गुप्त अभिलेख संग्रह (जे. एफ. फ्लीट - गुप्त सम्राटों एवं परवर्ती नरेशों के ऐतिहासिक पाषाण व ताम्रलेख)

142 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
equivalent to expired Śaka years. It is evident from the details given above regarding
Śaka-Saṁvat 1808 and 1809, that, according to the reckoning of the era as fixed by the
early astronomers, and as preserved to the present day, the true epoch is A.D. 77-78,
and the period from the 3rd March, A.D. 78, to the 20th February, A.D. 79, is in reality the
commencement, or first current year; and that, to obtain current Christian years,
equivalent to current Śaka years, the true additive quantity is 77-78. But, of
course, there is always the possibility that, if ever we obtain a date, with full details for
calculation, in a very early Śaka year, or in one of the very earliest of the regnal or
dynastic years which afterwards developed into the Śaka era, this exact equation may
not hold good; in consequence of the date belonging to a period anterior to the adoption
of the era by the astronomers.
The Śaka era is emphatically one of the eras that originated in an exten-
sion of regnal or dynastic years. The chief Hindu tradition about it, is, that it was
founded in celebration of a defeat of the Śaka king by the king Vikrama or Vikramâditya
who is also the supposed founder of the Vikrama era, commencing a hundred and thirty-
five years earlier.¹ This tradition is mentioned by Albêrûnî ;² but he saw through it so far
as to remark "since there is a long interval between the era which is called the era of
Vikramâditya and the killing of Śaka, we think that that Vikramâditya from whom the
era has got its name is not identical with that one who killed Śaka, but only a namesake
of his." And the tradition has now been quite exploded by the Bâdâmi cave inscrip-
tion of the Chalukya king Maṅgalîśa,³ which is specifically dated "when there have
expired five centuries of the years of the installation of the Śaka king (or kings) in the
sovereignty." It is certain, from this record, that the real historical starting-point of the
era, is the commencement of the reign of some particular king, or kings, of the Śaka
¹ Another tradition (e.g. Prinsep's Essays, Vol. II. Useful Tables, p. 154) is that the era dates
from the birth of Śâlivâhana, king of Pratishṭhâna, who opposed Vikramâditya, king of Ujjayinî.
But the introduction of the name of Śâlivâhana in connection with the era, is of comparatively
modern date, the earliest instance that I have succeeded in obtaining, being one of the thirteenth cen-
tury A.D. ; and the epigraphical instances speak of the year as having been 'established, settled,
or decided' (nirṇîta) by Śâlivâhana, but not as running from his birth (see Ind. Ant. Vol. XII. p.
214 f.).—A passage to the latter effect has been quoted by Prof. Max Müller (India ; What can it
teach us? p. 300 f.) from the Muhûrtabhuvanônmârtaṇḍa of Nârâyaṇa, which means "in the year
measured by three, the (nine) numerals, and the (fourteen) Indras, from the birth of Śâlivâhana (i.e.
in Śaka-Saṁvat 1493), in (the month) Tapas (Mâgha), this Mârtaṇḍa was composed."—As Prof.
Max Müller has pointed out, in his comments on this passage, it is not exactly wrong to speak of
the era as the Śâlivâhana-Śaka or Śâlivâhana era ; for there are ample instances in which the Hindus
give it that name, in epigraphical records of authority and of some antiquity. At the same time,
those instances shew that it was only in comparatively modern times that the name of Śâlivâhana
came to be connected with the era. And, in all discussions respecting early dates, it is an anachronism
and a mistake, to call the era by his name.
² Sachau's Albêrûnî's India, Translation, Vol. II. p. 6.
³ Ind. Ant. Vol. VI. p. 363 f., and Vol. X. p. 57 ff.

THE GUPTA INSCRIPTIONS; APPENDIX I. 143
tribe; and, therefore, that the years were originally regnal or dynastic years. Now,
such years must run on for a considerable time, before they can develop into a recog-
nised era; and this is undoubtedly the reason why we find the earlier years of every
such Hindu era quoted simply by the term varsha or saṁvatsara, 'a year,' without any
dynastic appellation. Again, such regnal or dynastic years can only come to be quoted
as expired years, when they have actually developed into an era which has become recog-
nised, or is sought to be applied, by astronomers for astronomical processes; up to that
point, the years, being wanted only for quasi-private dynastic purposes, would certainly
be quoted as current years. It is impossible to believe that the first Śaka king decreed,
immediately after his coronation, that a new era had been established from that event;
that it was to come at once into general use; and that, for the convenience of astrono-
mers, the first year, then running, was to be quoted as an expired year, which, in fact, it
would be rather difficult to do. If it were sought to fix the exact chronological position of
any public act performed in that first year, it might be referred to the expired years of an
earlier era; e.g. of that of the Kaliyuga. 'But, for any reference to the regnal year alone,
that act would be recorded as being performed "in the year one," "in the first year," or
"while the first year of the reign is current;" as, for instance, "in the first year; while
the Mahârâjâdhirâja, the glorious Tôramâṇa, is governing the earth," in line 1 f. of
the Êraṇ inscription, No. 36, page 158. This custom would continue as long as the
years were simply dynastic years; and perhaps, during the whole of that period, the
years might remain purely dynastic years, each of them having for its initial day the
anniversary of the original coronation from which they started, irrespective of the initial
day of the years of the astronomical era still continuing in use. When, however, astro-
nomers came to adopt them as an astronomical era, they would establish an exact epoch
by reckoning back from the dynastic year then current to the last year of the Kaliyuga
that had expired when the first current dynastic year commenced; in the course of
which, they would simplify matters by allotting to the dynastic years the same scheme,
as regards the starting-point of each year, and the arrangement of the fortnights of the
months, which belonged to the Kaliyuga in their part of the country. Thus they would
fix all the data necessary to enable them to use the new era for astronomical purposes.
All that would remain, would be to use its expired years, in accordance with the custom
and necessities of their science. The substitution of the Śaka era for the Kaliyuga, for
astronomical purposes, seems to have taken place after the time of Âryabhaṭa (born A.D.
476),¹ who used the Kaliyuga, and in or just before the time of Varâhamihira (died A.D.
587),² who used the Śaka era; and probably the apparent difference of one year in the
reckonings of the Śaka era will be found to have originated not far from Śaka-Saṁvat
500. Let us assume that this adoption of the Śaka era was made in Śaka-Saṁvat 500,
equivalent to A.D. 577-78. The astronomers would take it, at starting, as "Śaka-
¹ Jour. R. As. Soc. N. S. Vol. I. p. 405. ² Jour. R. As. Soc. N. S. Vol. I. p. 407.

144 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
Saṁvat 499, expired;" and, in quoting it and several subsequent years, would probably
be careful to connect with each year a word distinctly meaning "expired." In course of
time, however, such precision of expression would come to seem superfluous to them;
and, in issuing their almanacs, they would drop the word "expired," and would write,
for instance, simply "Śakē 510 Chaitra-māsa-śukla-pakshaḥ." It would cause no
difference or inconvenience to them; because any initiate would know that this really
designated the bright fortnight of the month Chaitra of Śaka-Saṁvat 511 current after
Śaka-Saṁvat 510 had expired. The people at large, however, including persons who
would use the almanacs for practical purposes without being properly initiated into the
application of them, would be thrown back in their reckoning by a year; and doubtless
at first a good deal of inconvenience and confusion might result. But this would soon
be forgotten; or might, for the sake of convenience, be intentionally put aside. And
thus they would very quickly arrive at the understanding, by which, in Northern and
Western India, Śaka-Saṁvat 1808 answers, popularly, as a current year, not as an expired
year, to the period from the 5th April, A.D. 1886, to the 24th March, A.D. 1887.
Postscript.
In connection with my remarks, on page 140 above, regarding the Sāyana-Pañchāṅg
for Śaka-Saṁvat 1807 and 1808, I would add that I had drawn Mr. Sh. B. Dikshit's
attention to the circumstances of the case; and I now find that, in their almanac for
Śaka-Saṁvat 1810 (expired) (A.D. 1888-89), the editors have reverted to the phraseology
used in their almanac for Śaka-Saṁvat 1807 (expired.)

THE GUPTA INSCRIPTIONS APPENDIX II 145
APPENDIX II.
A Method of Calculating the Week-Days of Hindu Tithis and the
corresponding English Dates.
By Shankar Balkrishna Dikshit; Bombay Educational Department.
In this paper I purpose to exhibit, according to the system laid down by the
late Professor Kero Lakshman Chhatre in his book entitled Graha-sâdhanâchîm
Kôshṭakêm, or "Tables for Calculating the Places of the Planets," the correct method by
which we may determine, for any given Hindu tithi or lunar day, the corresponding vâra
or week-day, and the equivalent English date according to either the Julian or the
Gregorian Calendar.
Before detailing, however, the steps of the process, I will explain the principal
technical terms which will be used, and which, for the sake of brevity and conciseness, will
be retained in their original Sanskrit forms.¹
Explanation of Technical Terms.
The abdapa, lit. 'lord of a year,' of any particular year, is the conventional term, —
in Prof. K. L. Chhatre's book, and others; but not universally, — for the time of the
Mêsha-Saṁkrânti, or 'entrance of the Sun into Aries, in that year. The abdapa
adopted by Prof. K. L. Chhatre is the time of the spashṭa or 'apparent,' — lit. 'clearly
perceived, distinctly visible,'² — Mêsha-Saṁkrânti, as ascertained by the method given in
the Sûrya-Siddhânta;³ whereas, in other Hindu works of the same kind, the term abdapa
¹ Most of these explanations are my own. Either to reduce the bulk of his book, or for some
other reason, Prof. K. L. Chhatre has used the technical terms without explanation, except in the case
of a few of the simpler ones; nor does he explain how he obtained certain figures for certain years
or the variation for a year.
² English astronomers use the word 'apparent' in all cases in which we use spashṭa. 'Apparent,'
therefore, is the proper translation of spashṭa.
³ There are three schools of astronomers in India. One follows the Sûrya-Siddhânta, and is
called Saurapaksha; another follows the Brahma-Siddhânta, and is named Brahmapaksha; while
the third follows the Ârya-Siddhânta, and is called Âryapaksha. The main point on which they
differ is the length of the year; but with differences, between each other, of only a few vipalas (a
vipala is the sixtieth part of a pala). Another point of difference is, that the number of revolutions
of the moon, planets, &c., in a certain period, — for instance in a Mahâyuga, — is generally different in
each of them. Prof. K. L. Chhatre has adopted, from the Sûrya-Siddhânta, only the length of the
year, and its starting-point, that is the Mêsha-Saṁkrânti; in almost every other respect he follows
none of these three authorities, but has based his Tables on European Tables of planets. As to his
Tables relative to tithis, however, in the part of his work called Kâla-sâdhana, see page 149 below,
note 11, and the text above note 1 on page 155.

146 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
is used as meaning the time of the Sun's entrance into Aries with reference to his
madhyama or 'mean' longitude. So, also, the length of the solar year adopted by him
is that of the Sûrya-Siddhânta, which is accepted, in the present day, in most parts of
India. It should, however, be borne in mind that the Tables of the sun and the moon, and
those of the planets, given by him, are based on European Tables; and that the places
of the sun and other heavenly bodies, obtained from his book, are reckoned from the
equinoctial point. The starting-point adopted by Hindu astronomers, for reckoning the
places of heavenly bodies, coincided, in their opinion, with the equinoctial point about
Śaka-Saṁvat 444 expired (A.D. 522-23). The interval in time between two successive
returns of the sun to the vernal equinox,—called "the tropical year,"—amounts at present
to 365 days, 14 ghaṭîs, and 31.972 palas; while the length of the year, according to the
Sûrya-Siddhânta, is 365 days, 15 ghaṭîs, and 31.523 palas. During this time, the sun's
motion amounts to one complete revolution from equinox to equinox, plus about 58.6881
seconds of arc. The starting-point, therefore, of the Hindu astronomers is at present a
little more than twenty-two degrees to the east of the vernal equinox. This difference is
called ayanâṁśas, lit. 'degrees of precession;' and the ayanâṁśas for the present year,
Śaka-Saṁvat 1809 expired (A.D. 1887-88), are 22 degrees, 45 minutes, according to the
Graha-Lâghava of Gaṇêśa Daivajña.¹ As the longitudes of heavenly bodies, reckoned
from the equinox, include these ayanâṁśas, they are called sâyana, lit. 'possessed of ayana
or precession.' And the places of heavenly bodies obtained by the method given in the
Sûrya-Siddhânta and other Hindu works, are called, for the sake of distinction, nirayaṇa,
lit. 'destitute of precession.' The places obtained from Prof. K. L. Chhatre's Tables are
sâyana. The tithi, however, obtained by either process, is the same; but this is not the
case with the nakshatra or 'lunar mansion,' and the yôga or 'addition of the longitudes of
the sun and the moon.'²
The figures for the abdapa are given in Table I. on pp. 10, 11, of Prof. K. L.
Chhatre's book, and are expressed in vâras, ghaṭîs, and palas. Of these, the vâra, or
week-day, sometimes also called dina or divasa, or 'solar day (and night),' is counted in
regular order from Sunday, as 1, up to Saturday, as 7 or 0; and it is always reckoned by
¹ The date of this work is Śaka-Saṁvat 1442 expired (A.D. 1520-21). At present, all the
Pañchâṅgs (Hindu calendars) in the Dekkan, and in some other parts of India, are prepared from this
authority, and from another small work, by the same author, entitled Tithi-Chintâmaṇi, containing
the necessary Tables.
² To calculate tithis, only the difference between the longitudes of the moon and of the sun is to
be taken. Therefore it matters not whether these longitudes are sâyana or nirayaṇa. To find a
nakshatra, the ayanâṁśas must be applied to the moon's longitude obtained from Prof. K. L. Chhatre's
Tables. The Sâyana-Pañchâṅg, annually published, from Śaka 1806, under the patronage of His
Highness the Mahârâja Hôlkar, by Mr. Visaji Raghunath Lele of Gwâlior, with the aid of Mr. Janardan
B. Modak, B.A. of the Bombay University, of myself, and of Mr. Krishnarao Raghunath Bhide of
Indôr, is based on the sâyana system.

THE GUPTA INSCRIPTIONS; APPENDIX II. 147
the Hindus from sunrise to sunrise. The vâra of the abdapa shews the week-day on
which the Mêsha-Saṁkrânti of the year fell. A ghaṭî, also ghaṭî and ghaṭikâ, is the
sixtieth division of a mean solar day and night; and it is, therefore, equal to twenty-four
English minutes. As a matter of convenience, the word ghaṭî is also used for the six-
tieth part of a tithi; but in that application it is not identical with the sixtieth division of a
solar day and night. A pala is the sixtieth division of a ghaṭî; and is, therefore, equal to
twenty-four English seconds. And the ghaṭîs and palas of the abdapa give the time after
sunrise, on the particular vâra, at which the Mêsha-Saṁkrânti took place. Thus, the abdapa
of Śaka-Saṁvat 0 expired, is given (opposite the entry Mêsha-Saṁkrântîchî vêḷ or
‘**time of the Mêsha-Saṁkrânti,’ p. 10) as 1 dina, 10 ghaṭîs, 10 palas; which indicates that
the Mêsha-Saṁkrânti then took place on Sunday, and 10 ghaṭîs and 10 palas, or four
hours and four minutes, after sunrise. The solar year adopted by Prof. K. L. Chhatre is
equal to 365 days, 15 ghaṭîs, 31·52 palas. Dividing 365 by 7 (the number of days in a
week), the remainder is 1. And so, if in one year the Sun enters Aries at the time of sun-
rise on a Sunday, then, in the following year, he will come to Aries on Monday, and
15 ghaṭîs, 31·5 palas, after sunrise. Therefore, the variation in the abdapa in one year
is given (p. 10, col. 3, under vâra) as 1 day, 15 ghaṭîs, 31·5 palas; the decimals being
supplied from column 2, in which is given the number of days, corresponding to the
number of years in column 1.
The word tithi denotes the thirtieth part of a lunation or lunar month; that is, as
applied to the ecliptic circle, it denotes exactly the one-thirtieth part of that circle, viz.
twelve degrees; but, taken as an apparent tithi, and applied to the period of a lunation, it
may be the exact thirtieth part of that period, or it may vary from fifty to sixty-six
ghaṭîs as subdivisions of a solar day. If the word tithi requires to be rendered into
English, it is best represented by ‘lunar day.’ Of the thirty tithis of each month, fifteen
belong to the bright fortnight, or period of the waxing moon, and fifteen to the dark
fortnight, or period of the waning moon. The fifteenth tithi of the bright fortnight is
called pûrṇimâ, pûrṇamâsî, or paurṇamâsî, lit. ‘that which has the full-moon, or that on
which the month is completed;’ and the fifteenth tithi of the dark fortnight is called
amâvâsyâ, lit. ‘that on which there is the dwelling-together (of the sun and moon).’ At
the end of the amâvâsyâ, the sun and the moon are together; that is, they have the same
longitude. When the moon, moving towards the east, leaves the sun behind by twelve
degrees of longitude, then ends the first tithi, which is technically called pratipad or
pratipadâ. So, a tithi is the time which the moon takes to out-go the sun by twelve
degrees. With the exception of the pratipadâ, the tithis are denoted by the regular
ordinal numerals, dvitîyâ, tṛitîyâ, &c., up to chaturdaśî, ‘the fourteenth.’ The pûrṇimâ
and amâvâsyâ are called sometimes by their own special names, and sometimes pañchadaśî,
‘the fifteenth;’ but the amâvâsyâ is generally entered in Pañchâṅgs as the thirtieth tithi,
19

even in Northern India, where the dark fortnight of the month precedes the bright.¹ In
Pañchâṅgs, the ghaṭīs and palas of tithis are given; and, by them, it is to be under-
stood that the tithis end so many ghaṭīs and palas after sun-rise. In general, the
expression tithi means the end of a tithi; not its beginning, or its duration.
The term tithi-śuddhi, lit. 'the subtraction of tithis,' denotes the number
of tithis that elapse from the beginning of the month Chaitra (March-April) up
to the time of the Mêsha-Saṁkrānti. In Prof. K. L. Chhatre's Tables, this term is
used to shew the number of tithis, calculated from the difference between the moon's
mean longitude and the sun's apparent longitude, that elapse from the beginning of
Chaitra to the time of the sun's spashṭa or 'apparent' Mêsha-Saṁkrānti.² Thus, for Śaka-
Saṁvat 0 expired, at the time of the Mêsha-Saṁkrānti the sun's mean longitude was
11 signs, 20 degrees, 46·1 minutes (p. 46); and the apparent longitude obtained from it,
according to the method given by Prof. K. L. Chhatre, is 11 signs, 22 degrees, 38·9
minutes. The moon's mean longitude at that time was 4 signs, 25 degrees, 42·4 minutes
(p. 87). The difference between the longitudes of the sun and the moon,—the sun's
longitude being subtracted from that of the moon,—is, therefore, 5 signs, 3 degrees (=
153 degrees), 3·5 minutes. Then 153° 3'·5 ÷ 12 = 12 + (9° 3'·5 ÷ 12) tithis; that is,
12 tithis, and about 45 ghaṭīs and 14 palas, had elapsed. This, therefore, is given as
the tithi-śuddhi for Saka-Saṁvat 0 expired. In one solar year, the mean tithis are 371,
and 3 ghaṭīs, 53·4 palas. Dividing 371 by 360, the remainder, 11 tithis, 3 ghaṭīs,
53·4 palas, is given as the variation in the tithi-śuddhi in one year (p. 10, col. 4).
The tithis obtained from the mean places and mean motions of both the sun and the
moon, are madhyama or 'mean' tithis. So, also, those calculated from the apparent
place and motion of the sun and the mean place and motion of the moon,—as in the case
of the tithi-śuddhi and the mean solar equivalents of tithis given in Table III. pp. 13-19,
col. 2,—may be called mean tithis and not apparent. But the tithis, &c., given in
our Pañchâṅgs are always spashṭa or 'apparent;'³ that is, they are calculated from the
apparent places and motions of the sun and the moon. The spashṭa-tithi differs from the
madhyama-tithi sometimes by nearly twenty-five ghaṭīs; and this is chiefly owing to the
fact that the moon's apparent longitude differs from her mean longitude sometimes by
about five degrees.⁴ Many corrections have to be applied to the mean place of the
¹ In astronomical works, it is always the Amânta southern arrangement of the lunar fortnights
that is actually taken for calculations.
² In the Siddhânta-Śirōmaṇi and other works, the term tithi-śuddhi is used in the sense of the
number of tithis, calculated from the mean places of the sun and the moon, that elapse from the
beginning of Chaitra to the time of the sun's madhyama or 'mean' Mêsha-Saṁkrānti.
³ Though not always in the strictest sense. I say so, because, in practice, extreme accuracy is
not, and cannot be, sought. But, in theory, they are required to be 'apparent' in the strictest sense.
⁴ According to European Tables, the difference is sometimes about eight degrees

THE GUPTA INSCRIPTIONS; APPENDIX II. 149
moon in order to find her apparent place; but only one of these, called phala-saṁskâra,
which is 'the equation of the centre,' is taken into account by Hindu astronomers:¹ and
this, as given by them, amounts to a little more than five degrees at the greatest. This
correction varies according to the moon's kêndra or 'anomaly;' which is taken to be her
distance from apogee.² From this correction is calculated the correction in time to be
applied to the mean tithi; it is named parâkhya; and it is given in Table IV., on p. 20,
in the column headed parâkhya. It evidently varies according to the moon's kêndra.
One revolution of the moon's kêndra is completed in 27 days, 33 ghaṭîs, 16·56 palas.
This period is called nîchôchcha-mâsa;³ and is known to English astronomers by the name
of the 'anomalistic month.' This period, converted into tithis,⁴ is equal to 27 tithis, 59
ghaṭîs, 33·36 palas; that is, nearly and practically, 28 tithis. It is converted into tithis
for the sake of convenience; since, the variation in the kêndra is one tithi of kêndra in
one tithi of time; and it is called tithi-kêndra, or 'the anomaly of the tithi, expressed
in tithis.' The moon's mean kêndra at the Mêsha-Saṁkrânti of Śaka-Saṁvat 0 expired,
was 10 signs, 19 degrees, 58·8 minutes (p. 87). This, converted into tithis, is equal to 24
tithis, 52 ghaṭîs, 50 palas;⁵ and this is given (p. 10) as the tithi-madhyama-kêndra, or
'mean anomaly of the tithi,' at the time of the Mêsha-Saṁkrânti of Śaka-Saṁvat 0 expired.
It shews that so many tithis and parts of a tithi had elapsed, up to that Mêsha-Saṁkrânti,
from the moon's preceding arrival at her apogee. The variation in the moon's kêndra, in
one solar year, is 3 signs, 2 degrees, 6·2 minutes (p. 87, col. 3). This, changed into tithis
by the rule of three, viz.—360° : 92° 6'·2 :: ti. 27, gh. 59, p. 33·36 : ti. 7, gh. 9, p. 42,—is
given, therefore, as the variation in the tithi-kêndra in one year (p. 10, col. 5).
A few other points and terms will be explained, as we proceed with the following
example.
To find the Week-Day of a given Tithi.
The process will be best illustrated, step by step, by actually working out an example.
And, at Mr. Fleet's request, I take, as my example, the date of Śaka-Saṁvat 406
¹ The amount of this correction, adopted by Prof. K. L. Chhatre, in finding out tithis in his Kâla-
sâdhana Tables (pp. 1 to 30 of his book), is nearly the same as that adopted by ancient Hindu astro-
nomers. Therefore, the tithis obtained by his method, as described above, should agree very closely
with those obtained from the methods prescribed in Sanskṛit works. But, in the abdapa and other
elements, the Sûrya-Siddhânta and other authorities themselves slightly differ, one from the other.
And, accordingly the difference will be sometimes about five or six ghaṭîs. There are, also, some
other minute causes of difference.
² In European astronomical works, the anomaly is reckoned from perigee or perihelion; but in
Hindu works it is reckoned from apogee or aphelion.
³ In this term nîcha means 'perigee;' and uchcha, 'apogee.' And nîchôchcha-mâsa is the
period in which the moon comes from perigee or apogee to the same point again.
⁴ One tithi is equal to 0·9843529572 of a mean solar day.
⁵ 360 : 319°58'·8 :: ti. 27, gh. 59, p. 33·36 : ti. 24, gh. 52, p. 50.

150 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
expired (A.D. 484-85); the month Âshâḍha (June-July); the bright fortnight;
the twelfth tithi.
From Table I. p. 10, write down (see Table V. on page 151 below), in three separ-
ate columns, three quantities, for Śaka-Saṁvat 0 expired, which are technically called
the kshêpaka or 'additive quantities;' viz. in (a) the abdapa, enter vâras 1, ghaṭîs 10,
palas 10; in (b) the tithi-śuddhi, enter tithis 12, ghaṭîs 45, palas 14; and in (c) the tithi-
madhyama-kêndra, enter tithis 24, ghaṭîs 52, palas 50. Below each of them respectively,
in its proper column, enter, from the same Table, the bhêda or 'variation' for the com-
ponent parts of the given Śaka year;¹ viz. for 400, in (a) vâras 6, gh. 30, p. 9.3, in (b)
tithis 15, gh. 55, p. 49.2, and in (c) tithis 9, gh. 24, p. 45; and for 6 years, in (a) vâras 0,
gh. 33, p. 9.1, in (b) tithis 6, gh. 23, p. 20.2, and in (c) tithis 14, gh. 58, p. 39.
Now, as the given year is anterior to Śaka-Saṁvat 1622 expired, a correction, to
be arrived at from Table II. p. 12, is to be applied, and is always to be added, in respect of
the tithi-śuddhi and the tithi-madhyama-kêndra. The reason for this correction, is this.
As explained above, the tithi-śuddhi and the tithi-madhyama-kêndra depend respective-
ly on the mean longitude and the mean anomaly of the moon. But the moon's mean
motion is not always the same. Therefore, to her mean longitude and mean anomaly,
obtained from the general Table of annual variation in them (Table III. p. 87 f., cols. 2, 3),
a correction (Table IV. p. 89 f., cols. 2, 3) is to be applied. Thus, for Śaka-Saṁvat 0
expired, the correction in the moon's mean longitude is 44 seconds, and that in the kêndra
is 2 degrees, 55 seconds (p. 90). These, turned into tithis, are 3 ghaṭîs, 40 palas, with
regard to the tithi-śuddhi; and 14 ghaṭîs, with regard to the tithi-kêndra. These figures,
therefore, are given as the correction in respectively the tithi-śuddhi and the tithi-kêndra
for Śaka-Saṁvat 0 expired. In the Table, this correction is given for intervals of 1000
years each. Taking first the tithi-śuddhi, the correction for Śaka-Saṁvat 0 expired, is
gh. 3, p. 40; and the correction for Śaka-Saṁvat 1000 expired, is p. 32. Therefore,
deducting the latter from the former, the difference, gh. 3, p. 8, or 188 palas, is the variation
of correction in 1000 years.² Then, by the Rule of Three, — 1000 years : 406 years :: 188
palas : 76 palas. And 76 palas are gh. 1, p. 16. As the quantities are decreasing ones,
this is to be subtracted from gh. 3, p. 40, for Śaka-Saṁvat 0 expired. And the remainder
gives us, as the sufficiently approximate correction for Śaka-Samvat 406 expired, gh. 2,
p. 24, to be added in (b). Similarly, the correction for the tithi-madhyama-kêndra,
worked out in the same way, is gh. 9, p. 8, to be added in (c).
¹ The decimals in the palas of (a) the abdapa are taken from the ahargaṇa, or total number of
solar days of the solar year, in col. 2.
² Properly speaking, this variation is for Śaka-Saṁvat 500 expired, midway between Śaka-
Saṁvat 0 and 1000. It should be reduced first for the year midway between Śaka-Saṁvat 0 and the
given year; in this instance 406. But there is no absolute necessity for such exact precision.

THE GUPTA INSCRIPTIONS; APPENDIX II. 151
TABLE V.
Calculation of the Week-day of a given Tithi.
Śaka-Saṁvat 406 expired = A.D. 484-85 current.
Âshâḍha (June-July); the bright fortnight; the 12th tithi; Suraguruvâra (Thursday).

152 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
Now add together the respective quantities in (a) (b) and (c), bearing in mind that, in
doing so, when the vâras in (a) the abdapa exceed 7, or any multiple of 7, only the
remainder, above 7 or its multiple, is to be brought to account, because there are 7 vâras
or week-days in each week; and that, when the tithis in (b) the tithi-śuddhi and in (c) the
tithi-madhyama-kêndra exceed 30 and 28 respectively, or any multiple of them, only the
remainders above 30 and 28, or their multiple, are to be taken notice of, because there are
30 tithis in one lunar month, and, as nearly as possible, 28 tithis in one revolution of the
tithi-kêndra.
We thus obtain in (a) the abdapa, vâras 1, gh. 13, p. 28·4. The first quantity, of the
days, shews that the week-day on which occurred the Mêsha-Saṁkrânti of the given year,
Śaka-Saṁvat 406 expired, was Sunday. And the remaining quantities shew that the
Mêsha-Saṁkrânti took place at the end of gh. 13, p. 28·4, after sunrise on that Sunday.
The small decimal which we have here, as also in (b) the tithi-śuddhi, under the palas,
may be disregarded in the following steps of the process.
In (b) the tithi-śuddhi, we obtain tithis 5, gh. 6, p. 47·4. From this we learn that,
when the Mêsha-Saṁkrânti of the given year, Śaka-Saṁvat 406 expired, occurred,
5 mean tithis of the month Chaitra were completed, and also 6 ghaṭîs and 47 palas of the
6th tithi had elapsed. The number of completed tithis, here 5, is technically called the
tithi-dhruva or ‘constant of the tithi;’ because, when it has been determined for any
given year, it remains uniform or constant in working out any example in that same year.
And the remainder, here gh. 6, p. 47, is called the bhukta-tithi or ‘elapsed portion of the
(current) tithi.’
Subtracting the bhukta-tithi, gh. 6, p. 47, from 1 tithi or 60 ghaṭîs, the remainder,
gh. 53, p. 13, gives the portion of the 6th tithi that was still to run. This is technically
called the bhôgya-tithi, lit. ‘(that portion of) the tithi which is still to be enjoyed.’
In (c), the tithi-madhyama-kêndra, we obtain tithis 21, gh. 25, p. 22. This gives us
the moon's kêndra, reduced to tithis, at the time of the Mêsha-Saṁkrânti of the given
year, Śaka-Saṁvat 406 expired.
To this, the bhôgya-tithi, viz. gh. 53, p. 13, is to be added. And the result, tithis 22,
gh. 18, p. 35, is the kêndra at the end of the 6th tithi of Chaitra. This is called the tithi-
spashṭa-kêndra or ‘apparent kêndra of the tithi.’
Next, by subtracting from the bhôgya-tithi, viz. gh. 53, p. 13, as many palas, 53, as
there are ghaṭîs in it,¹ we convert it into a mean solar day, with the result of gh. 52, p. 20.
¹ i.e. by subtracting the sixtieth part. This proportion is taken for the sake of easy calculation.
Properly speaking, to convert a tithi into a solar day, the sixty-fourth part should be subtract-
ed; because one mean tithi is equal to ·984353 of a solar day, i.e., as nearly as possible, sixty-three
sixty-fourths of a solar day. The difference, however, does not introduce any material error.

THE GUPTA INSCRIPTIONS: APPENDIX II. 153

Add this gh. 52, p. 20, to the ghaṭīs and palas only of (a) the abdapa. The result,
vāra 1, gh. 5, p. 48, shews that the 6th mean tithi of Chaitra ended with gh. 5, p. 48, after
sunrise on the following day, Monday, after the day of the Mêsha-Saṁkrānti, Sunday.
This quantity, vāra 1, gh. 5, p. 48, is called the tithi-bhōga, lit. 'the enjoyment or dura-
tion of the tithi;' and it is the end of the tithi-dhruva, increased by one tithi. It is, of
course, a mean tithi. And it shews that days 1, gh. 5, p. 48, had elapsed, from sunrise on
the day of the Mêsha-Saṁkrānti, up to the end of Chaitra śukla 6 as a mean tithi.
We have now to bring into consideration the number of tithis elapsed up to the
commencement of the given tithi. And, in doing this, we must of course take account of
any intercalary month that there may be, preceding the given tithi, in the given year.
In our example, however, the result in (b) the tithi-śuddhi, is less than nineteen tithis.
And a reference to Table VI. p. 22, — which would enable us to determine the inter-
calary month approximately, if there were one, — shews us that there was, therefore, no in-
tercalary month at all in the given year, Śaka-Saṁvat 406 expired. The explanation of
this, is, that, when the tithi-śuddhi is less than nineteen, it shews that the saṁkrānti in
Chaitra occurred within the first nineteen tithis of that month. And, as, generally, the
solar months are longer than the lunar months, the saṁkrāntis of the sun, i.e. his passage
from one sign of the zodiac into the next, occur continuously later in each successive
lunar month. But, when the saṁkrānti in Chaitra falls within the first nineteen tithis,
no saṁkrānti, up to the end of the year, can go beyond the thirtieth tithi of any lunar
month; and, therefore, no month will be intercalary.
Consequently, from the beginning of Chaitra, up to the commencement of the given
tithi, Âshâḍha śukla 12, there had elapsed only the usual number of one hundred and one
tithis; viz. in the bright fortnight of Chaitra, 15; in Vaiśâkha, 30; in Jyêshtha, 30; and
in Âshâḍha, 15 in the dark fortnight, and 11 in the bright.¹ From this number of tithis,
101, we subtract the tithi-dhruva, 5. And the remainder, 96, is the number of tithis
elapsed from the end of Chaitra śukla 5 up to the end of Âshâḍha śukla 11. But the tithi-
bhōga, which we have already arrived at, is the end of Chaitra śukla 6; and the same
number of tithis, 96, expire from the end of Chaitra śukla 6 up to the end of Âshâḍha śukla
12. Therefore, adding to the tithi-bhōga the solar equivalent, now to be introduced, the
result will bring us to the end of the given tithi, Âshâḍha śukla 12.
Turning to Table III. p 14, we find that the equivalent, in mean solar days, of
96 tithis, is days 94, gh. 17, p. 36. And, adding this to the tithi-bhōga, the result, days
95, gh. 23, p. 24, gives the interval that had elapsed, from sunrise on the day of the Mêsha-
Saṁkrānti up to the end of Âshadha śukla 12 as a mean tithi

¹ We are dealing with a northern date; and that is why the tithis are counted in this way. If we
were dealing with a southern date, the enumeration would be, in Chaitra, 30; in Vaiśâkha, 30:
in Jyêshṭha, 30; and in the bright fortnight of Âshâḍha, 11. The total is the same, 101, since a bright
fortnight is concerned.
20

154 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
Now, however, we have to determine the spashṭa-tithi, or apparent tithi. For this
purpose, we require the parâkhya-correction, which is to be ascertained through the tithi-
kêndra.
Turning again to Table III. p. 14, we find that the variation in the tithi-kêndra
for 96 tithis is tithis 12, gh. 1, p. 20. Enter this in (c), below tithis 22, gh. 18, p. 35,
which we have already arrived at as the tithi-kêndra at the end of Chaitra śukla 6. Add
the two quantities together; and the result, — excluding 28 tithis, as before, — is tithis 6,
gh. 19, p. 55; which is the tithi-spashṭa-kêndra at the end of the given tithi, Âshâḍha
śukla 12.
With this argument, we turn to Table IV. p. 20, for the parâkhya-correction. In
this Table, the correction is given for tithis and ghaṭîs, at intervals of ten ghaṭîs. Thus,
for the tithi-spashṭa-kêndra, tithis 6, gh. 10, the parâkhya is gh. 24, p. 10; and for tithis
6, gh. 20, it is gh. 24, p. 19. The difference, 9 palas, is shewn in the last column of the
Table, and would serve to calculate the exact parâkhya for the tithi-spashṭa-kêndra. But
here it is sufficiently close for our purposes to take the parâkhya as gh. 24, p. 19.
Under (a) the abdapa, enter this parâkhya below the sum of the tithi-bhôga and the
solar equivalent of 96 tithis, and, — as is indicated by the sign plus at the top of col. 1 in
Table IV., — add it to that sum.
The result, vâras 95, gh. 47, p. 43, gives the number of days, and parts of a day,
that had elapsed, from sunrise on the day of the Mêsha-Saṁkrânti, up to the end of
the apparent Âshâḍha śukla 12. To the days, 95, add 1, the week-day of the Mêsha-
Saṁkrânti. Divide the sum, 96, by 7, and the result is 13 weeks, and 5 days over; which
shews that the current week-day of Âshâḍha śukla 12 was the fifth day in the week; that
is Thursday. The remaining quantities, gh. 47, p. 43, shew the time after sunrise, on
that Thursday, on which the given tithi, Âshâḍha śukla 12, ended.
The Tables in Prof. K. L. Chhatre's book, however, are adapted to the meridian
of Bombay. The ghaṭîs and palas, therefore, of a tithi worked out by the method ex-
hibited above, are for Bombay; and are to be reckoned from mean sunrise at Bombay.
When the tithi is required for any other particular place, the difference of longitude in
time (one degree = ten palas) is to be added or subtracted, according as the place is east
or west of Bombay. In the present instance, as I learned after first working it out, the above
Śaka date was selected in consequence of its being the equivalent of the date, in Gupta-
Saṁvat 165, recorded in the pillar inscription of Budhagupta at Êraṇ in the Central
Provinces. We have therefore now to determine the tithi for Êraṇ itself. The longitude of
Bombay is 72° 51'; and that of Êraṇ is 78° 15'; both east of Greenwich. Êraṇ, therefore
is 5 degrees, 24 minutes, east from Bombay. Adding (5° 24' × 10 =) 54 palas to 47
ghaṭîs and 43 palas, which we have obtained above for Bombay, the tithi at Êraṇ is
gh. 48, p. 37, reckoned from mean sunrise on the same day, Thursday.

THE GUPTA INSCRIPTIONS; APPENDIX II. 155
The above result is sufficient for all practical purposes. But it is further to be noted
that the tithis in our Pañchângs are intended to be given from apparent sunrise. In
practice, however, so much minuteness is not always and everywhere attempted; at least,
in the present day, in the Dekkan. For this reason, it seems, Prof. K. L. Chhatre has
not noticed this point in his method exhibited above. But I will now give the tithi
in question from apparent sunrise at Êraṇ. Without going through the process, which is
rather too complicated to be given in the present paper, I will state only the result,
that the apparent sunrise at Êraṇ, on the day in question, took place gh. 1, p. 56, before
the mean sunrise; the latitude of Êraṇ used in the process, being 24° 5'. Adding, there-
fore, gh. 1, p. 56, to the above result from mean sunrise, we get gh. 50, p. 33, reckoned
from apparent sunrise, as the time at which the given tithi, Âshâḍha śukla 12, ended at
Êraṇ on the Thursday.
Before dismissing this part of the subject, I would point out that the calculation of a
tithi, by the above method, is not of necessity absolutely accurate, according to the pre-
sent absolutely accurate European Tables of the sun and the moon. Absolute accuracy,
in this sense, could be ensured only by working from the actual places or longitudes of the
sun and the moon, to be determined in strict accordance with the method prescribed for
that purpose. The tithi obtained by the method exhibited above, will differ, sometimes by
as much as ten ghaṭîs, from that which would be obtained from the apparent places of the
sun and the moon, actually calculated from Prof. K. L. Chhatre's Tables for the sun
and the moon. The difference, however, at full-moon and new-moon will be very small,
one ghaṭî at the greatest; and it reaches its maximum on the eighth tithi of the
bright and of the dark fortnight. But, in respect of this second possible method of Prof.
K. L. Chhatre, it must be stated that we have nothing to do with it in dealing with Hindu
tithis; for the reason that, with the exception of the phala-saṁskâra, the corrections
introduced by him in finding the apparent longitude of the moon, were not taken into account
by ancient Hindu astronomers.
And, on the other hand, the method exhibited above being in close agreement
with Hindu works, it may be claimed that the tithi obtained by it will differ but very little
from the tithi obtained by the method prescribed in the Sûrya-Siddhânta and other Hindu
works. The difference¹ will amount to five or six ghaṭîs at the utmost; and that in but
very few cases.
In order, however, that no room may be left for doubt, I have also calculated
the tithi in the present example actually by the Âryabhaṭîya or Ârya-Siddhânta of the first
Âryabhaṭa, the Brahma-Siddhânta of Brahmagupta, the Sûrya-Siddhânta, the Siddhânta-
¹ See page 149 above, note 1.
u 2

156 CORPUS INSCRIPTIONUM INDICARUM, VOL. III.
Śirōmaṇi, and the Ârya-Siddhânta of the second Âryabhaṭa.¹ I calculated it first for Ujjain,
i.e. for the Hindu meridian, reckoning from the mean sunrise there; and then turned it
into the tithi for Êraṇ. The longitude of Ujjain is 75° 43' east of Greenwich. I have
also calculated the ghaṭīs and palas from the apparent sunrise at Êraṇ; and all the results
are given in Table VI., on the upper part of page 157 below. From them we see that
the tithi fell on a Thursday,² according to all the authorities.
¹ i.e. the work that is usually called the Laghu-Ârya-Siddhânta.—There are two distinct
and separate works, each bearing the name of Âryabhaṭa as its author. The one published by Dr.
Kern contains one hundred and eighteen verses in the Âryâ metre, and is called Âryabhaṭīya in
general and by the author himself; but it may, and justly so, be called Ârya-Siddhânta; and it is
called so by many Hindu astronomers. The date of this work is Saka-Saṁvat 421 expired (A. D.
499-500). The other, which, I believe, has not yet been printed, contains about six hundred and
twenty-five verses in the Âryâ metre, divided into eighteen chapters. The date of it is not given.
But, from internal evidence, I find that it is later than the Brahma-Siddhânta of Brahmagupta
(Śaka-Saṁvat 550 expired; A.D. 628-29); and there is a reference to it in the Siddhânta-Śirōmaṇi
(Śaka-Saṁvat 1072 expired; A.D. 1150-51). Its date, therefore, lies somewhere between these two
limits. In the first verse of it, the author calls himself Âryabhaṭa; and his work, a Siddhânta, without
Laghu or any other epithet. In a manuscript copy of it, I find that it is named Mahâ-Siddhânta
at the end of some chapters, and Laghu-Ârya-Siddhânta at the end of others. For the sake of dis-
tinction and convenience, it is better to call the authors of these two works, the first and the second
Âryabhaṭa. The numbers of the revolutions of the planets &c. given, as belonging to the Ârya-
Siddhânta, in the Rev. E. Burgess' Translation of the Sûrya-Siddhânta and in some instances
those given by Prinsep (Prinsep's Essays, Vol. II. Useful Tables, n. 153) as belonging to the same
authority, belong really to the second Âryabhaṭa. Probably, when these gentlemen wrote, they had
not themselves seen the Siddhânta of the first Aryabhaṭa Gen. Sir A. Cunningham, also, was not,
it seems, aware of the two different Aryabhaṭas, when he wrote (Indian Eras, p. 88) "according to
Warren the number of days assigned by Âryabhaṭa to a Mahâyuga of 4,320,000 years is 1,577,917,500
in the south of India and 42 more in the MSS. preserved in Bengal." Of these two numbers, the
former belongs to the first, and the latter to the second, Âryabhaṭa.
² [i.e. on a Hindu Thursday.—It must be borne in mind that the Hindu week-day is reckoned, with
the solar day and night, from sunrise to sunrise, as stated at page 146 f. above; but the English week-
day, and the civil date coupled with it, from midnight to midnight. In comparing Hindu and English
dates, the only course is to take mean sunrise and mean midnight (6·0 A.M. and 12·0 P.M. respec-
tively), and to give, as the English equivalent, that week-day, with its civil date, which is actually
running during these eighteen hours, when of course the same week-day is running in India; i.e. the
week-day which is identical for the greater part by both the English and the Hindu reckonings. And,
if the difference in mean time between Greenwich and Ujjain, viz. 5 hours, 2 minutes, 52 seconds
(using the same longitude for Ujjain, 75° 43', taken from Keith Johnston's Atlas, that is used by Mr.
Sh. B. Dikshit for his calculations in this paper and for the Sâyana-Pañchâng) be taken into con-
sideration, of course the week-days of the two places are absolutely identical, except for the space
of 57 minutes, 8 seconds, or 2 ghaṭīs, 22·8 palas, at the end of the Hindu week-day; during that time,
while at Ujjain a Hindu Thursday for instance, is still running, at Greenwich the week-day will be
Friday. Owing to this, there may sometimes be a nominal discrepancy in the resulting English week-
day for a given tithi; but the instances will be few and far between, as very few tithis will be found
to end so late after sunrise; and the discrepancy will be confined mostly to such occurrences as the
heliacal rising of Jupiter (see, for instance, the double dates which it has been necessary to quote at
page 104 ff. above). In the present case, not one of the ending-points of the given tithi falls within
the debatable period; either for Ujjain, or for Êraṇ, which is appreciably to the east of Ujjain.—J. F. F.]

THE GUPTA INSCRIPTIONS; APPENDIX II. 157
TABLE VI.
The Times, after Sunrise, of the ending of a given Tithi.
Śaka-Saṁvat 406 expired = A.D. 484-85 current.
Âshâḍha (June-July); the bright fortnight; the 12th tithi; Suraguruvâra (Thursday).

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THE GUPTA INSCRIPTIONS; APPENDIX III. 161
APPENDIX III.
The Twelve-Year Cycle of Jupiter.
By Shankar Balkrishna Dikshit; Bombay Educational Department.
The names of the saṁvatsaras, or years, of the Twelve-Year Cycle of Jupiter,
are determined in accordance with the following rule in the Bṛihat-Saṁhitâ of Varâhamihira,
adhyâya viii. verse 1, — nakshatrêṇa sah=ôdayam upagachchhati yêna dêvapati-mantrî¹ |
tat-saṁjñaṁ vaktavyaṁ varshaṁ mâsa-kramêṇ=aiva ||, — "with whatever nakshatra
(Jupiter) the counsellor of (Indra) the lord of the gods attains (his) rising, the year is to
be spoken of (as) having the appellation of that (nakshatra), in accordance with the order
of the months."
Here, by the word udaya, 'rising,' we have to understand, not the daily rising of Jupi-
ter, but his heliacal rising. Jupiter becomes invisible for some days before and after
his conjunction with the sun. The sun's daily motion is faster than that of Jupiter. So,
when the sun in his course comes near Jupiter, the latter becomes invisible, on the west
side of the horizon; and he is then said to set. He remains from twenty-five to thirty-one
days in this state of invisibility. And, when he is left behind by the sun, he again becomes
visible, in the east; and then he is said to rise. Generally, in India, when the interval
between the daily settings or risings of the sun and Jupiter amounts to forty-four minutes
¹ The reading that I give, is from an old manuscript in my possession. But the commentator,
Utpala, explains the verse with the reading—nakshatrêṇa sah=ôdayam astaṁ vâ yêna yâti sura-
mantrî,—'with whatever nakshatra (Jupiter) the counsellor of the gods attains (his) rising or setting.'
It is curious that the text, in my manuscript copy, stands as given above. The copyists, however far
they might go wrong in ordinary transcribing, could not, unless intentionally, turn the letters mastaṁ
vâ yêna yâti sura, if they were original at all, into mupagachchhati yêna dêvapati. And Utpala him-
self gives the note—Ṛishiputr-âdibhiḥ udaya-nakshatra-mâsa-samjñâ-kramêṇa varshaṁ jñâtavyam
ity=uktam,—"it is said by Ṛishiputra and others, that the year is to be known according to the
order of the name of the month of the nakshatra of the rising (of Jupiter)." Also, all the other author-
ities, which I find giving the rule of naming the years of the cycle according to the risings of Jupiter,
make each year take its name from only the rising of the planet; not from its setting.—[The reading
in the text above, is the one adopted by Kern, in his edition of the Bṛihat-Saṁhitâ, p. 47. His
translation (Jour. R. As. Soc. N. S. Vol. V. p. 45) is—"each year (during which Jupiter completes a
twelfth part of his revolution) has to bear the name of the lunar mansion in which he rises; the years
follow each other in the same order as the lunar months." Both in his Various Readings, p. 6, and
in the note to his translation, he notices the reading sah=ôdayam astaṁ vâ yêna yâti sura-mantrî.
But he points out that "the comparison of the MSS leaves little doubt that this reading is a correc-
tion, suggested by the remark of Utpala, that, in case the planet should set in one and rise in another
nakshatra, only that name must be taken which agrees with the order of the month.—J. F. F.]

(of time), then the so-called setting or rising of Jupiter, i.e. his heliacal setting or rising,
takes place.
To such a system, as is taught in this verse of Varâhamihira, and by no less than
eleven other authorities, as will be seen further on, or determining the commencement of
a saṁvatsara of the Twelve-Year Cycle, and of naming it, from Jupiter's heliacal-rising,
I would give the name of the heliacal-rising system, in order to distinguish it from
the other system in which the duration and name of a saṁvatsara of the Twelve-Year
Cycle are determined from the particular sign of the zodiac in which Jupiter stands with
reference to his mean longitude, and which latter system, to be mentioned more full,
further on, I would name the mean-sign system.
Now, the years of the Sixty-Year Cycle of Jupiter and of the Twelve-Year Cycle ac-
cording to the mean-sign system, are determined by his mean longitude,¹ which sometimes
differs from his apparent longitude by as much as fifteen degrees. But, as the disappear-
ance or reappearance of Jupiter is no imaginary thing, it is evident that it can be calcu-
lated, and is to be calculated, only according to Jupiter's actual place, that is, his apparent
longitude (or right ascension), and not from his mean longitude. And, consequently, the
beginning of each saṁvatsara of the Twelve-Year Cycle according to the heliacal-rising
system, depends on Jupiter's apparent longitude at the time of his heliacal rising.
One revolution of Jupiter, in the zodiac, is completed in about twelve years; and, in
twelve years, there are twelve revolutions of the sun (that is, of the earth). So that, in
this period of about twelve years, there are only eleven conjunctions of the sun and Jupiter.
Therefore, in twelve years there are only eleven heliacal risings of Jupiter.² The interval
between two risings is generally 399 days. And thus, in each cycle of the heliacal-
rising system, there are only eleven saṁvatsaras in twelve years; the duration
of each being about 400 days, and one saṁvatsara, determined by the circumstances of
the particular cycle, being altogether omitted.
The names of the lunar months are used as the names of the saṁvatsaras of the
Twelve-Year Cycle of Jupiter. And the names of these months are given to the saṁvat-
saras, in accordance with the particular nakshatra in which Jupiter's heliacal rising takes
place. Of the twenty-seven nakshatras, two are assigned to each of nine of the twelve
months; and three to each of the remaining three months. The rule for this is given in
the Bṛihat-Saṁhitā, viii. 2, — varshâṇi Kârttik-âdîny=âgnêyâd bha-dvay-ânuyôgîni ।
kramaśas tri-bhaṁ tu pañchamam upântyam antyaṁ cha yad varsham ॥,—“ the years
Kârttika and others (that follow) combine two nakshatras, from (the nakshatra) belonging

¹ The mean longitude of a heavenly body is the longitude of an imaginary body, of the same
name, conceived to move uniformly with the mean motion of the real body.
² In his note on the Sûrya-Siddhânta, xiv. 17 (see the Rev. E. Burgess' Translation, p 271),
Prof. Whitney says that Jupiter “ would set and rise heliacally twelve times in each revolution, and

THE GUPTA INSCRIPTIONS; APPENDIX III 163
TABLE VIII.
Regulation of the Names of the Saṁvatsaras from the Nakshatras.