An Introduction to the Evidence of the Panchasiddhantika 

MICHAEL G. READE 

Michael Reade, D.S.C., a confectionery technologist, is also a specialist
in marine navigation. His earlier contributions to the Review have dealt
with astronomical records from Egypt.

This sixth-century Hindu astronomical manual gives anomalous synodic
periods for the planets Mercury, Venus, Jupiter and Saturn, as seen from
Earth. A preliminary study of the work reveals possible evidence for a
360-day year sometime in the past.

The Panchasiddhantika is a Hindu manual of astronomy compiled by Varaha
Mihira in about 550 AD. It is generally believed to employ an "epoch"
(base date for astronomical calculations) of 505 AD. A translation from
the original Sanskrit by G. Thibaut, Ph.D., and Mahamahopadhyaya Sudhakara
Dvivedi was published at Benares in 1889 and was based on "two
manuscripts, belonging to the Bombay government" [1]. The original text of
the better of these two manuscripts (in Sanskrit) is reproduced in
Thibaut's book, together with a second and "emended"  text, also in
Sanskrit. Thibaut comments:

"A comparison of the traditional text with the emended one ... will show
that the former had, in many cases, to be treated with great liberty ... a
few stanzas quoted by Bhattotpala and manifestly belonging to the
Panchasiddhantika, although not to be met with in our manuscripts, we have
inserted in the emended text ... In a few cases, notably in the last
chapter, where we were unable to emend the readings of the manuscripts in
any satisfactory way, Pandit Sudhakara has substituted for the traditional
text, rules founded on the general principles of Hindu astronomy; a
proceeding which will hardly be objected to, as the cases in question are
pointed out in the commentary, and as side by side with the substituted
text the traditional text is exhibited in full."

Another translation, essentially from the same original manuscripts and
accompanied by a detailed commentary by O. Neugebauer, was published by D.  
Pingree in 1970-71 [2]. Commenting on the work of Thibaut and Dvivedi,
Neugebauer and Pingree say:

"... the inherent difficulties of a technical text without a commentary
and the corruption of the manuscripts, while frequently overcome, at many
points obstructed their understanding of the work. It would serve no
purpose to discuss here in detail our disagreements with their
interpretations... the present edition of the Panchasiddhantika does not
solve all the remaining problems connected with this work. We suggest that
much will never be understood unless better manuscript material becomes
available." (Part I, pp. 18-19)

Varaha Mihira claims in his introduction to have reproduced the
astronomical rules taught by five earlier "astronomical manuals":* the
Paitamaha Siddhanta;  the Vasishtha Siddhanta; the Romaka Siddhanta; the
Paulisa Siddhanta and the Saura Siddhanta. (Of these, only the last-named
appears to have survived to the present day, where it is more commonly
referred to as the Surya Siddhanta; and it clearly differs at least
slightly from the original of the same or similar name which was known to
Varaha Mihira.)

[* Hence its name, Pancha-Siddhantika, "[composed] of five Siddhantas" -
Eds.]

The great bulk of the Panchasiddhantika deals with celestial arrangements
which are essentially similar to those known at the present day. Thibaut
derives various year lengths from the data cited by Varaha Mihira, for
instance, and none of them are at all far removed from 365 days 6 hours.
Thibaut also claims that quite a high proportion of the material in the
more "modern" of these Siddhantas consists essentially of the teachings of
Hipparchus (and/or Ptolemy)  grafted on to appreciably older Hindu
original matter. Neugebauer and Pingree trace not dissimilar Babylonian
associations. The only references to the "strange synodic periods" [3]
which have so far been identified, and which are of such interest to us,
occur in the last 16 stanzas of the 18th (and final) chapter of the
Panchasiddhantika (termed the 17th chapter in Neugebauer and Pingree's
translation). It is far from clear from where they originate, or even how
they fit into the pattern of the Panchasiddhantika as a whole, and it
needs to be emphasised that they occupy only two of the 103 pages which
comprise the whole translation (and from which Thibaut believes some
material to have been lost).  In addition to the original texts and the
translation, Thibaut's volume includes 55 pages of commentary by himself
(in English) and 110 pages of commentary by his Hindu co-worker (in
Sanskrit, which the present writer has unfortunately not so far been able
to read). On the subject of the "strange synodic periods", Thibaut
comments as follows:

"These amounts agree neither with those assigned to the synodical motions
in modern Hindu astronomy generally, nor, therefore, with the true periods
from which the periods implied in the teaching of the Siddhantas differ to
a very inconsiderable extent only. To meet in Hindu astronomy with a set
of numerical quantities widely differing from those generally accepted is
indeed so startling, that one at first feels strongly inclined to doubt
the soundness of the text, especially if one remembers the generally
unsatisfactory state of the two available manuscripts of the
Panchasiddhantika; but it so happens that just in the concluding portion
the text appears to be fairly correct, does at any rate not immediately
call for incisive emendations. Moreover, each figure is given twice over,
and that in two different forms, the text stating at first the length of
the synodical revolution in a fractional form (as f.i. 2752/7 = 393 1/7 in
Jupiter's case),

[note:

2752/7 = 393.14

synodic period for an Earth orbit of 360 days (4332 for Jupiter):  
4332*360/(4332-360) = 392.6

More accurate Earth orbit, if Jupiter was stable:
4332*360.43/(4332-360.43) = 393.1399 ]

and afterwards giving the details of motion for a certain number of
divisions of the synodical period ..."

There is also comment on this part of the Panchasiddhantika presented on
pp. 126-128 of Part II of Neugebauer and Pingree's work. This comment is
apparently intended to reconcile the "strange" figures with orthodox
teaching on the subject. It takes a great deal of unravelling, however, as
it is liberally spiced with quasi-scientific jargon, plagued with apparent
misprints in mathematical expressions, ornamented with tables and figures
which prove not to be entirely relevant to the arguments, and well
interlarded with red herrings. Nevertheless, when it is boiled down, it
actually becomes a rather neat demonstration - different from Thibaut's
but almost certainly preferable from most points of view - that Varaha
Mihira's data either (a) shows that the world was spinning at almost
exactly 360 revolutions per year or (b) was so shot through with mistaken
assumptions as to be worthless, except as an example of the relative
crudity of the work of early astronomers (this despite the fact that
Varaha Mihira's five sets of independent figures, each reproduced in two
different forms, all yield essentially the same result, an outcome which
would hardly have been expected had he been hopelessly misinformed as to
the true significance of the terms he employed, though Neugebauer and
Pingree would probably challenge the validity of this argument).

Neugebauer and Pingree proceed by way of a quantity which they term "mean
synodic arc" (= the mean travel of the Sun in longitude between one
conjunction with a planet and the next, as described on p. 110 of Part II
of their work, though they use a slightly modified form of it in their
actual calculations on pp. 126-128), claiming that this is what Varaha
Mihira records, rather than "mean synodic period" (as claimed by Thibaut).
Actually, the two quantities are numerically identical when the spin rate
of the Earth is 359.99 revolutions per tropical year (assuming normal
precession) but they diverge at other spin rates ("arc" being expressed in
degrees, "period" in days). This relationship between the two types of
reckoning is obscured in Neugebauer and Pingree's presentation, however,
largely because they divide most actual figures by 60 whenever opportunity
offers and also express all results in not entirely self-consistent
sexagesimal notation. Deviations in the value attributed to "mean synodic
arc"  cannot be interpreted in terms of spin rate change; insofar as there
are any discrepancies between Varaha Mihira's values and modern ones,
therefore, they have to be written off as "mistakes" (except insofar as
they could also be interpreted as indicating a change in the orbit of the
planet concerned).  Thibaut's "synodic period" interpretation, on the
other hand, permits such discrepancies to be evaluated in terms of
variation in the spin rate of the Earth (though only at the cost of
assuming no changes in planetary orbits).  Comparing Varaha Mihira's
figures with modern ones (planetary orbits being assumed the same in
ancient days as they are today) produces the following spin rates for the
Earth, as indicated by his data for the individual planets:*

Saturn 360.01 & 360.3 Jupiter 359.99 & 359.9 Mars 360.00 & 359.7 Venus
359.98 & 360.3 Mercury 359.98 & 359.3

[* The two columns of figures derive from the two apparently independent
sets of data cited for each planet. Some figures are also cited slightly
differently in the individual translations: the figures used to construct
this table derive from Neugebauer and Pingree's version. Thibaut's figures
change only the second value for Saturn, which becomes 362.3 revolutions
per year in place of 360.3.  Though Neugebauer and Pingree comment that it
is evident from the numerical values that Varaha Mihira is referring to
mean synodic arcs, they do not actually list the modern arcs (which have
been computed, for comparison purposes, by the present author). The Sun
has a mean travel in longitude of 359.99/N degrees per day, assuming
normal precession, N being the number of days in the (tropical)  year, so
that it traverses an arc of longitude L° in L x N/359.99 days. Note,
however, that although the mean travel of the Sun is 359.99/N degrees of
longitude per day, the actual rate at any instant varies slightly
according to the time of year, due to the non-circularity of the Earth's
orbit about the Sun.  Actual synodic periods also vary similarly. The
precision of the figures quoted in the various tables is therefore not
necessarily quite as good as it may appear; there is also a possibility
that the rate of precession could have been different in ancient days.
Varaha Mihira's figures actually give every indication of being the
average of several cycles - at least 29 in the case of Mercury - and it
would seem that they must have been derived from an unbroken period of
observation of at least 9 years, so that they should at least approximate
to mean values, whether of arc or of period.]

There is, of course, potential substance in Neugebauer and Pingree's basic
claim that if Varaha Mihira's figures (as in stanza 67, below) were
actually only angles (or "longitudes"), and not periods of time, then
there would be no reason to connect them in any way with the spin rate of
the Earth. Thibaut is quite clear that these figures must refer to periods
of time, however, and there seems very little room for assuming anything
else, especially as Varaha Mihira directly refers to the ahargana in
connection with them, and even Neugebauer and Pingree agree on the
definition of the ahargana as "lapsed savana days from a given epoch".
(Thibaut states this definition as "the sum of civil days which have
elapsed from an initial epoch up to a given date".)

The actual form in which the information appears in the Panchasiddhantika,
using the data for Mars as an example (Thibaut's translation, but there is
no difference of substance between the two translations at this point), is
as follows:

Chap. XVIII

67 Lessen [the ahargana] by 6329, multiply by 4 and divide by 3075. Divide
[the remainder] again by 4; the result are the days [which have elapsed]
since Mars was without degrees [i.e. had the same longitude as the Sun].
68 [Mars] becomes visible when less [in longitude than the Sun] by 15°,
within 36 days; then [passes] in 188 days [through] 60°; in 108 days
[through] 60°;  within 72 days [through] 90°. 69 In 68 days [through] 50°;
in 240 days [through] 70°. Then it sets; passes thereupon in 56 days [
through ] 15° and becomes niramsa.

The instructions of stanza 67 appear to be quite clear and unexceptionable
if Mars was observed to go through four complete (synodical) cycles in
3075 days.  The term "the remainder" may be confusing at first sight, but
if one forgets about modern calculating machines and goes back to longhand
division, it becomes obvious that division by 3075 will result in a
remainder of something between 0 and 3074; to turn this remainder into a
fraction of the synodic period - expressed in either days or degrees
(assumed interchangeable at this era) - one need only divide it by 4 once
more. If niramsa means "conjunction", as Thibaut claims (the term is not
used in Neugebauer and Pingree's work), then the sum of days between two
consecutive conjunctions (= one synodical period) indicated by stanzas 68
and 69 is 768, to be compared with the 768.75 days indicated by stanza 67
(and the 779.94 days of the modern equivalent). Moreover, it would seem
indisputable that stanzas 68 and 69 refer to periods of days, whether or
not stanza 67 does. Neugebauer and Pingree make their claim that the
figures of stanza 67 refer to longitudes more plausible by assuming that
these figures can legitimately be divided by 60 (to make degrees rather
than minutes**).

[**Neugebauer and Pingree also claim that the "days" of Varaha Mihira's
text should be understood as "degrees of solar motion". If the Spin rate
of the Earth was around 360 revolutions per year, however, the two are
essentially the same.  If it was not, one has no option but to accept that
Varaha Mihira simply did not understand what he was writing (there being
innumerable references to both "days"  and "degrees" in his text). This is
perhaps still a tenable conclusion, as it is admittedly strange that he
does not appear to recognise the contradiction between this part of his
work and the remainder; earlier in the same chapter, even, he cites
essentially "modern" synodic periods for the planets. He also claims that
his principal object was to list the teaching of the various Siddhantas,
however, whether "right" or "wrong". He does also have some comments on
the reliability of the individual Siddhantas, but it is unfortunately not
at all clear which Siddhanta - if any - actually yielded the present
figures.]

It appears probable that further study of both Thibaut's and Neugebauer
and Pingree's translations and commentaries will eventually yield more
evidence as to the state of the solar system at this (uncertain) ancient
era. Meanwhile, however, it can be observed that the second set of
"strange" figures (those of stanzas 68 and 69 above) can be re-arranged to
suggest that Mars was retrograde over a period of 72 days, back-tracking
18° of longitude during this period. The corresponding average figures for
present-day Mars are 72.7 days and 15.9° of longitude. Due to the
eccentricities of the orbits of the Earth and Mars, however, considerable
variation in the retrograde motion of Mars can occur from time to time,
the approximate range being from a minimum of 53.3 days in association
with 30.4° to a maximum of 90.2 days in association with 2.9°. It would
appear, therefore, that ancient Mars was behaving at least very similarly
to modern Mars.

The corresponding figures for the other external planets are as follows:

Saturn: Modern average: 137.6 days, 6.8° (range: from 135.2 days, 7.7° to
139.8 days, 6.0°); Thibaut's figures: 113 days, 7°. Jupiter: Modern
average: 120.6 days, 9.9° (range: from 117.4 days, 11.4° to 123.7 days,
8.6°); Thibaut's figures: 109 days, 11°.

The generally shorter durations of the retrograde motions derived from
Thibaut's figures (and Neugebauer and Pingree's figures yield the same
results) can rather tentatively suggest that the whole solar system may
have been slightly more compressed than it is at the present day, the
Earth and all the planets being rather closer to the Sun than they are at
present. Note also that Thibaut's figures for the durations in days could
be increased by about 1½% to give a modern equivalent if one were prepared
to allow that the absolute duration of a day was slightly longer at the
ancient era than it is today (in the ratio 365¼:360)  but that a further
such correction, and in the opposite sense, could be called for if the
solar system were actually rather more compressed than it is at present,
as has been suggested above.

It is more difficult to be precise in the case of the inner planets, Venus
and Mercury. Varaha Mihira's figures for Venus list only a half-cycle,
apparently from inferior conjunction to superior conjunction; he lists a
whole cycle for Mercury but it seems that one short sector (according to
Thibaut's data) may have been omitted (the travel between superior
conjunction and first visibility thereafter). In the case of the inner
planets, the retrograde travel is largely invisible (occurring at and
around the time of inferior conjunction) whereas in the case of the outer
planets it is wholly visible (occurring at and around the time of
opposition), a circumstance which would have made accurate recording of
the outer planets appreciably more easy than that of the inner planets.
Subject to some reserve as to what Thibaut's figures actually indicate,
therefore, the resultant comparisons of retrograde motions are as follows:

Venus: Modern average: 42.2 days, 16.2° (range: from 38.2 days, 20.8° to
45.1 days, 12.3°); Thibaut's figures: 40 days, 20°. Mercury: Modern
average: 22.9 days, 13.8° (range: from 18.5 days, 24.5° to 27.7 days,
2.4°); Thibaut's figures: 20 days, 4°.

The modern behaviour of Venus and Mercury (averaged) is presented in graph
form in Figs 1 and 2, together with the descriptions of their motions
given in the two translations. It does not take a great deal of inspection
of the detail to establish that there is almost certainly a difference
between Varaha Mihira's tabulations for Venus and Mercury. In the case of
Venus, it is evident that the observers recorded first visibility and
first stationary point, disregarding greatest elongation; in the case of
Mercury, however, with first visibility and first stationary point falling
close to one another, it seems clear that the first stationary point was
missed and that greatest elongation was recorded instead. This would mean
that the total retrogradation derived above for ancient Mercury - 4° - is
marginally less than the true value. (The difference would probably be too
small to be significant, having regard to the probable accuracy of the
observations; note also that the small total retrogradation observed would
have made separation of first visibility and first stationary point even
more difficult to detect than is suggested by Fig. 2.) "First visibility"
is in any case a rather uncertain measure at any time, being dependent on
many environmental circumstances (weather for the time being, latitude of
observer, season of the year etc.*)

[* The visibility/invisibility criteria used for the construction of Figs
1 and 2 were as follows: Venus at inferior conjunction - visible when
elongation exceeds 5° - at superior conjunction - visible when elongation
exceeds 9° Mercury at inferior conjunction - visible when elongation
exceeds 14° - at superior conjunction - visible when elongation exceeds
12°

These "average" elongations derive from various sources; but no very
satisfying basis of comparison has so far been discovered in the
literature and no special accuracy or authority is claimed for them. The
variation can be quite appreciable from one cycle to another and a brief
perusal of the correspondence columns of the Journal of the B.A.A. reveals
claims, especially from southern hemisphere observers, of occasions when
both Venus and Mercury were seen appreciably more easily than is usual.
One correspondent - in 1905 - speculated that Venus at one particular
conjunction might just have been seen by an ideally placed observer at
both sunrise and sunset on the same day (though he presumably meant the
same night rather than the same day, Venus being seen as an evening star
for the last time at dusk one evening and as a morning star for the first
time at dawn on a following morning). This was a conjunction at which the
difference in celestial latitude between the Sun and Venus was unusually
large, Venus being closer to the north celestial pole than the Sun. If any
reader knows of a comprehensive study of the visibilities of Venus and
Mercury, the writer would be glad to hear of it; ideally, of course, the
observations should relate to clear sky conditions and a relatively low
latitude of observer, as in these circumstances stars can often be seen
more easily than is usual in Great Britain.  The present writer has
certainly seen both Venus and the whole disk of the Sun simultaneously
(naked eye, in England), though not at a small elongation; the brightness
of both Venus and Mercury does also vary quite appreciably in the course
of each synodical cycle.]



FIG. 1. Progress of the elongation and celestial longitude of Venus,
starting from an inferior conjunction at zero longitude and zero time
(modern orbits of Earth and Venus, average values of elongation and
longitude, subject to slight change from cycle to cycle)

- to be compared with:

"Falling behind 9° within 5 days Venus rises in the east. It then falls
behind 21° within 15 days; after that 15° within 208 days; after that it
advances 5° (?) within 12 days (?) and disappears. Then it advances 10°
within 48 (?) days, and becomes niramsa. After that it moves in the
opposite direction, and rises in the west within the time which it had
taken to go to the niramsa position (?);  and again moving in the opposite
direction sets in the west." (Thibaut;  expressions such as "rises (sets)
in the east (west)" should presumably be understood as "appears
(disappears) in the east (west) as morning (evening) star".)

"In 5 (days) it is diminished by 9 (degrees) and rises in the east; in 15
(days)  diminished by 21 (degrees); in 208 (days) diminished by 15
(degrees); in 3 times 4 (= 12) (days) 5 (degrees) and it sets; in 6 times
8 (= 48) (days) 10 (degrees)  and it comes in conjunction (with the Sun);
then it goes in the reverse order in the west. After the time of
conjunction it rises, it stands still, it sets, and it comes (in
conjunction with) the Sun." (Pingree)  FIG. 2. Progress of the elongation
and celestial longitude of Mercury, starting from an inferior conjunction
at zero longitude and zero time (modern orbits of Earth and Mercury,
average values of elongation and longitude, subject to slight change from
cycle to cycle)

- to be compared with:

"Mercury having fallen behind (the Sun) by 12° - which takes place within
10 days - rises in the east, thereupon he falls behind by 10° in 14 days.
(Advancing thereupon) 9° within 18 days, he sets and again rises (in the
west) having advanced 13° within 30 days. Then he advances 9° within 18
days, and then, falling behind 8° within 16 days, he sets in the west.
After that, falling behind 9° in 8 days, he again becomes niramsa."
(Thibault; remarks as for Fig. 1 apply)

"In 10 (days) it is diminished by 12 (degrees) and rises in the east; in
14 (days)  (it is diminished by) 5°; in 18 (days) 14 (degrees); then it
sets; in 30 (days)  6° and it rises; in 18 (days) 14 (degrees); in 16
(days) it is diminished by 8 (degrees)  and sets in the west; in 8 (days)
Mercury is diminished by 9 (degrees) and comes into conjunction."
(Pingree; the note above re "risings" and "settings" also applies here.)

The detail of the figures for ancient Venus is remarkably consistent with
what a modern astronomer would have predicted, but with one exception;
this is that a point on the orbit is marked, about 60 days before superior
conjunction, which does not correspond with any noteworthy event in the
calendar of modern Venus.  Why this point was recorded must remain an open
question for the time being; if a viable reason can be found, it could be
of major interest. In this connection, it is perhaps also worth noting
that Varaha Mihira's tabulation covers only the morning star phase of
Venus's activities.

The data for Mercury is rather more difficult to assess, Mercury being
subject to a greater degree of variability than Venus, and the two
translations unfortunately differ; the one point they have in common,
almost whatever the "corrections"  one applies, seems to be that Mercury
was more easily seen, especially at superior conjunction, than it is at
the present day. (Pingree's translation actually suggests that Mercury was
seen at superior conjunction when its elongation was as little as 3° -
perhaps plus a little more if one allows that Mercury could have been
running somewhat to the north or south of the Sun at the era concerned -
but the present writer prefers the rather lesser, but still appreciable,
brightness implied by Thibaut's version.)

Again, the actual periods of retrogradation appear to have been rather
shorter than would have been expected, had the solar system then been as
it is now. If any planet was running "wild" at all, it would seem to have
been Mercury. (Its motions, as well as its brightness, seem to have been
slightly "irregular", at least by comparison with the other planets, but
the evidence for irregularities is admittedly more suggestive than
absolute.)

The modern motions cited here were computed by the author (using some
approximations) from the orbital data cited in the current (1980) Yearbook
of the British Astronomical Association; comparisons with historical
retrograde periods could prove preferable, however, as the more extreme
values are unlikely to recur in anything short of geological spans of
time.

The principal outcome of this study seems to be that a onetime 360-day
year is at least a possibility, even quite a strong probability (bearing
in mind that no other satisfying explanation seems to have been offered
for the origin of the manmade 360° circle), but it is still not an
absolutely proven one. There is clearly also a probability that the
earlier catastrophes - say, ones occurring before about 1300 BC - tended
to be more drastic and more far-reaching than the later ones. The later
ones - apparently centring around 850 - 700 BC - appear to have been
severe enough to produce at least one change in the spin rate of the
Earth, but not so severe as to cause major changes in the orbits of the
principal planets (with the possible exception of a small, maybe stepwise,
expansion in the dimensions of the solar system as a whole). To turn these
"probabilities"  into "certainties", however, will call for more research
yet. It also remains an open question, at least for the present writer,
just why the ancients of the 850 - 700 BC era attached so much importance
to the activities of Mars. Dr Velikovsky may well have been right in
claiming that Mars somehow reduced the capacity of Venus to interfere with
life on Earth, but there appears still to be no satisfying confirmation of
any conspicuously abnormal activity of Mars in the ancient astronomical
records which have so far been turned up.

References

1. Panchasiddhantika, translated by G. Thibaut, published by E. J. Lazarus
& Co., Medical Hall Press, Benares, 1889.

2. The Panchasiddhantika of Varaha Mihira, by O. Neugebauer and D.
Pingree, in two sections, published 1970 and 1971 as part of Historiske
Skrifter 6 by the K.  Danske Videnskabernes Selskab, Copenhagen. Note that
the chapter and stanza numbers in this translation differ slightly from
those in Thibaut's.

3. See letter from B. O'Gheoghan in SISR IV:4 (1980), p. 85. See also SISR
IV:2/3 (1980), p. 40 for the use made of these "strange periods" by Dr J.
Fermor and W in C II. viii: "The reforming of the calendar" for their use
by Dr Velikovsky.

Acknowledgements

The writer wishes to acknowledge the very considerable assistance of B.
O'Gheoghan and Dr J. Fermor in locating the published material and also
providing some comment on it. It may also be observed that the Society as
a whole can claim considerable credit for having made such joint efforts
possible.

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