated, even for a
longer period than that to which the observations of India extend.”

*

Pierre-Simon, marquis de Laplace (1749-1827), French astronomer and mathematician.


(according to the tables of Krishnapuram)


(according to the tables of Krishnapuram)
“The moon’s motion in 4383 years, 94 days, taken from those of
Chrisnabouram, is 3o, 2’, 10” less than in the tables of Tirvalore,
from which it is reasonable to conclude, with M. Bailly, that the
former are, in reality, more ancient than the latter…and hence also,
the tables of Chrisnabouram make the moon’s motion less than
Mayer’s, by the above mentioned interval, by 5o, 44’, 14”, which
therefore is, according to them, the quantity of the acceleration.”
 The angular motion of the moon predicted by the tables of
Krishnapuram for 4383 years, 94 days from the commencement of
the Kali Yuga, is 5o, 44’, 14”.
 The tables thus capture the subtle effects of the gravitational
force exerted by other planets on the motion of the moon.
 Antiquity of the Krishnapuram tables is probably even older than
those from Tiruvallur.

“…observations made in India, when all Europe was barbarous or
uninhabited, and investigations into the most subtle effects of
gravitation made in Europe, near five thousand years afterwards…
thus come in mutual support of one another.”

On the antiquity of the tables


Why it is highly probable that the tables go back to 1200
years before the commencement of the Kali-Yuga


 The values of three independent quantities, calculated back for the
beginning of the Kali-Yuga (3102 B.C.), using the theory of gravitation
and contemporary astronomical tables, show a slight mismatch
compared to the values assigned to them in the Indian tables.


 However, the same three quantities show a very good agreement

with the Indian tables when calculated back to 1200 years before the
commencement of the Kali-Yuga.



 The three quantities are:



1) The length of the tropical year,




2) The equation of the sun’s center, and




2) The equation of the sun’s center, and
3) The obliquity of the ecliptic.


The length of the tropical year


 365d, 5h, 50’, 35”, according to the tables from Tiruvallur.


 1’, 46” longer than contemporary records.


 But, “the tropical year was in reality longer at that time than it is at present,”
as it is affected by the precession of the equinoxes and is subjected to slow and
unequal alternations of diminution and increase.


 “If we suppose these observations to have been made in that period of 2400

years, immediately preceding the Calyougham,…we shall find, that, at the middle
of this period, or 1200 years before the beginning of the Calyougham, the length
of the year was 365d, 5h, 50’, 41”, almost precisely as in the tables of Tirvalore.”


 “If we suppose these observations to have been made in that period of 2400

years, immediately preceding the Calyougham,…we shall find, that, at the middle
of this period, or 1200 years before the beginning of the Calyougham, the length
of the year was 365d, 5h, 50’, 41”, almost precisely as in the tables of Tirvalore.”
 “And hence it is natural to conclude, that this determination of the solar year is

as ancient as the year 1200 before the Calyougham, or 4300 before the Christian
era.”


The equation of the sun’s center


 Fixed in the tables at 2o, 10’, 32”.


 Present value according to contemporary records: 1o, 551/₂’ , i.e., 15’ less than
in the Indian tables.


 But as shown by La Grange, the sun’s equation has been diminishing for many
ages. In 3102 B.C., it was therefore 2o, 6’, 281/₂”, i.e., only 4’ less than in the
Indian tables.


Indian tables.
 But if calculated for 1200 years before the commencement of the Kali-Yuga,
the agreement is still more exact.


Indian tables.
 But if calculated for 1200 years before the commencement of the Kali-Yuga,
the agreement is still more exact.
 “1200 years before the commencement of that period (Kali-Yuga)…it appears,

by computing from M. De La Grange’s formula, that the equation of the sun’s
center was actually 2o, 8’, 16”, so that if the Indian astronomy be as old as that
period, its error with respect to this equation is but of 2’. ”


Obliquity of the ecliptic


 The Indian tables take the obliquity of the ecliptic to be 24o. But contemporary
European records in the year 1700 take it to be 23o, 28’, 41”.


 But the obliquity itself, like the rate of precession of equinoxes, and the
equation of the sun’s center, slowly varies with time due to perturbing effects of
the other planets.


the other planets.
 Therefore, “M. De La Grange’s formula for the variation of the obliquity, gives

22’, 32”, to be added to its obliquity in 1700, … , in order to have that which took
place in the year 3102 before our era. This gives us 23o, 51’, 13”, which is 8’, 47”,
short of the determination of the Indian astronomers.”


the other planets.


 If the obliquity is calculated for 1200 years before the Kali-Yuga set in, the
agreement is still more exact.


the other planets.


 If the obliquity is calculated for 1200 years before the Kali-Yuga set in, the
agreement is still more exact.
 “But if we suppose, … , that the observations on which this determination is

founded, were made 1200 years before the Calyougham, we shall find that the
obliquity of the ecliptic was 23o, 53’, 45”, and that the error of the tables did not
much exceed 2’. ”


 Thus three entirely independent elements separately answer to belong
to an age 1200 years before the Kali-Yuga set in, or about 6300 years
from today. Too much to be a matter of chance!
 “This coincidence … cannot be the effect of chance…there is no other

alternative…but to acknowledge that the Indian astronomy is as ancient
as one, or other of the periods above mentioned.”

On the planets
(1)
The retrograde motion and the aphelion of Jupiter

On the planets
(1)
(2)

The equation of Saturn’s center

On the planets
(1)
(2)

(3)
The mutual interaction between Saturn and Jupiter

On the planets

Retrograde motion: The apparent occasional motion of a planet opposite to that
usually observed, due to the projection of the planet’s motion on the celestial
sphere.

Aphelion: The point in a
planet’s orbit which is farthest
from the sun. The motion is
slowest at this point.

Figure illustrating the retrograde motion of the planets

On the planets


On the planets

According to the Indian tables, Jupiter has a retrograde motion of 15o in 200,000
years, and thus, calculating back from Jupiter’s position in 1491 as given in the
Krishnapuram tables, its aphelion at the onset of Kali-Yuga was at 3s, 27o, 0’ from
the equinox.

On the planets

the equinox.
Calculating back from contemporary European records gives for the aphelion of
Jupiter in 3102 B.C. as 3s, 16o, 48’, 58”, “so that there would seem to be an error of

more than 10o in the tables of the Brahmins.”

On the planets

the equinox.
Calculating back from contemporary European records gives for the aphelion of
Jupiter in 3102 B.C. as 3s, 16o, 48’, 58”, “so that there would seem to be an error of

more than 10o in the tables of the Brahmins.”

But Jupiter’s orbit is subject to great disturbances from the action of Saturn, which
the above calculation ignores. Including these disturbances (by making use of a
formula due to La Grange), gives for the aphelion of Jupiter at the beginning of KaliYuga (3102 B.C.), as 3s, 26o, 50’, 40”, “which is but 10’, 40” different from the

tables of Krishnapuram.”

On the planets


On the planets


“The equation of Saturn’s center is an instance of the same kind. That
equation, at present, is according to M. De La Lande, 6o, 23’ 19”; and
hence, by means of one of the formulas above mentioned, M. Bailly
calculates, that, 3102 years before Christ, it was 7o, 41’, 22”. The tables
of the Brahmins make it 7o, 39’, 44”, which is less only by 1’, 38”, than
the preceding equation…”

On the planets


On the planets


“Since the publication of M. Bailly’s work*, two other instances of an exact
agreement, between the elements of these tables, and the conclusions
deduced from the theory of gravity, have been observed, and
communicated to him by M. De La Place, in a letter, inserted in the
Journal des Savans.”

*Traité

de l’astronomie Indienne et orientale”, M. Bailly, Paris, 1787.

On the planets


“Since the publication of M. Bailly’s work*, two other instances of an exact
agreement, between the elements of these tables, and the conclusions
deduced from the theory of gravity, have been observed, and
communicated to him by M. De La Place, in a letter, inserted in the
Journal des Savans.”
“M. De La Place has discovered, that there are inequalities belonging to
both these planets (Saturn and Jupiter), arising from their mutual action
on one another, which have long periods, one of them no less than 877
years; so that the mean motion must appear different, if it be determined
from observations made in different parts of those periods.”

*Traité

de l’astronomie Indienne et orientale”, M. Bailly, Paris, 1787.

On the planets


Says M. De La Place (as quoted in the article by Playfair):

On the planets



“Now I find, by my theory, that at the Indian epoch of 3102 years before
Christ, the apparent and annual mean motion of Saturn was 12o, 13’, 14”,
and the Indian tables make it 12o, 13’, 13”. ”

On the planets



“Now I find, by my theory, that at the Indian epoch of 3102 years before
Christ, the apparent and annual mean motion of Saturn was 12o, 13’, 14”,
and the Indian tables make it 12o, 13’, 13”. ”
“In like manner, I find, that the annual and apparent mean motion of
Jupiter at that epoch was 30o, 20’, 42”, precisely as in the Indian
astronomy.”

General conclusions

 Astronomy in Bharat-Varsha goes back to at least 6300 years ago, and
its development had no influence from any other systems of astronomy
anywhere else in the world.

“It is …certain that the astronomy of the Brahmins is neither derived from
that of the Greeks, the Arabians, the Persians or the Tartars.”

General conclusions

 The principles underlying the rules for calculating the positions of
heavenly objects implies a high level mathematical sophistication, and
much of the geometry attributed to the Greeks has originated in India.

General conclusions
 This is certainly true of the theorem attributed to Pythagoras,* for the
method of calculating the duration of a lunar or solar eclipse according to
the Indian tables is based on this theorem.

“These operations (for finding the duration of eclipses) are all founded on
a very distinct conception of what happens in the case of an eclipse, and
on the knowledge of this theorem, that, in a right angled triangle, the
square of the hypotenuse is equal to the squares of the other two sides. It
is curious to find the theorem of Pythagoras in India, where, for aught we
know, it may have been discovered, and from whence that philosopher
may have derived some of the solid, as well as the visionary speculations,
with which he delighted to instruct or amuse his disciples.”
The earliest mention of this theorem is to be found in the Sulbasutras credited to Bodhayana
(c. 800 B.C.), but it is clear from these astronomical tables that the knowledge of this theorem
was prevalent since much earlier.
*

General conclusions
 Regarding Pythagoras, it is interesting to note the following
remarks from another article:

“we know that he (Pythagoras) went to India to be instructed; but the
capacity of the learner determines his degree of proficiency, and if
Pythagoras on his return had so little knowledge in geometry as to
consider the forty-seventh of Euclid as a great discovery, he certainly was
entirely incapable of acquiring the Indian method of calculation, through
his deficiency of preparatory knowledge …
…each teacher, or head of sect that drew his knowledge from Indian
sources, might conceal his instructors to be reckoned an inventor.”
-Reuben Burrow, ‘Hints concerning the observatory at Benares’ c. 1783.

General conclusions

 The principles underlying the rules for using the tables also supposes
a deep knowledge of trigonometry and spherical trigonometry. That these
were carried to a high level of perfection in Bharat-Varsha, is amply
demonstrated by the ancient Sanskrit texts, the Surya Siddhanta.*

See, for example, John Playfair, “Observations on the trigonometrical tables of the Brahmins”,
Transactions of the Royal Society of Edinburgh, Vol. IV. (1798).
*

General conclusions
 The classic problem of the quadrature of the circle, or the ratio of the
circumference to the diameter of the circle (commonly known as π), was
solved in India much before the Greek mathematicians, and to much
greater precision.*

“…the Hindoos suppose the diameter of a circle to be to its circumference
as 1250 to 3927, and where the author, who knew that this was more
accurate than the proportion of Archimedes, (7 to 22), … expresses his
astonishment, that among so simple a people, there should be found a
truth, which, among the wisest and most learned nations, had been
sought for in vain.”

*

Also see, “On the Hindu quadrature of the circle, and the infinite series of the proportion of

the circumference to the diameter exhibited in the four Sastras, the Tantra Sangraham, Yucti
Bhasha, Carana Padhati, and Sadratnamala”, by Charles M. Whish, published in the
Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. III (1835).

Comparison of the days of the week in the Indian system
and the European system

(Table scanned from the book ‘Sketches chiefly relating the history,
religion, learning, and manners, of the Hindoos’, Vol. I, by Quentin
Craufurd, 1792. “the days are arranged exactly in the same order
that has been adopted by the Europeans.”)

“If to this we add the great extent of geometrical knowledge requisite to
combine this, and the other principles of their astronomy together and to
deduce from them the just conclusions; the possession of a calculus
equivalent to trigonometry; and, lastly, their approximation to the
quadrature of the circle, we shall be astonished at the magnitude of that
body of science, which must have enlightened the inhabitants of India in
some remote age, and which, whatever it may have communicated to the
western nations, appears to have received nothing from them.”

Final reflections
 It should be noted that Playfair’s article discussed in this presentation
sets only a lower limit for the antiquity of Indian astronomy, and in no
way rules out the possibility that it could be much more ancient. Indeed,
considering the level of mathematical and astronomical skills used in
making these tables, this certainly seems to be the case.

Final reflections
 In spite of there being abundant evidence concerning the achievements
of Bharat-Varsha in the mathematics and the sciences, all references to
Indian science and technology have been either eliminated or ignored;
not just by the world at large, but by the majority of Indians themselves.

Final reflections
 In spite of there being abundant evidence concerning the achievements
of Bharat-Varsha in the mathematics and the sciences, all references to
Indian science and technology have been either eliminated or ignored;
not just by the world at large, but by the majority of Indians themselves.
 It is time to undo this damage by learning about the true history and
contributions of Bharat-Varsha, instead of depending on borrowed
histories and identities.

Acknowledgements
A deep sense of appreciation is expressed to Shri Dharampal (19222006), for having compiled several publications on Indian science and
technology in the book, ‘Indian science and technology in the eighteenth
century’. The article discussed in this presentation is included in this
book.

This book, and several others, can be downloaded for free from the
website: www.samanvaya.com/dharampal/
Most of the references mentioned in this presentation are available for
free at www.books.google.com. The effort in making these documents
available to the general public is sincerely appreciated.

Astronomy in ancient india

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