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

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.

Introduction liii
mihira distinctly mentions the fact that in his time
the summer solstice took place at the beginning of
Cancer is a complete answer to the question we
have been discussing.
Dr. Thibaut, the learned editor of the
Panchasidhantika, however, finds difficulty in
accepting Saka 427, as the date of the composition
of the work. As far as I can gather from his
introduction to the work, his main objection lies in
a statement of Amraja, quoted by Bhau-Daji, that
“Varaha-mihira Acharya went to heaven in the
509th year of the Saka kala, i.e. A.D. 587”. For if
Varaha be supposed to have written the work in
Saka 427, he must have lived to the good old age
of eighty-two years after the composition; and
supposing him to have written it at the age of
twenty-years, he must have seen 102 summers.
Seeing that Varaha lived long enough to write
a very large number of works, there is nothing in
the statement intrinsically impossible. On the other
hand, if the Saka year were not the epoch of his
Karana, he defeated his own purpose. In deed, the
idea of a Karana-writer using a date borrowed from
some older Siddhanta, as has been supposed in this
case, and thus representing a time other than that
for which the work is written, is to say the least,
self-contradictory. Then again as has been pointed
out by Pandit Dwivedi, there is absolutely no proof
of the validity of Amraja’s assertion. Dr. Thibaut
admits that if Saka 427 be taken as the epoch of
Varaha’s work, several facts, not otherwise
explainable, become easy of explanation. All those
considerations, together with tradition about
Varaha’s time point to the conclusion we have
already arrived at by discussing ayanamsa.

liv सिद्धान्त दर्पण
Assuming then, that the zodiac at present in
use was fixed in Saka 427, let us explain a few facts
connected with it. First of all, we find that in 1389
years, which have elapsed since the date general
precession has amounted to 19°-23′-7″; while, had
the Surya-Siddhanta's year been in use throughout
the Sun must have been in error by 3°-15′-30″, thus
making the ayanamsa 22°-38′-37″ in Saka 1816 last.
The amount, however, differs from the observed by
28′,- a quantity too large to be neglected. On the
other hand, if Ganesa's length of the year be taken,
the error in the Sun amounts to 2°-9′-54″, making
the ayanamsa actually less by nearly 38′. It is
remarkable that the length of the year assumed by
Bhaskara makes the total 22°-12′21″, nearly the same
as is observed.¹
From the observed ayanamsa and the fact of its
commencement in Saka 427, the precessional rate
becomes 57″.45. This is lower than that obtained by
us from the Surya-Sidhanta's year, but nearly the
same as we obtained from Siromani's year. This
discrepancy between the rate calculated from
Surya-Siddhant and that found above can be
explained by supposing that the Surya-Siddhanta's
year may not have been in use throughout the large
interval of 1389 years, and that the rate of precession
is not known so acurately as may be sufficient for
the great length of time. The fact of Varaha's Surya-
Siddhanta giving a slightly shorter year (as shown
――――――――――――――――――――――――――――――――――

1. The formula 50-2411 t+0001134 t² is adopted for
   calculating the general precession. See Chauvenet's
   Astronomy. The precession constant appears to be a little
   too large, as is remarked by the author. A calculation
   made from Bessel's constant allowing for its variation,
   renders the general precession nearly 1′ less.

Introduction lv
by Dr. Thibaut), makes no sensible difference in the
result. Be the explanation what it may, it is to be
observed that Chandrasekhara gives 22°-26' as the
observed ayanamsa, which is 15' greater than what
we have assumed. This fact shows that somehow
or other, the length of the year in use was slightly
less than what we find it in the modern
Surya-Siddhanta.
At any rate, the vernal equinox in Revati did
not mark the beginning of the existing zodiac.
Ranganatha, a commentator of Surya-Siddhanta,
supports us by saying that the equinox fell
somewhere near Revati. For, the equinox fell on the
star in Saka 498, and during the period since
elapsed, the Sun has moved through an excess of
3°-5'-29", thus making the ayanamsa 21°-29'-18".
Even the Siddhanta's position of the star leaves a
difference of half a degree. So we must suppose
that the star Revati was nearly a degree to the East
of the initial point when the zodiac happened to
commence its existing fixity.
If any use is to be made of the recorded Dhruvas
of the stars, it appears to me to be the most
reasonable course to select those stars which are
situated near the ecliptic. For, as has been already
pointed out, the errors of observation by means of
rude appliances will increase with star-latitudes.
Accordingly, the following twelve stars situated
within five degrees on either side of the ecliptic are
given below for composition.

lvi
सिद्धान्त दर्पण
Stars | Longitude in Saka 1816 | Reduced longitude | Difference
Krittika (23 Tauri) | 58°-14′ | 38°-52′ | 19°-22′
Rohini | 68°-17′ | 47°-56′ | 20°-23′
Pushya (δCancri) | 127°-21′ | 106°-0′ | 21°-21′
Magha | 148°-21′ | 129°-0′ | 19°-21′
Chitra¹ | 202°-22′ | 181°-23′ | 21°-9′
Visakha² | | |
Anuradha | 241°-45′ | 224°-31′ | 17°-14′
Jyeshtha | 248°-17′ | 230°-41′ | 17°-36′
Purvashadha | 273°-6′ | 254°-30′ | 18°-36′
Uttarashadha | 278°-43′ | 260°-18′ | 18°-25′
Satatara | 340°-05′ | 319°-41′ | 20°-24′
Revati | 373°-34′ | 359°-50′ | 18°-34′
The mean of the differences is 19°-18′ carrying
us back to about 433 Saka. If it were possible to hit
upon the stars that were taken as fundamental stars
for the measurement of the Dhruvas, we could have
arrived at the correct result. We might fairly take
the star Magha as one of them, since it is a star of
the first magnitude, situated on the ecliptic, and
otherwise important in Sanskrit astronomy. It
would be seen from the precession of the star that
the Saka year 430 is reached.
As has been already pointed out, an allowance
must be made in the amount of precession deduced
from the recorded Dhruvas of stars. Still, the mean
general precession furnishes approximately the date
of their measurement. Dr. Thibaut has employed
the method to determine the "beginning of the
scientific period of Hindu astronomy". From Prof.
Whitney's discussion of the star-longitudes given in

1. Bhaskar's Dhruva of the star is assumed.
2. I am not certain of Visakha. If it be identified with α Libra
   the difference amounts to 22°-46′.

the modern Surya-Siddhanta Dr. Thibaut places the
period somewhere in the fifth century A.D. It is,
however, strange that he has omitted to discuss the
bearing of the Dhruvas of the seven stars he has
found in the old Surya-Siddhanta. Omitting Aslesha
which I cannot identify with certainty, the mean
general precession of the remaining six amounts to
about 24°. This implies, then, that astronomy as a
science began to be cultivated in India in the second
century A.D. at the latest.
We have already dwelt rather too long on the
determination of the starting point. But considering
the importance of the question in settling one of
the fundamental preliminaries to any possible
revision of on current almanacs, a word or two more
on the subject may be excused.
Chandrasekhara gives two methods for
determining the starting point. One is based upon
a verse of the modern Surya-Siddhanta, which he
takes to mean that the declination-circle passing
through the pole-star marks the junction between
Aries and Pisces (XII.61). He further cites his
authority for thus interpreting the verse from
Brahma-Jamala.
I am inclined to believe that he is mistaken in
thus interpreting the sloka of the Surya-Siddhanta,
though I have not seen Brahma-jamala, nor have I
an opportunity of doing so now. Taking, however,
the rule for granted it will be seen that the Dhruva
of the star was 21°-42' in Saka 1816. But from the
annual variation in the star's R.A. the change in its
Dhruva is nearly 3-3/4 minutes in a year. This fact
alone demolishes Chandrasekhar's view.
The other method suggested is an inverse

lviii सिद्धान्त दर्पण
application of the Sun's place, calculated after the
Surya-Siddhanta (S.D.VI.89). Assuming his
calculated place to be correct, it is easy to find the
beginning of the starting point of measurement. In
the absence of a better method, this remains the
only feasible course of procedure, and it is by this
method that we have got the present ayanamsa.
It will be noticed (S.D.XII.10) that
Chandrasekhara gives 5°N. as the polar latitude of
Revati. The fact is, he has given up the Revati of
our ancient astronomy for the simple reason that
the Siddhanta Revati is hardly recognisable, and
therefore practically useless. His Revati is not ζ
(zeta) but η (eta) Piscium. How far this innovation
will be acceptable to the public, remains to be seen.
Instances of giving old names to new stars are
not rare, and this appears to be one of the main
difficulties in identifying stars of different
Siddhantas. For example, the star Pushya of
Varaha-mihira's Surya-Siddhanta must have been a
different star from the one receiving the same name
in the current Surya-Siddhanta. For Pushya of old
appears to have been Proesepe of the Greeks, while
modern Pushya is δ Cancri.
It may be useful to knwo the stars to which
Chandrasekhara applies the Siddhanta names. For
this purpose I got a chart of the stars made by him
from which the following list is prepared.

1. Asvini ά Arietes
2. Bharani 41 Arietes
3. Krittika Tauri (Pleides)
4. Rohini ά Tauri (Aldebaran)
5. Mrigasira λ Orionis

Introduction lix
6. Ardra αOrionis
7. Punarvasu Geminorum
8. Pushya Proesepe (in Cancer)
9. Aslesha - Hydrae
10. Magha Leonis (Regulus)
11. Purvaphalguni δ Leonis
12. Uttaraphalguni Leonis
13. Hasta δ Corvi
14. Chitra αVirginis (Spica)
15. Svati αBootis (Arcturus)
16. Visakha αLibra
17. Anuradha δ Scorpionis
18. Jyeshtha αScorpionis (Antares)
19. Mula γScorpionis
20. Purvashadha δ Sagittarii
21. Uttarashadha Sagittarii
Abhijit Lyri (vega)
22. Sravana γAquiloe (Altair)
23. Dhanishtha β Delphinii
24. Satataraka λ Aquarii
25. Purvabhadrapada β Pegasi
26. Uttarabhadrapada αAndromeda
27. Revati ξ Piscium
Agni β Tauri
Brahmahridaya Aurigae (Capella)
Prajapati Aurigade
Ilvaka Orion's belt
Lubdhaka Sirius 8
Agastya Canopus 9

lx सिद्धान्त दर्पण
Kratu Ursa majoris
Pulaha Ditto
Pulastya Ditto
Atri Ditto
Angira Ditto
Vasishtha Ditto
Marichi Ditto
Apayavasu 5°N. of Spica; 5th mag.
Apamvatsa 6°N. of Apaya; 5th mag.
The polar longitude and latitude of
Chandraskhara's Revati , as given in the work, are
slightly incorrect. In a letter to me, he gave 359°10′
and 5°30′N. as the corrected longitude and latitude.
Accordingly, the precession of the star amounted
to 22°20′-41″ in Saka 1816. These correction were
thought necessary by him in order that the observed
ayanamsa might agree with the precession of the
star as he ascertained it. But we cannot but admit
that he has confounded ayanamsa (or
sun-precession) with star-precession.¹ For, it is
obvious the amount of ayanamsa we observe by the
Sun affects only the Sun's place in finding moment
of his crossing the equinoctial points, while the
general precession of the stars remains unaffected
by the slight increase assumed in the length of the
year. But credit must be given him for his
conssitency in the view, however wrong it may be
in the light of the facts disclosed.
Another important improvement made by

1. An instance of an error of this kind is currnt in our Bengali
   almanacs in which the equinoxes are puttwo days later
   on account of the ayanamsa being taken at 20°55′.

Introduction lxi
Chandrasekhara is in the Sun's parallax. The history
of his attempts at determining it is no less
interesting than the results he actually obtained.
The ancient astronomers of India were satisfied with
taking for the parallax of each planet, 1/5th part of
its mean daily motion. Accordingly, the Sun's
horizontal parallax was considered to be 3'56", and
that of the Moon 52' 42". With rough instruments
at their disposal, the ancient observers could not
but assign wrong values. But awkwardly enough,
the parallaxes of the Sun and Moon come into every
prediction of their eclipses. So while modern
astronomy increases the distance of the Sun to
something like 400 times the distance of the Moon,
our ancient astronomers placed him no farther than
14 times this distance. Chandrasekhara has removed
him to a distance of about 154 times the mean
distance of the Moon.
Chandrasekhara told me that before he got the
parallax, he had passed many an unhappy day. He
was then very young, but was anxious to know the
Sun's distance. He could not even imagined how
the distance could ever be known. The ancient
works were dogmatic in their assertions, and did
not say a word about the method employed. By his
failure in approaching the problem, he became
disheartened. One day thus dejected in mind, while
he was coming home, he noticed an image of the
Sun projected through a narrow aperture in a fence
of palm leaves close to his house. This phenomenon
well known to every tyro in physical science made
him reflect, and he thought that he had got a
solution of his problem. Could he but get the Sun's
real diameter. This similar triangles on the two sides
of the aperture, with their bases formed by the Sun

lxii सिद्धान्त दर्पण
and his image, would enable him to find the
distance. He hastened home, but before he looked
into any work for the Sun's true diameter, he saw
his error. For, was he not assuming the thing he
wanted to find out? And his exultation was followed
by despair.
At ths stage the problem rested for several
years, till he became acquainted with rules for
calculating eclipses. He found that the parallax of
the Sun assumed in the Siddhantas was too large
to make predictions of eclipses correct. The anular
and total eclipses, together with those in which the
Moon just grazes the shadow, were his landmarks
in the determination of the problem. He was for
sometime in a sea of doubt. Perhaps the Siddhantas
were wrong in the mater of parallax, as in many
other matters. Perhaps his observations were not
made with that degree of accuracy which was
demanded by such problems.
These questions were never out of his mind a
single day through a long year. An accident
favoured him. He was reading the Atharvan
Upanishad, and was surprised to find that the Sun's
diameter was not 6500 yojanas as given in the
Surya-Siddhanta, but 72000 yojanas. This gave him
a fresh basis for calculation, and he was satisfied
that this diameter did not shift his landmarks to
any extent which he could detect. This gave 22" for
the Sun's horizontal parallax, and with careful
calculation that of the Moon was found to be 56'28".
Repeated observations left no doubt in his mind
about the correctness of his figures. Besides, here
was an authority from Sruti itself.¹ Taking 72000

1. Lastely he learnt that European astronomers find the