The astronomy of the age of geometric altars Subhash C. Kak

Fire altars were an important part of ritual throughout the ancient
world. Geometric ritual, often a part of the re altars, was intimately
connected with problems of mathematics and astronomy. Manuals
of altar design from India explain the basis behind the reconciliation
of the lunar and the solar years. This astronomy is based on the
use of mean motions. Computation rules from Ved a _nga Jyotis
a, an.
astronomy manual from the latter part of the second millennium B.C.
that was used during the closing of the age of the altar ritual, are also
described.
1 Introduction
It has long been argued that science had an origin in ritual. According
to Plutarch (Epicurum IX) Pythagoras sacri ced an ox when he discovered
the theorem named after him. This legend is, in all probability, false since
Pythagoras was opposed to killing and sacri cing of animals, especially cat-
tle (van der Waerden 1961, page 100). Nevertheless, this story frames the
connection between ritual and science in the ancient world. Plutarch says
elsewhere (Quaestiones Convivii, VIII, Quaest. 2.4) that the sacri ce of
the bull was in connection with the problem of constructing a gure with

Department of Electrical & Computer Engineering, Louisiana State University, Baton
Rouge, LA 70803-5901, USA


the same area as another gure and a shape similar to a third gure. A.
Seidenberg (1962, 1978) has sketched persuasive arguments suggesting that
the birth of geometry and mathematics can be seen in the requirements
of geometric ritual of precisely this kind. He shows how geometric ritual
represented knowledge of the physical world through equivalences. But if ge-
ometric ritual represented a language that coded the knowledge of its times,
it should have been used for astronomical knowledge as well. This question
has recently been analyzed by the author (see Kak 1993a,b.c, 1994a,b) for
the Indian context.
It is generally accepted that Hipparchus discovered precession in 127 B.C.
The magnitude calculated by Hipparchus and accepted by Ptolemy was 1
degree in 100 years. The true value of this precession is about 1 degree in 72
years. Clearly the discovery of precession could not have been made based
on observations made in one lifetime. The ancient world marked seasons
with the heliacal rising of stars. So Hipparchus must have based his theory
regarding precession on an old tradition. That the ancients were aware of
the shift in the heliacal rising of stars with age was demonstrated by Giorgio
de Santillana and Hertha von Dechend (1969) in their famous book Hamlet's
Mill which appeared more than twenty ve years ago. By uncovering the
astronomical frames of myths from various ancient cultures, they showed that
man's earliest remembrance of astral events goes back at least ten thousand
years. Although, this does not mean that the principles behind the shifting
of the astronomical frame were known at this early time, the analysis of the
designs of Stonehenge, the pyramids, and other monuments establish that
the ancients made careful astronomical observations. But no corroborative
text, prior to the tablets from Babylon that date back to the middle of the
rst millennium B.C., is available.
Fire altars have been found in the third millennium cities of the Indus-
Sarasvati civilization (Rao 1991) of India. The texts that describe their de-
signs are conservatively dated to the rst millennium B.C., but their contents
appear to be much older. Basing his analysis on the Pythagorean triples in
Greece, Babylon, and India, Seidenberg (1978) concludes that the knowledge
contained in these texts|called Sulba S utras|goes back to at least 1700

B.C. More recent archaeological evidence, together with the astronomical
references in the texts (Sastry 1985), suggests that this knowledge belongs
to the third millennium B.C.
This article reviews the notions of equivalence by number and area that
lay at the basis of the geometric altars. Issues related to the altar designs, and
their astronomical signi cance, are summarized. We also consider Ved a _nga
Jyotis
a (VJ), an astronomical text that was in use during the times of the.
altar ritual. VJ has an internal date of c. 1350 B.C., give and take a couple of
centuries, (Sastry 1985), obtained from its assertion that the winter solstice
was at the asterism Sravis
.
t
h a (Delphini). Recent archaeological discoveries.
(see summary in Kak 1994b) support such an early date, and so this book
assumes great importance in the understanding of the earliest astronomy.
2 Altars and the Vedas
Fire altars were used extensively in several parts of Eurasia; for example, the
Greeks had re cults associated with Hephaistos and Hestia, whereas Rome
had the cult of Vesta. However, records giving details of the geometric altar
designs are available only from India. But as mentioned earlier, the drawing
of gures of the same areas was an important part of altar design in India
as well as Greece. Likewise, the altar ritual in Iran was very similar to the
Indian one.
There was also a connection between monumental architecture and as-
tronomy that can be seen from the temples and pyramids from Egypt, the
temples of Mesopotamia, and megalithic monuments such as Stonehenge.
Manuals of temple design from India spell this out most clearly. An Indic
temple was a representation of the universe; a striking example of this is the
temple at Angkor Wat.
Georges Dum ezil (1988) has drawn attention to several striking parallels
between the roles of the brahmin, the Indian re-priest, and
amen, his
Roman counterpart. The references by Plutarch regarding the signi cance of

drawing gures related in area, and the similarity between the o ces of the
re-priest in India and Greece and Rome, suggest that the ritual may have
been similar.
This brings us to the Vedic times of India. Veda means knowledge in
Sanskrit. The early Vedic times were characterized by the composition of
hymns that were collected together in four books. The oldest of these books
is the Rigveda; the one that deals with the performance of ritual is called
Ya jurveda.
What are the Vedas?
The central idea behind the Vedic system is the notion of connections between
the astronomical, the terrestrial, and the physiological. These connections
were described in terms of number or other characteristics. An example is
the 360 bones of the infant (which later fuse into the 206 bones of the adult)
and the 360 days of the year, and so on. Although the Vedic books speak
often about astronomical phenomena, it is only recently that the astronomical
substratum of the Vedas has been examined (Kak 1994b).
My own researches have outlined the astronomy of the Indian re altars
of the Vedic times and shown that this knowledge was also coded in the
organization of the Rigveda, which was taken to be a symbolic altar of hymns
(Kak 1994b). The examination of the Rigveda is of unique signi cance since
this ancient book has been preserved with incredible delity. This delity was
achieved by remembering the text not only as a sequence of syllables (and
words) but also through several di erent permutations of these syllables.
A.A. Macdonell, a major 19th century scholar of the Vedas, came to
the following conclusion after studying the Rigvedic text and its indexing
tradition:
[It is] one of the most remarkable facts in the history of literature
that a people ... have preserved its sacred book without adding
or subtracting a single word for 2300 years, and that too chie
y
by means of oral tradition. (Macdonell 1886, page xviii)

That the number of syllables and the verses of the Rigveda are according
to an astronomical plan is claimed in other books of nearly the same antiquity
such as the
Satapatha Br ahman. a (Kak 1992, 1994b). Rigveda may be con-
sidered an ancient word monument. It appears that the tradition, insisting
that not a single syllable of the Rigveda be altered, arose from an attempt
to be true to observed astronomical facts. Meanwhile, recent archaeological
discoveries have also pushed back the dating of the Rigveda. Its new esti-
mates of antiquity follow from the recent discoveries that date the drying up
of the river Sarasvati, the pre-eminent river of the Rigvedic era, to around
1900 B.C. In other words, the astronomical characteristics of the Rigveda
are to be dated to at least as early as 1900 B.C. The Ved a _nga Jyotis
a has.
been, on linguistic grounds, dated to about ve or six hundred years after
the Rigveda; its internal date is thus in accord with the new chronology of
the Rigveda.
3 Ritual and equivalence
Vedic ritual was generally performed at an altar. The altar design was based
on astronomical numbers related to the reconciliation of the lunar and solar
years. Vedic rites were meant to mark the passage of time. A considerable
part of the ritual deals with altar construction. The re altars symbolized
the universe and there were three types of altars representing the earth, the
space and the sky. The altar for the earth was drawn as circular whereas
the sky (or heaven) altar was drawn as square. The geometric problems of
circulature of a square and that of squaring a circle are a result of equating
the earth and the sky altars. As we know these problems are among the
earliest considered in ancient geometry.
Equivalence by number
The altar ground where special ritual was conducted was called the mahavedi.
This was an isosceles trapezoid having bases 24 and 30 and width 36. The
sum of these numbers is 90, which was chosen since it represents one-fourth
of the year. If the sum represents an example of equivalence by number, it
is not clear why the shape of a trapezoid, with its speci c dimensions, was
chosen. But this shape generates many Pythagorean triples. On the ma-

havedi six small altars, representing space, and a new High Altar, uttaravedi,
representing the sky were constructed.
The re altars were surrounded by 360 enclosing stones, of these 21 were
around the earth altar, 78 around the space altar and 261 around the sky
altar. In other words, the earth, the space, and the sky are symbolically
assigned the numbers 21, 78, and 261. Considering the earth/cosmos di-
chotomy, the two numbers are 21 and 339 since cosmos includes the space
and the sky.
The main altar was built in ve layers. The basic square shape was mod-
i ed to several forms, such as falcon and turtle (Figure 1). These altars were
built in ve layers, of a thousand bricks of speci ed shapes. The construc-
tion of these altars required the solution to several geometric and algebraic
problems (Sen and Bag 1983).
Equivalence by area
The main altar was an area of 7
1
2
units. This area was taken to be equivalent
to the nominal year of 360 days. Now, each subsequent year, the shape was
to be reproduced with the area increased by one unit.
The ancient Indians spoke of two kinds of day counts: the solar day,
and tithi, whose mean value is the lunar year divided into 360 parts. They
also considered three di erent years: (1) naks
atra, or a year of 324 days.
(sometimes 324 tithis) obtained by considering 12 months of 27 days each,
where this 27 is the ideal number of days in a lunar month; (2) lunar, which
is a fraction more than 354 days (360 tithis); and (3) solar, which is in excess
of 365 days (between 371 and 372 tithis). A well-known altar ritual says that
altars should be constructed in a sequence of 95, with progressively increasing
areas. The increase in the area, by one unit yearly, in building progressively
larger re altars is 48 tithis which is about equal to the intercalation required
to make the naks
atra year in tithis equal to the solar year in tithis. But there.
is a residual excess which in 95 years adds up to 89 tithis; it appears that after
this period such a correction was made. The 95 year cycle corresponds to the

tropical year being equal to 365.24675 days. The cycles needed to harmonize
various motions led to the concept of increasing periods and world ages.
4 The Rigvedic altar
The number of syllables in the Rigveda con rms the textual references that
the book was to represent a symbolic altar. According to various early texts,
the number of syllables in the Rigveda is 432,000, which is the number of
muhurtas (1 day = 30 muhurtas) in forty years. In reality the syllable count
is somewhat less because certain syllables are supposed to be left unspoken.
The verse count of the Rigveda can be viewed as the number of sky days
in forty years or 261 40 = 10; 440, and the verse count of all the Vedas is
261 78 = 20; 358. The detailed structure of the Rigveda also admits of other
astronomical interpretations (Kak 1994b); these include the fact that the sun
is about 108 sun-diameters and the moon is about 108 moon-diameters away
from the earth; this can be easily established by checking that the angular
size of a pole that is 108 lengths away from the observer equals that of the
sun or the moon. There also exists compelling evidence that the periods of
the planets had been obtained.
The number 108, the number of sun-steps away from the earth, assumed
great symbolic signi cance. The 108 beads in the rosary are these 108 steps
that represent the path from earth to heaven. In the temple of Angkor Wat
the complex is surrounded by a moat which is bridged by roads leading from
ve gates. \Each of these roads is bordered by a row of huge stone gures,
108 per avenue" (Santillana and Dechend 1969). \When measured in Khmer
hat, the 617-foot length of the bridge corresponds to the 432,000 years of an
age of decadence, and the 2,469 feet between the rst steps of the bridge and
the threshold of the temple's center represent the 1,728,000 years of a golden
age" (White 1982). So this temple complex merely represents an ancient
model.

5 The motions of the sun and the moon
The Ved a _nga Jyotis.a (VJ) (Sastry 1985) describes the mean motions of the
sun and the moon. This manual is available in two recensions: the earlier
Rigvedic VJ (RVJ) and the later Yajurvedic VJ (YVJ). RVJ has 36 verses
and YVJ has 43 verses.
The measures of time used in VJ are as follows:
1 lunar year = 360 tithis
1 solar year = 366 solar days
1 day  =  30 muh urtas
1 muh urta = 2 n ad.
ik as
1 n ad.
ik a = 10
1
20
kal as
1 day = 124 a _m sas (parts)
1 day = 603 kal as
Furthermore, ve years were taken to equal a yuga. A ordinary yuga
consisted of 1,830 days. An intercalary month was added at half the yuga
and another at the end of the yuga.
What are the reasons for the use of a time division of the day into 603
kal as? This is explained by the assertion VJ 29 that the moon travels through
1,809 naks
atras in a yuga. Thus the moon travels through one naks.
atra in.
1
7
603
sidereal days because
1; 809 1
7
603
= 1; 830:
Or the moon travels through one naks
atra in 610 kal as. Also note that.
603 has 67, the number of sidereal months in a yuga, as a factor The further
division of a kal a into 124 k as
.
t
h as was in symmetry with the division of a

yuga into 62 synodic months or 124 fortnights (of 15 tithis), or parvans. A
parvan is the angular distance travelled by the sun from a full moon to a new
moon or vice versa.
The ecliptic was divided into twenty seven equal parts, each represented
by a naks
atra or constellation. The VJ system is a coordinate system for the.
sun and the moon in terms of the 27 naks
atras. Several rules are given so.
that a speci c tithi and naks
.atra can be readily computed.
The number of risings of the asterism Sravis
.
t
h a in the  yuga  is the.
number of days plus ve (1830+5 = 1835). The number of risings
of the moon is the days minus 62 (1830-62 = 1768). The total
of each of the moon's 27 asterisms coming around 67 times in
the yuga equals the number of days minus 21 (1830-21 = 1809).
(YVJ 29)
The moon is conjoined with each asterism 67 times during a yuga.
The sun stays in each asterism 13
5
9
days. (RVJ 18, YVJ 39)
The explanations are straightforward. The sidereal risings equals the
1,830 days together with the ve solar cycles. The lunar cycles equal the
62 synodic months plus the ve solar cycles. The moon's risings equal the
risings of
Sravis
.
t
.h a minus the moon's cycles.
This indicates that the moon was taken to rise at a mean rate of
1; 830
1; 768
= 24 hours and 50:4864 minutes:
6 Computation of tithis, naks
atras, kal as.
Although we have spoken of a mean tithi related through the lunar year
equalling 360 tithis, the determination of a tithi each day is by a calculation
of a shift of 12

with respect to the sun. In other words, in 30 tithis it will
cover the full circle of 360

But the shift of 12 .

is in an irregular manner
and the duration of the tithi can vary from day to day. As a practical

method a mean tithi can be de ned by a formula. In terms of kal as a tithi is
approximately 593 kal as. VJ takes it to be 122 parts of the day divided into
124 parts (RVJ 22, YVJ 37, 40).
Since the calendar was calibrated by naks
atras, tithis were gured by a.
rule and not in a precise mean manner.
Each yuga was taken to begin with the asterism Sravis
.
t
h a and the synodic.
month of M agha, the solar month Tapas and the bright fortnight (parvan),
and the northward course of the sun and the moon (RVJ 5-6; YVJ 6-7). The
intercalary months were used in a yuga. But since the civil year was 366
days, or 372 tithis, it was necessary to do further corrections. As shown in
the earlier section, a further correction was performed at 95 year, perhaps at
multiples of 19 years.
The day of the lunar month corresponds to the tithi at sunrise. A tithi
can be lost whenever it begins and ends between one sunrise and the next.
Thus using such a mean system, the days of the month can vary in length.
Rule on end of parvan
The determination of the exact ending of the synodic fortnight (parvan) is
important from the point of view of the performance of ritual. Let p be the
parvans that have elapsed from the beginning of the yuga. Since each parvan
has 1,830 parts, the number of parts, b, remaining in the day at the end of p
parvans is:
b = 1830 p mod 124:
Now consider
p mod 4 = ;
and
1830 mod 31 = 1:
By multiplying the two modular equations, it can be easily shown that
b = (1829 + p)mod 124:


By substituting the values = 1; 2; 3 we get the YVJ 12 rule:
When = 1, b = p + 93 mod 124;
when = 2, b = p + 62 mod 124;
when = 3, b = p + 31 mod 124.
Rule on naks
atra parts.
The naks
:atra part of the sun at the end of the pth parvan, s, is clearly.
s = 135 p mod 124:
This is because 124 parvans equal the 135 naks
atra segments for the sun.
at the end of the yuga of 5 years. Let p = 12 q + r: Then we can write:
135 p mod 124 = 11 (12q + r) =  8q + 11r mod 124:
This is the rule described to compute the naks
,atras of the sun (RVJ 10.
YVJ 15).
If the moon is full, it will be in opposition to the sun and, therefore, 13
1
2
segments, or 13 naks
atras and 62 parts away. So the rule further states that.
for a full moon its naks
atra parts are computed by adding 62 to the parts.
obtained for the sun. This can be seen directly by noting that the naks
atra.
parts of the moon, m, will be according to:
m = 1809 p mod 124:
This leads to the equation:
m = 8q + 73r mod 124:
This is in excess from s by 62r mod 124, which is 62 when p is odd.
Moon naks
atra in kal as.
Since 124 parvans correspond to 1,809 or 67 27 naks
-atras, 17 parvans cor.
respond to 248 +
1
124
naks
atras. Now the moon passes through each naks.
atra.
in 610 kal as, therefore the 248 days correspond to
248 610
603
days; this equals
250 days and 530 kal as. If we assume that we are just one part short of the
16th parvan, we have its modular relationship with 530 kal as. For 8 naks
atra.
parts short, this corresponds to 530 8 mod 603 = 19 kal as. Each part is
-73 kal as. This rule is in RVJ 11 and YVJ 19.

Other rules and accuracy
There are other rules of a similar nature which are based on the use of
congruences. These include rules on hour angle of naks
atras, time of the day.
at the end of a tithi, time at the beginning of a naks
atra, correction for the.
sidereal day, and so on. But it is clear that the use of mean motions can lead
to discrepancies that need to be corrected at the end of the yuga.
The framework of VJ has approximations built into it such as considera-
tion of the civil year to be 366 days and the consideration of a tithi as being
equal to
122
124
of a day. The error between the modern value of tithi and its
VJ value is:
354:367
360
¡
122
124
which is as small as 5 10¡4
This leads to an error of less than a day in a .
yuga of ve years.
The constructions of the geometric altars as well as the Vedic books that
come centuries before VJ (Kak 1994b) con rm that the Vedic Indians knew
that the year was more than 365 days and less than 366 days. The VJ
system could thus have only served as a framework. It appears that there
were other rules of missing days that brought the calendar into consonance
with the reality of the naks
atras at the end of the ve year yuga and at the.
end of the 95 year cycle of altar construction.
Mean motion astronomy can lead to signi cant discrepancy between true
and computed values. The system of intercalary months introduced further
irregularity into the system. This means that the conjunction between the
sun and the moon that was assumed at the beginning of each yuga became
more and more out of joint until such time that the major extra-yuga cor-
rections were made. This is perhaps the reason why the Indian books do not
describe the location of the junction stars of the naks
.atras very accuratel

7 From altar astronomy to classical astron-
omy
Classical Indian astronomy arose after the close interaction between the In-
dians and the Greeks subsequent to the invasion of the borders of India by
Alexander the Great (323 B.C.). The existence of an independent tradition
of observation of planets and a theory thereof as shown by geometric altars,
the Rigvedic code, and the VJ helps explain the puzzle why classical Indian
astronomy uses many constants that are di erent from that of the Greeks.
This con rms the thesis that although classical Indian astronomy developed
in knowledge of certain Greek works, the reason why it retained its charac-
teristic form was because it was based on an independent, old tradition.
The astronomy of the third and the second millennium B.C. can provide
the context in which the developments of Babylonian, Chinese, Greek, and
the later Indian astronomy can be examined. It appears that certain features
of the earliest Babylonian astronomy can be derived from an altar astronomy
that may have been widely known in the ancient world, but whose records
are now available only in the Indian texts.
It also raises the question of an analysis of the altars and monuments
from Babylon, Greece, and Rome to examine their designs. Likewise, the
references in the Greek literature to geometric problems related to areas
need to be investigated further for their astronomical signi cance.
Later religious architecture, both in the east and the west, became more
abstract but its astronomical inspiration was never hidden. In Europe cathe-
drals were a representation of the vault of heavens. In India the temple
architecture, as spelt out in the manuals of the rst centuries A.D. (see, for
example, Kramrisch 1946), symbolizes the sky where in addition to equiva-
lence by number or area, equivalence by category was considered. The temple
platform was divided into 64 or 81 squares (Figure 2). In the case of the 64-
squared platform, the outer 28 squares represented the 28 lunar mansions of
the Indic astronomy. For the 81-squared platform, the outer 32 squares were

taken to represent the lunar mansions and the four planets who rule over the
equinoxial and solstitial points. Stella Kramrisch, the renowned scholar of
the Indian temple architecture, has also argued that another measure in the
temple was that of 25,920, the number of years in the period of the preces-
sion of the equinoxes. Whether the precessional gure was received by the
Indians from the Greeks or obtained independently is not clear.
8 On observation in the ancient world
As mentioned in Section 2, the ancients were aware of the parallels in the
astronomical, terrestrial and physiological phenomena. The Vedic books are
based on the ideology of such connections.
One can see a plausible basis behind the equivalences. Research has shown
that all life comes with its inner clocks. Living organisms have rhythms that
are matched to the periods of the sun or the moon. For example, the potato
has a variation in its metabolic processes that is matched to the sidereal day,
the 23-hour 56-minute period of rotation of the earth relative to the xed
stars. The cicadas come in many species including ones that appear yearly
in midsummer. The best-known amongst the others are those that have 13-
year and 17-year periods. There are quite precise biological clocks of 24-hour
(according to the day), 24 hour 50 minutes (according to the lunar day since
the moon rises roughly 50 minutes later every day) or its half representing
the tides, 29.5 days (the period from one new moon to the next), and the
year. Monthly rhythms, averaging 29.5 days, are re
ected in the reproductive
cycles of many marine plants and those of animals. It has been claimed that
there are others that correspond to the periods of the planets. There are
other biological periodicities of longer durations.
The use of the mean motions requires continual corrections. The con-
structions of the geometric altars indicates that such corrections were made
on a regular basis. The corrected mean motion astronomy of the geometric
altars and the VJ appears to have served a useful calendric purpose.


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