सिद्धान्तदर्पण (महामहोपाध्याय सामन्त चन्द्रशेखर सिंह - नग्न-नेत्र खगोल वेध एवं गणित महाकोश)

Introduction xxxiii
150 years B.C. It is an irregularity which may put
the Moon forward or backward over a degree.
Pandit Sudhakar Dwivedi informs us in his excellent
manual, called Ganaka-Tarangini, that Munjala
(A.D.933) had something like "’evecation’ in his
karana named Laghumanasa. He appears to have
been the oldest Hindu astronomer who detected the
irregularity, through curiously enough, his
successors including Bhaskara left it unnoticed.
The next large irregularity of the Moon, called
"variation," has a period of one month, and a
maximum of 39'31". This inequality does not affect
the time of an eclipse, and the fact sufficiently
explains its absence in Sanskrit Siddhantas.¹ It is
said to have been detected by an Arbian astronomer,
Aboul Wefa about the year 975 A.D., and re-
discovered by Tycho Brahe in the 16th century. The
last large inequality of the Moon's place is called
"annual equation," and has the maximum amount of
11'9". This was also discovered by Tycho Brahe.
It is singular that Chandrasekhara is the only
Indian astronomer who has detected all the three
important irregularities of the Moon. As has been
said before, he did not know English and had not
means of knowing of the existence of the
irregularities from any foreign source; and the
methods of applying the corrections together with
discrepancies between his values and those of
Europe leave no doubt in our mind that he must
be credited with their discovery. He has named the

1. From Dwivedi’s Ganaka-tarangini, it appears that
   Nityananda in 1639 A.D. used a correction called
   Pakshika. But it is not clear from the name alone, if it
   had any connection with "variation".

xxxiv सिद्धान्त दर्पण
inequalities, Tungantara, Pakshika and Digamsa, with
the maximum amounts of 2°40′, 38′12″, and 12′
respectively (S.D.VI.7).
For those who may be inclined to compare his
corrections, his method of applying them is briefly
described here. After the equation of centre
(maximum amount 5° 1′ 10″) has been applied to
the mean Moon, call the result Ist Moon M₁ Then
add to, or subtract from it, according as the anomaly
happens to be within the first six or the second six
signs.
160′ x Sin[A - (0+3)] Sin(M₁-0)
--------------------- x ---------
R R
Ist Moon motion
x ---------------
mean motion
where A stands for the Moon’s apogee, and 0
for the true Sun. The two signs (±) are to be taken
in the case of the light and dark halves of each lunar
month respectively. The result obtained is called the
2nd Moon, M₂.
To apply the Pakshika correction, take
M₂ - 0
------ = a+ b (say). Subtract b from 3 (Signs)
3 3 -
3
and take the less of two quantities, a and
(3 - b ) and say, it is y. Then sin2y is the

3 90
corection required. It should be noted that the
denominator (90) is not constant, but can be
obtained. The correction is to be added to, or
substracted from the 2nd Moon, as the latter lies
within the first or the second quadrature. Call the
Moon thus corrected, the third Moon, M₃-

Introduction xxxv
Now, take 1/10 of the Sun's equation and
mutiply by the first Moon's motion, dividing by its
mean motion. Add the result to the 3rd Moon, when
the Sun's equation is subtractive, and subtract from
it when the Sun's equation is additive. After this,
the other corrections (such as Bhujantara), common
to all the planets, are to be applied before the
Moon's place becomes apparent.¹
On a cursory view, the amount of Tungantara
inequality appears double of that of evection. But
if we take the Moon's greatest equation of centre
into account, the apparent discrepancy vanishes.
For, the amount of the greatest equation of the
Moon is 6° 18′, and the maximum evection 1° 20′
making the total of 7° 38′. Chandrasekhara has 5° 1′
as the greatest equation and 2° 40′ as the greatest
Tungantara, making the total of 7° 41′. The other
inequalities discovered by Chandrasekhara are
about the same as those in use in English
astronomy.
The astronomical constants adopted by our
India astronomers open up a large field for enquiry.
It is not my purpose to discuss their bearings upon
the antiquity of Indian astronomy. But I cannot but
remark that it is often reiterated in season and out
of season than substantiated, that the Indian
astronomers borrowed largely from Greek
astronomy. As far as I am aware, this assertion is
based upon (1) the identity of certain Sanskrit and
Greek astronomical terms, (2) the fact of Garga
praising the Yavanas for their proficiency in

1. It should be noted here that the last-named inequality
   (Digamsa) is also applied to the Moon's node in the
   above-mentioned manner (S.D. VIII, 33).

xxxvi सिद्धान्त दर्पण
astronomy, and (3) the presence in Sanskrit of
certain astronomical treatises words admittedly of
foreign origin. This is not the place to discuss the
subject throughly. But as far as my knowledge goes,
these arguments, when weighed against others, are
not at all convincing.
The presence of Greek terms in Sanskrit is
certainly a strong presumption in favour of the
theory. But when the uses of the terms are taken
into consideration, it is found that with the possible
exception to the word "Kendra'¹ meaning
"anomaly", all the rest properly belong to astrology.
Of course, when once the terms had been
introduced into Sanskrit, our astronomers did not
hesitate to use them freely in astronomy, side by
side with pure Sanskrit terms. The passage where
Garga extolled the Yavanas does not occur in an
astronomical work, but in a Samhita which was
rather a work on astrology than astronomy. Nor
can we logically infer from the passage that the
Yavanas were proficient in astronomy rather than
in astrology, or that the Hindus borrowed the
knowledge from them. The name Yavanacharya
occurs only in connection with Samhita. Next the
presence of Romaka² and Paulisa Siddhantas among
others of purely Indian origin proves nothing
beyond the bare fact of their presence and I do not
see how it can be taken to imply that the Hindus
had not obtained the constants of their Siddhantas

1. See, however Bhaskara's derivation of the word in his
   Siromani.
2. The Romaka-Siddhanta of Varaha-mihita is in no way
   superior to his Saura-Siddhanta, as was remarked by
   himself.

Introduction xxxvii
by independent observation. On the contrary, the
fact of their retaining the foreign name (at least the
name Romaka), shows that the foreign Siddhantas
were distinct from what were Indian, and there is
no proof that the Romaka or any other foreign
Siddhanta was ever in use in this country,
superseding the purely Indian productions. To give
an analogy, a Sanskrit translation of the British
Pharmacopoeia will no more prove the absence of
Charaka than an English translation of the later will
prove the absence of the former. All that we can
logically infer is that there was intimate intercourse
between the Hindus and the Yavanas, a fact
otherwise known from political history. The Hindus
may have been indirectly influenced by the
teachings of the Yavanas. But it must be admitted
that the Yavanas may have been also influenced by
their presence among the Hindus.
We can safely go so far, omitting all sorts of
possibilities and vague conjectures. Having regard
to our ignorance of the state of astronomy and its
gradual development in India anterior to
Aryabhatta, and our equal ignorance of the source
of the knowledge credited to Hiparchus, it is
hazardous to speak of the indebtedness of the one
nation to the other. If the Hindus learnt the science
of astronomy from the Yavanas, how is it that the
constants of sanskrit astronomy are so different
from those of Ptolemy ? How is it that the early
Hindu writers were unacquainted with such useful
as well as remarkable facts as the precesion of the
equinoxes, or the evection inequality of the Moon ?
On the contrary, the constants ought to have been
identical, not only becaus they are believed to have
a common origin, but also because they represent

xxxviii सिद्धान्त दर्पण
facts, as true for India as for Egypt or Greece. When
one learns a science from another, it is natural to
expect to find him copying not only the theory but
also the practice, write or wrong. Dr. Thibaut
repeats the suggestion made by Biot, that the early
Hindus learnt their astronomical theories probably
from Greek astrologers, and, the Doctor adds, also
from almanac-makers, whose knowledge was as
limited as that of there proto-types in the present
day.¹ This is certainly an ingenious hypothesis; but
as has been already remarked having regard to our
ignorance respecting the growth and development
of the science in India as well as in Egypt, we cannot
but regard the hypothesis as not proven. I am
inclined to believe that our knowledge of the source
of Hindu astronomy has not advanced beyond the
point where Colebrooke found it. All that can be
safely asserted is what the illustrious oriental scholar
said, that "the Hindus have certainly received and
welcomed communications from other rations on
topics of astrology and we add, that their
indebtedness to Greece for the knowledge of
scientific astronomy is still an open question.²
But this a digression. We have said that
Chandrasekhara had made a real advance upon
existing Hindu astronomy. But the best test of a
theory lies in facts. For the reason given below, it
is not easy, however, to satisfy oneself whether his

1. See, Introduction to Pancha-Siddhantika by Dr. Thibaut
   and Mm. Suddhakara Dwivedi.
2. It is much to be desired that some competent scholar
   would collect the arguments for and against the theory
   of indebtedness, and let us know how far, and in
   what-direction, Greek influence on Hindu astronomy
   extended.

Introduction xxxix
ephemeris is correct, and how much confidence may
be placed in it. For rough comparison, the places
of the planets computed by him are shown below,
toegther with their places, first according to the
British Nautical almanac, and next according to a
widely circulated Bengali almanac said to have been
computed after the Surya-Sidhanta. The correction
for precession in the places given in the Bengali
almanac has been made by adding 20°54′ to them
in accordance with the almanac. Besides these, I
append ephemeris for two other days from the
Nautical almanac and the Siddhanta-Darpana.
For various reasons, neither Chandrasekhara's,
nor for that matter - any other Hindu almanac can
coincide with the Nautical almanac. Neglecting
minor corrections unknown in our almanacs, there
is the determination of the exact amount of
precession, without which no comparison is
possible with European almanacs. To avoid it, the
relative distance of the planets may be taken. It will
be seen that while the Bengali almanac may be in
error by as much as 4°, the error in the ephemeris
by the Siddhanta-Darpana is limited to
half-a-degree. The reader will, however, remember
that a Hindu astronomer depends upon unaided
vision, and is satisfied if the longitudes are correct
to a half degree. Accepting this as our standard,
the ephemerides by the Siddhanta-Darpana
compare favourably with those of the N.A., while
the greater discrepancies in the Bengali almanac
conclusively prove how urgent has been the
question of the revision of our existing almanacs in
the light of observations now made. (See Table )

xl सिद्धान्त दर्पण
LONGITUDES OF PLANETS OF THE 31ST JANUARY 1893 (KATAK, 6 p.m.)

| Siddhanta-Darpana | Bengali Almanac | Dec. by | Dec. by |
| Nautical Almanac | From Fixed Aries | From Vernal Equinox | From Fixed Aries | From Vernal Equinox | N.A. | S.D. |
| (1) | (2) | (3) | (4) | (5) | (6) | (7) |
| Sun ... 10° 11° 54' | 9° 19° 31' | 10° 11° 56' | 9° 19° 40' | 10° 10° 34' | 17° 14' S | 17° 15' S |
| Moon ... 4 4 24 | 3 12 16 | 4 4 41 | 3 12 1 | 4 2 55 | 24 1 N | 24 15 N |
| Mars ... 0 22 48 | 0 0 7 | 0 22 22 | 11 29 43 | 0 20 37 | 9 14 S | 2 7 S |
| Mercury ... 10 0 33 | 9 8 23 | 10 0 58 | 9 12 16 | 10 3 10 | 21 45 S | 21 15 S |
| Jupiter ... 0 19 27 | 11 26 43 | 0 19 8 | 11 27 6 | 0 18 0 | 6 32 N | 6 50 N |
| Venus ... 9 19 19 | 8 26 48 | 9 19 12 | 8 27 46 | 9 18 40 | 22 7 S | 22 1 S |
| Saturn ... 6 12 43 | 5 20 38 | 6 13 3 | 5 22 14 | 6 13 8 | 2 41 S | 3 0 S |
20th February 1894 (Katak, 6 p.m.) Ist March 1894 (Katak, 6 p.m.)
N.A. S.D. N.A. S.D.
Sun ... 11 2° 8' 11° 2° 9' 11° 11° 10' 11° 11° 11'
Moon ... 0 21 40 0 21 46 4 27 7 4 27 4
Mars ... 1 6 5 1 5 30 1 12 3 1 11 27
Mercury ... 11 5 14 11 4 57 11 22 14 11 22 39
Jupiter ... 0 22 58 0 22 34 0 24 46 0 24 21
Venus ... 10 14 16 10 12 42 10 25 28 10 25 25
Saturn ... 6 12 1 6 12 23 6 11 13 6 11 56

• It may be mentioned here that Ayanamsa on that day according to Chandrasekhara's Siddhanta Darpana was 22° 25', while Lahiri's
  Ayanamsa as accepted by Govt. of India now, was 22° 21' 41" - EDITOR.

Introduction xli
But before any reformation is attempted, an
exact determination of the amount of precession
becomes a question of paramount importance. The
reader is a awre that in the Hindu system, the
longitudes of celestial bodies are measured from a
fixed point - say a star - in the ecliptic, instead of
from the movable vernal equinox as is the practice
in Europe. The question has therefore the same
bearing upon our calculations, as the position of the
so called First Point of Aries upon those of the
Nautical almanac.
Unfortunately, all attempts to solve the*
question have been practically fruitless. The reader
will hence understand the chaos into which our
almanacs have sunk. The gravity of the situation,
and the difficulty of escaping from it, demand a
fuller discussion than our space would permit. But
then this alone can give us an opportunity of
ascertaining Chandrasekhara's success in this
direction.
Promising, then, that we measure longitudes of
planets from a fixed point in the ecliptic, the
question resolves itself into a determination of the
point. In other words, what is that point, or what
is its longitude from the vernal equinoctical point ?
Whatever and wherever that point may be, it is the
starting point of our zodiac, and its longitude is
known as ayanamsa, which literally means amount
of solstices. For, we do not speak of the precession
of the equinoxes as often as we do of the precession
of the solstices. Hence, the ayana-chalanam of
Sanskrit astronomy is equivalent to the precession
of the equinoxes. To avoid ambiguity we shall use
the term ayanamsa rather than the amount of
precession.

xlii सिद्धान्त दर्पण
The exact amount of the ayanamsa may be
apparently determined in different ways. 1. The
Siddhantas furnish a rule for computing it, which
is in principle the same as the method of finding
the longitude of a star at any given date by applying
the amount of precession to its longitude at some
other date. 2. Defining the initial with the help of
other data, such as the recorded longitudes of stars,
its present longitude from the equinoctial point may
be ascertained. 3. Knowing the exact year when the
initial point was fixed, its present longitude
(ayanamsa) may be calculated from the known rate
of precession. But it so happens that the results
obtained by these three methods do not agree.
To begin with the first method, it will be seen
that the different Siddhantas do not agree, either
in the nature of precessional movement, or in its
annual rate. According to some, the equinoxes have
an oscillatory motion, turning to the right and to
the left of the initial point within certain limits, and
extending over a large interval of time; while others
maintain their continuous motion backwards.
Colebrooke compared the views of the libration and
revolution theorists, and gave the rate of precession
accordign to each. They are as follows:-
Liberation Theory | Annual Rate
Surya-Siddhanta | 54″
Soma¹ Siddhanta | 54″
Sakalya Siddhanta | 54″
Laghu-Vasishtha Siddhanta | 54″
Parasara-Siddhanta | 52″.35

1. Ranganatha, in his commentary to the Surya-Siddhanta,
   quotes Soma-Siddhanta in support of the reading.

Introduction xliii
Liberation Theory Annual Rate
Aryashta Satika
(quoted by Munisvara) 46″.25
Manjula (quoted by Bhaskara)¹ 59″.9
Bhasvati 60″
Grahalaghava 60″
The derangement of the equinox from the intial
point is 27° on either side according to the
Surya-Siddhanta, and this limit was probably
accepted by other libration theorists.
Aryashta-Satika, however, gives the limit of 24°
instead of 27' (Colebrooke).
We are not concerned here with the theories or
the limits of libration. Surya-Siddhanta and the
works based upon it, are now almost universally
adopted throughout India for computing an
almanac, though in some places Grahalaghava
occupies the field. Now from the first, we find the
ayanamsa for the first day of Saka 1816² (13th April
1894) was 20° 55′, and from the second, 22°50′ for
the same day. These amounts represent. the
minimum and maximum ayanamsa obtained from
Siddhanta rules.
Both the Surya-Siddhanta and Siddhanta-

1. Bhaskara did not believe in the liberation theory. For, as
   has been shown by Chandrasekhara (VI.100), he has
   directed us always to add the ayanamsa and never to
   subtract it.
2. In common parlanac, we speak of the year being Saka
   1816, though in reality we ought to say the first day of
   Saka 1817. The facts herein discussed were collected
   sometime ago, and as it immaterial which year is taken
   for illustration, they are not altered to suit the current
   year.

xliv सिद्धान्त दर्पण
Siromani, the standard Siddhantas at present in use,
give a rule to test the amount of ayanamsa by
observation. It consists in subtracting from 12 signs
the longitude of the Sun computed after the
Siddhantas for the moment when the Sun crosses
the vernal equinoctial point. Thus, on the 13th April
1894, the Sun’s longitude for Greenwich mean noon
was 23°32′7″, while an almanac gives us 1°21′25″
as the Sun’s longitude for the same instant, making
the ayanamsa 22°10′37″. It will be seen therefore
that not withstanding the higher precessional rate
of the Surya-Siddhanta, the calculated amount
becomes less than the actual by nearly 1°16′, while
the Grahalaghava makes it greater by about 39″.
To come to the second method, viz. to find the
true longitude of the initial point as defined by stars.
All the modern Siddhantas agree in the statement
that the longitudes of all heavenly bodies are to be
measured from the Star Revati situated on the
ecliptic. The star has been identified with Picicum.
Now, this is a star of the fifth magnitude and is
barely visible to the naked eye. The question is,
why was this particular star out of the many situated
on the ecliptic, - some of which are larger, - chosen
for marking the beginning of the zodiac? The answer
has been - and there is no doubt about the accuracy
of the answer - that the vernal equinox happened
near the star when the present system of
astronomical measurement came into vogue.
So far it is plant sailing. For, we can easily find
the longitude of the star from the vernal equinoctial
point and at once get the required ayanamsa. From
the Right Ascension and Declination of the star,
ζ Piscium, on the 13th April, 1894, we get 18°23′49″

Introduction xlv
as its longitude.¹ This is however, less than the
amount obtained above. It is clear therefore that
either the star Revati is not ζ Piscium, or it did not
mark the initial point. No doubt exists about the
identity of the star, neither can we suppose all the
Siddhantas false in their assertion.
The latitudes and longitudes of thirty-four
others stars are given in the Surya-Siddhanta. They
were not measured in the usual way. But from the
express instructions as to the method of
measurement adopted, we understand that the
so-called longitudes, Dhurvas, are the longitude of
the Right Ascensions of the stars supposing them
to be on the ecliptic, and that the latitudes
(Vikshepas) are the distances, north or south, of the
stars from the ecliptic, measured along the
declination circles passing through the stars. Now,
taking only the 27 stars of the zodiac, and
comparing their reduced Dhruvas with their present
longitude, we find ourselves confronted with a
curious result.² The difference between the recorded
and the present longitudes is not the same for every
star. Indeed, it varies from about 18° to 24°.
Omitting Visakha, of which I am not certain,
the mean difference of the longitudes of the stars
of the first half of the zodiac, i.e. from Asvini to
Chitra is 21°, while that of the stars of the second
half amounts to 19° only. So, while the mean
―――――

1. The modern Surya-Siddhanta gives 359°50′ as the
   longitude of the star Revati. This would make the present
   ayanamsa greater by 10′ only.
2. I understand that Prof. Whitney has discussed the point,
   and regret that I have no access to his edition of the
   Surya-Siddhanta at present.

xlvi सिद्धान्त दर्पण
precession for all is about 20°, that for Asvini
(ά Arietes) and Uttara-bhadrapada (άAndromeda), the
two stars on either side of Revati, is about 24' each.
The question becomes more perplexing when
remember that Bhaskara, who certainly made a few
corrections, has given in most cases the same
longitudes as we have in the current Surya-
Siddhanta.
Our identfication of some of the stars may be
doubtful.¹ But taking all the star-longitudes into
consideration, we are forced to admit that somehow
or other there were errors of observation. Possibly,
these errors were partly due to the rude means of
observation, increasing the errors with the increase
of the zenith distance of the stars, and partly to
azimuth errors. One or all of the explanations must
be accepted as true, as we cannot suppose that any
one having the slightest pretension to practical
astronomy, however rough the instrument might
be, could make such gross errors in the simple
measurements required. To take the mean of the
star-longitudes for a solution of our problem is,
therefore, a wrong procedure, in as much as, we
are entirely in the dark as to the weight to be
attached to the determinations. We shall however,
make use of them in another way later on.
We now proceed to consider the third method,
which consists in analysing the dates in which there
was no ayanamsa. For this purpose, we require not
only the dates but also the rates of precession

1. Thus, if Asvini be identified with β Arietes and
   Uttarabhadrapada with y Pegasi, the amount of percession
   deducted from them becomes nearly equal and agrees
   with that obtained from Magha.

Introduction xlvii
assigned by the astronomers. We have already seen
the various rates assigned by them. As the
Surya-Siddhanta's rate of 54 seconds per year is a
close approximation to the rate known at present,
let us take this first for consdieration.
Apparently, the rate of 54 seconds is hihger by
nearly 4 seconds than the actual. I say apparently
for the rate makes the ayanamsa less, instead of
increasing it in the proporiton of about 50 to 54. It
must be therefore really lower than the true rate.
To explain the anomaly we have to consider the
Siddhanta year. The length of the sidereal year -
the year used by us instead of the tropical year in
use in Europe - is 365.25875 mean solar days
according to the Surya-Siddhanta, and is thus
greater than the true length by 0.00238 days. Now,
taking the Sun's daily motion at 59'8", we find that
during the excess the Sun moves to the East through
a distance of 8".44 nearly. But the equinoctial point
moves to the West. It is therefore clear that the
precessional rate assigned by the Surya-Siddhanta
is practically 54"-8°.44, or 45".56 per year. It is for
this reason that the ayanamsa, calculated from the
Siddhanta, apparently higher rate, becomes less
than the amount observed. Hence if we accept the
Siddhanta's length of the year, we should make the
annual precessional rate 50" 24+8" 44 or 58".68. It
is remarkable that the rate of 45".56 per year is
almost equal to the precessional rate in Right
Asscension. Whether the Siddhanta writer meant
the rate to be what we find, I cannot say, though
there are certain reason favourable to our
conjecture. The point may be illustrated by taking
other Siddhantas. Bhaskara's year consists of
365.25843 days, and is therefore longer by 0.00206

xlviii सिद्धान्त दर्पण
days. Hence, accepting Bhaskara's year, we should
have 50° 24+7".31 or 57",55 as the precessional rate.
Bhaskara does not state in his Siromani any rate
deduced by him, but evidently supports Munjala's
rate of 59".9. Moreover, his adoption of the rate of
60" a year in his Karana-kutuhala leaves no doubt
in our mind as to the rate he used.
Similarly, Ganesa, author of Grahalaghava,
makes the year of 365.25856 mean solar days, which
is thus longer by 0.00219 days. He ought to have
therefore made the precessional rate 50".24+7".86
or 58".10 instead of 60" as he has done.
Chandrasekhara accepts the length of the year
of the Surya-Siddhanta, and gives 57".615 as the
precessional rate (S.D.VI.75). This is just less than
the rate we have assumed above by 58".68-57". 62
or 1"06. I was surprised to find this close
approximation, and could not but ask him the data
from which he derived the rate. In reply, he said
that finding the Surya-Siddhanta and the Siromani
make the ayanamsa either shorter or longer than
that observed, he was for some time in a fix about
the rate to be followed. Fortunately, while he was
studying Jata-karma-paddhati with the help of the
commentary called Jatakalankara written by
Suryadeva some 800 years ago, he accidentally
found a passage in which the commentator recorded
the ayanamsa as it was observed by him on a
particular day of a particular year. This find itself
is not less valuable than Chandrasekhara's rate. He
gives however, 22°26' as the ayanamsa for the year
we have taken for discussion.
Munjala appears to be the earliest writer who
has given the date of the year of no ayanmsa, as
well as the rate of precession observed by him. He

Introduction il
wrote his work in Saka 854,¹ and the precessional
rate assigned is 59".9 in a year. We also learn from
Dwivedi's Ganaka-tarangini that according to
Munjala, Saka 434 was without ayanamsa. Now
from 434 to 854 Saka, there were 420 years, during
which at the rate of 59".9 per year, the ayanamsa
had amounted to 6°59'18". However erroneous the
rate given by Munjala may be it will not be wrong
if we take 6°59'18" (say 7') as the ayanamsa in Saka
854. We know, however, that the rate of precession
could not have been greater than 58".68 per year.
Accordingly, the ayanamsa obtained above carries
us back through 429 years from Saka 854, the date
of Mujala's work. In other words, we find that
according to Munjala's data Saka 425 was the year
of no ayanamsa at the latest. From the ayanamsa
obtained from Munjala and that observed in Saka
1816, we note that the rate of precession amounts
to 56".8 in a year.
The next work we take is Bhasvati by
Satananda,² which is still regard as an authority for
the calculation of eclipses. He wrote his work in
Saka 1021, and according to him the rate of
precession is 60" per year. We also know that he
regarded the Saka year 450 as the year without
ayanamsa. Now, calculating the amount of

1. The date 584 in the Ganaka-tarangini seems to be a
   misprint, though it occurs thoughout the short notice of
   Munjala, excepting the passage wher Pandit Dwivedi
   quotes from Munjala, makign the date 854 Saka, adopted
   above.
2. Satananda was born in Orissa in Saka 990 (i.e., 1068 A.D.)
   and wrote the Bhasvati in Puri in Saka 1021 .(i.e., 1099
   A.D.) by adopting centmal system. Hence his name was
   Satananda as it was a pleasure to him to use centinals.

l · सिद्धान्त दर्पण
ayanamsa in Saka 1021 in the above manner, we
find that it was 9°31′ in that year. From this,
calculating backwards at 58″.68 a year, we come to
Saka 437 as the year of no ayanamsa. We also note
that the rate of precession reduced from the
ayanamsa in 1021, amounts to 57″.295 per year.
In his Karana-kutuhala, Bhaskara has roughly
given 11° as ayanamsa about Saka 1105. From this
at the rate of 60″ in a year, the rate assumed by
him, we are taken back to Saka 555, and, at the rate
of 57″.55 which we have found before, to Saka 423.
We also note that adopting his observation, the rate
amounts to 56″.624 per year.
To take another example, we mention Ganesa
who wrote his Grahalaghava in Saka 1442.
According to him the rate of precession is 60″ in a
year, and Saka 444 was without ayanamsa. These
premises lead us to infer that ayanamsa amounted
to 16°38′ in Saka 1442, which at the rate of 58″.1
carries us back to Saka 412, a-date which is the
earliest of all that we have hitherto obtained. But,
considering that the amount of ayanamsa reduced,
compared to the present amount, makes the rate at
low as 53″.4, we are lead to infer that his data were
wrong. It is to be observed, however, that the rate
of 58″.68 if adotped makes Saka 422 as the year
without ayanamsa.¹
We may almost guess the data from which
Ganesa derived his rate. It is likely he accepted 11°
as the amount of ayanamsa in 1105, as recorded by

1. An error in observing ayanamsa is not uncommon. One
   instance of it is furnished by Mahadeva who recorded
   13°45′ as the amount of ayanamsa in Saka 1238 (Dwivedi’s
   G.T.), thus making the rate since then 52″.5 in a year.

Introduction li
Bhaskara. From this he might have obtained the
Saka year 444 as the starting year, and also the rate
of 60 seconds a year, making the amount 16°37′ in
Saka 1442.
It remains to notice the data 421 Saka given by
some minor authors. This is obviously based upon
the Surya-Siddhanta's with its annual rate of 54,
making 20°55′ as the amount of present ayanamsa.
Chandrasekhara has not recorded the data
which furnished him the rate, knowing howerver,
his ayanamsa and rate, and calculating backwards,
we arrive at Saka 415.
So far we have obtained the following figures:
Sakabda Rate per year
Munjala 425 56″.828
Satananda 437 57.295
Bhaskara 423 56.624
Chandrasekhar 415 57.61
Mean 425 57.09
Having regard to the nature of the data ve
cannot expect a closer approximation than that
shown above. We may therefore fairly take the
mean of the dates, as well as the rates. On account
of obvious inaccuracy in the data of Ganesa the date
and rate obtained from him cannot be rightly
included. Neither will his date, if corrected increase
the weight to be attached to the mean to be
calculated from the above figures. We might,
however, include the data 421 Saka given by certain
weriters. Indeed, it will be seen that dates of the
starting years are of two classes; one somewhere
near 450 Saka, and the other, 421 Saka. But the

lii सिद्धान्त दर्पण
dates of the former class may be reduced to those
of the latter class. It will be further observed that
Munjala, the earliest writer of the set, makes the
closest approximation to the mean, Satananda
seems to have committed an error seeing that he
assigns Saka 450 as the starting year. Madhava
Misra, a commentator of his, makes the same
remark and suggests that the Saka year 421 ought-
to be taken incalculations.¹
From what has been described above the Saka
year 421 or 427 appears to mark the beginning of
the fixed zodiac. It is difficult to prefer one to the
other. The Saka year 421 is equivalent to the year
3600 of the Kali yuga, and the fact of its being so,
goes in its favour. Besides, it is the year in which
Aryabhtta composed his work, a work which
enjoyed a wide reputation. Lalla, whose Tantra
Bhaskara condescended to criticised adopted the
year 421 Saka, and the writer of modern
Surya-Siddhanta evidently held the same view while
writing the ayanamsa rule.
On the other hand, the great popularity of
Varaha-mihira leads us to suppose that the present
system of the zodiac had its beginning in Saka 427,
and I belive all the anomalies noticed before may
be better explained by accepting the year. We have
an authority for our view in the pancha-siddhantika
itself. The often-quoted verse in which Varaha-

1. The copy from which the remarks are made was printed
   in Benares in Samvat 1942. It is full of mistakes, but the
   sense is clear. It is curious to note that Satananda, while
   basing his work on the Surya-Siddhanta, gives a higher
   rate than what is given by the latter Madhava Misra tries
   to justify this rate by altering the rule.