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THE INDIAN CALENDAR.

PART I.

The Hindu Calendar.

1. In articles 118 to 134 below are detailed the various uses to which this work may be applied. Briefly speaking our chief objects are three; firstly, to provide simple methods for converting any Indian date—luni-solar or solar—falling between the years A.D. 300 and 1900 into its equivalent date A.D., and vice versâ, and for finding the week-day corresponding to any such date; secondly, to enable a speedy calculation to be made for the determination of the remaining three of the five principal elements of an Indian pañchâṅga (calendar), viz., the nakshatra, yoga, and karaṇa, at any moment of any given date during the same period, whether that date be given in Indian or European style; and thirdly, to provide an easy process for the verification of Indian dates falling in the period of which we treat.

2. For securing these objects several Tables are given. Table I. is the principal Table, the others are auxiliary. They are described in Part III. below. Three separate methods are given for securing the first of the above objects, and these are detailed in Part IV.

All these three methods are simple and easy, the first two being remarkably so, and it is these which we have designed for the use of courts and offices in India. The first method (A) (Arts. 135, 136) is of the utmost simplicity, consisting solely in the use of an eye-table in conjunction with Table I., no calculation whatever being required. The second (B) is a method for obtaining approximate results by a very brief calculation (Arts. 137, 138) by the use of Tables I., III. and IX. The result by both these methods is often correct, and it is always within one or two days of the truth, the latter rarely. Standing by itself, that is, it can always, provided that the era and the original bases of calculation of the given date are known, be depended on as being within two days of the truth, and is often only one day out, while as often it is correct. When the week-day happens to be mentioned in the given date its equivalent, always under the above proviso, can be fixed correctly by either of these methods.^[1] The third method (C) is a method by which entirely correct results may be obtained by the use of Tables I. to XI. (Arts. 139 to 160), and though a little more complicated is perfectly simple and easy when once studied and understood. From these results the nakshatra, yoga, and karaṇa can be easily calculated.

3. Calculation of a date may be at once begun by using Part IV. below, but the process will be more intelligible to the reader if the nature of the Indian calendar is carefully explained to him beforehand, for this is much more intricate than any other known system in use.

Elements and Definitions.

4. The pañchâṅga. The pañchâṅga (calendar), lit. that which has five (pañcha) limbs (aṅgas), concerns chiefly five elements of time-division, viz., the vâra, tithi, nakshatra, yoga and karaṇa.

5. The vâra or week-day. The natural or solar day is called a sâvana divasa in Hindu Astronomy. The days are named as in Europe after the sun, moon, and five principal planets,^[2] and are called vâras (week-days), seven of which compose the week, or cycle of vâras. A vâra begins at sunrise. The week-days, with their serial numbers as used in this work and their various Sanskrit synonyms, are given in the following list. The more common names are given in italics. The list is fairly exhaustive but does not pretend to be absolutely so.

Days of the Week.

1.	Sunday.	Âdi,^[3] Aditya, Ravi, Ahaskara, Arka, Aruṇa, Bhaṭṭaraka, Aharpati, Bhâskara, Bradhna, Bhânu etc.
2.	Monday.	Soma, Abja, Chandramas, Chandra, Indu, Nishpati, Kshapâkara, etc.
3.	Tuesday.	Maṅgala, Aṅgâraka, Bhauma, Mahîsuta, Rohitâṅga.
4.	Wednesday.	Budha, Baudha, Rauhiṇeya, Saumya.
5.	Thursday.	Guru, Áṅgirasa, Bṛihaspati, Dhishaṇa, Surâchârya, Vâchaspati, etc.
6.	Friday.	Śukra, Bhârgava, Bhṛigu, Daityaguru, Kâvya, Uśanas, Kavi.
7.^[4]	Saturday.	Śani, Sauri, Manda.

Time-Divisions.

6. The Indian time-divisions. The subdivisions of a solar day (sâvana divasa) are as follow:

A	prativipala (sura) is equal to 0.006̣ of a second.
60	prativipalas	make 1 vipala (para, kâshṭha-kalâ) = 0.4 of a second.
60	vipalas	makedo. 1 pala (vighaṭî, vinâḍî) = 24 seconds.
60	palas	makedo. 1 ghaṭikâ (ghaṭi, daṇḍa, nâḍî, nâḍîkâ) = 24 minutes.
60	ghatikas	makedo. 1 divasa (dina, vâra, vâsara) = 1 solar day.
Again
10	vipalas	makedo. 1 prâṇa = 4 seconds.
6	prâṇas	makedo. 1 pala = 24 seconds.

7. The tithi, amâvâsyâ, pûrṇimâ. The moment of new moon, or that point of time when the longitudes of the sun and moon are equal, is called amâvâsyâ (lit. the "dwelling together" of the sun and moon). A tithi is the time occupied by the moon in increasing her distance from the sun by 12 degrees; in other words, at the exact point of time when the moon (whose apparent motion is much faster than that of the sun), moving eastwards from the sun after the amâvâsyâ, leaves the sun behind by 12 degrees, the first tithi, which is called pratipadâ or pratipad, ends; and so with the rest, the complete synodic revolution of the moon or one lunation occupying 30 tithis for the 360 degrees. Since, however, the motions of the sun and moon are always varying in speed^[5] the length of a tithi constantly alters. The variations in the length of a tithi are as follow, according to Hindu calculations:

	gh.	pa.	vipa.	h.	m.	s.
Average or mean length	59	3	40.23	23	37	28.092
Greatest length	65	16	0	26	6	24
Least length	53	56	0	21	34	24

The moment of full moon, or that point of time when the moon is furthest from the sun,—astronomically speaking when the difference between the longitudes of the sun and moon amounts to 180 degrees—is called pûrṇimâ. The tithi which ends with the moment of amâvâsyâ is itself called "amâvâsyâ", and similarly the tithi which ends with the moment of full moon is called "pûrṇimâ." (For further details see Arts. 29, 31, 32.)

8. The nakshatra. The 27th part of the ecliptic is called a nakshatra, and therefore each nakshatra occupies (360°/27 =) 13° 20′. The time which the moon (whose motion continually varies in speed) or any other heavenly body requires to travel over the 27th part of the ecliptic is also called a nakshatra. The length of the moon's nakshatra is:

	gh.	pa.	vipa.	h.	m.	s.
Mean	60	42	53.4	24	17	9.36
Greatest	66	21	0	26	32	24
Least	55	56	0	22	22	24

It will be seen from this that the moon travels nearly one nakshatra daily. The daily nakshatra of the moon is given in every pañchâṅg (native almanack) and forms one of its five articles. The names of the 27 nakshatras will be found in Table VIII., column 7. (See Arts. 38, 42.)

9. The yoga. The period of time during which the joint motion in longitude, or the sum of the motions, of the sun and moon is increased by 13° 20′, is called a yoga, lit. "addition". Its length varies thus:

	gh.	pa.	vipa.	h.	m.	s.
Mean	56	29	21.75	22	35	44.7
Greatest	61	31	0	24	36	24
Least	52	12	0	20	52	48

The names of the 27 yogas will be found in Table VIII., col. 12. (See Art. 39.)

10. The karaṇa. A karaṇa is half a tithi, or the time during which the difference of the longitudes of the sun and moon is increased by 6 degrees. The names of the karaṇas are given in Table VIII., cols. 4 and 5. (See Art. 40.)

11. The paksha. The next natural division of time greater than a solar day is the paksha (lit. a wing^[6]) or moon's fortnight. The fortnight during which the moon is waxing has several names, the commonest of which are śukla or śuddha (lit. "bright", that during which the period of the night following sunset is illuminated in consequence of the moon being above the horizon). The fortnight during which the moon is waning is called most commonly kṛishṇa or bahula or vadya (lit. "black", "dark", or the fortnight during which the portion of the night following sunset is dark in consequence of the moon being below the horizon). The first fortnight begins with the end of amâvâsyâ and lasts up to the end of pûrṇimâ; the second lasts from the end of pûrṇimâ to the end of amâvâsyâ. The words "pûrva" (former or first) and "apara" (latter or second) are sometimes used for śukla and kṛishṇa respectively. "Śudi" (or "sudi") is sometimes used for śukla, and "vadi" or "badi" for kṛishṇa. They are popular corruptions of the words "śuddha" and "vadya" respectively.

12. Lunar months. The next natural division of time is the lunation, or lunar month of two lunar fortnights, viz., the period of time between two successive new or full moons. It is called a chândra mâsa, or lunar month, and is the time of the moon's synodic revolution.^[7]

The names of the lunar months will be found in Table II., Parts i. and ii., and Table III., col. 2, and a complete discussion on the luni-solar month system of the Hindus in Arts. 41 to 51. (For the solar months sec Arts. 22 to 24.)

13. Amânta and pûrṇimânta systems. Since either the amâvâsyâ or pûrṇimâ, the new moon or the full moon, may be taken as the natural end of a lunar month, there are in use in India two schemes of such beginning and ending. By one, called the amânta system, a month ends with the moment of amâvâsyâ or new moon; by the other it ends with the pûrṇimâ or full moon, and this latter is called a pûrṇimânta month. The pûrṇimânta scheme is now in use in Northern India, and the amânta scheme in Southern India. There is epigraphical evidence to show that the pûrṇimânta scheme was also in use in at least some parts of Southern India up to about the beginning of the 9^th century A.D.^[8] The Mârvâḍis of Northern India who, originally from Mârwâr, have come to or have settled in Southern India still use their pûrṇimânta arrangement of months and fortnights; and on the other hand the Dakhanis in Northern India use the scheme of Amânta fortnights and months common in their own country.

14. Luni-solar month names. The general rule of naming the lunar months so as to correspond with the solar year is that the amanta month in which the Mêsha saṅkrânti or entrance of the sun into the sign of the zodiac Mesha, or Aries, occurs in each year, is to be called Chaitra, and so on in succession. For the list and succession see the Tables. (See Arts. 41—43.)

15. The solar year—tropical, sidereal, and anomalistic. Next we come to the solar year, or period of the earth's orbital revolution, i.e., the time during which the annual seasons complete their course. In Indian astronomy this is generally called a varsha, lit. "shower of rain", or "measured by a rainy season".

The period during which the earth makes one revolution round the sun with reference to the fixed stars,^[9] is called a sidereal year.

The period during which the earth in its revolution round the sun passes from one equinox or tropic to the same again is called a tropical year. It marks the return of the same season to any given part of the earth's surface. It is shorter than a sidereal year because the equinoxes have a retrograde motion among the stars, which motion is called the precession of the equinoxes. Its present annual rate is about 50″.264.^[10]

Again, the line of apsides has an eastward motion of about 11″.5 in a year; and the period during which the earth in its revolution round the sun comes from one end of the apsides to the same again, i. e., from aphelion to aphelion, or from perihelion to perihelion, is called an anomalistic year.^[11]

The length of the year varies owing to various causes, one of which is the obliquity of the ecliptic,^[12] or the slightly varying relative position of the planes of the ecliptic and the equator. Leverrier gives the obliquity in A.D. 1700 as 23° 28′ 43″.22, in A.D. 1800 as23° 27′ 55″.63, and in A.D. 1900 as 23° 17′ 08″.03. The various year-lengths for A.D. 1900, as calculated by present standard authorities, are as follows:

	d.	h.	m.	s.
Mean Sidereal solar year	365	6	9	9.29
Do. AnomalisticTropical yeardo.	365	5	48	45.37
Do. Anomalistic yeardo.	365	6	13	48.61

16. Kalpa. Mahâyuga. Yuga. Julian Period. A kalpa is the greatest Indian division of time. It consists of 1000 mahâyugas. A mahâyuga is composed of four yugas of different lengths, named Kṛita, Tretâ, Dvâpara, and Kali. The Kali-yuga consists of 432,000 solar years. The Dvâpara yuga is double the length of the Kali. The Tretâ-yuga is triple, and the Kṛita-yuga quadruple of the Kali. A mahâyuga therefore contains ten times the years of a Kali-yuga, viz., 4,320,000. According to Indian tradition a kalpa is one day of Brahman, the god of creation. The Kaliyuga is current at present; and from the beginning of the present kalpa up to the beginning of the present Kali-yuga 4567 times the years of a Kali-yuga have passed. The present Kaliyuga commenced, according to the Sûrya Siddhânta, an authoritative Sanskrit work on Hindu astronomy, at midnight on a Thursday corresponding to 17th—18th February, 3102 B. C., old style; by others it is calculated to have commenced on the following sunrise, viz., Friday, 18th February. According to the Sûrya and some other Siddhântas both the sun and moon were, with reference to their mean longitude, precisely on the beginning point of the zodiacal sign Aries, the Hindu sign Mesha, when the Kali-yuga began.

European chronologists often use for purposes of comparison the 'Julian Period' of 7980 years, beginning Tuesday 1st January, 4713 B. C. The 18th February, 3102 B. C., coincided with the 588,466th day of the Julian Period.

17. Siddhanta year-measurement. The length of the year according to different Hindu authorities is as follows:

Siddhântas.	Hindu reckoning.					European reckoning.
	days.	gh.	pa.	vipa.	pra. vi.	days.	h.	mins.	sec.
The Vedâṅga Jyotisha	366	0	0	0	0	366	0	0	0
The Paitâmaha Siddhânta^[13]	365	21	25	0	0	365	8	34	0
The Romaka Siddhanta„	365	14	48	0	0	365	5	55	12
The Pauliśa^[14] Siddhanta„	365	15	30	0	0	365	6	12	0
The original Sûrya Siddhânta	365	15	31	30	0	365	6	12	36
The Present Sûrya, Vâsishṭha, Śâkalya-Brahma, Romaka, & Soma Siddhântas	365	15	31	31	24	365	6	12	36.56
The first Ârya Siddhânta^[15] (A. D. 499)	365	15	31	15	0	365	6	12	30
The Brahma Siddhânta by Brahma-gupta (A. D. 628)	365	15	30	22	30	365	6	12	9
The second Ârya Siddhânta	365	15	31	17	6	365	6	12	30.84
The Parâśara Siddhânta^[16]	365	15	31	18	30	365	6	12	31.6
Râjamṛigâṅka^[17] Siddhanta„ (A. D. 1042)	365	15	31	17	17.3	365	6	12	30.915

It will be seen that the duration of the year in all the above works except the first three approximates closely to the anomalistic year; and is a little greater than that of the sidereal year. In some of these works theoretically the year is sidereal; in the case of some of the others it cannot be said definitely what year is meant; while in none is it to be found how the calculations were made. It may, however, be stated roughly that the Hindu year is sidereal for the last 2000 years.

18. The year as given in each of the above works must have been in use somewhere or another in India at some period; but at present, so far as our information goes, the year of only three works is in use, viz., that of the present Sûrya Siddhânta, the first Ârya Siddhânta. and the Râjamṛigâṅka.

The Siddhântas and other astronomical works.

19. It will not be out of place here to devote some consideration to these various astronomical works; indeed it is almost necessary to do so for a thorough comprehension of the subject. Many other Siddhântas and Karaṇas are extant besides those mentioned in the above list. We know of at least thirty such works, and some of them are actually used at the present day in making calculations for preparing almanacks.^[18] Many other similar works must, it is safe to suppose, have fallen into oblivion, and that this is so is proved by allusions found in the existing books. Some of these works merely follow others, but some contain original matter. The Karaṇas give the length of the year, and the motions and places at a given time of the sun, moon, and planets, and their apogees and nodes, according to the standard Siddhânta. They often add corrections of their own, necessitated by actual observation, in order to make the calculations agree. Such a correction is termed a bîja. Generally, however, the length of the year is not altered, but the motions and places are corrected to meet requirements.

As before stated, each of these numerous works, and consequently the year-duration and other elements contained in them, must have been in use somewhere or another and at some period or another in India. At the present time, however, there are only three schools of astronomers known; one is called the Saura-paksha, consisting of followers of the present Sûrya Siddhânta; another is called the Ârya-paksha, and follows the first Ârya Siddhânta; and the third is called the Brahma-paksha, following the Râjamṛigâṅka, a work based on Brahmagupta's Brahma Siddhânta, with a certain bîja. The distinctive feature of each of these schools is that the length of the year accepted in all the works of that school is the same, though with respect to other elements they may possibly disagree between themselves. The name Râjamṛigâṅka is not now generally known, the work being superseded by others; but the year adopted by the present Brâhma-school is first found, so far as our information goes, in the Râjamṛigâṅka, and the three schools exist from at least A. D. 1042, the date of that work.

20. It is most important to know what Siddhântas or Karaṇas were, or are now, regarded as standard authorities, or were, or are, actually used for the calculations of pañchâṅgs (almanacks) during particular periods or in particular tracts of country.^[19] for unless this is borne in mind we shall often go wrong when we attempt to convert Indian into European dates. The sketch which follows must not, however, be considered as exhaustive. The original Sûrya-Siddhânta was a standard work in early times, but it was superseded by the present Sûrya-Siddhânta at some period not yet known, probably not later than A.D. 1000. The first Ârya-Siddhânta, which was composed at Kusumapura (supposed to be Patṇâ in Bengal), came into use from A.D. 499.^[20] Varâhamihira in his Pañchasiddhântikâ (A.D. 505) introduced a bîja to Jupiter's motion as given in the original Sûrya-Siddhânta, but did not take it into account in his rule (see Art. 62 below) for calculating a samvatsara. Brahmagupta composed his Brahma-Siddhânta in A. D. 628. He was a native of Bhillamâla (the present Bhinmâl), 40 miles to the north-west of the Abu mountains. Lalla, in his work named Dhî-vṛiddhida, introduced a bîja to three of the elements of the first Ârya-Siddhânta, namely, the moon, her apogee, and Jupiter, i.e., three out of the six elements with which we are concerned. Lalla's place and date are not known, but there is reason to believe that he flourished about A.D. 638. The date and place of the second Ârya-Siddhânta are also not known, but the date would appear to have been about A.D. 950. It is alluded to by Bhâskarâchârya (A.D. 1150), but does not seem to have been anywhere in use for a long time. The Râjamṛigâṅka (A.D. 1042) follows the Brahma-Siddhânta,^[21] but gives a correction to almost all its mean motions and places, and even to the length of the year. The three schools—Saura, Ârya and Brahma—seem to have been established from this date if not earlier, and the Brahma-Siddhânta in its orginal form must have then dropped out of use. The Karaṇa-prakâśa, a work based on the first Ârya-Siddhânta as corrected by Lalla's bîja, was composed in A.D. 1092, and is considered an authority even to the present day among many Vaishnavas of the central parts of Southern India, who are followers of the Ârya-Siddhânta. Bhâskarâchârya's works, the Siddhânta Śiromaṇi (A.D. 1150) and the Karaṇa-Kutûhala (A.D. 1183) are the same as the Râjamṛigâṅka in the matter of the calculation of a pañchâṅg. The Vâkkya-Karaṇa, a work of the Arya school, seems to have been accepted as the guide for the preparation of solar pañchâṅgs in the Tamil and Malayalam countries of Southern India from very ancient times, and even to the present day either that or some similar work of the Ârya school is so used. A Karaṇa named Bhâsvatî was composed in A.D. 1099, its birthplace according to a commentator being Jagannâtha (or Purî) on the east coast. The mean places and motions given in it are from the original Sûrya-Siddhânta as corrected by Varâhamihira's bîja,^[22] and it was an authority for a time in some parts of Northern India. Vâvilâla Kochchanna, who resided somewhere in Telingaṇa, composed a Karaṇa in 1298 A.D. He was a strict follower of the present Sûrya-Siddhânta, and since his day the latter Siddhânta has governed the preparation of all Telugu luni-solar calendars. The Makaranda, another Karaṇa, was composed at Benares in A.D. 1478, its author following the present Sûrya-Siddhânta, but introducing a bîja. The work is extensively used in Northern India in the present day for pañchâṅga calculations. Bengalis of the present day are followers of the Saura school, while in the western parts of Northern India and in some parts of Gujarât the Brâhma school is followed. The Graha-lâghava, a Karaṇa of the Saura school, was composed by Ganesa Daivjña of Nandigrâma (Nândgâm), a village to the South of Bombay, in A.D. 1520. The same author also produced the Bṛihat and Laghutithichintâmaṇis in A.D. 1525, which may be considered as appendices to the Graha-lâghava. Gaṇeśa adopted the present Sûrya Siddhânta determinations for the length of the year and the motions and places of the sun and moon and their apogees, with a small correction for the moon's place and the sun's apogee; but he adopted from the Ârya Siddhânta as corrected by Lalla the figures relating to the motion and position of Jupiter.

The Graha-lâghava and the Laghutithichintâmaṇi were used, and are so at the present day, in preparing pañchâṅgs wherever the Mahrathi language was or is spoken, as well as in some parts of Gujarât, in the Kanarese Districts of the Bombay and Madras Presidencies, and in parts of Haidarâbâd, Maisûr, the Berars, and the Central Provinces. Mahratha residents in Northern India and even at Benares follow these works.

21. It may be stated briefly that in the present day the first Ârya-Siddhânta is the authority in the Tamil and Malayâḷam countries of Southern India;^[23] the Brâhma-paksha obtains in parts of Gujarât and in Râjputâna and other western parts of Northern India; while in almost all other parts of India the present Sûrya-Siddhânta is the standard authority. Thus it appears that the present Sûrya-Siddhânta has been the prevailing authority in India for many centuries past down to the present day, and since this is so, we have chiefly followed it in this work.^[24]

The bîja as given in the Makaranda (A. D. 1478) to be applied to the elements of the Sûrya-Siddhânta is generally taken into account by the later followers of the Sûrya-Siddhânta, but is not met with in any earlier work so far as our information goes. We have, therefore, introduced it into our tables after A.D. 1500 for all calculations which admit of it. The bîja of the Makaranda only applies to the moon's apogee and Jupiter, leaving the other four elements unaffected.

Further details. Contents of the Pañchâṅga.

22. The Indian Zodiac. The Indian Zodiac is divided, as in Europe, into 12 parts, each of which is called râśi or "sign". Each sign contains 30 degrees, a degree being called an aṁśa. Each aṁśa is divided into 60 kalâs (minutes), and each kala into 60 vikalâs (seconds). This sexagesimal division of circle measurement is, it will be observed, precisely similar to that in use in Europe.^[25]

23. The Saṅkrânti. The point of time when the sun leaves one zodiacal sign and enters another is called a saṅkrânti. The period between one saṅkrânti and another, or the time required for the sun to pass completely through one sign of the zodiac, is called a saura mâsa, or solar month. Twelve solar months make one solar year. The names of the solar months will be found in Table II., Part ii., and Table III., col. 5. A saṅkrânti on which a solar month commences takes its name from the sign-name of that month. The Mesha saṅkrânti marks the vernal equinox, the moment of the sun's passing the first point of Aries. The Karka saṅkrânti, three solar months later, is also called the dakshinâyana ("southward-going") saṅkrânti: it is the point of the summer solstice, and marks the moment when the sun turns southward. The Tulâ saṅkrânti, three solar months later, marks the autumnal equinox, or the moment of the sun's passing the first point of Libra. The Makara saṅkrânti, three solar months later still, is also called the uttarâyana saṅkrânti ("northward-going"). It is the other solstitial point, the point or moment when the sun turns northward. When we speak of "saṅkrântis" in this volume we refer always to the nirayana saṅkrântis, i.e., the moments of the sun's entering the zodiacal signs, as calculated in sidereal longitude—longitude measured from the fixed point in Aries—taking no account of the annual precession of the equinoxes—(nirayana = "without movement", excluding the precession of the solstitial—ayana—points). But there is also in Hindu chronology the sayana saṅkrânti (sa-ayana = "with movement", including the movement of the ayana points), i.e., a saṅkrânti calculated according to tropical longitude—longitude measured from the vernal equinox, the precession being taken into account. According to the present Sûrya-Siddhânta the sidereal coincided with the tropical signs in K. Y. 3600 expired, Śaka 421 expired, and the annual precession is 54″. By almost all other authorities the coincidence took place in K. Y. 3623 expired, Śaka 444 expired, and the annual precession is (1′) one minute. (The Siddhânta Śiromaṇi, however, fixes this coincidence as in K. Y. 3628). Taking either year as a base, the difference in years between it and the given year, multiplied by the total amount of annual precession, will shew the longitudinal distance by which, in the given year, the first point of the tropical (sâyana) sign precedes the first point of the sidereal (nirayana) sign. Professor Jacobi (Epig. Ind., Vol. 1, p. 422, Art. 39) points out that a calculation should be made "whenever a date coupled with a saṅkrânti does not come out correct in all particulars. For it is possible that a sâyana saṅkrânti may be intended, since these saṅkrântis too are suspicious moments."

We have, however, reason to believe that sâyana saṅkrântis have not been in practical use for the last 1600 years or more. Dates may be tested according to the rule given in Art. 160(a).

It will be seen from cols. 8 to 13 of Table II., Part ii., that there are two distinct sets of names given to the solar months. One set is the set of zodiac-month-names ("Mesha" etc.), the other has the names of the lunar months. The zodiac-sign-names of months evidently belong to a later date than the others, since it is known that the names of the zodiacal signs themselves came into use in India later than the lunar names, "Chaitra" and the rest.^[26] Before sign-names came into use the solar months must have been named after the names of the lunar months, and we find that they are so named in Bengal and in the Tamil country at the present day.^[27]

24. Length of months. It has been already pointed out that, owing to the fact that the apparent motion of the sun and moon is not always the same, the lengths of the lunar and solar months vary. We give here the lengths of the solar months according to the Sûrya and Ârya-Siddhântas.

Serial No.	Name of the month.			Duration of each month.
	Sign-name.	Tamil name.	Bengâli name.	By the Ârya-Siddhânta.							By the Sûrya-Siddhânta.
	Sign-name.	Tamil name.	Bengâli name.	days	gh.	pa.	days	hrs.	mn.	sec.	days	gh.	pa.	days	hrs.	mn.	sec.
1	Mesha	Śittirai (Chittirai)	Vaiśâkha	30	55	30	30	22	12	0	30	56	7	30	22	26	48
2	Vṛishabha	Vaigâśi, or Vaiyâśi	Jyeshṭha	31	24	4	31	9	37	36	31	25	13	31	10	5	12
3	Mithuna	Âni	Âshâḍha	31	36	26	31	14	34	24	31	38	41	31	15	28	24
4	Karka	Âḍi	Srâvaṇa	31	28	4	31	11	13	36	31	28	31	31	11	24	24
5	Siṁha	Âvaṇi	Bhâdrapada	31	2	5	31	0	50	0	31	1	7	31	0	26	48
6	Kanyâ	Puraṭṭâdi, or Puraṭṭâśi	Âśvina	30	27	24	30	10	57	36	30	26	29	30	10	35	36
7	Tulâ	Aippaśi, or Arppiśi, or Appiśi	Kârttika	29	54	12	29	21	40	48	29	53	36	29	21	26	24
8	Vṛiśchika	Kârttigai	Mârgaśîrsha	29	30	31	29	12	12	24	29	29	25	29	11	46	0
9	Dhanus	Mârgaḷi	Pausha	29	21	2	29	8	24	48	29	19	4	29	7	37	36
10	Makara	Tai	Mâgha	29	27	24	29	10	57	36	29	26	53	29	10	45	12
11	Kumbha	Mâśi	Phâlguna	29	48	30	29	19	24	0	29	49	13	29	19	41	12
12	Mîna	Paṅguni	Chaitra	30	20	19¼	30	8	7	42	30	21	12.52	30	8	29	0.56
				365	15	31¼	365	6	12	30	365	15	31.52	365	6	12	36.56

For calculation of the length by the Sûrya-Siddhânta the longitude of the sun's apogee is taken as 77° 16′, which was its value in A. D. 1137, a date about the middle of our Tables. Even if its value at our extreme dates, i.e., either in A. D. 300 or 1900, were taken the lengths would be altered by only one pala at most. By the Ârya-Siddhânta the sun's apogee is taken as constantly at 78°.^[28]

The average (mean) length in days of solar and lunar months, and of a lunar year is as follows:

	Sûrya-Siddhânta	Modern science
Solar month (1/12 of a sidereal year)	30.438229707	30.438030.
Lunar month	29.530587946	29.530588.
Lunar year (12 lunations)	354.36705535	354.367056.

25. Adhika mâsas. Calendar used. A period of twelve lunar months falls short of the solar year by about eleven days, and the Hindus, though they use lunar months, have not disregarded this fact; but in order to bring their year as nearly as possible into accordance with the solar year and the cycle of the seasons they add a lunar month to the lunar year at certain intervals. Such a month is called an adhika or intercalated month. The Indian year is thus either solar or luni-solar. The Muhammadan year of the Hijra is purely lunar, consisting of twelve lunar months, and its initial date therefore recedes about eleven days in each year. In luni-solar calculations the periods used are tithis and lunar months, with intercalated and suppressed months whenever necessary. In solar reckoning solar days and solar months are alone used. In all parts of India luni-solar reckoning is used for most religious purposes, but solar reckoning is used where it is prescribed by the religious authorities. For practical civil purposes solar reckoning is used in Bengal and in the Tamil and Malayalam countries of the Madras Presidency; in all other parts of the country luni-solar reckoning is adopted.

26. True and mean saṅkrântis. Śodhya. When the sun enters one of the signs of the zodiac, as calculated by his mean motion, such an entrance is called a mean saṅkrânti; when he enters it as calculated by his apparent or true motion, such a moment is his apparent or true^[30] saṅkrânti. At the present day true saṅkrântis are used for religious as well as for civil purposes. In the present position of the sun's apogee, the mean Mesha saṅkrânti takes place after the true saṅkrânti, the difference being two days and some ghaṭikâs. This difference is called the śodhya. It differs with different Siddhântas, and is not always the same even by the same authority. We have taken it as 2d. 10 gh. 14 p. 30 vipa. by the Sûrya-Siddhânta, and 2d. 8 gh. 51 p. 15 vipa. by the Ârya-Siddhânta. The corresponding notion in modern European Astronomy is the equation of time. The śodhya is the number of days required by the sun to catch up the equation of time at the vernal equinox.

27. It must be remembered that whenever we use the word "saṅkrânti" alone, (e.g., "the Mesha-saṅkrânti") the apparent and not the mean nirayana saṅkrânti is meant.

28. The beginning of a solar month. Astronomically a solar month may begin, that is a saṅkrânti may occur, at any moment of a day or night; but for practical purposes it would be inconvenient to begin the month at irregular times of the day. Suppose, for example, that a Makara-saṅkrânti occurred 6 hours 5 minutes after sunrise on a certain day, and that two written agreements were passed between two parties, one at 5 hours and another at 7 hours after sunrise. If the month Makara were considered to have commenced at the exact moment of the Makara-saṅkrânti, we should have to record that the first agreement was passed on the last day of the month Dhanus, and the second on the first day of Makara, whereas in fact both were executed on the same civil day. To avoid such confusion, the Hindus always treat the beginning of the solar month as occurring, civilly, at sunrise. Hence a variation in practice.

(1) (a) In Bengal, when a saṅkrânti takes place between sunrise and midnight of a civil day the solar month begins on the following day; and when it occurs after midnight the month begins on the next following, or third, day. If, for example, a saṅkrânti occurs between sunrise and midnight of a Friday, the month begins at sunrise on the next day, Saturday; but if it takes place after midnight of Friday^[31] the month begins at sunrise on the following Sunday. This may be termed the Bengal Rule. (b) In Orissa the solar month of the Amli and Vilayati eras begins civilly on the same day as the saṅkrânti, whether this takes place before midnight or not. This we call the Orissa Rule.

(2) In Southern India there are two rules, (a) One is that when a saṅkrânti takes place after sunrise and before sunset the month begins on the same day, while if it takes place after sunset the month begins on the following day; if, for example, a saṅkrânti occurs on a Friday between sunrise and sunset the month begins on the same day, Friday, but if it takes place at any moment of Friday night after sunset the month begins on Saturday.^[32] (b) By another rule, the day between sunrise and sunset being divided into five parts, if a saṅkrânti takes place within the first three of them the month begins on the same day, otherwise it begins on the following day. Suppose, for example, that a saṅkrânti occurred on a Friday, seven hours after sunrise, and that the length of that day was 12 hours and 30 minutes; then its fifth part was 2 hours 30 minutes, and three of these parts are equal to 7 hours 30 minutes. As the saṅkrânti took place within the first three parts, the month began on the same day, Friday; but if the saṅkrânti had occurred 8 hours after sunrise the month would have begun on Saturday. The latter (b) rule is observed in the North and South Malayâḷam country, and the former (a) in other parts of Southern India where the solar reckoning is used, viz., in the Tamil and Tinnevelly countries.^[33] We call a. the Tamil Rule: b. the Malabar Rule.

29. Pañchâṅgs. Before proceeding we revert to the five principal articles of the pañchâṅg. There are 30 tithis in a lunar month, 5 to each fortnight. The latter are generally denoted by the ordinary numerals in Sanskrit, and these are used for the fifteen tithis of each fortnight. Some tithis are, however, often called by special names. In pañchâṅgs the tithis are generally particularized by their appropriate numerals, but sometimes by letters. The Sanskrit names are here given.^[34]

Tithis.	Sanskrit Names.	Vulgar Names.	Tithis.	Sanskrit Names.	Vulgar Names.
1	Pratipad, Pratipadâ, Prathamâ	Pâdvâ, Pâdyami	9	Navamî
2	Dvitîyâ	Bîja, Vidiya	10	Daśamî
3	Tritiyâ	Tija, Tadiya	11	Ekâdaśî
4	Chaturthî	Chauth, Chauthi	12	Dvâdaśî	Bâras
5	Pañchamî		13	Trayôdaśî	Teras
6	Shashṭhî	Saṭh	14	Chaturdaśî
7	Saptamî		15	Pûrṇimâ, Paurṇimâ, Pûrṇamâsi, Pañchadaśi	Punava, Punnamî
8	Ashtamî		30	Amâvâsyâ, Darśa, Pañchadaśî

The numeral 30 is generally applied to the amâvâsyâ (new moon day) in pañchâṅgs, even in Northern India where according to the pûrṇimânta system the dark fortnight is the first fortnight of the month and the month ends with the moment of full moon, the amâvâsyâ being really the 15th tithi.

30. That our readers may understand clearly how a Hindu pañchâṅg is prepared and what information it contains, we append an extract from an actual pañchâṅg for Śaka 1816, expired, A. D. 1894—95, published at Poona in the Bombay Presidency.^[35]

Extract from an actual Pañchâṅga.

Śaka 1816 expired (1817 current) (A. D. 1894) amânta Bhâdrapada, śukla-paksha. Solar months Siṁha and Kanyâ; Muhamaddan months Śafar and Rabî-ul-awwal. English months August and September.

Tithi.

Vâra.

gh.

pa.

Nakshatra.

gh.

pa.

Yoga.

gh.

pa.

Karaṇa.

gh.

pa.

Moon's phase.

Length Day.

Solar date.

Muhammadan date.

Date A. D.

OTHER PARTICULARS.

iukla-pakslia. Solar month. " Sn'iika

</PARAGRAPH></PARAGRAPH>1 Vfira. Fri. gl-. !»■ Kalisliatra. b'!"- jia. Yoga. gh. l-a. Karaua. b'l'- pa. i 1 s

</PARAGRAPH></PARAGRAPH>"3 S i S 1

</PARAGRAPH></PARAGRAPH>43 59 Pui-TaPhalguni: 40 16 Siddha 31 22 Kiiiistagbna 16 30 Sii!iha*15 gh. pa. 30 59 16 29 31

</PARAGRAPH></PARAGRAPH>2 Sat. 39 47 Uttara Phalguni : 37 57 Sidhya 25 23 Baiava 11 53 Kauj-a 30 57 17 30 1

</PARAGRAPH></PARAGRAPH>3 Sun. 36 31 Hasta 36 29 Subha 19 31 Taitila 8 9 Kanya 30 54 18 1 2

</PARAGRAPH></PARAGRAPH>4 >Ion. 34 23 Chitra 36 7 Sukla 14 50 Vauij 5 27 Kanya 6 30 52 19 2 3

</PARAGRAPH></PARAGRAPH>5 Tues. 33 26 Svati 36 52 Brahman 11 7 Bava 3 54 Tula 30 49 20 3 4

</PARAGRAPH></PARAGRAPH>6 ■Wed. 33 58 Vis&kha 38 58 Aindra .8 24 Kaulava 3 42 Tula 23 30 45 21 4 5

</PARAGRAPH></PARAGRAPH>7 Thurs. 35 29 Anuradia 42 19 Vaidhriti 6 36 Gara 4 44 Vrischi: 30 44 22 5 6

</PARAGRAPH></PARAGRAPH>8 Fri. 38 16 Jyeshthu 46 48 Visbkambha 5 49 Visbti 6 53 Vris:47 30 41 23 6 7

</PARAGRAPH></PARAGRAPH>9 Sat. 42 9 MOla 52 13 Priti 6 3 Baiava 10 13 Dbanus 30 38 24 7 8

</PARAGRAPH></PARAGRAPH>10 Sun. 46 48 Pflrva Ashudha 58 11 Ayushmat 6 53 Taitila 14 28 Dbanus 30 36 25 8 9

</PARAGRAPH></PARAGRAPH>11 Mon. 51. 43 Uttara AshSdha 60 </LINE> Saubhfigya 8 1 Vanij 19 16 Uba:15 30 33 26 9 10

</PARAGRAPH></PARAGRAPH>12 Tues. 56 44 Uttara Ashadhu 4 35 Sobbana 9 29 Bava 24 14 Makara 30 30 27 10 11

</PARAGRAPH></PARAGRAPH>13 Wed. 60 </LINE> Sravaua 10 59 Atiganila 10 58 Kaulava 29 3 Maka ; 44 30 28 28 11 12

</PARAGRAPH></PARAGRAPH>13 Thurs. 1 23 Dhanishthu 16 45 Sukavman 11 54 Taitila 1 23 Kumbha 30 25 1 29 12 13

</PARAGRAPH></PARAGRAPH>U Fri. 5 18 Satabhishaj 21 52 Dbriti 12 26 Vanij 5 18 Kumbha 30 22 1 30 13 14

</PARAGRAPH></PARAGRAPH>15 Sal. 8 11 Pfirva Hhudru: 26 4 Sula 12 7 Bava 8 11 Kum:10 30 20 1 31 14 15

</PARAGRAPH></PARAGRAPH>Aiitanta Bhadrapada krisltnapaksha. Thurs. Fri. 26 17 Bbarani Robiui Mrigasiras Ardra Mugha Uttara I'bniguni Vyaghttta Vajra Vyatipaia Vanvas Parigha Siva 50 54 52 5 24 52 31 44 35

tirre iKt numbers arc inserted

ulumn it mn»t l» 38 IC nnJer^t. Vauij Vauy NAga 7 26 26 17 Mitlm:l Karka: Siiiiha Siiii: 14 30 17 29 47 the i-in" during ihe whole ilri Page:Sewell Dikshit The Indian Calendar (1896) proc.djvu/31 The above extract is for the amânta month Bhâdrapada or August 31st to September 29th, 1894. The month is divided into its two fortnights. The uppermost horizontal column shews that the first tithi, "pratipadâ", was current at sunrise on Friday, and that it ended at 43 gh. 59 p. after sunrise. The moon was 12 degrees to the east of the sun at that moment, and after that the second tithi, "dvitîyâ", commenced. The nakshatra Pûrva-Phalgunî ended and Uttara-Phalgunî commenced at 40 gh. 16 p. after sunrise. The yoga Siddha ended, and Sâdhya began, at 31 gh. 22 p. after sunrise; and the karaṇa Kiṁstughna ended, and Bava began, at 16 gh. 30 p. after sunrise. The moon was in the sign Siṁha up to 15 gh. after sunrise and then entered the sign Kanyâ. The length of the day was 30 gh. 59 pa. (and consequently the length of the night was 29 gh. 1 pa.). The solar day was the 16th of Siṁha.^[36] The Muhammadan day was the 29th of Śafar, and the European day was the 31st of August. This will explain the bulk of the table and the manner of using it.

Under the heading "other particulars" certain festival days, and some other information useful for religious and other purposes, are given. To the right, read vertically, are given the places of the sun and the principal planets at sunrise of the last day of each fortnight in signs degrees, minutes, and seconds, with their daily motions in minutes and seconds. Thus the figures under "sun" shew that the sun had, up to the moment in question, travelled through 4 signs, 29 degrees, 27 minutes, and 9 seconds; i.e., had completed 4 signs and stood in the 5th, Siṁha,—had completed 29 degrees and stood in the 30th, and so on; and that the rate of his daily motion for that moment was 58 minutes and 30 seconds. Below are shown the same in signs in the horoscope. The ahargaṇa, here 34—227, means that since the epoch of the Grahalâghava,^[37] i.e., sunrise on amânta Phâlguna kṛishṇa 30th of Śaka 1441 expired, or Monday 19th March, A.D. 1520, 34 cycles of 4016 days each, and 227 days, had elapsed at sunrise on Saturday the 15th of the bright half of Bhâdrapada. The horoscope entries are almost always given in pañchâṅgs as they are considered excessively important by the Hindus.

31. Tithis and solar days. Solar or civil days are always named after the week-days, and where solar reckoning is in use are also counted by numbers, e.g., the 1st, 2nd, etc., of a named solar month. But where solar reckoning does not prevail they bear the names and numerals of the corresponding tithis. The tithis, however, beginning as they do at any hour of the day, do not exactly coincide with solar days, and this gives rise to some little difficulty. The general rule for civil purposes, as well as for some ordinary religious purposes for which no particular time of day happens to be prescribed, is that the tithi current at sunrise of the solar day gives its name and numeral to that day, and is coupled with its week-day. Thus Bhâdrapada śukla chaturdaśî Śukravâra (Friday the 14th of the first or bright fortnight of Bhâdrapada) is that civil day at whose sunrise the tithi called the 14th śukla is current, and its week-day is Friday. Suppose a written agreement to have been executed between two parties, or an ordinary religious act to have been performed, at noon on that Friday at whose sunrise Bhâdrapada śukla chaturdaśî of Śaka 1816 expired was current, and which ended (see the table) 5 gh. 18 p., (about 2 h. 7 m.) after sunrise, or at about 8.7 a.m. Then these two acts were actually done after the chaturdaśî had ended and the pûrṇimâ was current, but they would be generally noted as having been done on Friday śukla chaturdaśî. It is, however, permissible, though such instances would be rare, to state the date of these actions as "Friday pûrṇimâ;" and sometimes for religious purposes the date would be expressed as "chaturdaśî yukta pûrṇimâ" (the 14th joined with the pûrṇimâ). Where, however, successive regular dating is kept up, as, for instance, in daily transactions and accounts, a civil day can only bear the name of the tithi current at its sunrise.

Some religious ceremonies are ordered to be performed on stated tithis and at fixed times of the day. For example, the worship of the god Gaṇeśa is directed to take place on the Bhâdrapada śukla chaturthî during the third part (madhyâhna) of the five parts of the day. A śrâddha, a ceremony in honour of the pitṛis (manes), must be performed during the 4th (aparâhṇa) of these five periods. Take the case of a Brâhmaṇa, whose father is dead, and who has to perform a śrâddha on every amâvâsyâ. In the month covered by our extract above the amâvâsyâ is current at sunrise on Saturday. It expired at 11 gh. 40 p. after sunrise on Saturday, or at about 10.40 a.m. Now the aparâhṇa period of that Saturday began, of course, later than that hour, and so the amâvâsyâ of this Bhadrapada was current during the aparâhṇa, not of Saturday, but of the previous day, Friday. The śrâddha ordered to be performed on the amâvâsyâ must be performed, not on Saturday, but on Friday in this case. Again, suppose a member of the family to have died on this same Friday before the end of the tithi kṛishṇa chaturdaśî, and another on the same day but after the end of the tithi. A śrâddha must be performed in the family every year, according to invariable Hindu custom, on the tithi on which each person died. Therefore in the present instance the śrâddha of the first man must be performed every year on the day on which Bhâdrapada kṛishṇa chaturdaśî is current, during the aparâhṇa; while that of the second must take place on the day on which the amâvâsyâ of that month is current during the aparâhṇa, and this may be separated by a whole day from the first. Lengthy treatises have been written on this subject, laying down what should be done under all such circumstances.^[38]

At the time of the performance of religious ceremonies the current tithi, vâra, and all other particulars have to be pronounced; and consequently the tithi, nakshatra, etc., so declared may differ from the tithi, etc., current at sunrise. There is a vrata (observance, vow) called Saṅkashtanâśana-chaturthî, by which a man binds himself to observe a fast on every kṛishṇa chaturthî up to moonrise, which takes place about 9 p.m. on that tithi, but is allowed to break the fast afterwards. And this has of course to be done on the day on which the chaturthî is current at moonrise. From the above extract the evening of the 18th September, Tuesday, is the day of this chaturthî, for though the 3rd tithi, tṛitîyâ, of the kṛishṇa paksha was current at sunrise on Tuesday it expired at 9 gh. 35 pa. after sunrise, or about 9.50 a.m. If we suppose that this man made a grant of land at the time of breaking his fast on this occasion, we should find him dating his grant "kṛishṇa chaturthî, Tuesday," though for civil purposes the date is kṛishṇa tṛitîyâ, Tuesday.

The general rule may be given briefly that for all practical and civil purposes, as well as for some ordinary religious purposes, the tithi is connected with that week-day or solar day at whose sunrise it is current, while for other religious purposes, and sometimes, though rarely, even for practical purposes also, the tithi which is current at any particular moment of a solar day or week-day is connected with that day.

32. Adhika and kshaya tithis. Twelve lunar months are equal to about 354 solar days (see Art. 24 above), but there are 360 tithis during that time and it is thus evident that six tithis must somehow be expunged in civil (solar) reckoning. Ordinarily a tithi begins on one day and ends on the following day, that is it touches two successive civil days. It will be seen, however, from its length (Art. 7 above) that a tithi may sometimes begin and end within the limits of the same natural day; while sometimes on the contrary it touches three natural days, occupying the whole of one and parts of the two on each side of it.

A tithi on which the sun does not rise is expunged. It has sustained a diminution or loss (kshaya), and is called a kshaya tithi, On the other hand, a tithi on which the sun rises twice is repeated. It has sustained an increase (vṛiddhi), and is called an adhika, or added, tithi. Thus, for example, in the pañchâṅg extract given above (Art. 30) there is no sunrise during kṛishṇa saptamî (7th), and it is therefore expunged. Kṛishṇa shashṭhî (6th) was current at sunrise on Friday, for it ended 16 palas after sunrise; while kṛishṇa saptamî began 16 palas after that sunrise and ended before the next sunrise; and kṛishṇa ashtami (8th) is current at sunrise on the Saturday. The first day is therefore named civilly the (6th) shashṭhî, Friday, and the second is named (8th) ashtami, Saturday; while no day is left for the saptamî, and it has necessarily to be expunged altogether, though, strictly speaking, it was current for a large portion of that Friday. On the other hand, there are two sunrises on Bhâdrapada śukla trayôdaśî (śukla 13th), and that tithi is therefore repeated. It commenced after 56 gh. 44. pa. on Tuesday, i.e., in European reckoning about 4.20 a.m. on the Wednesday morning, was current on the whole of Wednesday, and ended on Thursday at 1 gh. 23 pa. after sunrise, or about 6.33 am. It therefore touched the Tuesday (reckoned from sunrise to sunrise) the Wednesday and the Thursday; two natural civil days began on it; two civil days, Wednesday and Thursday, bear its numeral (13); and therefore it is said to be repeated.^[39]

In the case of an expunged tithi the day on which it begins and ends is its week-day. In the case of a repeated tithi both the days at whose sunrise it is current are its week-days.

A clue for finding when a tithi is probably repeated or expunged is given in Art. 142.

Generally there are thirteen expunctions (kshayas) and seven repetitions (vṛiddhis) of tithis in twelve lunar months.

The day on which no tithi ends, or on which two tithis end, is regarded as inauspicious. In the pañchâṅg extract above (Art. 30) Bhâdrapada śukla trayôdaśî Wednesday, and Bhâdrapada kṛishṇa shashṭhî, Friday (on which the saptamî was expunged), were therefore inauspicious.

33. It will be seen from the above that it is an important problem with regard to the Indian mode of reckoning time to ascertain what tithi, nakshatra, yoga, or karaṇa was current at sunrise on any day, and when it began and ended. Our work solves this problem in all cases.

34. Variation on account of longitude. The moment of time when the distance between the sun and moon amounts to 12, or any multiple of 12, degrees, or, in other words, the moment of time when a tithi ends, is the same for all places on the earth's surface; and this also applies to nakshatras, yogas, and karaṇas. But the moment of sunrise of course varies with the locality, and therefore the ending moments of divisions of time such as tithis, when referred to sunrise, differ at different places. For instance, the tithi Bhâdrapada śukla pûrṇimâ (see above Art. 30) ended at Poona at 8 gh. 11 pa. after sunrise, or about 9.16 am. At a place where the sun rose 1 gh. earlier than it does at Poona the tithi would evidently have ended one ghaṭikâ later, or at g gh. 11 pa. after sunrise, or at about 9.40 a.m. On the other hand, at a place where the sun rose 1 gh. later than at Poona the tithi would have ended when 7 gh. 11 pa. had elapsed since the sunrise at that place, or at about 8.52 a.m.

35. For this reason the expunction and repetition of tithis often differs in different localities. Thus the nakshatra Pûrvâshâḍhâ (see pañchâṅg extract Art. 30) was 58 gh. 11 pa.^[40] at Poona on Sunday, śukla 10th. At a place which is on the same parallel of latitude, but 12 degrees eastward, the sun rises 2 gh. earlier than at Poona, and there this nakshatra ended (58 gh. 11 pa. + 2 gh =) 60 gh. 11 pa. after sunrise on Sunday, that is at 11 pa. after sunrise on Monday. It therefore touches three natural days, and therefore it (Pûrvâshâḍhâ) is repeated, whereas at Poona it is Uttarâshâḍhâ which is repeated. On the other hand, the nakshatra Maghâ on Kṛishṇa 13th was 3 gh. 4 pa., and Pûrva-phalgunî was (3 gh. 4 pa. + 56 gh.^[41] 51 pa. =) 59 gh. 55 pa. at Poona. At a place which has the same latitude as Poona, but is situated even at so short a distance as 1 degree to the east, the nakshatra Pûrva-phalgunî ended 60 gh. 5 pa after sunrise on Thursday, that is 5 pa. after sunrise on Friday; and therefore there will be no kshaya of that nakshatra at that place, but the following nakshatra Uttara phalgunî will be expunged there.

36. True or apparent, and mean, time. The sun, or more strictly the earth in its orbit, travels, not in the plane of the equator, but in that of the ecliptic, and with a motion which varies every day; the length of the day, therefore, is not always the same even on the equator. But for calculating the motions of the heavenly bodies it is evidently convenient to have a day of uniform length, and for this reason astronomers, with a view of obtaining a convenient and uniform measure of time, have had recourse to a mean solar day, the length of which is equal to the mean or average of all the apparent solar days in the year. An imaginary sun, called the mean sun, is conceived to move uniformly in the equator with the mean angular velocity of the true sun. The days marked by this mean sun will all be equal, and the interval between two successive risings of the mean sun on the equator is the duration of the mean solar day, viz., 24 hours or 60 ghaṭikâs. The time shown by the true sun is called true or apparent time, and the time shown by the mean sun is known as mean time. Clocks and watches, whose hands move, at least in theory, with uniform velocity, evidently give us mean time. With European astronomers "mean noon" is the moment when the mean sun is on the meridian; and the "mean time" at any instant is the hour angle of the mean sun reckoned westward from 0 h. to 24 h., mean noon being 0 h. for astronomical purposes.

Indian astronomers count the day from sunrise, to sunrise, and give, at least in theory, the ending moments of tithis in time reckoned from actual or true sunrise. The true or apparent time of a place, therefore, in regard to the Indian pañchâṅg, is the time counted from true (i.e., actual) sunrise at that place. For several reasons it is convenient to take mean sunrise on the equator under any given meridian to be the mean sunrise at all places under the same meridian. The mean sunrise at any place is calculated as taking place at 0 gh. or 0 h.—roughly 6 a.m. in European civil reckoning; and the mean time of a place is the time counted from 0 gh. or 0 h.

The moment of true sunrise is of course not always the same at all places, but varies with the latitude and longitude. Even at the same place it varies with the declination of the sun, which varies every day of the year. And at any given place, and on any given day of the year, it is not the same for all years. The calculation, therefore, of the exact moment of true sunrise at any place is very complicated—too complicated to be given in this work,^[42] the aim of which is extreme simplicity and readiness of calculation, and therefore mean time at the meridian of Ujjain^[43] or Lanka is used throughout what follows.

All ending moments of tithis calculated by our method C (Arts. 139 to 160) are in Ujjain mean time; and to convert Ujjain mean time into that of any other given place the difference of longitude in time— 4 minutes (10 palas) to a degree—should be added or subtracted according as the place is east or west of Ujjain. Table XI. gives the differences of longitude in time for some of the most important places of India.

The difference between the mean and apparent (true) time of any place in India at the present day varies from nil (in March and October) to 26 minutes (in January and June) in the extreme southern parts of the peninsular. It is nowhere more than 65 minutes.

37. Basis of calculation for the Tables. All calculations made in this work in accordance with luni-solar reckoning are based on the Sûrya-Siddhânta, and those for solar reckoning on the Sûrya and Ârya Siddhântas. The elements of the other authorities being somewhat different, the ending moments of tithis etc., or the times of sankrantis as calculated by them may sometimes differ from results obtained by this work; and it must never be forgotten that, when checking the date of a document or record which lays down, for instance, that on a certain week-day there fell a certain tithi, nakshatra, or yoga, we can only be sure of accuracy in our results if we can ascertain the actual Siddhânta or other authority used by the author of the calendar which the drafter of the document consulted. Prof. Jacobi has given Tables for several of the principal Siddhântas in the Epigraphica Indica (Vol. II., pp. 403 et seq.), and these may be used whenever a doubt exists on the point.

Although all possible precautions have been taken, there, must also be a slight element of uncertainty in the results of a calculation made by our Tables owing to the difference between mean and apparent time, independently of that arising from the use of different authorities. Owing to these two defects it is necessary sometimes to be cautious. If by any calculation it is found that a certain tithi, nakshatra, yoga, or karaṇa ended nearly at the close of a solar day—as, for example, 55 ghaṭikâs after mean sunrise on a Sunday, i.e., 5 ghaṭikâs before sunrise on the Monday—it is possible that it really ended shortly after true sunrise on the Monday. And, similarly, if the results shew that a certain tithi ended shortly after the commencement of a solar day,—for instance, 5 ghaṭikâs after mean sunrise on a Sunday,—it is possible that it really ended shortly before the true termination of the preceding day, Saturday. Five ghaṭikâs is not the exact limit, nor of course the fixed limit. The period varies from nil to about five ghaṭikâs, rarely more in the case of tithis, nakshatras, and karanas; but in the case of yogas it will sometimes reach seven ghatikas.

Calculations made by our method C will result in the finding of a "tithi index" ( $t$ .), or a nakshatra or yoga-index ( $n$ . or $y$ .), all of which will be explained further on; but it may be stated in this connection that when at any ascertained mean sunrise it is found that the resulting index is within 30 of the ending index of the tithi, (Table VIII., col. 3), nakshatra or karaṇa (id. col. 8, 9, 10), or within 50 of the ending index of a yoga (id. col. 11), it is possible that the result may be one day wrong, as explained above. The results arrived at by our Tables, however, may be safely relied on for all ordinary purposes.

38. Nakshatras There are certain conspicuous stars or groups of stars in the moon's observed path in the heavens, and from a very remote age these have attracted attention. They are called in Sanskrit "Nakshatras". They were known to the Chaldœans and to the ancient Indian Âryas. Roughly speaking the moon makes one revolution among the stars in about 27 days, and this no doubt led to the number^[44] of nakshatras being limited to 27.

The distance between the chief stars, called yôga-târâs, of the different nakshatras is not uniform. Naturally it should be 13° 20′, but, in some cases it is less than 7°, while in others it is more than 20°. It is probable that in ancient times the moon's place was fixed merely by stating that she was near a particular named nakshatra (star) on a certain night, or on a certain occasion. Afterwards it was found necessary to make regular divisions of the moon's path in her orbit, for the sake of calculating and foretelling her position; and hence the natural division of the ecliptic, consisting of twenty-seven equal parts, came into use, and each of these parts was called after a separate nakshatra (see Art. 8). The starry nakshatras, however, being always in view and familiar for many centuries, could not be dispensed with, and therefore a second and unequal division was resorted to. Thus two systems of nakshatras came into use. One we call the ordinary or equal-space system, the other the unequal-space system. The names of the twenty-seven stellar nakshatras are given to both sets. In the equal-space system each nakshatra has 13° 20′ of space, and when the sun, the moon, or a planet is between 0°, i.e., no degrees, and 13° 20′ in longitude it is said to be in the first nakshatra Aśvinî, and so on. The unequal-space system is of two kinds. One is described by Garga and others, and is called here the "Garga system." According to it fifteen of the nakshatras are held to be of equal average (mean) length—i.e., 13° 20′,—but six measure one and-a-half times the average—i.e., 20°, and six others only half the average, viz., 6° 40′. The other system is described by Brahmagupta and others, and therefore we call it the "Brahma-Siddhânta" system. In its leading feature it is the same with Garga's system, but it differs a little from Garga's in introducing Abhijit in addition to the twenty-seven ordinary nakshatras. The moon's daily mean motion,—13 degrees, 10 minutes, 35 seconds,—is taken as the average space of a nakshatra. And as the total of the spaces thus allotted to the usual twenty-seven nakshatras, on a similar arrangement of unequal spaces, amounts to only 355 degrees, 45 minutes, 45 seconds, the remainder,—4 degrees, 14 minutes, 15 seconds,—is allotted to Abhijit, as an additional nakshatra placed between Uttara-Ashâḍhâ and Śravaṇa.

The longitude of the ending points of all the nakshatras according to these three systems is given below. The entries of "½" and "1½" in subcolumn 3 mark the variation in length from the average.

The nakshatras by any of these systems, for all years between 300 and 1900 A. D., can be calculated by our Tables (see method "C", Arts. 139 to 160). The indices for them, adapted to our Tables, are given in Table VIII., cols. 8, 9, 10.

The ordinary or equal-space system of nakshatras is in general use at the present day, the unequal-space systems having almost dropped out of use. They were, however, undoubtedly prevalent to a great extent in early times, and they were constantly made use of on important religious occasions.^[45]

**Longtitudes of the Ending-points of the Nakshatras.**
Order of the Nakshatras.		System of Equal Spaces.		Systems of Unequal Spaces.
Order of the Nakshatras.		System of Equal Spaces.		Garga System.				Brahma-Siddhânta System.
1		2		3	4			4
		Deg.	Min.		Deg.	Min.	Sec.	Deg.	Min.	Sec.
1	Asvini	13°	20′	. . . .	13°	20′	0	13°	10′	35½″
2	Bharaṇî	26	40	½	20	0	0	19	45	52½
3	Kṛittikâ	40	0	. . . .	33	20	0	32	56	27½
4	Rohiṇî	53	20	1½	53	20	0	52	42	20½
5	Mṛigaśiras	66	40	. . . .	66	40	0	65	52	55½
6	Ârdrâ	80	0	½	73	20	0	72	28	12½
7	Punarvasu	93	20	½	93	20	0	92	14	5½
8	Pushya	106	40	. . . .	106	40	0	105	24	40½
9	Aśleshâ	120	0	½	113	20	0	111	59	57½
10	Maghâ	133	20	. . . .	126	40	0	125	10	32½
11	Pûrva-Phalgunî	146	40	. . . .	140	0	0	138	21	7½
12	Uttara-Phalgunî	160	0	1½	160	0	0	158	7	0½
13	Hasta	173	20	. . . .	173	20	0	171	17	35½
14	Chitrâ	186	40	. . . .	186	40	0	184	28	10½
15	Svâti	200	0	½	193	20	0	191	3	27½
16	Viśâkhâ	213	20	1½	213	20	0	210	49	20½
17	Anurâdhâ	226	40	. . . .	226	40	0	223	59	55½
18	Jyeshṭhâ	240	0	½	233	20	0	230	35	12½
19	Mûla	253	20	. . . .	246	40	0	243	45	47½
20	Pûrva-Ashâdha	266	40	. . . .	260	0	0	256	56	22½
21	Uttara-Ashâḍhâ	280	0	1½	280	0	0	276	42	15½
	(Abhijit)	. . .		(Balance)	. . . .			280	56	30½
22	Śravaṇa	293	20	. . . .	293	20	0	294	7	5½
23	Dhanishṭhâ	306	40	. . . .	306	40	0	307	17	40½
24	Śatatârakâ	320	0	½	313	20	0	313	52	57½
25	Pûrva-Bhadrapadâ	333	20	. . . .	326	40	0	327	3	32½
26	Uttara-Bhadrapadâ	346	40	1½	346	40	0	346	49	25½
27	Revatî	360	0	. . . .	360	0	0	360	0	0½

39. Auspicious Yogas. Besides the 27 yogas described above (Art. 9), and quite different from them, there are in the Indian Calendar certain conjunctions, also called yogas, which only occur when certain conditions, as, for instance, the conjunction of certain vâras and nakshatras, or vâras and tithis, are fulfilled. Thus, when the nakshatra Hasta falls on a Sunday there occurs an amṛita siddhiyoga. In the pañchâṅg extract (Art. 30) given above there is an amṛita siddhiyoga on the 2nd, 5th and 18th of September. It is considered an auspicious yoga, while some yogas are inauspicious.

40. Karaṇas. A karaṇa being half a tithi, there are 60 karaṇas in a lunar month. There are seven karaṇas in a series of eight cycles—total 56—every month, from the second half of śukla pratipadâ (1st) up to the end of the first half of kṛishṇa chaturdaśî (14th). The other four karaṇas are respectively from the second half of kṛishṇa chaturdaśî (14th) to the end of the first half of śukla pratipadâ.^[46]

Table VIII., col. 4, gives the serial numbers and names of karaṇas for the first half, and col. 5 for the second half, of each tithi.

40a. Eclipses. Eclipses of the sun and moon play an important part in inscriptions, since, according to ancient Indian ideas, the value of a royal grant was greatly enhanced by its being made on the occasion of such a phenomenon; and thus it often becomes essential that the moments of their occurrence should be accurately ascertained. The inscription mentions a date, and an eclipse as occurring on that date. Obviously we shall be greatly assisted in the determination of the genuineness of the inscription if we can find out whether such was actually the case. Up to the present the best list of eclipses procurable has been that published by Oppolzer in his "Canon der Finsternisse" (Denkschriften der Kaiserl. Akademie der Wissenschaften. Vienna, Vol. LII.), but this concerns the whole of our globe, not merely a portion like India; the standard meridian is that of Greenwich, requiring correction for longitude; and the accompanying maps are on too small a scale to be useful except as affording an approximation from which details can be worked out. Our object is to save our readers from the necessity of working out such complicated problems. Prof. Jacobi's Tables in the Indian Antiquary (Vol. XVII.) and Epigraphia Indica (Vol. II.) afford considerable help, but do not entirely meet the requirements of the situation. Dr. Schram's contribution to this volume, and the lists prepared by him, give the dates of all eclipses in India and the amount of obscuration observable at any place. His article speaks for itself, but we think it will be well be add a few notes.

Prof. Jacobi writes (Epig. Ind., II., p. 422):—"The eclipses mentioned in inscriptions are not always actually observed eclipses, but calculated ones. My reasons for this opinion are the following: Firstly, eclipses are auspicious moments, when donations, such as are usually recorded in inscriptions, are particularly meritorious. They were therefore probably selected for such occasions, and must accordingly have been calculated beforehand. No doubt they were entered in pañchâṅgs or almanacs in former times as they are now. Secondly, even larger eclipses of the sun, up to seven digits, pass unobserved by common people, and smaller ones are only visible under favourable circumstances. Thirdly, the Hindus place implicit trust in their Śâstras, and would not think it necessary to test their calculations by actual observation. The writers of inscriptions would therefore mention an eclipse if they found one predicted in their almanacs."

Our general Table will occasionally be found of use. Thus a lunar eclipse can only occur at the time of full moon (pûrṇimâ), and can only be visible when the moon is above the horizon at the place of the observer; so that when the pûrṇimâ is found by our Tables to occur during most part of the daytime there can be no visible eclipse. But it is possibly visible if the pûrṇimâ is found, on any given meridian, to end within 4 ghaṭikâs after sunrise, or within 4 ghaṭikâs before sunset. A solar eclipse occurs only on an amâvâsyâ or new moon day. If the amâvâsyâ ends between sunset and sunrise it is not visible. If it ends between sunrise and sunset it may be visible, but not of course always.

41. Lunar months and their names. The usual modern system of naming lunar months is given above (Art. 14), and the names in use will be found in Tables II. and III. In early times, however, the months were known by another set of names, which are given below, side by side with those by which they are at present known.

	Ancient names.	Modern names.		Ancient names.	Modern names.
1.	Madhu	Chaitra	7.	Isha	Âśvina
2.	Mâdhava	Vaiśâkha	8.	Ûrja	Kârttika
3.	Śukra	Jyeshṭha	9.	Sahas	Mârgaśîrsha
4.	Śuchi	Âshâḍha	10.	Sahasya	Pausha
5.	Nabhas	Srâvaṇa	11.	Tapas	Mâgha
6.	Nabhasya	Bhâdrapada	12.	Tapasya	Phâlguna

The names "Madhu" and others evidently refer to certain seasons and may be called season-names^[47] to distinguish them from "Chaitra" and those others which are derived from the nakshatras. The latter may be termed sidereal names or star-names. Season-names are now nowhere in use, but are often met with in Indian works on astronomy, and in Sanskrit literature generally.

The season-names of months are first met with in the mantra sections, or the Saṁhitâs, of both the Yâjur-Vedas, and are certainly earlier than the sidereal names which are not found in the Saṁhitâs of any of the Vedas, but only in some of the Brâhmaṇas, and even there but seldom.^[48]

42. The sidereal names "Chaitra", etc., are originally derived from the names of the nakshatras. The moon in her revolution passes about twelve times completely through the twenty-seven starry nakshatras in the course of the year, and of necessity is at the full while close to some of them. The full-moon tithi (pûrṇimâ), on which the moon became full when near the nakshatra Chitrâ, was called Chaitrî; and the lunar month which contained the Chaitrî pûrṇimâ was called Chaitra and so on.

43. But the stars or groups of stars which give their names to the months are not at equal distances from one another; and as this circumstance,—together with the phenomenon of the moon's apparent varying daily motion, and the fact that her synodic differs from her sidereal revolution—prevents the moon from becoming full year after year in the same nakshatra, it was natural that, while the twenty-seven nakshatras were allotted to the twelve months, the months themselves should be named by taking the nakshatras more or less alternately. The nakshatras thus allotted to each month are given on the next page.

44. It is clear that this practice, though it was natural in its origin and though it was ingeniously modified in later years, must often have occasioned considerable confusion; and so we find that the months gradually ceased to have their names regulated according to the conjunction of full moons and nakshatras, and were habitually named after the solar months in which they occurred. This change began to take place about 1400 B. C., the time of the Vedâṅga-jyotisha; and from the time when the zodiacal-sign-names, "Mesha" and the rest, came into use till the present day, the general rule has been that that amânta lunar month in which the Mesha saṅkrânti occurs, is called Chaitra, and the rest in succession.

**Derivation of the Names of the Lunar Months from the Nakshatras.**
Names and Grouping of the Nakshatras.	Names of the Months.
Kṛittikâ; Rohiṇî	Kârttika.
Mṛigaśiras; Ardrâ	Mârgaśirsha.
Punarvasu; Pushya	Pausha.
Aśleshâ; Maghâ	Mâgha.
Pûrva-Phalgunî; Uttara-Phalgunî; Hasta	Phâlguna.
Chitrâ; Svâti	Chaitra.
Viśâkhâ; Anurâdhâ	Vaiśâkha.
Jyeshṭhâ; Mûla	Jyeshṭha.
Pûrva-Ashâdha; Uttara-Ashâḍhâ; (Abhijit)	Âshâḍha.
(Abhijit); Śravaṇa; Dhanishṭhâ	Śrâvaṇa.
Śatatârakâ; Pûrva-Bhadrapadâ; Uttara-Bhadrapadâ	Bhâdrapada.
Revatî; Aśvinî; Bharaṇi	Aśvina.

45. Adhika and kshaya mâsas. It will be seen from Art. 24 that the mean length of a solar month is greater by about nine-tenths of a day than that of a lunar month, and that the true length of a solar month, according to the Sûrya-Siddhânta, varies from 29 d. 7 h. 38 m. to 31 d. 15 h. 28 m. Now the moon's synodic motion, viz., her motion relative to the sun, is also irregular, and consequently all the lunar months vary in length. The variation is approximately from 29 d. 7 h. 20 m. to 29 d. 19 h. 30 m., and thus it is clear that in a lunar month there will often be no solar saṅkrânti, and occasionally, though rarely, two. This will be best understood by the following table and explanation. (See p. 26.)

We will suppose (see the left side of the diagram, cols. 1, 2.) that the sun entered the sign Mesha,—that is, that the Mesha saṅkrânti took place, and therefore the solar month Mesha commenced,—shortly before the end of an amânta lunar month, which was accordingly named "Chaitra" in conformity with the above rule (Art. 14, or 44); that the length of the solar month Mesha was greater than that of the following lunar month; and that the sun therefore stood in the same sign during the whole of that lunar month, entering the sign Vṛishabha shortly after the beginning of the third lunar month, which was consequently named Vaiśâkha because the Vṛishabha saṅkrânti took place, and the solar month Vṛishabha commenced, in it,—the Vṛishabha saṅkrânti being the one next following the Mesha saṅkrânti. Ordinarily there is one saṅkrânti in each lunar month, but in the present instance there was no saṅkrântiwhatever in the second lunar month lying between Chaitra and Vaiśâkha.

The lunar month in which there is no saṅkrânti is called an adhika (added or intercalated) month; while the month which is not adhika, but is a natural month because a saṅkrânti actually occurred in it, is called nija, i.e., true or regular month.^[49] We thus have an added month between natural Chaitra and natural Vaiśâkha.

The next peculiarity is that when there are two saṅkrântis in a lunar month there is a kshaya mâsa, or a complete expunction of a month. Suppose, for instance, that the Vṛiśchika saṅkrânti took place shortly after the beginning of the amânta lunar month Kârttika (see the lower half of the diagram col. 2); that in the next lunar month the Dhanus-saṅkrânti took place

Amanta lunar months.	Solar months; saṅkrânti to saṅkrânti.	Fortnights.	Pûrṇimânta lunar months.^[50]
			By one system.	By another system.
1	2	3	4	5
Chaitra.	—Mesha saṅkrânti	Śukla	½ Chaitra	½ Chaitra
		Kṛishṇa	Vaiśâkha	First Vaiśâkha
Adhika Vaiśâkha	Intercalated period.	Śukla	Adhika Vaiśâkha
		Kṛishṇa		Second Vaiśâkha
Nija Vaiśâkha	—Vṛishabha saṅkrânti	Śukla	Vaiśâkha
		Kṛishṇa	½ Jyeshṭha	½ Jyeshṭha
(Several months are omitted here.)
Kârttika	—Vṛiśchika saṅkrânti	Śukla	½ Kârttika	½ Kârttika
		Kṛishṇa	Mârgaśîrsha	Mârgaśîrsha
Mârgaśîrsha (Pausha suppressed)	—Dhanus saṅkrânti	Śukla
	—Makara saṅkrânti	Kṛishṇa	(Pausha suppressed) Mâgha	(Pausha suppressed) Mâgha
Mâgha	—Kumbha saṅkrânti	Śukla
		Kṛishṇa	½ Phâlguna	½ Phâlguna

shortly after it began, and the Makara-saṅkrânti shortly before it ended, so that there were two saṅkrântis in it; and that in the third month the Kumbha-saṅkrânti took place before the end of it. The lunar month in which the Kumbha-saṅkrânti occurred is naturally the month Mâgha. Thus between the natural Kârttika and the natural Mâgha there was only one lunar month instead of two, and consequently one is said to be expunged.

46. Their names. It will be seen that the general brief rule (Art. 44) for naming lunar months is altogether wanting in many respects, and therefore rules had to be framed to meet the emergency. But different rules were framed by different teachers, and so arose a difference in practice. The rule followed at present is given in the following verse.

Mînâdistho Ravir yeshâm âraṁbha-prathame kshaṇe | bhavet te 'bde Chândra mâsâś chaîtrâdyâ dvâdaśa smṛitâh." ‖

"The twelve lunar months, at whose first moment the sun stands in Mîna and the following [signs], are called Chaitra, and the others [in succession]."

According to this rule the added month in the above example (Art. 45) will be named Vaiśâkha, since the sun was in Mesha when it began; and in the example of the expunged month the month between the natural Kârttika and the natural Mâgha will be named Mârgaśirsha, because the sun was in Vṛiśchika when it commenced, and Pausha will be considered as expunged.

This rule is given in a work named Kâlatatva-vivechana, and is attributed to the sage Vyâsa. The celebrated astronomer Bhâskarâchârya (A. D. 1150) seems to have followed the same rule,^[51] and it must therefore have been in use at least as early as the 12th century A. D. As it is the general rule obtaining through most part of India in the present day we have followed it in this work.

There is another rule which is referred to in some astronomical and other works, and is attributed to the Brâhma-Siddhânta.^[52] It is as follows:

"Meshâdisthe Savitari yo yo mâsaḥ prapûryate chândraḥ | Chaitrâdyaḥ sa jñeyaḥ pûrtidvitve 'dhimâso 'ntyaḥ." ‖

"That lunar month which is completed when the sun is in [the sign] Mesha etc., is to be known as Chaitra, etc. [respectively]; when there are two completions, the latter (of them] is an added month."

It will be seen from the Table given above (p. 26) that for the names of ordinary months both rules are the same, but that they differ in the case of added and suppressed months. The added month between natural Chaitra and natural Vaiśâkha, in the example in Art. 45, having ended when the sun was in Mesha, would be named "Chaitra" by this second rule, but "Vaiśâkha" by the first rule, because it commenced when the sun was in Mesha. Again, the month between natural Kârttika and natural Mâgha, in the example of an expunged month, having ended when the sun was in Makara, would be named "Pausha" by this second rule, and consequently Mârgaśirsha would be expunged; while by the first rule it would be named "Mârgaśirsha" since it commenced when the sun was in Vṛiśchika, and Pausha would be the expunged month. It will be noticed, of course, that the difference is only in name and not in the period added or suppressed.^[53] Both these rules should be carefully borne in mind when studying inscriptions or records earlier than 1100 A. D.

47. Their determination according to true and mean systems. It must be noted with regard to the intercalation and suppression of months, that whereas at present these are regulated by the sun's and moon's apparent motion,—in other words, by the apparent length of the solar and lunar months—and though this practice has been in use at least from A. D. 1100 and was followed by Bhâskarâchârya, there is evidence to show that in earlier times they were regulated by the mean length of months. It was at the epoch of the celebrated astronomer Śrîpati,^[54] or about A. D. 1040, that the change of practice took place, as evidenced by the following passage in his Siddhânta Śekhara, (quoted in the Jyotisha-darpaṇa, in A. D. 1557.)

Madhyama-Ravi-saṅkrânti-praveśa-rahito bhaved adhikaḥ
Madhyaś Chândro mâso madhyâdhika-lakshaṇam chaitat ‖
Vidvâṁsas-tv-âchâryâ nirasya madhyâdhikam mâsaṁ
Kuryuḥ sphuṭa-mânena hi yato 'dhikaḥ spashṭa eva syât. ‖

"The lunar month which has no mean sun's entrance into a sign shall be a mean intercalated month. This is the definition of a mean added month. The learned Âchâryas should leave off [using] the mean added months, and should go by apparent reckoning, by which the added month would be apparent (true)."

It is clear, therefore, that mean intercalations were in use up to Śrîpatis time. In the Vedâṅga Jyotisha only the mean motions of the sun and moon are taken into account, and it may therefore be assumed that at that time the practice of regulating added and suppressed months by apparent motions was unknown. These apparent motions of the sun and moon are treated of in the astronomical Siddhântas at present in use, and so far as is known the present system of astronomy came into force in India not later than 400 A. D.^[55] But on the other hand, the method of calculating the ahargana (a most important matter), and of calculating the places of planets, given in the Sûrya and other Siddhântas, is of such a nature that it seems only natural to suppose that the system of mean intercalations obtained for many centuries after the present system of astronomy came into force, and thus we find Śrîpati's utterance quoted in an astronomical work of the 15th century. There can be no suppression of the month by the mean system, for the mean length of a solar month is longer than that of a mean lunar month, and therefore two mean saṅkrântis cannot take place in a mean lunar month.

The date of the adoption of the true (apparent) system of calculating added and suppressed months is not definitely known. Bhâskarâchârya speaks of suppressed months, and it seems from his work that mean intercalations were not known in his time (A. D. 1150.) We have therefore in our Tables given mean added months up to A. D. 1100. and true added and suppressed months for the whole period covered by our Tables.^[56]

48. For students more familiar with solar reckoning we will give the rules for the intercalation and suppression of months in another form. Ordinarily one lunar month ends in each solar month. When two lunar months end in a solar month the latter of the two is said to be an adhika (added or intercalated) month, and by the present practice it receives the name of the following natural lunar month, but with the prefix adhika. Thus in the Table on p. 25, two lunar months end during the solar month Mesha, the second of which is adhika and receives, by the present practice, the name of the following natural lunar month. Vâiśakha. When no lunar month ends in a solar month there is a kshaya mâsa, or expunged or suppressed month; i.e., the name of one lunar month is altogether dropped, viz., by the present practice, the one following that which would be derived from the solar month. Thus, in the Table above, no lunar month ends in the solar month Dhanus. Mârgaśirsha is the name of the month in which the Dhanus saṅkrânti occurs; the name Pausha is therefore expunged.

The rule for naming natural lunar months, and the definition of, and rule for naming, added and suppressed months, may be summed up as follows. That amânta lunar month in which the Mesha saṅkrânti occurs is called Chaitra, and the rest in succession. That amânta lunar month in which there is no saṅkrânti is adhika and receives the name (i) of the preceding natural lunar month by the old Brâhma-Siddhânta rule, (2) of the following natural lunar month by the present rule. When there are two saṅkrântis in one amânta lunar month, the name which would be derived from the first is dropped by the old Brâhma-Siddhânta rule, the name which would be derived from the second is dropped by the present rule.

49. Different results by different Siddhântas. The use of different Siddhântas will sometimes create a difference in the month to be intercalated or suppressed, but only when a saṅkrânti takes place very close^[57] to the end of the amâvâsyâ. Such cases will be rare. Our calculations for added and suppressed months have been made by the Sûrya-Siddhânta, and to assist investigation we have been at the pains to ascertain and particularize the exact moments (given in tithi-indices, and tithis and decimals) of the saṅkrântis preceding and succeeding an added or suppressed month, from which it can be readily seen if there be a probability of any divergence in results if a different Siddhânta be used. The Special Tables published by Professor Jacobi in the Epigraphia Indica (Vol., II., pp. 403 ff.) must not be relied on for calculations of added and suppressed months of Siddhântas other than the Sûrya-Siddhânta. If a different Siddhânta happened to have been used by the original computor of the given Hindu date, and if such date is near to or actually in an added or suppressed month according to our Table I., it is possible that the result as worked out by our Tables may be a whole month wrong. Our mean intercalations from A. D. 300 to 1100 are the same by the original Sûrya-Siddhânta, the present Sûrya-Siddhânta, and the first Ârya-Siddhânta.

50. Some peculiarities. Certain points are worth noticing in connection with our calculations of the added and suppressed months for the 1600 years from A. D. 300 to 1900 according to the Sûrya-Siddhânta.

(a) Intercalations occur generally in the 3rd, 5th, 8th, 11th. 14th, 16th and 19th years of a cycle of 19 years. (b) A month becomes intercalary at an interval of 19 years over a certain period, and afterwards gives way generally to one of the months preceding it, but sometimes, though rarely, to the following one. (c) Out of the seven intercalary months of a cycle one or two are always changed in the next succeeding cycle, so that after a number of cycles the whole are replaced by others. (d) During our period of 1600 years the months Mârgaśîrsha, Pausha, and Mâgha are never intercalary. (e) The interval between years where a suppression of the month occurs is worth noticing. In the period covered by our Tables the first suppressed month is in A.D. 404, and the intervals are thus: 19, 65, 38, 19, 19, 46, 19, 141, 122, 19, 141, 141, 65, 19, 19, 19, 19, 46, 76, 46, 141, 141, and an unfinished period of 78 years. At first sight there seems no regularity, but closer examination shews that the periods group themselves into three classes, viz., (i.) 19, 38, 76; (ii.) 141; and (iii.) 122, 65 and 46 years; the first of which consists of 19 or its multiples, the second is a constant, and the third is the difference between (ii.) and (i.) or between 141 and a multiple of 19. The unfinished period up to 1900 A.D. being 78 years, we are led by these peculiarities to suppose that there will be no suppressed month till at earliest (122 years =) A.D. 1944, and possibly not till (141 years =) A.D. 1963.^[58] (</) Mâgha is only once suppressed in Saka 1398 current, Mârgaśîrsha is suppressed six times, and Pausha 18 times. No other month is suppressed.

Bhâskarâchârya lays down^[59] that Kârttika, Mârgaśîrsha and Pausha only arc liable to be suppressed, but this seems applicable only to the Brâhma-Siddhânta of which Bhâskarâchârya was a follower. He further states, "there was a suppressed month in the Śaka year 974 expired, and there will be one in Śaka 1115, 1256 and 1378 all expired", and this also seems applicable to the Brâhma-Siddhânta only. By the Sûrya-Siddhânta there were suppressed months in all these years except the last one, and there was an additional suppression in Saka 1180 expired.

Gaṇeśa Daivajña, the famous author of the Grahalâghava (A.D. 1520), as quoted by his grandson, in his commentary on the Siddhânta-Siromaṇi, says, "By the Sûrya-Siddhânta there will be a suppressed month in Śaka 1462, 1603, 1744, 1885, 2026, 2045, 2148, 2167, 2232, 2373, 2392, 2514, 2533, 2655, 2674, 2796 and 2815, and by the Ârya-Siddhânta^[60] there will be one in 1481, 1763, 1904, 2129, 2186, 2251 (all expired)." The first four by Sûrya calculations agree with our results.

51. By the pûrṇimânta scheme. Notwithstanding that the pûrṇimânta scheme of months is and was in use in Northern India, the amânta scheme alone is recognized in the matter of the nomenclature and intercalation of lunar months and the commencement of the luni-solar year. The following is the method adopted—first, the ordinary rule of naming a month is applied to an amânta lunar month, and then, by the pûrṇimânta scheme, the dark fortnight of it receives the name of the following month. The correspondence of amânta and pûrṇimânta fortnights for a year is shown in Table II., Part i., and it will be observed that the bright fortnights have the same name by both schemes while the dark fortnights differ by a month, and thus the pûrṇimânta scheme is always a fortnight in advance of the amânta scheme.

The saṅkrântis take place in definite amânta lunar months, thus the Makara-saṅkrânti invariably takes place in amânta Pausha, and in no other month; but when it takes place in the kṛishṇapaksha of amânta Pausha it falls in pûrṇimânta Magha, because that fortnight is said to belong to Magha by the pûrṇimânta scheme. If, however, it takes place in the sukla paksha, the month is Pausha by both schemes. Thus the Makara-sankranti, though according to the amânta scheme it can only fall in Pausha, may take place either in Pausha or Magha by the pûrṇimânta scheme; and so with the rest.

The following rules govern pûrṇimânta intercalations. Months are intercalated at first as if there were no pûrṇimânta scheme, and afterwards the dark fortnight preceding the intercalated month receives, as usual, the name of the month to which the following natural bright fortnight belongs, and therefore the intercalated month also receives that name. Thus, in the example given above (Art. 45), intercalated amânta Vaiśâkha (as named by the first rule) lies between natural amânta Chaitra and natural amânta Vaiśâkha. But by the pûrṇimânta scheme the dark half of natural amânta Chaitra acquires the name of natural Vaiśâkha; then follow the two fortnights of adhika Vaiśâkha; and after them comes the bright half of the (nija) natural pûrṇimânta Vaiśâkha. Thus it happens that half of natural pûrṇimânta Vaiśâkha comes before, and half after, the intercalated month.^[61]

Of the four fortnights thus having the name of the same month the first two fortnights are sometimes called the "First Vaiśâkha," and the last two the "Second Vaiśâkha."

It will be seen from Table II., Part i., that amânta Phalguna kṛishṇa is pûrṇimânta Chaitra kṛishṇa. The year, however, does not begin then, but on the same day as the amânta month, i.e., with the new moon, or the beginning of the next bright fortnight.

Having discussed the lesser divisions of time, we now revert to the Hindu year. And, first, its beginning.

Years and Cycles.

52. The Hindu New-year's Day.—In Indian astronomical works the year is considered to begin, if luni-solar, invariably with amânta Chaitra śukla 1st,—if solar with the Mesha saṅkrânti; and in almost all works mean Mesha saṅkrânti is taken for convenience of calculations, very few works adopting the apparent or true one. At present in Bengal and the Tamil country, where solar reckoning is in use, the year, for religious and astronomical purposes, commences with the apparent Mesha-saṅkrânti, and the civil year with the first day of the month Mesha, as determined by the practice of the country (See above Art. 28). But since mean Mesha-saṅkrânti is taken as the commencement of the solar year in astronomical works, it is only reasonable to suppose that the year actually began with it in practice in earlier times, and we have to consider how long ago the practice ceased.

In a Karaṇa named Bhâsvatî (A. D. 1099) the year commences with apparent Mesha saṅkrânti, and though it is dangerous to theorize from one work, we may at least quote it as shewing that the present practice was known as early as A. D. 1100. This date coinciding fairly well with Śrîpati's injunction quoted above (Art. 47) we think it fair to assume for the present that the practice of employing the mean Mesha saṅkrânti for fixing the beginning of the year ceased about the same time as the practice of mean intercalary months.

The luni-solar Chaitrâdi^[62] year commences, for certain religious and astrological purposes, with the first moment of the first tithi of Chaitra, or Chaitra śukla pratipadâ and this, of course, may fall at any time of the day or night, since it depends on the moment of new moon. But for the religious ceremonies connected with the beginning of a samvatsara (year), the sunrise of the day on which Chaitra śukla pratipadâ is current at sunrise is taken as the first or opening day of the year. When this tithi is current at sunrise on two days, as sometimes happens, the first, and when it is not current at any sunrise (i.e., when it is expunged) then the day on which it ends, is taken as the opening day. For astronomical purposes the learned take any convenient moment,—such as mean sunrise, noon, sunset, or midnight, but generally the sunrise,—on or before Chaitra śukla pratipadâ, as their starting-point.^[63] Sometimes the beginning of the mean Chaitra śukla pratipadâ is so taken.

When Chaitra is intercalary there seems to be a difference of opinion whether the year in that case is to begin with the intercalated (adhika) or natural (nija) Chaitra. For the purposes of our Table I. (cols. 19 to 25) we have taken the adhika Chaitra of the true system as the first month of the year.

But the year does not begin with Chaitra all over India. In Southern India and especially in Gujarât the years of the Vikrama era commence in the present day with Kârttika śukla pratipadâ. In some parts of Kâṭhiâvâḍ and Gujarât the Vikrama year commences with Âshâḍha śukla pratipadâ.^[64] In a part of Ganjam and Orissa, the year begins on Bhâdrapada śukla 12th. (See under Oṅko reckoning, Art. 64.) The Amli year in Orissa begins on Bhâdrapada śukla 12th, the Vilâyatî year, also in general use in Orissa, begins with the Kanyâ saṅkrânti; and the Fasli year, which is luni-solar in Bengal, commences on pûrṇimânta Âśvina kṛi. 1st (viz., 4 days later than the Vilâyatî).

In the South Malayâḷam country (Travancore and Cochin), and in Tinnevelly, the solar year of the Kollam era, or Kollam âṇḍu, begins with the month Chiṅgam (Siṁha), and in the North Malayâḷam tract it begins with the month Kanni (Kanyâ). In parts of the Madras Presidency the Fasli year originally commenced on the 1st of the solar month Âḍî (Karka), but by Government order about A.D. 1800 it was made to begin on the 13th of July, and recently it was altered again, so that now it begins on 1st July. In parts of the Bombay Presidency the Fasli year begins when the sun enters the nakshatra Mṛigaśîrsha, which takes place at present about the 5th or 6th of June.

Alberuni mentions (A.D. 1030) a year commencing with Mârgaśîrsha as having been in use in Sindh, Multân, and Kanouj, as well as at Lahore and in that neighbourhood; also a year commencing with Bhâdrapada in the vicinity of Kashmir.^[65] In the Mahâbhârata the names of the months are given in some places, commencing with Mârgaśîrsha. (Anuśâsana parva adhyâyas 106 and 109). In the Vedâṅga Jyotisha the year commences with Mâgha śukla pratipadâ.

53. The Sixty-year cycle of Jupiter.^[66] In this reckoning the years are not known by numbers, but are named in succession from a list of 60 names, often known as the "Bṛihaspati samvatsara chakra,"^[67] the wheel or cycle of the years of Jupiter. Each of these years is called a "samvatsara." The word "samvatsara" generally means a year, but in the case of this cycle the year is not equal to a solar year. It is regulated by Jupiter's mean motion; and a Jovian year is the period during which the planet Jupiter enters one sign of the zodiac and passes completely through it with reference to his mean motion. The cycle commences with Prabhava. See Table I., cols. 6, 7, and Table XII.

54. The duration of a Bârhaspatya samvatsara, according to the Sûrya-Siddhânta, is about 361.026721 days, that is about 4.232 days less than a solar year. If, then, a samvatsara begins exactly with the solar year the following samvatsara will commence 4.232 days before the end of it. So that in each successive year the commencement of a samvatsara will be 4.232 days in advance, and a time will of course come when two samvatsaras will begin during the same solar year. For example, by the Surya-Siddhanta with the bîja, Prabhava (No. 1) was current at the beginning of the solar year Saka 1779. Vibhava (No. 2) commenced 3.3 days after the beginning of that year, that is after the Mesha saṅkrânti; and Śukla (No. 3) began 361.03 days after Vibhava, that is 364.3 days after the beginning of the year. Thus Vibhava and Śukla both began in the same solar year. Now as Prabhava was current at the beginning of Śaka 1779, and Śukla was current at the beginning of Śaka 1780, Vibhava was expunged in the regular method followed in the North. Thus the rule is that when two Bârhaspatya samvatsaras begin during one solar year the first is said to be expunged, or to have become kskaya; and it is clear that when a samvatsara begins within a period of about 4.232 days after a Mesha saṅkrânti it will be expunged.

By the Sûrya-Siddhânta 85 65/211 solar years are equal to 86 65/211 Jovian years. So that one expunction is due in every period of 85 65/211 solar years. But since it really takes place according to the rule explained above, the interval between two expunctions is sometimes 85 and sometimes 86 years.

55. Generally speaking the samvatsara which is current at the beginning of a year is in practice coupled with all the days of that year, notwithstanding that another samvatsara may have begun during the course of the year. Indeed if there were no such practice there would be no occasion for an expunction. Epigraphical and other instances, however, have been found in which the actual samvatsara for the time is quoted with dates, notwithstanding that another samvatsara was current at the beginning of the year.^[68]

56. Variations. As the length of the solar year and year of Jupiter differs with different Siddhântas it follows that the expunction of samvatsaras similarly varies.

57. Further, since a samvatsara is expunged when two samvatsaras begin in the same year, these expunctions will differ with the different kinds of year. Where luni-solar years are in use it is only natural to suppose that the rule will be made applicable to that kind of year, an expunction occurring when two samvatsaras begin in such a year; and there is evidence to show that in some places at least, such was actually the case for a time. Now the length of an ordinary luni-solar year (354 days) is less than that of a Jovian year (361 days), and therefore the beginning of two consecutive samvatsaras can only occur in those luni-solar years in which there is an intercalary month. Again, the solar year sometimes commences with the mean Mesha-saṅkrânti, and this again gives rise to a difference.^[69]

The Jyotisha-tattva rule (given below Art. 59) gives the samvatsara current at the time of the mean, not of the apparent, Mesha-saṅkrânti, and hence all expunctions calculated thereby must be held to refer to the solar year only when it is taken to commence with the mean Meshasaṅkrânti.^[70] It is important that this should be remembered.

58. To find the current samratsara. The samvatsaras in our Table I., col. 7, are calculated by the Sûrya-Siddhânta without the bîja up to A.D. 1500, and with the bîja from AD. 1 501 to 1900; and are calculated from the apparent Mesha-saṅkrânti. If the samvatsara current on a particular day by some other authority is required, calculations must be made direct for that day according to that authority, and we therefore proceed to give some rules for this process.

59. Rules for finding the Bârhaspatya samvatsara current on a particular day.^[71]

a. By the Sûrya-Siddhânta.^[72] Multiply the expired Kali year by 211. Subtract 108 from the product. Divide the result by 18000. To the quotient, excluding fractions, add the numeral of the expired Kali year plus 27. Divide the sum by 60. The remainder, counting from Prabhava as 1, is the samvatsara current at the beginning of the given solar year, that is at its apparent Mesha-saṅkrânti. Subtract from 18000 the remainder previously left after dividing by 18000. Multiply the result by 361, and divide the product by 18000. Calculate for days, ghaṭikâs, and palas. Add 15 palas to the result. The result is then the number of days, etc., elapsed between the apparent Mesha-saṅkrânti and the end of the samvatsara current thereon. By this process can be found the samvatsara current on any date.

Example 1.—Wanted the samvatsara current at the beginning of Saka 233 expired and the date on which it ended. Śaka 233 expired = (Table I.) Kali 3412 expired. 3412 × 214 − 108/18000 = 39 17824/18000. 39 + 3412 + 27 = 3478. 3478/60 = 57 58/60. The remainder is 58; and we have it that No. 58 Raktâkshin (Table XII.) was the samvatsara current at the beginning (apparent Mesha-saṅkrânti) of the given year. Again; 18000 − 17824 = 176. 176 × 361/18000 = 3 d. 31 gh. 47.2 p. Adding 15 pa. we have 3 d. 32 gh. 2.2 pa. This shews that Raktâkshin will end and Krodhana (No. 59) begin 3 d. 32 gh. 2.2 pa. after the apparent Meska saṅkrânti. This last, by the Sûrya Siddhânta, occurred on 17th March, A.D. 311, at 27 gh. 23 pa. (see Table I., col. 13, and the Table in Art. 96), and therefore Krodhana began on the 20th March at 59 gh. 25.2 pa., or 34.8 palas before mean sunrise on 2 1st March. We also know that since Krodhana commences within four days after Mesha it will he expunged (Art. 54 above.)

b. By the Ârya Siddhânta. Multiply the expired Kali year by 22. Subtract 11 from the product. Divide the result by 1875. To the quotient excluding fractions add the expired Kali year + 27. Divide the sum by 60. The remainder, counted from Prabhava as 1, is the samvatsara current at the beginning of the given solar year. Subtract from 1875 the remainder previously left after dividing by 1875. Multiply the result by 361. Divide the product by 1875. Add 1 gh. 45 pa. to the quotient. The result gives the number of days, etc., that have elapsed between the apparent Mesha-saṅkrânti and the end of the samvatsara current thereon.

Example 2.—Required the samvatsara current at the beginning of Śaka 230 expired, and the time when it ended.

Śaka 230 expired = Kali 3409 expired. 3409 × 22 − 11/1875 = 39 1862/1875. 39 + 3409 + 27 = 3475, which, divided by 60, gives the remainder 55. Then No. 55 Durmati (Table XII.) was current at the beginning of the given year. Again; 1875 − 1862 = 13. 13 × 361/1875 = 2 d. 30 gh. 10.56 pa. Adding 1 gh. 45 pa., we get 2 d. 31 gh. 55.56 pa. Add this to the moment of the Mesha saṅkrânti as given in Table I., cols. 13—16, viz., 16th March, 308 A.D., Tuesday, at 41 gh. 40 p., and we have 19th March, Friday, 13 gh. 35.56 p. after mean sunrise as the moment when Durmati ends and Dundubhi begins. Here again, since Dundubhi commences within four days of the Mesha saṅkrânti, it will be expunged.

c. By the Sûrya-Siddhânta with the bîja (to be used for years after about 1500 A.D.). Multiply the expired Kali year by 117. Subtract 60 from the product. Divide the result by 10000. To the figures of the quotient, excluding fractions, add the number of the expired Kali year plus 27. Divide the sum by 60. And the remainder, counted from Prabhava as 1, is the samvatsara current at the beginning of the given solar year. Subtract from 10000 the remainder left after the previous division by 10000. Multiply the difference by 361, and divide the product by 10000. Add 15 pa. The result is the number of days, etc., that have elapsed between the apparent Mesha sankranti and the end of the samvatsara current thereon.^[73] Example.— Required the samvatsara current at the beginning of Śaka 1436 expired, and the moment when it ends. Śaka 1436 expired = Kali 4615 expired (Table I.), 4615 × 117 − 60/10000 = 53 9895/10000. 53 + 4615 + 27/60 = 78 15/60. The remainder 15 shews that Vṛisha was current at the Mesha-saṅkrânti. (10000 − 9895) 361/10000 + 15 p. = 3 d. 47 gh. 25.8 p. + 15 p. = 3 d. 47 gh. 40.8 p. Table I. gives the Meshasaṅkrânti as March 27th, 44 gh. 25 p., Monday. 27 d. 44 gh. 25 p. + 3 d. 47 gh. 40.8 p. = 31 d. 32 gh. 5.8 p.; and this means that Vṛisha ended at 32 gh. 5.8 p. after mean sunrise at Ujjain on Friday, 31st March. At that moment Chitrabhânu begins, and since it began within four days of the Mesha-saṅkrânti, it is expunged.

d. Bṛihatsaṁhitâ and Jyotishatattva Rules. The rules given in the Bṛihatsaṁhitâ and the Jyotishatattva seem to be much in use, and therefore we give them here. The Jyotishatattva rule is the same as that for the Ârya-Siddhânta given above, except that it yields the year current at the time of mean Mesha-saṅkrânti, and that it is adapted to Śaka years. The latter difference is merely nominal of course, as the moment of the beginning of a samvatsara is evidently the same by both.^[74] We have slightly modified the rules, but in words only and not in sense.

The Jyotishatattva rule is this. Multiply the current Śaka year by 22. Add 4291. Divide the sum by 1875. To the quotient excluding fractions add the number of the current Śaka year. Divide the sum by 60. The remainder, counted from Prabhava as 1, is the samvatsara current at the beginning of the given year. Subtract the remainder left after previously dividing by 1875 from 1875. Multiply the result by 361. And divide the product by 1875. The result gives the number of days by which, according to the Ârya-Siddhânta, the samvatsara ends after mean Meshasaṅkrânti. The mean^[75] Mesha-saṅkrânti will be obtained by adding 2 d. 8 gh. 51 pa. 15 vipa. to the time given in Table I., cols. 13 to 18.

Work out by this rule the example given above under the Ârya-Siddhânta rule, and the result will be found to be the same by both.

The Bṛihatsaṁhitâ rule. Multiply the expired Śaka year by 44. Add 8589. Divide the sum by 3750. To the quotient, excluding fractions, add the number of the expired Śaka year plus 1. Divide the sum by 60. The remainder, counted from Prabhava as 1, is the samvatsara current at the beginning of the year. Subtract from 3750 the remainder obtained after the previous division by 3750. Multiply the result by 361, and divide the product by 3750. This gives the number of days by which the samvatsara current at the beginning of the year will end after the Mesha saṅkrânti.^[76]

60. List of Expunged Samvatsaras. The following is a comparative list of expunged samvatsaras as found by different authorities, taking the year to begin at the mean Mesha saṅkrânti.

**List of Expunged Samvatsaras.**^[77]
First Ârya-Siddhânta, Bṛihatsaṁhitâ, Ratnamâlâ, Jyotishatattva Rules.			Sûrya-Siddhânta Rule without bîja up to 1500 A.D., and with bîja afterwards.			First Ârya-Siddhânta, Bṛihatsaṁhitâ, Ratnamâlâ, Jyotishatattva Rules.			Sûrya-Siddhânta Rule without bîja up to 1500 A.D., and with bîja afterwards.
Śaka year current.	A. D.	Expunged Samvatsara.	Śaka year current.	A. D.	Expunged Samvatsara.	Śaka year current.	A. D.	Expunged Samvatsara.	Śaka year current.	A. D.	Expunged Samvatsara.
232	309–10	57 Rudhirodgârin	234	311–12	59 Krodhana	1084	1161–62	19 Pârthiva	1087	1164–65	22 Sarvadhârin
317	394–95	23 Virodhin	319*	396–97	25 Khara	1169	1246–47	34 Virodhakṛit	1172*	1249–50	48 Ânanda
402	479–80	49 Râkshasa	404*	481–82	51 Piṅgala	1254	1331–32	11 Îśvara	1258	1335–36	15 Vṛisha
487	564–65	15 Vṛisha	490	567–68	18 Târaṇa	1340	1417–19	38 Krodhin	1343	1420–21	41 Plavaṅga
572	649–50	41 Plavaṅga	575*	652–53	44 Sâdhâraṇa	1425	1502–03	4 Pramoda	1437	1514–15	16 Chitrabhânu
658	735–36	8 Bhâva	660*	737–38	10 Dhâtṛi	1510	1587–88	39 Durmukha	1522*	1599–1600	42 Kîlaka
743	820–21	34 Śârvari	746	823–24	37 Śobhana
828	905–06	60 Kshaya	831	908–09	3 Śukla	1595	1672–73	56 Dundubhi	1608	1685–86	9 Yuvan
913	990–91	26 Nandana	916*	993–94	29 Manmatha	1680	1757–58	22 Sarvadhârin	1693*	1770–71	35 Plava
999	1076–77	53 Siddhârthin	1002	1079–80	56 Dundubhi	1766	1843–44	49 Râkshasa	1779	1856–57	2 Vibhava

If we take the years to commence with the apparent Mesha-saṅkrânti the samvatsaras expunged by Sûrya Siddhânta calculation will be found in Table I., col. 7; and those by the Ârya Siddhânta can be found by the rule for that Siddhânta given in Art. 59 above.

61. The years of Jupiter's cycle are not mentioned in very early inscriptions. They are mentioned in the Sûrya-Siddhânta. Dr. J. Burgess states that he has reason to think that they were first introduced about A.D. 349, and that they were certainly in use in A.D. 530. We have therefore given them throughout in Table I.

62. The southern (luni-solar) sixty-year cycle. The sixty-year cycle is at present in daily use in Southern India (south of the Narmadâ), but there the samvatsaras are made to correspond with the luni-solar year as well as the solar; and we therefore term it the luni-solar 60-year cycle in contradistinction to the more scientific Bârhaspatya cycle of the North. There is evidence^[78] to show that the cycle of Jupiter was in use in Southern India before Saka 828 (A.D. 905-6); but from that year, according to the Ârya Siddhânta, or from Saka 831 (A.D. 908-9) according to the Sûrya-Siddhânta, the expunction of the samvatsaras was altogether neglected, with the result that the 60-year cycle in the south became luni-solar from that year. At present the northern samvatsara has advanced by 12 on the southern. There is an easy rule for finding the samvatsara according to the luni-solar cycle, viz., add 11 to the current Saka year, and divide by 60; the remainder is the corresponding luni-solar cycle year. It must not be forgotten that the samvatsaras of Jupiter's and the southern cycle, are always to be taken as current years, not expired.

63. The twelve-year cycle of Jupiter. There is another cycle of Jupiter consisting of twelve samvatsaras named after the lunar months. It is of two kinds. In one, the samvatsara begins with the heliacal rising^[79] of Jupiter and consists of about 400 solar days, one samvatsara being expunged every 12 years or so.^[80] In the other, which we have named the "twelve-year cycle of Jupiter of the mean-sign system", the years are similar in length to those of the sixty-year cycle of Jupiter just described, and begin at the same moment. Both kinds, though chiefly the former, were in use in early times, and the latter is often employed in modern dates, especially in those of the Kollam era. The samvatsaras of this heliacal rising system can only be found by direct calculations according to some Siddhânta. The correspondence of the samvatsaras of the mean-sign system with those of the sixty-year cycle are given in Table XII. They proceed regularly.

64. The Graha-parivṛitti and Oṅko cycles. There are two other cycles, but they are limited to small tracts of country and would perhaps be better considered as eras. We however give them here.

The southern inhabitants of the peninsula of India (chiefly of the Madura district) use a cycle of 90 solar years which is called the Graha-parivṛitti. Warren has described the cycle, deriving his information from the celebrated Portuguese missionary Beschi, who lived for over forty years in Madura. The cycle consists of 90 solar years, the length of one year being 365 d. 15 gh. 31 pa. 30 vi., and the year commences with Mesha. Warren was informed by native astronomers at Madras that the cycle consisted of the sum in days of 1 revolution of the sun, 15 of Mars, 22 of Mercury, 11 of Jupiter, 5 of Venus and 29 of Saturn, though this appears to us quite meaningless. The length of this year is that ascertained by using the original Sûrya-Siddhânta; but from the method given by Warren for finding the beginning of the years of this cycle it appears that astronomers have tried to keep it as nearly as possible in agreement with calculations by the Ârya-Siddhânta, and in fact the year may be said to belong to the Ârya-Siddhânta. The cycle commenced with Kali 3079 current (B. C. 24) and its epoch, i.e., the Graha-parivṛitti year 0 current^[81] is Kali 3078 current (B. C. 25).

To find the year of the Graha-parivṛitti cycle, add 72 to the current Kali-year, 11 to the current Śaka year, or 24 or 23 to the A.D. year, viz., 24 from Mesha to December 31st, and 23 from January 1st to Mesha; divide by 90 and the remainder is the current year of the cycle.

The Oṅko^[82] cycle of 59 luni-solar years is in use in part of the Ganjam district of the Madras Presidency. Its months are pûrṇimânta, but it begins the year on the 12th of Bhâdrapada-śuddha,^[83] calling that day the 12th not the 1st. In other words, the year changes its numerical designation every 12th day of Bhâdrapada-śuddha. It is impossible as yet to say decidedly when the Oṅko reckoning commenced. Some records in the temple of Jagannātha at Purī (perfectly valueless from an historical point of view) show that it commenced with the reign of Subhānideva in 319 A.D., but the absurdity of this is proved by the chronicler's statement that the great Mughal invasion took place in 327 A.D. in the reign of that king's successor.^[84] Some say that the reckoning commenced with the reign of Chōḍagaṅga or Chōrgaṅga, the founder of the Gāṅgavaṁśa, whose date is assigned usually to 1131-32 A.D., while Sutton in his History of Orissa states that it was introduced in 1580 A.D. In the zamindari tracts of Parlakimeḍi, Peddakimeḍi and Chinnakimeḍi the Oṅko Calendar is followed, but the people there also observe each a special style, only differing from the parent style and from one another in that they name their years after their own zamindars. A singular feature common to all these four kinds of regnal years is that, in their notation, the years whose numeral is 6, or whose numerals end with 6 or 0 (except 10), are dropped.^[85] For instance, the years succeeding the 5th and 19th Oṅkos of a prince or zamindar are called the 7th and 21st Oṅkos respectively. It is difficult to account for this mode of reckoning; it may be, as the people themselves allege, that these numerals are avoided because, according to their traditions and śâstras, they forebode evil, or it may possibly be, as some might be inclined to suppose, that the system emanated from a desire to exaggerate the length of each reign. There is also another unique convention according to which the Oṅko years are not counted above 59, but the years succeeding 59 begin with a second series, thus "second 1", "second 2", and so on. It is also important to note that when a prince dies in the middle of an Oṅko year, his successor's 1st Oṅko which commences on his accession to the throne, does not run its full term of a year, but ends on the 12th day of Bhâdrapada-śuddha following; consequently the last regnal year of the one and the first of the other together occupy only one year, and one year is dropped in effect. To find, therefore, the English equivalent of a given Oṅko year, it will be necessary first to ascertain the style to which it relates, i.e., whether it is a Jagannātha Oṅko or a Parlakimeḍi Oṅko, and so on; and secondly to value the given year by excluding the years dropped (namely, the 1st—possibly, the 6th, 16th, 20th, 26th, 30th, 36th, 40th, 46th, 50th, 56th). There are lists of Orissa princes available, but up to 1797 A.D. they would appear to be perfectly inauthentic.^[86] The list from that date forwards is reliable, and below are given the names of those after whom the later Oṅko years have been numbered, with the English dates corresponding to the commencement of the 2nd Oṅkos of their respective reigns.

Oṅko 2 of Mukundadeva	September 2, 1797.	(Bhâdrapada śukla 12th.)
Onko 2 ofDo. Râmachandradcva	September 22, 1817.	(BhadrapadaDo. sukla 12th.)Do.
Onko 2 ofDo. Vîrakeśvaradeva	September 4, 1854.	(BhadrapadaDo. sukla 12th.)Do.
Onko 2 ofDo. Divyasiṁhadeva	September 8, 1859.	(BhadrapadaDo. sukla 12th.)Do.

PART II.

The Various Eras.

65. General remarks. Different eras have, from remote antiquity, been in use in different parts of India, having their years luni-solar or solar, commencing according to varying practice with a given month or day; and in the case of luni-solar years, having the months calculated variously according to the amânta and pûrṇimânta system of pakshas. (Art. 12 above). The origin of some eras is well known, but that of others has fallen into obscurity. It should never be forgotten, as explaining at once the differences of practice we observe, that when considering "Indian" science we are considering the science of a number of different tribes or nationalities, not of one empire or of the inhabitants generally of one continent.

66. If a number of persons belonging to one of these nationalities, who have been in the habit for many years of using a certain era with all its peculiarities, leave their original country and settle in another, it is natural that they should continue to use their own era, notwithstanding that another era may be in use in the country of their adoption; or perhaps, while adopting the new era, that they should apply to it the peculiarities of their own. And vice versâ it is only natural that the inhabitants of the country adopted should, when considering the peculiarities of the imported era, treat it from their own stand-point.

67. And thus we actually find in the pañchâṅgs of some provinces a number of other eras embodied, side by side with the era in ordinary use there, while the calendar-makers have treated them by mistake in the same or nearly the same manner as that of their own reckoning. For instance, there are extant solar pañchâṅgs of the Tamil country in which the year of the Vikrama era is represented as a solar Meshâdi year. And so again Śaka years are solar in Bengal and in the Tamil country, and luni-solar in other parts of the country. So also we sometimes find that the framers of important documents have mentioned therein the years of several eras, but have made mistakes regarding them. In such a case we might depend on the dates in the document if we knew exactly the nationality of the authors, but very often this cannot be discovered, and then it is obviously unsafe to rely on it in any sense as a guide. This point should never be lost sight of.

68. Another point to be always borne in mind is that, for the sake of convenience in calculation a year of an era is sometimes treated differently by different authors in the same province, or indeed even by the same author. Thus, Gaṇeśa Daivajña makes Śaka years begin with Chaitra śukla pratipadâ in his Grahalâghava (A.D. 1520), but with mean Mesha saṅkrânti in his Tithichintāmaṇi (A.D. 1525.)

69. It is evident therefore that a certain kind of year, e.g., the solar or luni-solar year, or a certain opening month or day, or a certain arrangement of months and fortnights and the like, cannot be strictly defined as belonging exclusively to a particular era or to a particular part of India. We can distinctly affirm that the eras whose luni-solar years are Chaitrâdi (i.e., beginning with Chaitra śukla pratipadâ) are always Meshâdi (beginning with the Mesha saṅkrânti) in their corresponding solar reckoning, but beyond this it is unsafe to go.

70. Current and expired years. It is, we believe, now generally known what an "expired" or "current" year is, but for the benefit of the uninitiated we think it desirable to explain the matter fully. Thus; the same Śaka year (A.D. 1894) which is numbered 1817 vartamâna, or astronomically current, in the pañchâṅgs of the Tamil countries of the Madras Presidency, is numbered 1816 gata ("expired") in other parts of India. This is not so unreasonable as Europeans may imagine, for they themselves talk of the third furlong after the fourth mile on a road as "four miles three furlongs" which means three furlongs after the expiry of the fourth mile, and the same in the matter of a person's age; and so September, A.D. 1894, (Śaka 1817 current) would be styled in India "Śaka 1816 expired, September", equivalent to "September after the end of Śaka 1816" or "after the end of 1893 A.D". Moreover, Indian reckoning is based on careful calculations of astronomical phenomena, and to calculate the planetary conditions of September, 1894, it is necessary first to take the planetary conditions of the end of 1893, and then add to them the data for the following nine months. That is, the end of 1893 is the basis of calculation. It is always necessary to bear this in mind because often the word gata is omitted in practice, and it is therefore doubtful whether the real year in which an inscription was written was the one mentioned therein, or that number decreased by one.^[87]

In this work we have given the corresponding years of the Kali and Śaka eras actually current, and not the expired years. This is the case with all eras, including the year of the Vikrama^[88] era at present in use in Northern India.

71. Description of the several eras. In Table II., Part iii., below we give several eras, chiefly those whose epoch is known or can be fixed with certainty, and we now proceed to describe them in detail.

The Kali-Yuga.—The moment of its commencement has been already given (Art. 16 above). Its years are both Chaitrâdi (luni-solar) and Meshâdi (solar.) It is used both in astronomical works and in pañchâṅgs. In the latter sometimes its expired years, sometimes current years are given, and sometimes both. It is not often used in epigraphical records.^[89]

Saptarshi-Kala.—This era is in use in Kashmir and the neighbourhood. At the time of Alberuni (1030 A.D.), it appears to have been in use also in Multân and some other parts. It is the only mode of reckoning mentioned in the Râja-Taraṅgiṇi. It is sometimes called the "Laukika-Kâla" and sometimes the "Śâstra-Kâla". It originated on the supposition that the seven Ṛishis (the seven bright stars of Ursa Major) move through one nakshatra (27th part of the ecliptic) in 100 years, and make one revolution in 2700 years; the era consequently consists of cycles of 2700 years. But in practice the hundreds are omitted, and as soon as the reckoning reaches 100, a fresh hundred begins from 1. Kashmirian astronomers make the era, or at least one of its cycles of 2700 years, begin with Chaitra śukla 1st of Kali 27 current. Disregarding the hundreds we must add 47 to the Saptarshi year to find the corresponding current Saka year, and 24—25 for the corresponding Christian year. The years are Chaitrâdi. Dr. F. Kielhorn finds^[90] that they are mostly current years, and the months mostly pûrṇimânta.

The Vikrama era.—In the present day this era is in use in Gujarât and over almost all the north of India, except perhaps Bengal.^[91] The inhabitants of these parts, when migrating to other parts of India, carry the use of the era with them. In Northern India the year is Chaitrâdi, and its months pûrṇimânta, but in Gujarat it is Kârttikâdi and its months are amânta. The settlers in the Madras Presidency from Northern India, especially the Mârvâḍis who use the Vikrama year, naturally begin the year with Chaitra śukla pratipadâ and employ the pûrṇimânta scheme of months; while immigrants from Gujarât follow their own scheme of a Kârttikâdi amânta year, but always according to the Vikrama era. In some parts of Kâṭhiâvâḍ and Gujarât the Vikrama era is Âshâḍhâdi^[92] and its months amânta. The practice in the north and south leads in the present day to the Chaitrâdi pûrṇimânta Vikrama year being sometimes called the "Northern Vikrama," and the Kârttikâdi amânta Vikrama year the "Southern Vikrama."

The correspondence of these three varieties of the Vikrama era with the Śaka and other eras, as well as of their months, will be found in Table II., Parts ii. and iii.

Prof. F. Kielhorn has treated of this era at considerable length in the Ind. Antiq., vols. XIX. and XX., and an examination of 150 different dates from 898 to 1877 of that era has led him to the following conclusions (ibid., XX., p. 398 ff.).

(1) It has been at all times the rule for those who use the Vikrama era to quote the expired years, and only exceptionally^[93] the current year.

(2) The Vikrama era was Kârttikâdi from the beginning, and it is probable that the change which has gradually taken place in the direction of a more general use of the Chaitrâdi year was owing to the increasing growth and influence of the Śaka era. Whatever may be the practice in quite modern times, it seems certain that down to about the 14th century of the Vikrama era both kinds of years, the Kârttikâdi and the Chaitrâdi, were used over exactly the same tracts of country, but more frequently the Kârttikâdi.

(3) While the use of the Kârttikâdi year has been coupled with the pûrṇimânta as often as with the amânta scheme of months, the Chaitrâdi year is found to be more commonly joined with the pûrṇimânta scheme: but neither scheme can be exclusively connected with either the Kârttikâdi or Chaitrâdi year.

The era was called the "Mâlava" era from about A.D. 450 to 850. The earliest known date containing the word "Vikrama" is Vikrama-samvat 898 (about A.D. 840); but there the era is somewhat vaguely described as "the time called Vikrama"; and it is in a poem composed in the Vikrama year 1050 (about A.D. 992) that we hear for the first time of a king called Vikrama in connection with it. (See Ind. Antiq., XX., p. 404).

At the present day the Vikrama era is sometimes called the "Vikrama-samvat", and sometimes the word "samvat" is used alone as meaning a year of that era. But we have instances in which the word "samvat" (which is obviously an abbreviation of the word samvatsara, or year) is used to denote the years of the Śaka, Siṁha, or Valabhi eras^[94] indiscriminately.

In some native pañchâṅgs from parts of the Madras presidency and Mysore for recent years the current Vikrama dates are given in correspondence with current Śaka dates; for example, the year corresponding to A.D. 1893—94 said to be Śaka 1816, or Vikrama 1951. (See remarks on the Śaka era above.)

The Christian era. This has come into use in India only since the establishment of the English rule. Its years at present are tropical solar commencing with January 1st, and are taken as current years. January corresponds at the present time with parts of the luni-solar amânta months Mârgaśîrsha and Pausha, or Pausha and Mâgha. Before the introduction of the new style, however, in 1752 A.D., it coincided with parts of amânta Pausha and Mâgha, or Mâgha and Phâlguna. The Christian months, as regards their correspondence with luni-solar and solar months, are given in Table II., Part ii.

The Śaka era.—This era is extensively used over the whole of India; and in most parts of Southern India, except in Tinnevelly and part of Malabar, it is used exclusively. In other parts it is used in addition to local eras. In all the Karaṇas, or practical works on astronomy it is used almost exclusively.^[95] Its years are Chaitrâdi for luni-solar, and Meshâdi for solar, reckoning. Its months are pûrṇimânta in the North and amânta in Southern India. Current years are given in some pañchâṅgs, but the expired years are in use in most^[96] parts of India.

The Chedi or Kalachuri era.—This era is not now in use. Prof. F. Kielhorn, examining the dates contained in ten inscriptions of this era from 793 to 934,^[97] has come to the conclusion that the 1st day of the 1st current Chedi year corresponds to Âśvina śukla pratipadâ of Chaitrâdi Vikrama 306 current, (Śaka 171 current, 5th Sept., A. D. 248); that consequently its years are Âśvinâdi; that they are used as current years; that its months are pûrṇimânta; and that its epoch, i.e., the beginning of Chedi year 0 current, is A. D. 247—48.

The era was used by the Kalachuri kings of Western and Central India, and it appears to have been in use in that part of India in still earlier times.

The Gupta era.—This era is also not now in use. Dr. Fleet has treated it at great length in the introduction to the Corpus. Inscrip. Ind. (Vol. III, "Gupta Inscriptions"), and again in the Indian Antiquary (Vol. XX., pp. 376 ff.) His examination of dates in that era from 163 to 386 leads him to conclude that its years are current and Chaitrâdi; that the months are pûrṇimânta; and that the epoch, i.e., the beginning of Gupta Samvat 0 current, is Śaka 242 current (A. D. 319—20). The era was in use in Central India and Nepal, and was used by the Gupta kings.

The Valabhi era.—This is merely a continuation of the Gupta era with its name changed into "Valabhi." It was in use in Kâṭhiâvâḍ and the neighbourhood, and it seems to have been introduced there in about the fourth Gupta century. The beginning of the year was thrown back from Chaitra śukla ist to the previous Kârttika śukla 1st, and therefore its epoch went back five months, and is synchronous with the current Kârttikâdi Vikrama year 376 (A. D. 318—19, Śaka 241—42 current). Its months seem to be both amânta and pûrṇimânta.

The inscriptions as yet discovered which are dated in the Gupta and Valabhi era range from the years 82 to 945 of that era.

The Bengali San.—An era named the "Bengali San" (sometimes written in English "Sen") is in use in Bengal. It is a solar year and runs with the solar Śaka year, beginning at the Mesha saṅkrânti; but the months receive lunar month names, and the first, which corresponds with the Tamil Chaitra, or with Mesha according to the general reckoning, is here called Vaiśâkha, and so on throughout the year, their Chaitra corresponding with the Tamil Phâlguna, or with the Mîna of our Tables. We treat the years as current ones. Bengali San 1300 current corresponds with Śaka 1816 current (A. D. 1893—94.) Its epoch was Śaka 516 current, A. D. 593—94. To convert a Bengali San date into a Śaka date for purposes of our Tables, add 516 to the former year, which gives the current Śaka solar year, and adopt the comparison of months given in Table II., Part, ii., cols. 8, 9.

The Vilâyatî year.—This is another solar year in use in parts of Bengal, and chiefly in Orissa; it takes lunar-month names, and its epoch is nearly the same as that of the "Bengali San", viz., Śaka 515—16 current, A.D. 592—93, But it differs in two respects. First, it begins the year with the solar month Kanyâ which corresponds to Bengal solar Âśvina or Âssin. Secondly, the months begin on the day of the saṅkrânti instead of on the following (2nd) or 3rd day (see Art. 28, the Orissa Rule).

The Amli Era of Orissa—This era is thus described in Giriśa Chandra's "Chronological Tables" (preface, p. xvi.): "The Amli commences from the birth of Indradyumna, Râjâ of Orissa, on Bhâdrapada śukla 12th, and each month commences from the moment when the sun enters a new sign. The Amli San is used in business transactions and in the courts of law in Orissa."^[98] It is thus luni-solar with respect to changing its numerical designation, but solar as regards the months and days. But it seems probable that it is really luni-solar also as regards its months and days.

The Kanyâ saṅkrânti can take place on any day from about 11 days previous to lunar Bhâdrapada śukla 12th to about 18 days after it. With the difference of so many days the epoch and numerical designation of the Amli and Vilâyatî years are the same.

The Fasali year.—This is the harvest year introduced, as some say, by Akbar, originally derived from the Muhammadan year, and bearing the same number, but beginning in July. It was, in most parts of India, a solar year, but the different customs of different parts of India caused a divergence of reckoning. Its epoch is apparently A. H. 963 (A. D. 1556), when its number coincided with that of the purely lunar Muhammadan year, and from that date its years have been solar or luni-solar. Thus (A. H.) 963 + 337 (solar years) = 1300, and (A. D.) 1556 + 337 = 1893 A.D., with a part of which year Fasali 1300 coincides, while the same year is A. H. 1310. The era being purely official, and not appealing to the feelings of the people of India, the reckoning is often found to be loose and unreliable. In Madras the Fasali year originally commenced with the 1st day of the solar month Âḍî (Karka), but about the year 1800 A.D. the British Government, finding that this date then coincided with July 13th, fixed July 13th as the permanent initial date; and in A.D. 1855 altered this for convenience to July 1st, the present reckoning. In parts of Bombay the Fasali begins when the sun enters the nakshatra Mṛigaśîrsha, viz., (at present) about the 5th or 6th June. The Bengâli year and the Vilâyatî year both bear the same number as the Fasali year.

The names of months, their periods of beginning, and the serial number of days are the same as in the Hijra year, but the year changes its numerical designation on a stated solar day. Thus the year is already a solar year, as it was evidently intended to be from its name. But at the present time it is luni-solar in Bengal, and, we believe, over all North-Western India, and this gives rise to a variety, to be now described.

The luni-solar Fasali year.—This reckoning, though taking its name from a Muhammadan source, is a purely Hindu year, being luni-solar, pûrṇimânta, and Âśvinâdi. Thus the luni-solar Fasali year in Bengal and N. W. India began (pûrṇimânta Âśvina kṛishṇa pratipadâ, Śaka 1815 current =) Sept. 7th, 1882. A peculiarity about the reckoning, however, is that the months are not divided into bright and dark fortnights, but that the whole runs without distinction of pakshas, and without addition or expunction of tithis from the 1st to the end of the mouth, beginning with the full moon. Its epoch is the same as that of the Vilâyatî year, only that it begins with the full moon next preceding or succeeding the Kanyâ saṅkrânti, instead of on the saṅkrânti day.

In Southern India the Fasali year 1302 began on June 5th, 1892, in Bombay, and on July 1st, 1892, in Madras. It will be seen, therefore, that it is about two years and a quarter in advance of Bengal.

To convert a luni-solar Bengali or N. W. Fasali date, approximately, into a date easily workable by our Tables, treat the year as an ordinary luni-solar pûrṇimânta year; count the days after the 15th of the month as if they were days in the śukla fortnight, 15 being deducted from the given figure; add 515 to make the year correspond with the Saka year, for dates between Âśvina 1st and Chaitra 15th (= amânta Bhâdrapada kṛishṇa 1st andamânta amanta Phâlguna kṛishṇa 30th)—and 516 between Chaitra 15th and Âśvina 1st. Thus, let Chaitra 25th 1290 be the given date. The 25th should be converted into śukla 10th; adding 516 to 1290 we have 1806, the equivalent Śaka year. The corresponding śukla date is therefore amânta Chaitra śukla 10th, 1806 current. From this the conversion to an A. D. date can be worked by the Tables. For an exact equivalent the saṅkrânti day must be ascertained.

The Mahratta Sûr-san or Shahûr-san.—This is sometimes called the Arabi-san. It was extensively used during the Mahratta supremacy, and is even now sometimes found, though rarely. It is nine years behind the Fasali of the Dakhan, but in other respects is just the same; thus, its year commences when the sun enters the nakshatra Mṛigaśîrsha, in which respect it is solar, but the days and months correspond with Hijra reckoning. It only diverged from the Hijra in A.D. 1344, according to the best computation, since when it has been a solar year as described above. On May 15th, AD. 1344, the Hijra year 745 began. But since then the Shahûr reckoning was carried on by itself as a solar year. To convert it to an A.D. year, add 599.

The Harsha-Kâla.—This era was founded by Harshavardhana of Kanauj,^[99] or more properly of Thaṇeśar. At the time of Alberuni (A.D. 1030) it was in use in Mathurâ (Muttra) and Kanauj. Its epoch seems to be Śaka 529 current, A.D. 606—7. More than ten inscriptions have been discovered in Nepal^[100] dated in the first and second century of this era. In all those discovered as yet the years are qualified only by the word "samvat".

The Mâgi-San.—This era is current in the District of Chittagong. It is very similar to the Bengali-san, the days and months in each being exactly alike. The Mâgi is, however, 45 years behind the Bengali year,^[101] e.g., Magi 1200 = Bengali 1245.

The Kollam era, or era of Paraśurâma.—The year of this era is known as the Kollam âṇḍu. Kollam (anglicé Quilon) means "western", âṇḍu means "a year". The era is in use in Malabar from Mangalore to Cape Comorin, and in the Tinnevelly district. The year is sidereal solar. In North Malabar it begins with the solar month Kanni (Kanyâ), and in South Malabar and Tinnevelly with the month Chiṅgam (Siṁha). In Malabar the names of the months are sign-names, though corrupted from the original Sanskṛit; but in Tinnevelly the names are chiefly those of lunar months, also corrupted from Sanskṛit, such as Śittirai or Chittirai for the Sanskrit Chaitra, corresponding with Mesha, and so on. The sign-names as well as the lunar-month names are given in the pañchâṅgs of Tinnevelly and the Tamil country. All the names will be found in Table II., Part ii. The first Kollam âṇḍu commenced in Kali 3927 current, Śaka 748 current, A.D. 825—26, the epoch being Śaka 747—48 current, A.D. 824—25. The years of this era as used are current years, and we have treated them so in our Tables.

The era is also called the "era of Paraśurâma", and the years run in cycles of 1000. The present cycle is said to be the fourth, but in actual modern use the number has been allowed to run on over the 1000, A.D. 1894—95 being called Kollam 1070. We believe that there is no record extant of its use earlier than A.D. 825, and we have therefore, in our Table I., left the appropriate column blank for the years A.D. 300—825. If there were really three cycles ending with the year 1000, which expired A.D. 824—25, then it would follow that the Paraśurâma, or Kollam, era began in Kali 1927 current, or the year 3528 of the Julian period.^[102]

The Nevâr era. This era was in use in Nepal up to A.D. 1768, when the Śaka era was introduced.^[103] Its years are Kârttikâdi, its months amânta, and its epoch (the beginning of the Nevâr year 0 current) is the Kârttikâdi Vikrama year 936 current, Śaka 801—2 current, A.D. 878—79. Dr. F. Kielhorn, in his Indian Antiquary paper on the "Epoch of the Newâr era"^[104] has come to the conclusion that its years are generally given in expired years, only two out of twenty-five dates examined by him, running from the 235th to the 995th year of the era, being current ones. The era is called the "Nepâl era" in inscriptions, and in Sanskrit manuscripts; "Nevâr" seems to be a corruption of that word. Table II., Part iii., below gives the correspondence of the years with those of other eras.

The Châlukya era. This was a short-lived era that lasted from Śaka 998 (A.D. 1076) to Śaka 1084 (A.D. 1162) only. It was instituted by the Chalukya king Vikramâditya Tribhuvana Malla, and seems to have ceased after the defeat of the Eastern Châlukyas in A.D. 1162 by Vijala Kalachuri. It followed the Śaka reckoning of months and pakshas. The epoch was Śaka 998—99 current, A.D. 1075—76.

The Simha Samvat.—This era was in use in Kâṭhiâvâḍ and Gujarat. From four dates in that era of the years 32, 93, 96 and 151, discussed in the Indian Antiquary (Vols. XVIII. and XIX. and elsewhere), we infer that its year is luni-solar and current; the months are presumably amânta, but in one instance they seem to be pûrṇimânta, and the year is most probably Âshâḍhâdi. It is certainly neither Kârttikâdi nor Chaitrâdi. Its epoch is Śaka 1036—37 current, A.D. 1113—14.

The Lakshmaṇa Sena era.—This era is in use in Tirhut and Mithila, but always along with the Vikrama or Śaka year. The people who use it know little or nothing about it. There is a difference of opinion as to its epoch. Colebrooke (A.D. 1796) makes the first year of this era correspond with A.D. 1105; Buchanan (A.D. 1810) fixes it as A.D. 1105 or 1106; Tirhut almanacs, however, for the years between A.D. 1776 and 1880 shew that it corresponds with A.D. 1108 or 1109. Buchanan states that the year commences on the first day after the full moon of the month Âshâḍha, while Dr. Râjendra Lâl Mitra (A.D. 1878) and General Cunningham assert that it begins on the first Mâgha badi (Mâgha kṛishṇa 1st).^[105] Dr. F. Kielhorn, examining six independent inscriptions dated in that era (from A.D. 11 94 to 1551), concludes^[106] that the year of the era is Kârttikâdi; that the months are amânta; that its first year corresponds with A.D. 1119—20, the epoch being A.D. 1118—19, Śaka 1041—42 current; and that documents and inscriptions are generally dated in the expired year. This conclusion is supported by Abul Fazal's statement in the Akbarnâma (Śaka 1506, A.D. 1584). Dr. Kielhorn gives, in support of his conclusion, the equation "Laksh: sam: 505 = Śaka sam: 1546" from a manuscript of the Smṛititattvâmṛita, and proves the correctness of his epoch by other dates than the six first given.

The Ilâhi era.—The "Târîkh-i Ilâhi," that is "the mighty or divine era," was established by the emperor Akbar. It dates from his accession, which, according to the Tabakât-i-Akbari, was Friday the 2nd of Rabî-uś-śânî, A.H. 963, or 14th February,^[107] 1556 (O. S.), Śaka 1478 current. It was employed extensively, though not exclusively on the coins of Akbar and Jahângîr, and appears to have fallen into disuse early in the reign of Shâh-Jahân. According to Abûl Fazal, the days and months are both natural solar, without any intercalations. The names of the months and days correspond with the ancient Persian. The months have from 29 to 30 days each. There are no weeks, the whole 30 days being distinguished by different names, and in those months which have 32 days the two last are named roz o shab (day and night), and to distinguish one from another are called "first" and "second".^[108] Here the lengths of the months are said to be "from 29 to 30 days each", but in the old Persian calendar of Yazdajird they had 30 days each, the same as amongst the Parsees of the present day. The names of the twelve months are as follow.—

1	Farwardîn	5	Mirdâd	9	Ader
2	Ardi-behisht	6	Shariûr	10	Dêi
3	Khurdâd	7	Mihir	11	Bahman
4	Tîr	8	Abân	12	Isfandarmaz

The Mahratta Râja Śaka era.—This is also called the "Râjyâbhisheka Śaka". The word "Śaka" is used here in the sense of an era. It was established by Śivajî, the founder of the Mahratta kingdom, and commenced on the day of his accession to the throne, i.e., Jyeshṭha śukla trayodaśî (13th) of Śaka 1596 expired, 1597 current, the Ânanda samvatsara. The number of the year changes every Jyeshṭha śukla trayodaśî; the years are current; in other respects it is the same as the Southern luni-solar amânta Śaka years. Its epoch is Śaka 1596—97 current, A.D. 1673—74. It is not now in use.

72. Names of Hindi and N. W. Fasali months.—Some of the months in the North of India and Bengal are named differently from those in the Peninsula. Names which are manifestly corruptions need not be noticed, though "Bhâdûn" for Bhâdrapada is rather obscure. But "Kuar" for Âśvina, and "Âghân", or "Aghrân", for Mârgaśîrsha deserve notice. The former seems to be a corruption of Kumârî, a synonym of Kanyâ (= Virgo, the damsel), the solar sign-name. If so, it is a peculiar instance of applying a solar sign-name to a lunar month. "Âghân" (or "Aghrân") is a corrupt form of Âgrahâyaṇa, which is another name of Mârgaśîrsha.

PART III.

Description and Explanation of the Tables.

73. Table I.—Table I. is our principal and general Table, and it forms the basis for all calculations. It will be found divided into three sections. (1) Table of concurrent years; (2) intercalated and suppressed months; (3) moments of commencement of the solar and luni-solar years. All the figures refer to mean solar time at the meridian of Ujjain. The calculations are based on the Sûrya-Siddhânta, without the bîja up to 1500 A.D. and with it afterwards, with the exception of cols. 13 to 17 inclusive for which the Ârya-Siddhânta has been used. Throughout the table the solar year is taken to commence at the moment of the apparent Mesha saṅkrânti or first point of Aries, and the luni-solar year with amânta Chaitra śukla pratipadâ. The months are taken as amânta.

74. Cols. 1 to 5.—In these columns the concurrent years of the six principal eras are given. (As to current and expired years see Art. 70 above.) A short description of eras is given in Art. 71. The years in the first three columns are used alike as solar and luni-solar, commencing respectively with Mesha or Chaitra. (For the beginning point of the year see Art. 52 above.) The Vikrama year given in col. 3 is the Chaitrâdi Vikrama year, or, when treated as a solar year which is very rarely the case, the Meshâdi year. The Âshâḍhâdi and Kârttikâdi Vikrama years are not given, as they can be regularly calculated from the Chaitrâdi year, remembering that the number of the former year is one less than that of the Chaitrâdi year from Chaitra to Jyeshṭha or Âśvina (both inclusive), as the case may be, and the same as the Chaitrâdi year from Âshâḍha or Kârttika to the end of Phâlguna.

Cols. 4 and 5. The eras in cols. 4 and 5 are described above (Art. 71.) The double number is entered in col. 4 so that it may not be forgotten that the Kollam year is non-Chaitrâdi or non-Meshâdi, since it commences with either Kanni (Kanyâ) or Chiṅgam (Siṁha). In the case of the Christian era of course the first year entered corresponds to the Kali, Śaka or Chaitrâdi Vikrama year for about three-quarters of the latter's course, and for about the last quarter the second Christian year entered must be taken. The corresponding parts of the years of all these eras as well as of several others will be found in Table II., Parts ii. and iii.

75. Cols. 6 and 7.—These columns give the number and name of the current samvatsara of the sixty-year cycle. There is reason to believe that the sixty-year luni-solar cycle (in use mostly in Southern India) came into existence only from about A. D. 909; and that before that the cycle of Jupiter was in use all over India. That is to say, before A. D. 909 the samvatsaras in Southern India were the same as those of the Jupiter cycle in the North. If, however, it is found in any case that in a year previous to A.D. 908 the samvatsara given does not agree with our Tables, the rule in Art. 62 should be applied, in order to ascertain whether it was a luni-solar samvatsara.

The samvatsara given in col. 7 is that which was current at the time of the Mesha saṅkrânti of the year mentioned in cols, i to 3. To find the samvatsara current on any particular day of the year the rules given in Art. 59 should be applied. For other facts regarding the samvatsaras, see Arts. 53 to 63 above.

76. Cols. 8 to 12, and 8a to 12a. These concern the adhika (intercalated) and kshaya (suppressed) months. For full particulars see Arts. 45 to 51. By the mean system of intercalations there can be no suppressed months, and by the true system only a few. We have given the suppressed months in italics with the sufifix "Ksh" for "kshaya." As mean added months were only in use up to A.D. 1100 (Art. 47) we have not given them after that year.

77. The name of the month entered in col. 8 or 8a is fixed according to the first rule for naming a lunar month (Art. 46), which is in use at the present day. Thus, the name Âshâḍha, in cols. 8 or 8a, shows that there was an intercalated month between natural Jyeshṭha and natural Âshâḍha, and by the first rule its name is "Adhika Âshâḍha", natural Âshâḍha being "Nija Âshâḍha."

By the second rule it might have been called Jyeshṭha, but the intercalated period is the same in either case. In the case of expunged months the word "Pausha", for instance, in col. 8 shows that in the lunar month between natural Kârttika and natural Mâgha there were two saṅkrântis; and according to the rule adopted by us that lunar month is called Mârgaśîrsha, Pausha being expunged.

78. Lists of intercalary and expunged months are given by the late Prof K. L. Chhatre in a list published in Vol. I., No. 12 (March 1851) of a Mahrâṭhi monthly magazine called Jñanaprasâraka, formerly published in Bombay, but now discontinued; as well as in Cowasjee Patell's "Chronology", and in the late Gen. Sir A. Cunningham's "Indian Eras,"^[109] But in none of these three works is a single word said as to how, or following what authority, the calculations were made, so that we have no guide to aid us in checking the correctness of their results.

79. An added lunar month being one in which no saṅkrânti of the sun occurs, it is evident that a saṅkrânti must fall shortly before the beginning, and another one shortly after the end, of such a month, or in other words, a solar month must begin shortly before and must end shortly after the added lunar month. It is further evident that, since such is the case, calculation made by some other Siddhânta may yield a different result, even though the difference in the astronomical data which form the basis of calculation is but slight. Hence we have deemed it essential, not only to make our own calculations afresh throughout, but to publish the actual resulting figures which fix the months to be added and suppressed, so that the reader may judge in each case how far it is likely that the use of a different authority would cause a difference in the months affected. Our columns fix the moment of the saṅkrânti before and the saṅkrânti after the added month, as well as the saṅkrânti after the beginning, and the saṅkrânti before the end, of the suppressed month; or in other words, determine the limits of the adhika and kshaya masas. The accuracy of our calculation can be easily tested by the plan shewn in Art. 90 below. (See also Art. 88 below.) The moments of time are expressed in two ways, viz., in lunation-parts and tithis, the former following Prof. Jacobi's system as given in Ind. Ant., Vol. XVII.

80. Lunation-parts or, as we elsewhere call them, "tithi-indices" (or " $t$ ") are extensively used throughout this work and require full explanation. Shortly stated a lunation-part is 1/10000th of an apparent synodic revolution of the moon (see Note 2, Art. 12 above). It will be well to put this more clearly. When the difference between the longitude of the sun and moon, or in other words, the eastward distance between them, is nil, the sun and moon are said to be in conjunction; and at that moment of time occurs (the end of) amâvâsyâ, or new moon. (Arts. 7.29 above.) Since the moon travels faster than the sun, the difference between their longitudes, or their distance from one another, daily increases during one half and decreases during the other half of the month till another conjunction takes place. The time between two conjunctions is a synodic lunar month or a lunation, during which the moon goes through all its phases. The lunation may thus be taken to represent not only time but space. We could of course have expressed parts of a lunation by time-measure, such as by hours and minutes, or ghaṭikâs and palas, or by space-measure, such as degrees, minutes, or seconds, but we prefer to express it in lunation-parts, because then the same number does for either time or space (see Art. 89 below). A lunation consists of 30 tithis. 1/30th of a lunation consequently represents the time-duration of a tithi or the space-measurement of 12 degrees. Our lunation is divided into 10,000 parts, and about 333 lunation-parts (1/10000ths) go to one tithi, 667 to two tithis, 1000 to three and so on. Lunation-parts are therefore styled "tithi-indices", and by abbreviation simply " $t$ ". Further, a lunation or its parts may be taken as apparent or mean. Our tithi-, nakshatra-, and yoga-indices are apparent and not mean, except in the case of mean added months, where the index, like the whole lunation, is mean.

Our tithi-index, or " $t$ ", therefore shows in the case of true added months as well as elsewhere, the space-difference between the apparent, and in the case of mean intercalations between the mean, longitudes of the sun and moon, or the time required for the motions of the sun and moon to create that difference, expressed in 10,000ths of a unit, which is a circle in the case of space, and a lunation or synodic revolution of the moon in the case of time. Briefly the tithi-index " $t$ " shews the position of the moon in her orbit with respect to the sun, or the time necessary for her to gain that position., e.g., "0" is new moon, "5000" full moon, "10,000" or "0" new moon; "50" shews that the moon has recently (i.e., by 50/10000ths, or 3 hours 33 minutes—Table X.. col. 3) passed the point or moment of conjunction (new moon); 9950 shews that she is approaching new-moon phase, which will occur in another 3 hours and 33 minutes.

81. A lunation being equal to 30 tithis, the tithi-index, which expresses the 10,000th part of a lunation, can easily be converted into tithi-notation, for the index multiplied by 30 (practically by 3), gives, with the decimal figures marked off, the required figure in tithis and decimals. Thus if the tithi-index is 9950, which is really 0.9950, it is equal to (0.9950 × 30 =) 29.850 tithis, and the meaning is that 9950/10000ths of the lunation, or 29.850 tithis have expired. Conversely a figure given in tithis and decimals divided by 30 expresses the same in 10,000ths parts of a lunation.

82. The tithi-index or tithi is often required to be converted into a measure of solar time, such as hours or ghaṭikâs. Now the length of an apparent lunation, or of an apparent tithi, perpetually varies, indeed it is varying at every moment, and consequently it is practically impossible to ascertain it except by elaborate and special calculations; but the length of a mean lunation, or of a mean tithi, remains permanently unchanged. Ignoring, therefore, the difference between apparent and mean lunations, the tithi-index or tithi can be readily converted into time by our Table X.. which shews the time-value of the mean lunation-part (1/10000th of the mean lunation), and of the mean tithi-part (1/1000th of the mean tithi). Thus, if $t=50$ , Table X. gives the duration as 3 hours 33 minutes; and if the tithi-part^[110] is given as 0.150 we have by Table X. (2 h. 22 m. + 1 h. 11 min. = ) 3 h. 33 m.

It must be understood of course that the time thus given is not very accurate, because the tithi-index ( $t$ ) is an apparent index, while the values in Table X. are for the mean index. The same remark applies to the nakshatra ( $n$ ) or yoga ( $y$ ) indices, and if accuracy is desired the process of calculation must be somewhat lengthened. This is fully explained in example 1 in Art. 148 below. In the case of mean added months the value of ( $t$ ) the tithi-index is at once absolutely accurate.

83. The saṅkrântis preceding and succeeding an added month, as given in our Table I., of course take place respectively in the lunar month preceding and succeeding that added month.

84. To make the general remarks in Arts. 80, 81, 82 quite clear for the intercalation of months we will take an actual example. Thus, for the Kali year 3403 the entries in cols. 9 and 11 are 9950 and 287, against the true added month Âśvina in col. 8. This shews us that the saṅkrânti preceding the true added, or Adhika, Âśvina took place when 9950 lunation-parts of the natural month Bhâdrapada (preceding Adhika Âśvina) had elapsed, or when (10,000 − 9950 =) 50 parts had to elapse before the end of Bhâdrapada, or again when 50 parts had to elapse before the beginning of the added month; and that the saṅkrânti succeeding true Adhika Âśvina took place when 287 parts of the natural month Nija Âśvina had elapsed, or when 287 parts had elapsed after the end of the added month Adhika Âśvina.

85. The moments of the saṅkrântis are further given in tithis and decimals in cols. 10, 12, 10a and 12a. Thus, in the above example we find that the preceding saṅkrânti took place when 29.850 tithis of the preceding month Bhâdrapada had elapsed, i.e., when (30 − 29.850 =) 0.150 tithis had still to elapse before the end of Bhâdrapada; and that the succeeding saṅkrânti took place when 0.861 of a tithi of the succeeding month, Âśvina, had passed.

To turn these figures into time is rendered easy by Table X. We learn from it that the preceding saṅkrânti took place (50 lunation parts or 0.150 tithi parts) about 3 h. 33 m. before the beginning of Adhika Âśvina; and that the succeeding saṅkrânti took place (287 lunation parts, or .861 tithi parts) about 20 h. 20 m. after the end of Adhika Âśvina. This time is approximate. For exact time see Arts. 82 and 90.

The tithi-indices here shew (see Art. 88) that there is no probability of a different month being intercalated if the calculation be made according to a different authority.

86. To constitute an expunged month we have shewn that two saṅkrântis must occur in one lunar month, one shortly after the beginning and the other shortly before the end of the month; and in cols. 9 and 10 the moment of the first saṅkrânti, and in cols. 11 and 12 that of the second saṅkrânti, is given. For example see the entries against Kali 3506 in Table I. As already stated, there can never be an expunged month by the mean system

87. In the case of an added month the moon must be waning at the time of the preceding, and waxing at the time of the succeeding saṅkrânti, and therefore the figure of the tithi-index must be approaching 10,000 at the preceding, and over 10,000, or beginning a new term of 10,000, at the succeeding, saṅkrânti. In the case of expunged months the case is reversed, and the moon must be waxing at the first, and waning at the second saṅkrânti; and therefore the tithi-index must be near the beginning of a period of 10,000 at the first, and approaching 10,000 at the second, saṅkrânti.

88. When by the Sûrya-Siddhânta a new moon (the end of the amâvâsyâ) takes place within about 6 ghaṭikâs, or 33 lunation-parts, of the saṅkrânti, or beginning and end of a solar month, there may be a difference in the added or suppressed month if the calculation be made according to another Siddhânta. Hence when, in the case of an added month, the figure in col. 9 or 9a is more than (10,000 − 33 =) 9967, or when that in col. 11 or 11a is less than 33; and in the case of an expunged month when the figure in col. 9 is less than 33, or when that in col. 11 is more than 9967, it is possible that calculation by another Siddhânta will yield a different month as intercalated or expunged; or possibly there will be no expunction of a month at all. In such cases fresh calculations should be made by Prof. Jacobi's Special Tables (Epig. Ind., Vol. II.) or direct from the Siddhânta in question. In all other cases it may be regarded as certain that our months are correct for all Siddhântas. The limit of 33 lunation-parts here given is generally sufficient, but it must not be forgotten that where Siddhântas are used with a bîja correction the difference may amount to as much as 20 ghaṭikâs, or 113 lunation-parts (See above, note to Art. 49).

In the case of the Sûrya-Siddhânta it may be noted that the added and suppressed months are the same in almost all cases, whether the bîja is applied or not.

89. We have spared no pains to secure accuracy in the calculation of the figures entered in cols. 9 to 12 and 9a to 12a, and we believe that they may be accepted as finally correct, but it should be remembered that their time-equivalent as obtained from Table X. is only approximate for the reason given above (Art. 82.) Since Indian readers are more familiar with tithis than with lunation-parts, and since the expression of time in tithis may be considered desirable by some European workers, we have given the times of all the required saṅkrântis in tithis and decimals in our columns, as well as in lunation-parts; but for turning our figures into time-figures it is easier to work with lunation-parts than with tithi-parts. It may be thought by some readers that instead of recording the phenomena in lunation-parts and tithis it would have been better to have given at once the solar time corresponding to the moments of the saṅkrântis in hours and minutes. But there are several reasons which induced us, after careful consideration, to select the plan we have finally adopted. First, great labour is saved in calculation; for to fix the exact moments in solar time at least five processes must be gone through in each case, as shewn in our Example I. below (Art. 148) It is true that, by the single process used by us, the time-equivalents of the given lunation-parts are only approximate, but the lunation-parts and tithis are in themselves exact. Secondly, the time shewn by our figures in the case of the mean added months is the same by the Original Sûrya, the Present Sûrya, and the Ârya-Siddhânta, as well as by the Present Sûrya-Siddhânta with the bîja, whereas, if converted into solar time, all of these would vary and require separate columns. Thirdly, the notation used by us serves one important purpose. It shews in one simple figure the distance in time of the saṅkrântis from the beginning and end of the added or suppressed month, and points at a glance to the probability or otherwise of there being a difference in the added or suppressed month in the case of the use of another authority. Fourthly, there is a special convenience in our method for working out such problems as are noticed in the following articles.

90. Supposing it is desired to prove the correctness of our added and suppressed months, or to work them out independently, this can easily be done by the following method: The moment of the Mesha saṅkrânti according to the Sûrya-Siddhânta is given in cols. 13, 14 and 15a to 17a for all years from A.D. 1100 to 1900, and for other years it can be calculated by the aid of Table D. in Art. 96 below. Now we wish to ascertain the moment of two consecutive new moons connected with the month in question, and we proceed thus. The interval of time between the beginning of the solar year and the beginning or end of any solar month according to the Sûrya-Siddhânta, is given in Table III., cols. 8 or 9; and by it we can obtain by the rules in Art. 151 below, the tithi-index for the moment of beginning and end of the required solar month, i.e., the moments of the solar saṅkrântis, whose position with reference to the new moon determines the addition or suppression of the luni-solar month. The exact interval also in solar time between those respective saṅkrântis and the new moons (remembering that at new moon " $t$ " = 10,000) can be calculated by the same rules. This process will at once shew whether the moon was waning or waxing at the preceding and succeeding saṅkrântis, and this of course determines the addition or suppression of the month. The above, however, applies only to the apparent or true intercalations and suppressions. For mean added months the Śodhya (2 d. 8 gh. 51 p. 15 vi.) must be added (see Art. 26) to the Mesha-saṅkrânti time according to the Ârya-Siddhânta (Table I., col. 15), and the result will be the time of the mean Mesha saṅkrânti. For the required subsequent saṅkrântis all that is necessary is to add the proper figures of duration as given in Art. 24, which shews the mean length of solar months, and to find the " $a$ " for the results so obtained by Art. 151. Then add 200 to the totals and the result will be the required tithi-indices.

91. It will of course be asked how our figures in Table I. were obtained, and what guarantee we can give for their accuracy. It is therefore desirable to explain these points. Our calculations for true intercalated and suppressed months were first made according to the method and Tables published by Prof. Jacobi (in the Ind. Ant., Vol. XVII., pp. 145 to 181) as corrected by the errata list printed in the same volume. We based our calculations on his Tables 1 to 10, and the method given in his example 4 on pp. 152—53,^[111] but with certain differences, the necessity of which must now be explained. Prof Jacobi's Tables 1 to 4, which give the dates of the commencement of the solar months, and the hour and minute, were based on the Ârya-Siddhânta, while Tables 5 to 10 followed the Sûrya-Siddhânta, and these two Siddhântas differ. In consequence several points had to be attended to. First, in Prof. Jacobi's Tables l to 4 the solar months are supposed to begin exactly at Ujjain mean sunset, while in fact they begin (as explained by himself at p. 147) at or shortly after mean sunset. This state of things is harmless as regards calculations made for the purpose for which the Professor designed and chiefly uses these Tables, but such is not the case when the task is to determine an intercalary month, where a mere fraction may make all the difference, and where the exact moment of a saṅkrânti must positively be ascertained. Secondly, the beginning of the solar year, i.e., the moment of the Mesha-saṅkrânti, differs when calculated according to those two Siddhântas, as will be seen by comparing cols. 15 to 17 with cols. 15a to 17a of our Table I., the difference being nil in A.D. 496 and 6 gh 23 pa. 41.4 pra. vi. in 1900 A.D. Thirdly, even if we suppose the year to begin simultaneously by both Siddhântas, still the collective duration of the months from the beginning of the year to the end of the required solar month is not the same,^[112] as will be seen by comparing cols. 6 or 7 with cols. 8 or 9 of our Table III. We have applied all the corrections necessitated by these three differences to the figures obtained from Prof Jacobi's Tables and have given the final results in cols. 9 and 11. We know of no independent test which can be applied to determine the accuracy of the results of our calculations for true added and suppressed months; but the first calculations were made exceedingly carefully and were checked and rechecked. They were made quite independently of any previously existing lists of added and suppressed months, and the results were afterwards compared with Prof. Chhatre's list; and whenever a difference appeared the calculations were completely re-examined. In some cases of expunged months the difference between the two lists is only nominal, but in other cases of difference it can be said with certainty that Prof. Chhatre's list is wrong. (See note to Art. 46.) Moreover, since the greatest possible error in the value of the tithi-index that can result by use of Prof. Jacobi's Table is 7 (see his Table p. 164), whenever the tithi-index for added and suppressed months obtained by our computation fell within 7 of 10,000, i.e., whenever the resulting index was below 7 or over 9993, the results were again tested direct by the Sûrya-Siddhânta.^[113]

As regards mean intercalations every figure in our cols, 9a to 12a was found correct by independent test. The months and the times of the saṅkrântis expressed in tithi-indices and tithis were calculated by the present Sûrya-Siddhânta, and the results are the same whether worked by that or by the Original Surya-Siddkanta, the First Arya-Siddhanta, or the Present SuryaSiddhanta with the bija. We think, therefore, that the list of true added and suppressed months and that of the mean added months as given by us is finally reliable.

92. Cols. 13 to 17 or to 17a. The solar year begins from the moment of the Mesha saṅkrânti and this is taken as apparent and not mean. We give the exact moment for all years from A.D. 300 to 1900 by the Ârya-Siddhânta, and in addition for years between A.D. 1100 and 1900 by the Sûrya-Siddhântas as well. (See also Art. 96). Every figure has been independently tested, and found correct. The week-day and day of the month A.D. as given in cols. 13 and 14 are applicable to both the Siddhântas, but particular attention must be paid to the footnote in Table I., annexed to A.D. 1117—18 and some other subsequent years. The entries in cols. 15 and 15a for Indian reckoning in ghaṭikâs and palas, and in cols. 17 and 17a for hours and minutes, imply that at the instant of the sankranti so much time has elapsed since mean sunrise at Ujjain on the day in question. Ujjain mean sunrise is generally assumed to be 6.0 a.m.

93. The alteration of week-day and day of the month alluded to inthe footnote mentioned in the last paragraph (Table I., A.D. 1117—18) is due to the difference resulting from calculations made by the two Siddhântas, the day fixed by the Sûrya-Siddhânta being sometimes one later than that found by the Ârya-Siddhânta. It must be remembered, however, that the day in question runs from sunrise to sunrise, and therefore a moment of time fixed as falling between midnight and sunrise belongs to the preceding day in Indian reckoning, though to the succeeding day by European nomenclature. For example, the Mesha saṅkrânti in Śaka 1039 expired (A.D. 1117) took place, according to the Ârya-Siddhânta on Friday 23rd March at 58 gh. 1 p. after Ujjain mean sunrise (23 h. 12 m. after sunrise on Friday, or 5.12 a.m. on Saturday morning, 24th); while by the Sûrya-Siddhânta it fell on Saturday 24th at 0 gh. 51 pa. (= 0 h. 20 m. after sunrise or 6.20 a.m.). This only happens of course when the saṅkrânti according to the Ârya-Siddhânta falls nearly at the end of a day, or near mean sunrise.

94. In calculating the instant of the apparent Mesha-saṅkrântis, we have taken the śodhya at 2 d. 8 gh. 51 pa. 15 vipa. according to the Ârya-Siddhânta, and 2d. 10 gh. 14 pa. 30 vipa. according to the Sûrya-Siddhânta. (See Art. 26.)

95. The figure given in brackets after the day and month in cols. 13 and 19 is the number of that day in the English common year, reckoning from January 1st. For instance, 75 against 16th March shows that 16th March is the 75th day from January 1st inclusive. This figure is called the "date indicator", or shortly ( $d$ ), in the methods of computation "B" and "C" given below (Part IV.), and is intended as a guide with reference to Table IX., in which the collective duration of days is given in the English common year.

96. The fixture of the moments of the 1600 Mesha-sankrantis noted in this volume will be found advantageous for many purposes, but we have designed it chiefly to facilitate the conversion of solar dates as they are used in Bengal and Southern India.^[114] We have not given the moments of Mesha-saṅkrântis according to the Sûrya-Siddhânta prior to A.D. 1100, so that the Ârya-Siddhânta computation must be used for dates earlier than that, even those occurring in Bengal. There is little danger in so doing, since the difference between the times of the Mesha-saṅkrântis according to the two Siddhântas during that period is very slight, being nil in A.D. 496, and only increasing to 1 h. 6 m. at the most in 1100 A.D. It is, however, advisable to give a correction Table so as to ensure accuracy, and consequently we append the Table which follows, by which the difference for any year lying between A.D. 496 and 1100 A.D. can be found. It is used in the following manner. First find the interval in years between the given year and A.D. 496. Then take the difference given for that number of years in the Table, and subtract or add it to the moment of the Mesha-saṅkrânti fixed by us in Table 1. by the Ârya-Siddhânta, according as the given year is prior or subsequent to A.D. 496. The quotient gives the moment of the Mesha-saṅkrânti by the Sûrya-Siddhânta.

**TABLE**
**Shewing the difference between the moments of the Mesha-saṅkrânti as calculated by the Present Sûrya and the first Ârya-Siddhântas; the difference in A.D. 496 (Śaka 496 current) being 0.**
No. of years.	Difference Expressed in			No. of years.	Difference Expressed in			No. of years.	Difference Expressed in
No. of years.	gh.	pa.	minutes.	No. of years.	gh.	pa.	minutes.	No. of years.	gh.	pa.	minutes.
1	0	0.3	0.1	10	0	2.7	1.1	100	0	27.3	10.9
2	0	0.5	0.2	20	0	5.5	2.2	200	0	54.6	21.9
3	0	0.8	0.3	30	0	8.2	3.3	300	1	22.0	32.8
4	0	1.1	0.4	40	0	10.9	4.4	400	1	49.3	43.7
5	0	1.4	0.5	50	0	13.7	5.5	500	2	16.6	54.7
6	0	1.6	0.7	60	0	16.4	6.6	600	2	44.0	65.6
7	0	1.9	0.8	70	0	19.1	7.7	700	3	11.3	76.5
8	0	2.2	0.9	80	0	21.9	8.7	800	3	38.6	87.5
9	0	2.5	1.0	90	0	24.6	9.8	900	4	6.0	98.4

Example. Find the time of the Mesha saṅkrânti by the Sûrya-Siddhânta in A.D. 1000. The difference for (1000 − 496 =) 504 years is (2 gh. 16.6 pa. + 1.1 pa. =) 2 gh. 17.7 pa. Adding this to Friday, 22nd March, 42gh. 5pa., i.e., the time fixed by the Ârya-Siddhânta (Table I., cols. 14, 15), we have 44 gh. 22.7 pa. from sunrise on that Friday as the actual time by the Sûrya-Siddhânta.

97. Cols. 19 to 25. The entries in these columns enable us to convert and verify Indian luni-solar dates. They were first calculated, as already stated, according to the Tables published by Prof. Jacobi in the Indian Antiquary^[115] (Vol. XVII.). The calculations were not only most carefully made, but every figure was found to be correct by independent test. As now finally issued, however, the figures are those obtained from calculations direct from the Sûrya-Siddhânta, specially made by Mr. S. Bâlkṛishṇa Dîkshit. The articles $a$ , $b$ , $c$ , in cols. 23 to 25 are very important as they form the basis for all calculations of dates demanding an exact result. Their meaning is fully described below (Art. 102.).

The meaning of the phrase "moon's age" (heading of cols. 21, 22) in the Nautical Almanack is the mean time in days elapsed since the moon's conjunction with the sun (amâvâsyâ, new moon). For our purposes the moon's age is its age in lunation-parts and tithis, and these have been fully explained above.

98. The week-day and day of the month A.D. given in cols. 19 and 20 shew the civil day on which Chaitra śukla pratipadâ of each year, as an apparent tithi, ends.^[116] The figures given in cols. 21 to 25 relate to Ujjain mean sunrise on that day.

99 When an intercalary Chaitra occurs by the true system (Arts. 45 etc. above) it must be remembered that the entries in cols. 19 to 25 are for the śukla-pratipadâ of the intercalated, not the true, Chaitra.

100. The first tithi of the year (Chaitra śukla pratipadâ) in Table I., cols. 19 to 25, is taken as an apparent, not mean, tithi, which practice conforms to that of the ordinary native pañchâṅgs. By this system, as worked out according to our methods A and B, the English equivalents of all subsequent tithis will be found as often correct as if the first had been taken as a mean tithi;—probably more often.

101. The figures given in cols. 21 and 22, except in those cases where a minus sign is found prefixed (e.g., Kali 4074 current), constitute a first approximation showing how much of chaitra śukla pratipadâ had expired on the occurrence of mean sunrise at Ujjain on the day given in cols. 19 and 20. Col. 21 gives the expired lunation-parts or tithi-index, and col. 22 shews the same period in tithi-parts, i.e., decimals of a tithi. The meaning of both of these is explained above (Arts. 80 and 81). We differ from the ordinary pañchâṅgs in one respect, viz., that while they give the portion of the tithi which has to run after mean sunrise, we have given, as in some ways more convenient, the portion already elapsed at sunrise. Thus, the entry 286 in col. 21 means that 286 lunation-parts of Chaitra śukla 1st had expired at mean sunrise. The new moon therefore took place 286 lunation-parts before mean sunrise, and by Table X., col. 3, 286 lunation-parts are equal to (14 h. 10 m. + 6 h. 6 m. =) 20 h. 16 m. The new moon therefore took place 20 h. 16 m. before sunrise, or at 9.44 a.m. on the previous day by European reckoning. The ending-moment of Chaitra śukla pratipadâ can be calculated in the same way, remembering that there are 333 lunation-parts to a tithi.

We allude in the last paragraph to those entries in cols. 21 and 22 which stand with a minus sign prefixed. Their meaning is as follows:—Just as other tithis have sometimes to be expunged so it occasionally happens that Chaitra śukla 1st has to be expunged. In other words, the last tithi of Phâlguna, or the tithi called amâvâsyâ, is current at sunrise on one civil day and the 2nd tithi of Chaitra (Chaitra śukla dvitîyâ) at sunrise on the following civil day. In such a case the first of these is the civil day corresponding to Chaitra śukla 1st; and accordingly we give this civil day in cols. 19 and 20. But since the amâvâsyâ-tithi (the last tithi of Phalguna) was actually current at sunrise on that civil day we give in cols. 21 and 22 the lunation-parts and tithiparts of the amâvâsyâ-tithi which have to run after sunrise with a minus sign prefixed to them. Thus, "−12" in col. 21 means that the tithi-index at sunrise was 10,000 − 12 = or 9988, and that the amâvâsyâ-tithi (Phâlguna Kṛishṇa 15 or 30) (Table VIII., col. 3) will end 12 lunation-parts after sunrise, while the next tithi will end 333 lunation-parts after that.

102. ( $a$ , $b$ , $c$ , cols. 23, 24, 25). The moment of any new moon, or that moment in each lunation when the sun and moon are nearest together, in other words when the longitudes of the sun and moon are equal, cannot be ascertained without fixing the following three elements,—( $a$ ) The eastward distance of the moon from the sun in mean longitude, ( $b$ ) the moon's mean anomaly (Art. 15 and note), which is here taken to be her distance from her perigee in mean longitude, ( $c$ ) the sun's mean anomaly, or his distance from his perigee in mean longitude. And thus our " $a$ ", " $b$ ", " $c$ ", have the above meanings; " $a$ " being expressed in 10,000ths of a circle reduced by 200.6 for purposes of convenience of use, all calculations being then additive, " $b$ " and " $c$ " being given in 1000ths of the circle. To take an example. At Ujjain mean sunrise on Chaitra śukla pratipadâ of the Kali year 3402 (Friday. 8th March, A.D. 300), the mean longitudes calculated direct from the Sûrya-Siddhânta were as follow: The sun, 349° 22′ 27″.92. The sun's perigee, 257° 14′ 22″.86. The moon, 355° 55′ 35″.32. The moon's perigee, 33° 39′ 58″.03. The moon's distance from the sun therefore was (355° 55′ 35″.32 − 349° 22′ 27″.92 =) 6° 33′ 7″.4 = .0182 of the orbit of 360". This (1.0182) reduced by 0.0200,6 comes to 0.99814; and consequently " $a$ " for that moment is 9981.41. The moon's mean anomaly " $b$ " was (355° 55′ 35″.32 − 33° 39′ 58″.03 =) 322° 15′ 37″.29 = 895.17. And the sun's mean anomaly " $c$ " was (349° 22′ 27″.92 − 257° 14′ 22″.86 =) 92° 8′ 5″.06 = 255.93.^[117] We therefore give $a=9981$ , $b=895$ , $c=256$ . The figures for any other year can if necessary be calculated from the following Table, which represents the motion. The increase in $a$ , $b$ , $c$ , for the several lengths of the luni-solar year and for 1 day, is given under their respective heads; the figures in brackets in the first column representing the day of the week, and the first figures the number of days in the year.

**Increase of a, b, c, in one year, and in one day.**
Number of days in the year.	a.	b. without bija.	b. with bija.	c.
354(4)	9875.703337	847.2197487	847.220646	969.1758567
355(5)	214.335267	883.5113299	883.512230	971.9136416
383(5)	9696.029305	899.675604	899.676575	48.57161909
384(8)	34.661235	935.967185	935.968158	51.3094039
385(0)	373.293166	972.258766	972.259742	54.04789
001(1)	338.63193033	36.291581211	36.291583746	2.737784906

103. Table II., Part i., of this table will speak for itself (see also Art. 51 above). In the second part is given, in the first five columns, the correspondence of a cycle of twelve lunar months of a number of different eras with the twelve lunar months of the Śaka year 1000,^[118] which itself corresponds exactly with Kali 4179, Chaitrâdi Vikrama 1135, and Gupta 738. Cols. 8 to 13 give a similar concurrence of months of the solar year Śaka 1000. The concurrence of parts of solar months and of parts of the European months with the luni-solar months is given in cols. 6 and 7, and of the same parts with the solar months in cols. 14 and 15. Thus, the luni-solar amânta month Âshâḍha of the Chaitrâdi Śaka year 1000 corresponds with amânta Âshâḍha of Kali 4179, of Chaitrâdi Vikrama 1135, and of the Gupta era 758; of the Âshâḍhâdi Vikrama year 1135, and of the Chedi or Kaḷachuri 828; of the Karttikâdi Vikrama year 1134, and of the Nêvâr year 198. Parts of the solar months Mithuna and Karka, and parts of June and July of 1077 A.D. correspond with it; in some years parts of the other two Christian months noted in col. 7 will correspond with it. In the year Śaka 1000, taken as a Meshâdi solar year, the month Siṁha corresponds with the Bengali Bhâdrapada and the Tamil Âvaṇi of the Meshâdi Kali 4179, and Meshâdi Vikrama 1135; with Âvaṇi of the Siṁhadi Tinnevelly year 253; with Chingam of the South Malayâḷam Siṁhadi Kollam âṇḍu 253, and of the North Malayâḷam Kanyâdi Kollam âṇḍu 252. Parts of the lunar months Śrâvaṇa and Bhâdrapada correspond with it, as well as parts of July and August of the European year 1077 A.D; in some years parts of August and September will correspond with it.

All the years in this Table are current years, and all the lunar months are amânta.

It will be noticed that the Tuḷu names of lunar months and the Tamil and Tinnevelly names of solar months are corruptions of the original Sanskrit names of lunar months; while the north and south Malayâḷam names of solar months are corruptions of the original Sanskṛit sign-names. Corruptions differing from these are likely to be found in use in many parts of India. In the Tamil Districts and the district of Tinnevelly the solar sign-names are also in use in some places.

104. Table II.. Part iii. This portion of the Table, when read with the notes printed below would seem to be simple and easy to be understood, but to make it still clearer we give the following rules:—

I. Rule for turning into a Chaitrâdi or Meshâdi year (for example, into a luni-solar Śaka, or solar Śaka, year) a year of another era, whether earlier or later, which is non-Chaitrâdi or non-Meshâdi.

(a) For an earlier era. When the given date falls between the first moment of Chaitra or Mesha and the first moment of the month in which, as shewn by the heading, the year of the given earlier era begins, subtract from the given year the first, otherwise the second, of the double figures given under the heading of the earlier era along the line of the year 0 of the required Chaitrâdi or Meshâdi era (e.g., the Śaka).

Examples. (1) To turn Vaiśâkha Śukla 1st of the Âshâḍhâdi Vikrama year 1837, or Srâvaṇa śukla 1st of the Kârttikâdi Vikrama year 1837 to corresponding Śaka reckoning. The year is (1837 − 134 =) 1703 Śaka. The day and month are the same in each case. (2) To turn Mâgha śukla 1st of the Kârttikâdi Vikrama samvat 1838 into the corresponding Śaka date. The year is (1838 − 135 =) 1703 Śaka. The day and month are the same. (3) Given 1st December, 1822 A.D. The year is (1822 − 77 =) 1745 Śaka current. (4) Given 2nd January, 1823 A.D. The year is (1823 − 78 =) 1745 Śaka current.

(b) For a later era. When the given day falls between the first moment of Chaitra or Mesha and the first moment of the month in which, as shewn by the heading, the later era begins, add to the number of the given year the figure in the Table under the heading of the required Chaitrâdi or Meshâdi era along the line of the year 0/1 of the given later era. In the reverse case add that number reduced by one.

Examples. (1) To turn the 1st day of Mithuna 1061 of the South Malayâḷam Kollam Âṇḍu into the corresponding Śaka date. The year is (1061 + 748 =) Śaka 1809 current. The day and month are the same. (2) To turn the 1st day of Makara 1062 of the South Malayâḷam Kollum Âṇḍu into the corresponding Śaka date. The year is (1062 + 747 =) 1809 Śaka current. The day and month are the same.

II. Rule for turning a Chaitrâdi or Meshâdi (e.g., a Śaka) year into a non-Chaitrâdi or non-Meshâdi year of an earlier or later era.

(a) For an earlier era. When the given day falls between the first moment of Chaitra or Mesha and the first moment of the month in which, as shown by the heading, the year of the earlier era begins, add to the given Chaitrâdi or Meshâdi year the first, otherwise the second, of the double figures given under the heading of the earlier era along the line of the year 0 of the Chaitrâdi or Meshâdi era given.

Examples. (1) To turn Bhâdrapada kṛishṇa 30th of the Śaka year 1699 into the corresponding Kârttikâdi Vikrama year. The year is (1699 + 134 =) 1833 of the Kârttikâdi Vikrama era. The day and month are the same. (2) To turn the same Bhâdrapada kṛishṇa 30th, Śaka 1699, into the corresponding Âshâḍhâdi Vikrama year. The year is (1699 + 135 =) 1834 of the Âshâḍhâdi Vikrama era. The day and month are the same.

(b) For a later era. When the given day falls between the first moment of Chaitra or Mesha and the first moment of the month in which, as shown by the heading, the later era begins, subtract from the given year the number under the heading of the given Chaitrâdi or Meshâdi era along the line of the year 0/1 of the given later era; in the reverse case subtract that number reduced by one.

Examples. (1) To turn the 20th day of Siṁha Śaka 1727 current into the corresponding North Malayâḷam Kollam Âṇḍu date. The day and month are the same. The era is a Kanyâdi era, and therefore the required year is (1727 − 748 =) 979 of the required era. (2) To turn the 20th day of Siṁha Śaka 1727 current into the corresponding South Malayâḷam (Tinnevelly) Kollam Âṇḍu date. The day and month are the same. The era is Siṁhâdi, and therefore the required year is (1727 − 747 =) 980 of the required era.

III. Rule for turning a year of one Chaitrâdi or Meshâdi era into one of another Chaitrâdi or Meshâdi era. This is obviously so simple that no explanations or examples are required.

IV. Rule for turning a year of a non-Chaitrâdi or non-Meshâdi era into one of another year equally non-Chaitrâdi or non-Meshâdi These are not required for our methods, but if any reader is curious he can easily do it for himself.

This Table must be used for all our three methods of conversion of dates.

105. Table III.—The numbers given in columns 3a and 10 are intended for use when calculation is made approximately by means of our method "B" (Arts. 137, 138).

It will be observed that the number of days in lunar months given in col. 3a is alternately 30 and 29; but such is not always the case in actual fact. In all the twelve months it occurs that the number of days is sometimes 29 and sometimes 30. Thus Bhâdrapada has by our Table 29 days, whereas it will be seen from the pañchâṅg extract printed in Art. 30 above that in A.D. 1894 (Śaka 1816 expired) it had 30 days.

The numbers given in col. 10 also are only approximate, as will be seen by comparing them with those given in cols. 6 to 9.

Thus all calculations made by use of cols. 3a and 10 will be sometimes wrong by a day. This is unavoidable, since the condition of things changes every year, so that no single Table can be positively accurate in this respect; but, other elements of the date being certain, calculations so made will only be wrong by one day, and if the week-day is given in the document or inscription concerned the date may be fixed with a fair pretence to accuracy. If entire accuracy is demanded, our method "C" must be followed. (See Arts. 2 and 126.)

The details in cols. 3, and 6 to 9, are exactly accurate to the unit of a pala, or 24 seconds. The figure in brackets, or week-day index (w), is the remainder after casting out sevens from the number of days; thus, casting out sevens from 30 the remainder is 2, and this is the (w) for 30. To guard against mistakes it may be mentioned that the figure "2" does not of course mean that the Mesha or Vṛishabha saṅkrânti always takes place on (2) Monday.

106. Tables IV. and V. These tables give the value of (w) (week-day) and (a) (b) and ( $c$ ) for any required number of civil days, hours, and minutes, according to the Sûrya Siddhânta. It will be seen that the figures given in these Tables are calculated by the value for one day given in Art. 102.

Table IV. is Prof. Jacobi's Indian Antiquary (Vol. XVII.) Table 7, slightly modified to suit our purposes; the days being run on instead of being divided into months, and the figures being given for the end of each period of 24 hours, instead of at its commencement. Table V. is Prof. Jacobi's Table 8.

107. Tables VI. and VII. These are Prof. Jacobi's Tables 9 and 10 re-arranged. It will be well that their meaning and use should be understood before the reader undertakes computations according to our method "C". It will be observed that the centre column of each columntriplet gives a figure constituting the equation for each figure of the argument from 0 to 1000, the centre figure corresponding to either of the figures to right or left. These last are given only in periods of 10 for convenience, an auxiliary Table being added to enable the proper equation to be determined for all arguments. Table VI. gives the lunar equation of the centre, Table VII. the solar equation of the centre. (Art. 75 note 3 above). The argument-figures are expressed in 1000ths of the circle, while the equation-figures are expressed in 10,000ths to correspond with the figures of our " $a$ ," to which they have to be added. Our ( $b$ ) and ( $c$ ) give the mean anomaly of the moon and sun for any moment, ( $a$ ) being the mean longitudinal distance of the moon from the sun. To convert this last ( $a$ ) into true longitudinal distance the equation of the centre for both moon and sun must be discovered and applied to ( $a$ ) and these Tables give the requisite quantities. The case may perhaps be better understood if more simply explained. The moon and earth are constantly in motion in their orbits, and for calculation of a tithi we have to ascertain their relative positions with regard to the sun. Now supposing a railway train runs from one station to another twenty miles off in an hour. The average rate of running will be twenty miles an hour, but the actual speed will vary, being slower at starting and stopping than in the middle. Thus at the end of the first quarter of an hour it will not be quite five miles from the start, but some little distance short of this, say $m$ yards. This distance is made up as full speed is acquired, and after three-quarters of an hour the train will be rather more than 15 miles from the start, since the speed will be slackened in approaching the station,—say $n$ yards more than the 15 miles. These distances of $m$ yards and $n$ yards, the one in defect and the other in excess, correspond to the "Equation of the Centre" in planetary motion. The planetary motions are not uniform and a planet is thus sometimes behind, sometimes in front of, its mean or average place. To get the true longitude we must apply to the mean longitude the equation of the centre. And this last for both sun (or earth) and moon is what we give in these two Tables. All the requisite data for calculating the mean anomalies of the sun and moon, and the equations of the centre for each planet, are given in the Indian Siddhântas and Karaṇas, the details being obtained from actual observation; and since our Tables generally are worked according to the Sûrya Siddhânta, we have given in Tables VI. and VII. the equations of the centre by that authority.

Thus, the Tables enable us to ascertain ( $a$ ) the mean distance of moon from sun at any moment, ( $b$ ) the correction for the moon's true (or apparent) place with reference to the earth, and ( $c$ ) the correction for the earth's true (or apparent) place with reference to the sun; and with these corrections applied to the ( $a$ ) we have the true(or apparent) distance of the moon from the sun, which marks the occurrence of the true (or apparent) tithi; and this result is our tithi-index, or ( $t$ ). From this tithi-index ( $t$ ) the tithi current at any given moment is found from Table VIII.. and the time equivalent is found by Table X. Full explanation for actual work is given in Part IV. below (Arts. 139—160).

The method for calculating a nakshatra or yoga is explained in Art. 133.

108. Since the planet's true motion is sometimes greater and sometimes less than its mean motion it follows that the two equations of the centre found from ( $b$ ) and ( $c$ ) by our Tables VI. and VII. have sometimes to be added to and sometimes subtracted from the mean longitudinal distance ( $a$ ), if it is required to find the true (or apparent) longitudinal distance ( $t$ ). But to simplify calculation it is advisable to eliminate this inconvenient element, and to prepare the Tables so that the sum to be worked may always be one of addition. Now it is clear that this can be done by increasing every figure of each equation by its largest amount, and decreasing the figure ( $a$ ) by the sum of the largest amount of both, and this is what has been done in the Tables. According to the Sûrya Siddhânta the greatest possible lunar equation of the centre is 5° 2′ 47″.17 (= .0140,2 in our tithi-index computation), and the greatest possible solar equation of the centre is 2° 10′ 32″.35 (= .0060,4). But the solar equation of the centre, or the equation for the earth, must be introduced into the figure representing the distance of the moon from the sun with reversed sign, because a positive correction to the earth's longitude implies a negative correction to the distance of moon from sun. This will be clear from a diagram.

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Let $S$ be the sun, $M$ the moon, $E$ the earth, $P$ the direction of perigee. Then the angle $SEM$ represents the distance of moon from sun. But if we add a positive correction to (i.e., increase) the earth's longitude $PSE$ and make it $PSE'$ (greater than $PSE$ by $ESE'$ ) we thereby decrease the angle $SEM$ to $SE'M'$ , and we decrease it by exactly the same amount, since the angle $SEM=\angle SE'M'+\angle ESE'$ , as may be seen if we draw the line $EX$ parallel to $E'S$ ; for the angle $SEX=\angle ESE'$ by Euclid.

Every figure of each equation is thus increased in our Tables VI. and VII. by its greatest value, i.e., that of the moon by 140,2 and that of the sun by 60,4, and every figure of ( $a$ ) is decreased by the sum of both, or (140,2 + 60,4 =) 200,6.^[119]

In conclusion, Table VI. yields the lunar equation of the centre calculated by the Sûrya Siddhânta, turned into 10,000ths of a circle, and increased by 140.2; and Table VII. yields the solar equation of the centre calculated by the Sûrya Siddhânta, with sign reversed, converted into 10,000ths of a circle, and increased by 60.4.^[120] This explains why for argument 0 the equation given is lunar 140 and solar 60. If there were no such alteration made the lunar equation for Arg. 0 would be ±0, for Arg. 250 (or 90°) +140, for Arg. 500 (180°) ±0, and for Arg. 750 (or 270°) −140, and so on.

109. The lunar and solar equations of the centre for every degree of anomaly are given in the Makaranda, and from these the figures given by us for every 1/100th of a circle, or 10 units of the argument of the Tables, are easily deduced.

110. The use of the auxiliary Table is fully explained on the Table itself.

111. Table VIII. This is designed for use with our method C, the rules for which are given in Arts. 139—160. As regards the tithi-index, see Art. 80. The period of a nakshatra or yoga is the 27th part of a circle, that is 13° 20′ or 10000/27 = 370 10/27. Thus, the index for the ending point of the first nakshatra or yoga is 370 and so on.^[121] Tables VIII.A. and VIII.B. speak for themselves. They have been inserted for convenience of reference.

112. Table IX. is used in both methods B and C. See the rules for work.

113. Table X. (See the rules for work by method C.) The mean values in solar time of the several elements noted herein, as calculated by the Sûrya-Siddhânta, are as follow:—

A tithi	=	1417.46822	minutes.
A lunation	=	42524.046642	minutes.do.
A sidereal month	=	39343.21	minutes.do.
A yoga-chakra	=	36605.116	minutes.do.

From these values the time-equivalents noted in this Table^[122] have been calculated. (See also note to Art. 82.)

114. Table XI. This Table enables calculations to be made for observations at different places in India. (See Art. 36, and the rules for working by our method C.)

115. Table XII. We here give the names and numbers of the samvatsaras, or years of the sixty-year cycle of Jupiter, with those of the twelve-year cycle corresponding thereto. (See the description of these cycles given above, Arts. 53 to 63.)

116. Table XIII. This Table was furnished by Dr. Burgess and is designed to enable the week-day corresponding to any European date to be ascertained. It explains itself. Results of calculations made by all our methods may be tested and verified by the use of this Table.

117. Tables XIV. and XV. are for use by our method A (see the rules), and were invented and prepared by Mr. T. Lakshmiah Naidu of Madras.

Table XVI. is explained in Part V.

PART IV.

Use of the Tables.

118. The Tables now published may be used for several purposes, of which some are enumerated below.

(1) For finding the year and month of the Christian or any Indian era corresponding to a given year and month in any of the eras under consideration.

(2) For finding the samvatsara of the sixty-year cycle of Jupiter, whether in the southern (luni-solar) or northern (mean-sign) scheme, and of the twelve-year cycle of Jupiter, corresponding to the beginning of a solar (Meshâdi) year, or for any day of such a year.

(3) For finding the added or suppressed months, if any, in any year.

But the chief and most important use of them are;

(4) The conversion of any Indian date—luni-solar (tithi) or solar—into the corresponding date A.D. and vice versâ, from A.D. 300 to 1900, and finding the week-day of any such date;

(5) Finding the karaṇa. nakshatra, and yoga for any moment of any Indian or European date, and thereby verifying any given Indian date;

(6) Turning a Hindu solar date into a luni-solar date, and vice versâ.

(7) Conversion of a Muhammadan Hijra date into the corresponding date A.D., and vice versâ. This is fully explained in Part V. below.

119. (1) For the first purpose Table I., cols, 1 to 5. or Table II., must be used, with the explanation given in Part III. above. For eras not noted in these two Tables see the description of them given in Art. 71. In the case of obscure eras whose exact nature is not yet well known, the results will only be approximate.

(N.B.—It will be observed that in Table II., Part ii., portions of two solar months or of four^[123] Christian months are made to correspond to a lunar month and vice versa, and therefore that if this Table only be used the results may not be exact).

The following note, though not yielding very accurate results, will be found useful for finding the corresponding parts of lunar and solar months. The tithi corresponding to the Mesha-saṅkrânti can be approximately^[124] found by comparing its English date (Table I., col. 13) with that of the luni-solar Chaitra śukla 1st (Table I., col. 19); generally the saṅkrântis from Vṛishabha to Tulâ fall in successive lunar months, either one or two tithis later than the given one. Tulâ falls about 10 tithis later in the month than Mesha; and the saṅkrântis from Vṛischika to Mîna generally fall on the same tithi as that of Tulâ. Thus, if the Mesha saṅkrânti falls on śukla pañchami (5th) the Vṛishabha saṅkrânti will fall on śukla shasṭhî (6th) or śukla saptamî (7th), the Mithuna saṅkrânti on śukla ashṭamî (8th) or navamî (9th). and so on.

120. (2) For the samvatsara of the southern sixty-year cycle see col. 6 of Table I., or calculate it by the rule given in Art. 62. For that of the sixty-year cycle of Jupiter of the mean sign system, according to Sûrya Siddhânta calculations, current at the beginning of the solar year, i.e., at the true (or apparent) Mesha saṅkrânti, see col. 7 of Table I.; and for that current on any day in the year according to either the Sûrya or Ârya Siddhântas, use the rules in Art. 59. To find the samvatsara of the twelve-year cycle of the mean-sign system corresponding to that of the Jupiter sixty-year cycle see Table XII.

121. (2) To find the added or suppressed month according to the Sûrya Siddhânta by the true (apparent) system see col. 8 of Table I. throughout; and for an added month of the mean system according to either the Original or Present Sûrya Siddhântas, or by theÂrya Siddhântas, see col. 8a of Table I. for any year from A. D. 300 to 1100.

122. (4) For conversion of an Indian date into a date A.D. and vice versâ, and to find the week day of any given date, we give below three methods, with rules and examples for work.

123. The first method A (Arts. 135, 136), the invention of Mr. T. Lakshmiah Naidu of Madras, is a method for obtaining approximate results without any calculation by the careful use of mere eye-tables, viz., Tables XIV. and XV. These, with the proper use of Table I., are alone necessary. But it must never be forgotten that this result may differ by one, or at the utmost two, days from the true one, and that it is not safe to trust to them unless the era and bases of calculation of the given date are clearly known. (See Art. 126 below.)

124. By our second method B (Arts. 137, 138), which follows the system established by Mr. W. S. Kṛishṇasvâmi Naidu of Madras, author of "South Indian Chronological Tables"

(Madras 1889), and which is intended to enable an approximation to be made by a very simple calculation, a generally accurate correspondence of dates can be obtained by the use of Tables I., III., and IX. The calculation is so easy that it can be done in the head after a little practice. It is liable to precisely the same inaccuracies as method A, neither more nor less.

125. Tables II. and III. will also be sometimes required for both these methods.

126. The result obtained by either of these methods will thus be correct to within one or two days, and as often as not will be found to be quite correct; but there must always be an element of uncertainty connected with their use. If, however, the era and original bases of calculation of the given date are certainly known, the result arrived at from the use of these eye-Tables may be corrected by the week-day if that has been stated; since the day of the month and year will not be wrong by more than a day, or two at the most, and the day of the week will determine the corresponding civil day. Suppose, for instance, that the given Hindu date is Wednesday, Vaiśâkha śukla 5th, and it is found by method A or method B that the corresponding day according to European reckoning fell on a Thursday, it may be assumed, presuming that all other calculations for the year and month have been correctly made, that the civil date A.D. corresponding to the Wednesday is the real equivalent of Vaiśâkha śukla 5th. But these rough methods should never be trusted to in important cases. For a specimen of a date where the bases of calculation are not known see example xxv., Art. 160 below.

127. When Tables XIV. and XV. are once understood (and they are perfectly simple) it will probably be found advisable to use method A in preference to method B.

128. As already stated, our method "C" enables the conversion of dates to be made with precise accuracy; the exact moments of the beginning and ending of every tithi can be ascertained; and the corresponding date is obtained, simultaneously with the week-day, in the required reckoning.

129. The weekday for any European date can be found independently by Table XIII.. which was supplied by Dr. Burgess.

131^[125] (5) To find the karaṇa, nakshatra, or yoga current on any Indian or European date; and to verify any Indian date.

Method C includes calculations for the karaṇa, nakshatra and yoga current at any given moment of any given day, as well as the instants of their beginnings and endings; but for this purpose, if the given date is other than a tithi or a European date, it must be first turned into one or the other according to our rules (Art. 139 to 152.)

132. It is impossible, of course, to verify any tithi or solar date unless the week-day, nakshatra. karaṇa, or yoga, or more than one of these, is also given; but when this requirement is satisfied our method C will afford proof as to the correctness of the date. To verify a solar date it must first be turned into a tithi or European date. (Art. 134 or 149.)

133. For an explanation of the method of calculating tithis and half-tithis (karanas) see Art. 107 above. Our method of calculation for nakshatras and yogas requires a little more explanation. The moon's nakshatra (Arts. 8, 38) is found from her apparent longitude. By our method C we shew how to find $t$ (= the difference of the apparent longitudes of sun and moon), and equation^[126] $c$ (= the solar equation of the centre) for any given moment. To obtain ( $t$ ) the sun's apparent longitude is subtracted from that of the moon, so that if we add the sun's apparent longitude to ( $t$ ) we shall have the moon's apparent longitude. Our ( $c$ ) (Table 1., last column) is the sun's mean anomaly, being the mean sun's distance from his perigee. If we add the longitude of the sun's perigee to ( $c$ ), we have the sun's mean longitude, and if we apply to this the solar equation of the centre (+ or −) we have the sun's apparent longitude.^[127] According to the Sûrya Siddhânta the sun's perigee has only a very slight motion, amounting to 3′ 5″.8 in 1600 years. Its longitude for A.D. 1100, the middle of the period covered by our Tables, was 257° 15′ 55″.7 or .7146,3 of a circle, and therefore this may be taken as a constant for all the years covered by our Tables.

Now, true or apparant sun = mean sun + equation of centre. But we have not tabulated in Table VII., col. 2, the exact equation of the centre; we have tabulated a quantity (say $x$ ) the value of which is expressed thus;—

$x=60,4-{\text{equation of centre}}$ (see Art. 108).
So that ${\text{equation of centre}}=60,4-x$ .
Hence, ${\text{apparent sun}}={\text{mean sun}}+60,4-x$ .
But ${\text{mean sun}}=c+{\text{perigee}}$ , (which is 7146,3 in tithi-indices.)
But mean sun.. $=c+7146,3$ .
Hence apparent sun (which we call $s$ )	$=c+7146,3+60,4-x$ .
	$=c+7206,7-x$ ; or, say, $=c+7207-x$

where $x$ is, as stated, the quantity tabulated in col. 2, Table VII.

( $c$ ) is expressed in 1000ths, while 7207 and the solar equation in Table VII. are given in 10000ths of the circle, and therefore we must multiply ( $c$ ) by 10. $t+s={\text{apparent moon}}=n$ (the index of a nakshatra.) This explains the rule given below for work (Art. 156).

For a yoga, the addition of the apparent longitude of the sun ( $s$ ) and moon ( $n$ ) is required. $s+n=y$ (the index of a yoga.) And so the rule in Art. 159.

134. (6) To turn a solar date into its corresponding luni-solar date and vice versa.

First turn the given date into its European equivalent by either of our three methods and then turn it into the required one. The problem can be worked direct by anyone who has thoroughly grasped the principle of these methods.

Method A.

Approximate computation of dates by use of the eye-table.

This is the method invented by Mr. T. Lakshmiah Naidu, nephew of the late W. S. Kṛishṇasvâmi Naidu of Madras, author of "South Indian Chronological Tables."

Results found by this method may be inaccurate by as much as two days, but not mure. If the era and bases of calculation of the given Hindu date are clearly known, and if the given date mentions a week-day, the day found by the Tables may be altered to suit it. Thus, if the Table yield result Jan. 10th, Thursday, but the inscription mentions the week-day as "Tuesday", then Tuesday, January 8th, may be assumed to be the correct date A.D. corresponding to the given Hindu date, if the principle on which the Hindu date was fixed is known. If not, this method must not be trusted to.

135. (A.) Conversion of a Hindu solar date into the corresponding date A.D. Work by the following rules, always bearing in mind that when using the Kaliyuga or Śaka year Hindus usually give the number of the expired year, and not that astronomically current, {e.g., Kaliyuga 4904 means in full phrase "after 4904 years of the Kaliyuga had elapsed") — but when using the name of the cyclic year they give that of the one then current. All the years given in Table I. are current years. The Table to work by is Table XIV.

Rule I. From Table I., cols, i to 7, and Table II., as the case may be, find the year (current) and its initial date, and week-day (cols. 13, 14, Table I.). But if the given Hindu date belongs to any of the months printed in italics at the head of Table XIV., take the next follow- ing initial date and weekday in cols. 13, 14 of Table I. The months printed in the heading in capitals are the initial months of the years according to the different reckonings.

Rule II. For either of the modes of reckoning given at the left of the head-columns of months, find the given month, and under it the given date.

Rule III. From the given date so found, run the eye to the left and find the week-day in the same line under the week-day number found by Rule I. This is the required week-day.

Rule IV. Note number in brackets in the same line on extreme left.

Rule V. In the columns to left of the body of the Table choose that headed by the bracket-number so found, and run the eye down till the initial date found by Rule I. is obtained.

Rule VI. From the month and date in the upper columns (found by Rule II.) run the eye down to the point of junction (vertical and horizontal lines) of this with the initial date found by Rule V. This is the required date A. D.

Rule VII. If the date A. D. falls on or after 1st January in columns to the right, it belongs to the next following year. If such next following year is a leap-year (marked by an asterisk in Table I.) and the date falls after February 28th in the above columns, reduce the date by one day.

N.B. — The dates A.D. obtained from this Table for solar years are Old Style dates up to 8th April, 1753, inclusive.

Example. Find date A.D. corresponding to 20th Paṅguni of the Tamil year Rudhirodgâri, Kali 4904 expired.

By Rule I. Kali 4905 current, 2 (Monday), 11th April, 1803.

By Rule„ II. Tamil Panguni 20.

By Rule„ III. (under "2") Friday.

By Rule„ IV. Bracket-number (5).

By Rule„ V. [Under (5)]. Run down to April 11th.

By Rule„ VI. (Point of junctions) March 31st.

By Rule„ VII. March 30th. (1804 is a leap year.)

Answer.—Friday, March 30th, 1804 N.S. (See example 11, p. 74.)

(B.) Conversion of a date A.D. into the corresponding Hindu solar date. (See Rule V.. method B, Art. 137, p. 70.) Use Table XIV.

Rule I. From Tables I., cols, i to 7 and 13, 14, and Table II., as the case may be. find the Hindu year, and its initial date and week-day, opposite the given year A. D. If the given date falls before such initial date, take the next previous Hindu year and its initial date and week-day A.D.

Rule II. From the columns to the left of the body of Table XIV. find that initial date found by Rule I. which is in a line, when carrying the eye horizontally to the right, willi the given A.D. date, and note point of junction. Rule III. Note the bracket-figure at head of the column on left so selected.

Rule IV. From the point of junction (Rule II.) run the eye vertically up to the Hindu date-columns above, and select that date which is in the same horizontal line as the bracket-figure on the extreme left corresponding with that found by Rule III. This is the required date.

Rule V. If the given date falls in the columns to the right after the 28th February in a leap-year (marked with an asterisk in Table I.), add 1 to the resulting date.

Rule VI. From the date found by Rule IV. or V., as the case may be, carry the eye horizontally to the weekday columns at the top on the left, and select the day which lies under the week-day number found from Table I. (Rule I.). This is the required week-day.

Rule VII. If the Hindu date arrived at falls under any of the months printed in italics in the Hindu month-columns at head of Table, the required year is the one next previous to that given in Table I. (Rule I.).

Example. Find the Tamil solar date corresponding to March 30th, 1804 (N.S.).

(By Rule I.) Rudhirodgâri, Kali 4905 current. 2 (Monday) April 11th. (March 30th precedes April 11th.)

(By Rules II., III.) The point of junction of March 30th (body of Table), and April 11th, (columns on left) is under "(4)." Other entries of April nth do not correspond with any entry of March 30).

(By Rule IV.) The date at the junction of the vertical column containing this "March 30th" with "(4)" horizontal is 19th Paṅguni.

(By Rule V.) (1804 is a leap-year) 20th Paṅguni.

(By Rule VI.) Under "2" (Rule I.), Friday.

Answer.—Friday, 20th Paṅguni, of Rudhirodgâri, Kali 4905 current. (See example 15, p. 76.

136. (A.) Conversion of a Hindu luni-solar date into the corresponding date A.D. Work by the following rules, using Tables XV.A., and XV.B.

Rule I. From Table I. find the current year and its initial day and week-day in A.D. reckoning, remembering that if the given Hindu date falls in one of the months printed in italics at the head of Table XV. the calculation must be made for the next following A.D. year. (The months printed in capitals are the initial months of the years according to the different reckonings enumerated in the column to the left.)

Rule II. (a.) Find the given month, and under it the given date, in the columns at the head of Table XV., in the same line with the appropriate mode of reckoning given in the column to the left. The dates printed in black type are kṛishṇa, or dark fortnight, dates.

(b.) In intercalary years (cols. 8 to 12, 8a to 12a of Table I.), if the given month is itself an adhika mâsa (intercalary month), read it, for purpose of this Table, as if it were not so; but if the given month is styled nija, or if it falls after a repeated month, but before an expunged one (if any), work in this Table for the month next following the given one, as if that and not the given month had been given. If the given month is preceded by both an intercalated and a suppressed month, work as if the year were an ordinary one.

Rule III. From the date found by Rule II. carry the eye to the left, and find the week-day in the same horizontal line, but directly under the initial week-day found by Rule I.

Rule IV. Note the number in brackets on the extreme left opposite the week-day last found.

Rule V. In the columns to the left of the body of the Table choose that headed by the bracket-number so found, and run the eye down till the initial date found by Rule I. is obtained.

Rule VI. From the Hindu date found by Rule II. run the eye down to the point of junction, (vertical and horizontal lines) of this date with the date found by Rule V. The result is the required date A.D.

Rule VII (a.) If the date A.D. falls on or after January 1st in the columns to the right, it belongs to the next following year A.D.

(b.) If it is after February 28th in a leap-year (marked by an asterisk in col. 5, Table I.) reduce the date by one day, except in a leap-year in which the initial date (found in Table I.) itself falls after February 28th.

(c.) The dates obtained up to April 3rd, A.D. 1753, are Old Style dates.

Example. To find the date A. D. corresponding to amânta Kârttika kṛishṇa 2nd of Kali 4923 expired, Ŝaka 1744 expired, Kârttikâdi Vikrama 1878 expired, Chaitrâdi Vikrama 1879 expired (1880 current), "Vijaya" in the Bṛihaspati cycle, "Chitrabhânu" in the luni-solar 60-year cycle.

(By Rule I.) (Kali 4924 current), 11Sunday, March 24th, 1822.

(By Rule II.) (Kârttika, the 8th month, falls after the repeated month, 7 Âśvina, and before the suppressed month, 10 Pausha), Mârgaśîrsha kṛishṇa 2nd.

(By Rule III.) (Under "1"), i Sunday.

(By Rule IV.) Bracket-number (1).

(By Rule V.) Under (1) run down to March 24th (Rule I.)

(By Rule VI.) (Point of junction) December 1st.

Answer.—Sunday, December 1st, 1822.

(B.) Conversion of a date A. D. into the corresponding luni-solar Hindu date. (See Rule V. method B, p. 67 below). Use Tables XV.A., XV.B.

Rule I. From Table I. find the Hindu year, and its initial date and week-day, using also Table II., Parts ii., iii. If the given date falls before such initial date take the next previous Hindu year, and its initial date and weekday.

Rule II. In the columns to the left of the body of Table XV. note the initial date found by Rule I., which is in the same horizontal line with the given date in the body of the Table.

Rule III. Carrying the eye upwards, note the bracket-figure at the head of the initial date-column so noted.

Rule IV. From the given date found in the body of the Table (Rule 11.) run the eye upwards to the Hindu date-columns above, and select the date which is in the same horizontal line as the bracket-figure in the extreme left found by Rule III. This is the required Hindu date.

Rule V. Note in Table I. if the year is an intercalary one (cols. 8 to 12, and 8a to 12a). If it is so, note if the Hindu month found by Rule IV. (a) precedes the first intercalary month, (b) follows one intercalated and one suppressed month, (c) follows an intercalated, but precedes a suppressed month, (d) follows two intercalated months and one suppressed month. In cases (a) and (b) work as though the year were a common year, i.e., make no alteration in the date found by Rule IV. In cases (c) and (d) if the found month immediately follows the intercalated month, the name of the required Hindu month is to be the name of the intercalated month with the prefix "nija," and not the name of the month actually found; and if the found month does not immediately follow the intercalated month, then the required Hindu month is the month immediately preceding the found month. If the found month is itself intercalary, it retains its name, but with the prefix "adhika." If the found month is itself suppressed, the required month is the month immediately preceding the found month. Rule VI. If the given date A.D. falls after February 29th in the columns to the right, in a leap-year (marked with an asterisk in Table I.), add 1 to the resulting Hindu date.

Rule VII. From the date found by Rule IV. carry the eye horizontally to the week-day columns on the left, and select the day which lies under the initial week-day number found by Rule I. This is the required week-day.

Rule VIII. If the Hindu date arrived at falls under any of the months printed in italics in the Hindu month-columns at head of the table, the required year is the one next previous to that given by Table I. (Rule I. above.)

Example. Find the Telugu luni-solar date corresponding to Sunday, December 1st, 1822.

(By Rule I.) A.D. 1822—23, Sunday, March 24th, Kali 4923 expired, Śaka 1744 expired, Chitrabhânu samvatsara in the luni-solar 60-year or southern cycle reckoning, Vijaya in the northern cycle.

(By Rules II., III.) (Bracket-figure) 1.

(By Rule IV.) Mârgaśîrsha kṛishṇa 2nd.

(By Rule Vc.) (Âśvina being intercalated and Pausha suppressed in that year), Kârttika kṛishṇa 2nd.

(By Rule VI.) The year was not a leap-year.

(By Rule VII.) Sunday.

(By Rule VIII.) Does not apply.

Answer.—Sunday, Kârttika kṛishṇa 2nd, Kali 4923 expired, Śaka 1744 expired. (This can be applied to all Chaitrâdi years.) (See example 12 below, p. 75.)

Method B.

Approximate computation of dates by a simple process.

This is the system introduced by Mr. W. S. Kṛishṇasvâmi Naidu of Madras into his "South-Indian Chronological Tables."

137. (A.) Conversion of Hindu dates into dates A.D. (See Art. 135 above, para. 1.)

Rule I. Given a Hindu year, month and date. Convert it if necessary by cols. 1 to 5 of Table I., and by Table II., into a Chaitrâdi Kali or Śaka year, and the month into an amânta month. (See Art. 104.) Write down in a horizontal line ( $d$ ) the date-indicator given in brackets in col. 13 or 19 of Table I., following the names of the initial civil day and month of the year in question as so converted, and ( $w$ ) the week-day number (col. 14 or 20) corresponding to the initial date A.D. given in cols. 13 or 19. To both ( $d$ ) and ( $w$ ) add, from Table III., the collective duration of days from the beginning of the year as given in cols. 3a or 10 as the case may be, up to the end of the month preceding the given month, and also add the number of given Hindu days in the given month minus 1. If the given date is luni-solar and belongs to the kṛishṇa paksha, add 15 to the collective duration and proceed as before.

Rule II. From the sum of the first addition find in Table IX. (top and side columns) the required English date, remembering that when this is over 365 in a common year or 366 in a leap-year the date A.D. falls in the ensuing A.D. year.

Rule III. From the sum of the second addition cut out sevens. The remainder shews the required day of the week.

Rule IV. If the Hindu date is in a luni-solar year where, according to cols. 8 to 12, there was an added (adhika) or suppressed (kshaya) month, and falls after such month, the addition or suppression or both must be allowed for in calculating the collective duration of days; i.e., add 30 days for an added month, and deduct 30 for a suppressed month.

Rule V. The results are Old Style dates up to, and New Style dates from, 1752 A.D. The New style in England was introduced with effect from after 2nd September, 1752. Since the initial dates of 1752, 1753 only are given, remember to apply the correction (+ 11 days) to any date between 2nd September, 1752, and 9th April, 1753, in calculating by the Hindu solar year, or between 2nd September, 1752, and 4th April, 1753, in calculating by the Hindu lunisolar year, so as to bring out the result in New Style dates A.D. The day of the week requires no alteration.

Rule VI. If the date A.D. found as above falls after February 29th in a leap-year, it must be reduced by one day.

(a) Luni-Solar Dates.

Example 1. Required the A.D. equivalent of (luni-solar) Vaiśâkha śukla shashṭhî (6th), year Śârvari, Śaka 1702 expired, (1703 current).

The A.D. year is 1780 (a leap-year). The initial date $(d)=$ 5th April (96), and $(w)=4$ Wednesday, (Table I., cols. 5, 19, 20).

	d.		w.
State this accordingly	96	4
Collective duration (Table III., col. 3a)	30	30
Given date (6) − 1	5	5
	131
	1	(Rule VI.)
	130		39	÷ 7 = Rem. 4

The result gives 130 (Table IX.) = May 10th, and 4 = Wednesday. The required date is therefore Wednesday, May 10th, A.D. 1780.

Example 2. Required the A.D. equivalent of (luni-solar) Kârttika śukla paṅchamî (5th) Śaka 1698 expired (1699 current).

The A.D. year is 1776, and the initial date is $(d)=$ 20th March (80), $(w)=$ Wednesday (4). This is a leap-year, and the Table shews us that the month (6) Bhâdrapada was intercalated. So there is both an adhika Bhâdrapada and a nija Bhâdrapada in this year, which compels us to treat the given month Kârttika as if it were the succeeding month Mârgaśîrsha in order to get at the proper figure for the collective duration.

	d.		w.
The given figures are	80	4
Collective duration (Table III.) for Mârgaśîrsha	236	236
Given date (5) − 1	4	4
	320
	− 1	(Rule VI.)
	319		244	÷ 7 = Rem. 6
319 = (Table IX.) November 15th. 6 = Friday

Answer.—Friday, November 15th, A.D. 1776.

Example 3. Required the A.D. equivalent of Kârttika kṛishṇa paṅchamî (5th) of the same luni-solar year.

	d.		w.
As before	80	4
Collective duration (Table III., col. 3a)	236	236
Given date (5 + 15) − 1	19	19
	335
	− 1	(Rule VI.)
	334		259	÷ 7 = Rem. 0
334 = (Table IX.) November 30th. 0 = Saturday.

Answer.—Saturday, November 30th, A.D. 1776.

Example 4. Required the A.D. equivalent of Kârttika kṛishṇa pâdyami (1st) of K.Y. 4923 expired (4924 current). This corresponds (Table I., col. 5) to A.D. 1822, the Chitrabhânu samvatsara, and col. 8 shews us that the month Âśvina was intercalated (adhika), and the month Pausha suppressed (kshaya). We have therefore to add 30 days for the adhika month and subtract 30 days for the kshaya month, since Mâgha comes after Pausha. Hence the relative place of the month Mâgha remains unaltered,

Table I. gives 24th March (83), (1) Sunday, as the initial day.

	d.		w.
Initial date	83	1
Collective duration (Table III., col. 3a)	295	295
Given date (1 + 15) − 1	15	(Rule I.)	15
	393		311	÷ 7 = Rem. 3.
3 = Tuesday. 393 —January 28th of the following A.D. year (Table IX.).

Answer.—Tuesday, January 28th, A.D. 1823. This is correct by the Tables, but as there happened to be an expunged tithi in Mâgha śukla, the first fortnight of Mâgha, the result is wrong by one day. The corresponding day was really Monday, January 27th, and to this we should have been guided if the given date had included the mention of Monday as the week-day. That is, we should have fixed Monday, January 27th, as the required day A.D. because our result gave Tuesday, January 28th, and we knew that the date given fell on a Monday,

Example 5. Required the A.D. equivalent of Pausha śukla trayôdaśî (13th) K.Y. 4853 expired, Aṅgiras samvatsara in luni-solar or southern reckoning. This is K. Y. 4854 current.

The year (Table I., col. 5) is A.D. 1752, a leap-year. The initial date (cols. 19, 20) is 5th March (65), (5) Thursday. The month Âshâḍha was intercalated. Therefore the given month (Pausha) must be treated, for collective duration, as if it were the succeeding month Mâgha.

	d.		w.
Initial date	65	5
Collective duration (Table III., col. 3a)	295	295
Given date (13) − 1	12	12
	372
	− 1	(Rule VI.)
	371		312	÷ 7, Rem. 4.

We must add eleven days to the amount 371 to make it a New Style date, because it falls after September 2nd, 1752, and before 4th April, 1753, (after which all dates will be in New Style by the Tables). 371 + 11 = 382 = January 17th (Table IX.). 4 = Wednesday.

Answer.—Wednesday, January 17th, A.D. 1753.

Example 6. Required the A.D. equivalent of Vikrama samvatsara 1879 Âshâḍha kṛishṇa dvitîyâ (2nd). If this is a southern Vikrama year, as used in Gujarât, Western India, and countries south of the Narmadâ, the year is Kârttikâdi and amânta, i.e., the sequence of fortnights makes the month begin with śukla 1st. The first process is to convert the date by Table II., Part iii., col. 3, Table II., Part ii., and Table I., into a Chaitrâdi year and month. Thus—Âshâḍha is the ninth month of the year and corresponds to Âshâḍha of the following Chaitrâdi Kali year, so that the given month Âshâḍha of Vikrama 1879 corresponds to Âshâḍha of Kali 4924. Work as before, using Table I. for Kali 4924. Initial date, 24th March (83), (1) Sunday.

	d.		w.
Initial date	83	1
Collective duration (Table III., col. 3a)	89	89
Given date (2 + 15) − 1	16	16
	188		106	÷ 7 Rem. 1
188 (Table IX.) = July 7th. 1 = Sunday.

Answer.—Sunday, July 7th, A.D. 1822.^[128]

If the year given be a northern Vikrama year, as used in Mâlwa, Benares, Ujjain, and countries north of the Narmadâ, the Vikrama year is Chaitrâdi and corresponds to the Kali 4923, except that, being pûrṇimânta, the sequence of fortnights differs (see Table II., Part i.). In such a case Âshâḍha kṛishṇa of the Vikrama year corresponds to Jyeshṭha kṛishṇa in amânta months, and we must work for Kali 4923 Jyeshṭha kṛishṇa 2nd. By Table I. the initial date is April 3rd (93), (3) Tuesday. The A.D. year is 1821—22.

	d.		w.
	93	3
Collective duration (Table III., col. 3a)	59	59
Given date (2 + 15) − 1	16	16
	168		78	÷ 7, Rem. 1.
168 = June 17th. 1 = Sunday.

Answer.—Sunday. June 17th, A.D. 1821.

(b) Solar Dates.

Example 7. Required the date A.D. corresponding to the Tamil (solar) 18th Puraṭṭâśi of Rudhirodgârin = K.Y. 4904 expired, or 4905 current.

Table I., cols. 13 and 14, give $(d)=$ April 11th (101), $(w)=$ (2) Monday, and the year A.D. 1803.

	d.		w.
Initial date	101	2
Collective duration (Table III., col. 10)	156	156
Given date (18) − 1	17	17
	274		175	÷ 7, Rem. 0.
274 (Table IX.) gives October 1st. 0 = Saturday.

Answer.— Saturday, October 1st, A.D. 1803.

Example 8. Required the equivalent A.D. of the Tinnevelly Âṇḍu 1024, 20th Âvaṇi. The reckoning is the same as the Tamil as regards months, but the year begins with Âvaṇi. Âṇḍu 1024 = K.Y. 4950. It is a solar year beginning (see Table I.) 11th April (102), (3) Tuesday, A.D. 1848 (a leap-year).

	d.		w.
Initial date	102	3
Tables II., Part ii., cols. 10 & y, and III., col. 10.	125	125
Given date (20) − 1	19	19
	246
	− 1	(Rule VI.)
	245		147	÷ 7, Rem. 0.
0 = Saturday; 245 = (Table IX.) September 2nd.

Answer.—Saturday, September 2nd, A.D. 1848.

Example 9. Required the equivalent date A.D. of the South Malayâḷam Âṇḍu 1024, 20th Chiṅgam. The corresponding Tamil month and date (Table II., Part ii., cols. 9 and 11) is 20th Âvaṇi K.Y. 4950, and the answer is the same as in the last example.

Example 10. Required the equivalent date A.D. of the North Malayâḷam (Kollam) Âṇḍu 1023, 20th Chiṅgam. This (Chiṅgam) is the 12th month of the Kollam Âṇḍu year which begins with Kanni. It corresponds with the Tamil 20th Âvaṇi K.Y. 4950 (Table II., Part ii., cols. 9, 12, and Table II., Part iii.), and the answer is similar to that in the two previous examples.

[The difference in the years will of course be noted. The same Tamil date corresponds to South Malayâḷam Âṇḍu 1024, 20th Chiṅgam, and to the same day of the month in the North Malayâḷam (Kollam) Âṇḍu 1023, the reason being that in the former reckoning the year begins with Chiṅgam, and in the latter with Kanni.]

Example 11. Required the A.D. equivalent of the Tamil date, 20th Paṅguni of Rudhirodgârin, K.Y. 4905 current (or 4904 expired.)

Table I. gives ( $d$ ) 11th April (101), 1803 A.D. as the initial date of the solar year, and its week-day ( $w$ ) is (2) Monday.

	d.		w.
Initial date	101	2
Collective duration (Table III., col. 10)	335	335
Given date (20) − 1	19	19
	455
	− 1	(Rule VI.)
	454		356	÷ 7, Rem. 6.
6 = Friday; 454 = (Table IX.) March 30th in the following A.D. year, 1804.

Answer.—Friday, March 30th, 1804. (See example i, above.)

138. (B.) Conversion of dates A.D. into Hindu dates. (See Art. 135 above, par. 1.)

Rule I. Given a year, month, and date A.D. Write down in a horizontal line ( $d$ ) the date-indicator of the initial date [in brackets (Table I., cols. 13 or 19, as the case may be)] of the corresponding Hindu year required, and ( $w$ ) the week-day number of that initial date (col. 14 or 20), remembering that, if the given date A.D. is earlier than such initial date, the ( $d$ ) and ( $w$ ) of the previous Hindu year must be taken. Subtract the date-indicator from the date number of the given A.D. date in Table IX., remembering that, if the previous Hindu year has been taken down, the number to be taken from Table IX. is that on the right-hand side of the Table and not that on the left. From the result subtract (Table III., col. 3a or 10) the collective-duration-figure which is nearest to, but lower than, that amount, and add 1 to the total so obtained; and to the ( $w$ ) add the figure resulting from the second process under ( $d$ ), and divide by 7. The result gives the required week-day. The resulting ( $d$ ) gives the day of the Hindu month following that whose collective duration was subtracted.

Rule II. Observe (Table I., cols. 8 or 8a) if there has been an addition or suppression of a month prior to the month found by Rule I. and proceed accordingly.

An easy rule for dealing with the added and suppressed month is the following. When the intercalated month (Table I., col. 8 or 8a) precedes the month immediately preceding the one found, such immediately preceding month is the required month; when the intercalated month immediately precedes the one found, such immediately preceding month with the prefix "nija," natural, is the required month; when the intercalated month is the same as that found, such month with the prefix "adhika" is the recjuircd month. When a suppressed month precedes the month found, the required month is the same as that found, because there is never a suppression of a month without the intercalation of a previous month, which nullifies the suppression so far as regards the collective duration of preceding days. But if the given month falls after two intercalations and one suppression, act as above for one intercalation only.

Rule III. See Art. 137 (A) Rule V. (p. 70), but subtract the eleven days instead of adding.

Rule IV. If the given A.D. date falls in a leap-year after 29th February, or if its date-number (right-hand side of Table IX.) is more than 365, and the year next preceding it was a leap-year, add 1 to the date-number of the given European date found by Table IX., before subtracting the figure of the date-indicator

Rule V. Where the required date is a Hindu luni-solar date the second total, if less than 15, indicates a śukla date. If more than 15, deduct 15, and the remainder will be a kṛishṇa date. Kṛishṇa 15 is generally termed kṛishṇa 30; and often śukla 15 is called "pûrṇimâ" (fullmoon day), and kṛishṇa 15 (or "30") is called amâvâsyâ (new-moon day).

(a) Luni-Solar Dates.

Example 12. Required the Telugu or Tuḷu equivalent of December 1st, 1822. The luni-solar year began 24th March (83) on (1) Sunday (Table I., cols. 19 and 20.)

	d.			w.
( $d$ ) and ( $w$ ) of initial date (Table I.)	83			1
(Table IX.) 1st December (335)	(335 − 83=)	252		252
(Table III.) Collective duration to end of Kârttika		−236
Add 1 to remainder		16	+ 1 = 17	253	÷ 7, Rem. 1.

17 indicates a kṛishṇa date. Deduct 15. Remainder 2. The right-hand remainder shews (1) Sunday.

The result so far is Sunday Mârgaśîrsha kṛishṇa 2nd. But see Table I., col. 8. Previous to this month Âśvina was intercalated. (The suppression of Pausha need not be considered because that month comes after Mârgaśîrsha.) Therefore the required month is not Mârgaśîrsha, but Kârttika; and the answer is Sunday Kârttika kṛishṇa 2nd (Telugu), or Jarde (Tuḷu), of the year Chitrabhânu, K.Y. 4923 expired, Śaka 1744 expired. (See the example on p. 69.)

(Note.) As in example 6 above, this date is actually wrong by one day, because it happened that in Kârttika śukla there was a tithi, the 12th, suppressed, and consequently the real day corresponding to the civil day was Sunday Kârttika kṛishṇa 3rd. These differences cannot possibly be avoided in methods A and B, nor by any method unless the duration of every tithi of every year be separately calculated. (See example xvii., p. 92.)

Example 13. Required the Chaitrâdi Northern Vikrama date corresponding to April 9th 1822. By Table I. A.D. 1822—23 = Chaitrâdi Vikrama 18S0 current. The reckoning is luni-solar. Initial day ( $d$ ) March 24th (83), ( $w$ ) 1 Sunday

	d.		w.
From Table I.	83			1
(Table IX.) April 9th (99)	99 − 83	= 16		16
Add		1
		17
For śukla dates		−15
		2		17	÷ 7, Rem. 3.

This is Tuesday, amânta Chaitra kṛishṇa 2nd.^[129] But it should be converted into Vaiśâkha kṛishṇa 2nd, because of the custom of beginning the month with the full-moon (Table II., Part i.).

Since the Chaitrâdi Vikrama year begins with Chaitra, the required Vikrama year is 1880 current, 1879 expired. But if the required date were in the Southern reckoning, the year would be 1878 expired, since 1879 in that reckoning does not begin till Kârttika.

(b) Solar Dates.

Example 14. 1. Required the Tamil equivalent of May 30th, 1803 A.D.

Table I. gives the initial date April 11th (101), and week-day number 2 Monday.

	d.		w.
From Table I.	101		2
(Table IX.) May 30th (150)	150 − 101	= 49	49
(Table III.) Collective duration to end of Śittirai (Mesha)		− 31
		18
Add 1		+ 1
		19	51	÷ 7, Rem. 2.

The day is the 19th; the month is Vaiyâśi, the month following Śittirai; the week-day is (2) Monday.

Answer.—Monday, 19th Vaiyâśi of the year Rudhirodgârin, K.Y. 4904 expired, Śaka 1725 expired.

Example 15. Required the Tamil equivalent of March 30th, 1804. The given date precedes the initial date in 1804 A.D. (Table 1., col. 13) April 10th, so the preceding Hindu year must be taken. Its initial day is 11th April (101), and the initial week-day is (2) Monday. 1804 was a leap-year.

	d.		w.
From Table I.	101		2
(Table IX.) (March 30th) 454 + 1 for leap-year,	455 − 101	= 354	49
(Table III., col. 10) Collective duration to end of Mâśi = Kumbha (Table II., Part ii.)		− 335
		19
Add 1		+ 1
		20	356	÷ 7, Rem. 6.

Answer.—Friday 20th Paṅguṇi of the year Rudhirodgârin K.Y. 4904 expired, Śaka 1725 expired. (See the example on p. 67.)

Example 16. Required the North Malayâḷam Âṇḍu equivalent of September 2nd, 1848. Work as by the Chaitrâdi year. The year is solar. 1848 is a leap-year.

	d.		w.
From Table I.	102		3
(Table IX.) September 2nd (245) + 1 for leap year	246 − 102	= 144	144
Coll. duration to end of Karka		− 125
		19
Add 1		+ 1
		20	147	÷ 7, Rem. 0.

Answer.—Saturday 20th Chiṅgam. This is the 12th month of the North Malayâḷam Âṇḍu which begins with Kanni. The year therefore is 1023.

If the date required had been in South Malayâḷam reckoning, the date would be the same, 20th Chiṅgam, but as the South Malayâḷis begin the year with Chiṅgam as the first month, the required South Malayâḷam year would be Âṇḍu 1024.

Method C.

Exact calculation of dates.

(A.) Conversion of Hindu luni-solar dates into dates A.D.

139. To calculate the week-day. the equivalent date A.D., and the moment of beginning or ending of a tithi. Given a Hindu year, month, and tithi.—Turn the given year into a Chaitrâdi Kali, Śaka, or Vikrama year, and the given month into an amânta month (if they are not already so) and find the corresponding year A.D., by the aid of columns 1 to 5^[130] of Table I., and Table II., Parts i., ii., iii. Referring to Table I., carry the eye along the line of the Chaitrâdi year so found, and write down^[131] in a horizontal line the following five quantities corresponding to the day of commencement (Chaitra śukla pratipadâ) of that Chaitrâdi-year, viz., ( $d$ ) the date-indicator given in brackets after the day and month A.D. (Table I., col. 19), ( $w$ ) the week-day number (col. 20), and ( $a$ ), ( $b$ ), ( $c$ ) (cols. 23, 24, 25). Find the number of tithis which have intervened between the initial day of the year (Chaitra śukla pratipadâ), and the given tithi, by adding together the number of tithis (collective duration) up to the end of the month previous to the given one (col. 3, Table III.), and the number of elapsed tithis of the given month (that is the serial number of the given tithi reduced by one), taking into account the extra 15 days of the śukla paksha if the tithi belongs to the kṛishṇa paksha, and also the intervening intercalary month,^[132] if any, given in col. 8 (or 8a) of Table I. This would give the result in tithis. But days, not tithis, are required. To reduce the tithis to days, reduce the sum of the tithis by its 60th part,^[133] taking fractions larger than a half as one, and neglecting half or less The result is the ( $d$ ), the approximate number of days which have intervened since the initial day of the Hindu year. Write this number under head ( $d$ ), and write under their respective heads, the ( $w$ ), ( $a$ ), ( $b$ ), ( $c$ ) for that number of days from Table IV. Add together the two lines of five quantities, but in the case of ( $w$ ) divide the result by 7 and write only the remainder, in the case of ( $a$ ) write only the remainder under 10000, and in the case of ( $b$ ) and ( $c$ ) only the remainder under 1000.^[134] Find separately the equations to arguments ( $b$ ) and ( $c$ ) in Tables VI. and VII. respectively, and add them to the total under ( $a$ ). The sum ( $t$ ) is the tithi-index, which, by cols. 2 and 3 of Table VIII., will indicate the tithi current at mean sunrise on the week-day found under ( $w$ ). If the number of the tithi so indicated is not the same as that of the given one, but is greater or less by one (or by two in rare cases), subtract one (or two) from, or add one (or two) to, both ( $d$ ) and ( $w$ );^[135] subtract from, or add to, the ( $a$ ) ( $b$ ) ( $c$ ) already found, their value for one (or two) days (Table IV.); add to ( $a$ ) the equations for ( $b$ ) and ( $c$ ) (Tables VI. and VII.) and the sum ( $t$ ) will then indicate the tithi. If this is the same as given (if not, proceed again as before till it corresponds), the ( $w$ ) is its week-day, and the date shewn in the top line and side columns of Table IX. corresponding with the ascertained ( $d$ ) is its equivalent date A.D. The year A.D. is found on the line of the given Chaitrâdi year in col. 5, Table I. Double figures are given in that column; if ( $d$ ) is not greater than 365 in a common year, or 366 in a leap-year, the first, otherwise the second, of the double figures shows the proper A.D. year.

140. For all practical purposes and for some ordinary religious purposes a tithi is connected with that week-day at whose sunrise it is current. For some religious purposes, however, and sometimes even for practical purposes also, a tithi which is current at any particular moment of a week-day is connected with that week-day. (See Art. 31 above.)

141. In the case of an expunged tithi, the day on which it begins and ends is its weekday and equivalent. In the case of a repeated tithi, both the civil days at whose sunrise it is current,^[136] are its week-days and equivalents.

142. A clue for finding when a tithi is probably repeated or expunged. When the tithi-index corresponding to a sunrise is greater or less, within 40, than the ending index of a tithi, and when the equation for ( $b$ ) (Table VI.) is decreasing, a repetition of the same or another tithi takes place shortly after or before that sunrise; and when the equation for ( $b$ ) is increasing an expunction of a tithi (different from the one in question) takes place shortly before or after it.

143. The identification of the date A.D. with the week-day arrived at by the above method, may be verified by Table XIII. The verification, however, is not in itself proof of the correctness of our results.

144. To find the moment of the ending of a tithi. Find the difference between the ( $t$ ) on the given day at sunrise and the ( $t$ ) of the tithi-index which shews the ending point of that tithi (Table VIII.). With this difference as argument find the corresponding time either in ghaṭikâs and palas, or hours and minutes, according to choice, from Table X. The given tithi ends after the given sunrise by the interval of time so found. But this interval is not always absolutely accurate. (See Art. 82). If accuracy is desired add the ( $a$ ) ( $b$ ) ( $c$ ) for this interval of time (Table V.) to the ( $a$ ) ( $b$ ) ( $c$ ) already obtained for sunrise. Add as before to ( $a$ ) the equations for ( $b$ ) and ( $c$ ) from Tables VI. and VII., and find the difference between the ( $t$ ) thus arrived at and the ( $t$ ) of the ending point of the tithi (Table VIII.). The time corresponding to that difference, found from Table X., will show the ending of the tithi before or after the first found time. If still greater accuracy is desired, proceed until ( $t$ ) amounts exactly to the ( $t$ ) of the ending point (Table VIII.) For ordinary purposes, however, the first found time, or at least that arrived at after one more process, is sufficiently accurate.

145. The moment of the beginning of a tithi is the same as the moment of ending of the tithi next preceding it; and this can be found either by calculating backwards from the ( $t$ ) of the same tithi, or independently from the ( $t$ ) of the preceding tithi.

146. The moment of beginning or ending of tithis thus found is in mean time, and is applicable to all places on the meridian of Ujjain, which is the same as that of Laṅkâ. If the exact mean time for other places is required, apply the correction given in Table XI., according to the rule given under that Table. If after this correction the ending time of a tithi is found to fall on the previous or following day the ( $d$ ) and ( $w$ ) should be altered accordingly.

Mean time is used throughout the parts of the Tables used for these rules, and it may sometimes differ from the true, used, at least in theory, in Hindu paṅchâṅgs or almanacks.

The ending time of a tithi arrived at by these Tables may also somewhat differ from the ending time as arrived at from authorities other than the Sûrya Siddhânta which is used by us. The results, however, arrived at by the present Tables, may be safely relied on for all ordinary purposes.^[137]

147. N.B. i. Up to 1100 A.D. both mean and true intercalary months are given in Table I. (see Art. 47 above). When it is not certain whether the given year is an expired or current year, whether it is a Chaitrâdi year or one of another kind, whether the given month is amânta or pûrṇimânta, and whether the intercalary month, if any, was taken true or mean, the only course is to try all possible years and months.

N.B. ii. The results are all Old Style dates up to, and New Style dates from, 1753 A.D The New Style was introduced with effect from after 2nd September, 1752. Since only the initial dates of 1752 and 1753 are given, remember to apply the correction (+ 11 days) to any date between 2nd September, 1752, and 9th April, 1753, in calculating by the Hindu solar year, and between 2nd September, 1752, and 4th April, 1753, in calculating by the Hindu luni-solar year, so as to bring out the result in New Style dates A.D. The day of the week requires no alteration.

N.B. iii. If the date A.D. found above falls after February 28th in a leap-year, it must be reduced by 1.

N.B. iv. The Hindus generally use expired (gata) years, while current years are given throughout the Tables. For example, for Śaka year 1702 "expired" 1703 current is given.

148. Example 1. Required the week-day and the A.D. year, month, and day corresponding to Jyeshṭha śukla paṅchâmî (5th), year Śârvari, Śaka year 1702 expired (1703 current), and the ending and beginning time of that tithi.

The given year is Chaitrâdi (see N.B. ii., Table II., Part iii.). It does not matter whether the month is amânta or pûrṇimânta, because the fortnight belongs to Jyeshṭha by both systems (see Table II., Part i.). Looking to Table I. along the given current Śaka year 1703, we find that its initial day falls in A.D. 1780 (see note 1 to Art. 139), a leap-year, on the 5th April, Wednesday; and that $d$ (col. 19), $w$ (col. 20), $a$ (col. 23), $b$ (col. 24) and $c$ (col. 25) are 96, 4, 1, 657 and 267 respectively. We write them in a horizontal line (see the working of the example below). From Table I., col. 8, we find that there is no added month in the year. The number therefore of tithis between Chaitra ś. 1 and Jyeshṭha ś. 5 was 64, viz., 60 up to the end of Vaiśâkha (see Table III., col. 3), the month preceding the given one, and 4 in Jyeshṭha. The sixtieth part of 64 (neglecting the fraction 4/60 because it is not more than half) is 1. Reduce 64 by one and we have 63 as the approximate number of days between Chaitra ś. 1 and Jyeshtha ś. 5. We write this number under ( $d$ ). Turning to Table IV. with the argument 63 we find under ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) the numbers 0, 1334, 286, 172, respectively, and we write them under their respective heads, and add together the two quantities under each head. With the argument ( $b$ ) (943) we turn to Table VI. for the equation. We do not find exactly the number 943 given, but we have 940 and 950 and must see the difference between the corresponding equation figures and fix the appropriate figure for 943. The auxiliary table given will fix this, but in practice it can be easily calculated in the head. (The full numbers are not given so as to avoid cumbrousness in the tables.) Thus the equation for ( $b$ ) (943) is found to be 90, and from Table VII. the equation for ( $c$ ) is found to be 38. Adding 90 and 38 to ( $a$ ) (1335) we get 1463, which is the required tithi-index ( $t$ ). Turning with this to Table VIII., col. 3, we find by col. 2 that the tithi current was śukla 5, i.e., the given date. Then ( $w$ ) 4, Wednesday, was its week-day; and the tithi was current at mean sunrise on the meridian of Ujjain on that week-day. Turning with ( $d$ ) 159 to Table IX., we find that the equivalent date A.D. was 8th June; but as this was after 28th February in a leap-year, we fix 7th June, A.D. 1780, (see N.B. iii.. Art. 147) as the equivalent of the given tithi. As ( $t$ ) is not within 40 of 1667, the ( $t$ ) of the 5th tithi (Table VIII.), there is no probability of an expunction or repetition shortly preceding or following (Art. 142). The answer therefore is Wednesday, June 7th, A.D. 1780.

To find the ending time of the tithi. ( $t$ ) at sunrise is 1463; and Table VIII., col. 3, shews that the tithi will end when ( $t$ ) amounts to 1667. (1667 − 1463 =) 204 = (Table X.) 14 hours, 27 minutes, and this process shews us that the tithi will end 14 hours, 27 minutes, after sunrise on Wednesday, June 7th. This time is, however, approximate. To find the time more accurately we add the increase in ( $a$ ) ( $b$ ) ( $c$ ) for 14 h. 27 m. (Table V.) to the already calculated ( $a$ ) ( $b$ ) ( $c$ ) at sunrise; and adding to ( $a$ ) as before the equations of ( $b$ ) and ( $c$ ) (Tables VI. and VII.) we find that the resulting ( $t$ ) amounts to 1686. 1686 − 1667 = 19 = 1 hour and 21 minutes (Table X.). But this is a period beyond the end of the tithi, and the amount must be deducted from the 14 h. 27 m. first found to get the true end. The true end then is 13 h. 6 m. after sunrise on June 7th. This time is accurate for ordinary purposes, but for still further accuracy we proceed again as before. We may either add the increase in ( $a$ ) ( $b$ ) ( $c$ ) for 13 h. 6 m. to the value of ( $a$ ) ( $b$ ) ( $c$ ) at sunrise, or subtract the increase of ( $a$ ) ( $b$ ) ( $c$ ) for 1 h. 21 m. from their value at 14 h. 27 m. By either process we obtain ( $t$ ) = 1665. Proceed again. 1667 − 1665 = 2 = (Table X.) 9 minutes after 13 h. 6 m. or 13 h. 15 m. Work through again for 13 h. 15 m. and we obtain ( $t$ ) = 1668. Proceed again. 1668 − 1667 = 1 = (Table X.) 4 minutes before 13 h. 15 m. or 13 h. 11 m. Work for 13 h. 11 m., and we at last have 1667, the known ending point. It is thus proved that 13 h. 11 m. after sunrise is the absolutely accurate mean ending time of the tithi in question by the Sûrya-Siddhânta.

To find the beginning time of the given tithi. We may find this independently by calculating as before the ( $t$ ) at sunrise for the preceding tithi, (in this case śukla 4th) and thence finding its ending time. But in the example given we calculate it from the ( $t$ ) of the given tithi. The tithi begins when ( $t$ ) amounts to 1333 (Table VIII.). or (1463 − 1333) 130 before sunrise on June 7th. 130 is (Table X.) 9 h. 13 m. Proceed as before, but deduct the ( $a$ ) ( $b$ ) ( $c$ ) instead of adding, and (see working below) we eventually find that ( $t$ ) amounts exactly to 1333 and therefore the tithi begins at 8 h. 26 m. before sunrise on June 7th, that is 15 h. 34 m. after sunrise on Tuesday the 6th. The beginning and ending times are by Ujjain or Laṅkâ mean time. If we want the time, for instance, for Benares the difference in longitude in time, 29 minutes, should be added to the above result (See Table XI.). This, however, does not affect the day.

It is often very necessary to know the moments of beginning and ending of a tithi. Thus our result brings out Wednesday, June 7th, but since the 5th tithi began 15 h. 34 m. after sunrise on Tuesday, i.e., about 9 h. 34 m. p.m.. it might well happen that an inscription might record a ceremony that took place at 10 p.m., and therefore fix the day as Tuesday the 5th tithi, which, unless the facts were known, would appear incorrect.

From Table XII. we find that 7th June, A.D. 1780, was a Wednesday, and this helps to fix that day as current.

We now give the working of Example 1.

Working of Example 1.

(a) The day corresponding to Jyeshṭha śukla 5th.

	d.	w.	a.	b.	c.
Śaka 1703 current, Chaitra śukla 1st, (Table I., cols. 19, 20, 23, 24, 25)	96	4	1	657	267
Approximate number of days from Chaitra śukla 1st to Jyeshṭha śuk. 5th, (64 tithis reduced by a 60th part, neglecting fractions, = 63,) with its ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) (Table IV.)	63	0	1334	286	172
	159	4	1335	943	439
Equation for ( $b$ ) (943) (Table VI.)			90
Equation forDo. ( $c$ ) (439) (Table VII.)			38
			1463	$=t$ .
( $t$ ) gives śukla 5th (Table VIII., cols. 2, 3) (the same as the given tithi).
$(d)-1$ , (N.B. iii., Art. 147), or the number of days elapsed from January 1st, =\|\|158

158 = June 7th (Table IX.). A.D. 1780 is the corresponding year, and 4 ( $w$ ) Wednesday is the week-day of the given tithi.

Answer.—Wednesday, June 7th, 1780 A.D.

(b) The ending of the tithi Jyeshṭha śuk. 5. (Table VIII.) 1667 − 1463 = 204 = (14 h. 10 m. + 0 h. 17 m.) = 14h. 27 m. (Table X.). Therefore the tithi ends at 14h. 27 m. after mean sunrise on Wednesday. For more accurate time we proceed as follows:

	a.	b.	c.
At sunrise on Wednesday (see above)	1335	943	439
For 14 hours (Table V.)	198	21	2
For 27 minutes, (Do.)	6	1	0
	1539	965	441
Equation for ( $b$ ) (965) (Table VI.)	109
Equation forDo. ( $c$ ) (441) (Table VII.)	38
	1686	$=t$ .

1686 − 1667 (Table VIII.) = 19 = 1 h. 21m.; and 1 h. 21 m. deducted from 14 h. 27 m. gives 13 h. 6 m. after sunrise on Wednesday as the moment when the tithi ended. This is sufficient for all practical purposes. For absolute accuracy we proceed again.

	a.	b.	c.
For sunrise (as before)	1335	943	439
For 13 hours (Table V.)	183	20	1
For 6 minutes (Do.)	1	0	0
	1519	963	440
Equation for ( $b$ ) (963) (Table VI.)	108
Equation forDo. ( $c$ ) (440) (Table VII.)	38
	1665	$=t$ .

1667 − 1665 = 2 = 9 m. after 13 h. 6 m. = 13 h. 15 m.

	a.	b.	c.
Again for sunrise (as before)	1335	943	439
For 13 hours (Table V.)	183	20	1
For 15 minutes (Do.)	4	0	0
	1522	963	440
Equation for ( $b$ ) (963) (Table VI.)	108
Equation forDo. ( $c$ ) (440) (Table VII.)	38
	1668	$=t$ .

1668 − 1667 = 1 = 4 m. before 13 h. 15 m. = 13 h. 11 m.

	a.	b.	c.
Again for sunrise (as before)	1335	943	439
For 13 hours (Table V.)	183	20	1
For 11 minutes (Do.)	3	0	0
	1521	963	440
Equation for ( $b$ ) (963) (Table VI.)	108
Equation forDo. ( $c$ ) (440) (Table VII.)	38
Actual end of the tithi	1667	$=t$ .

Thus 13 h. 11 m. after sunrise is the absolutely accurate ending time of the tithi.

(c) The beginning of the tithi, Jyeshṭha śuk. 5. Now for the beginning. 1463 (the original $t$ . as found) − 1333 (beginning of the tithi, (Table VIII.) = 130 = (Table X.) (7 h. 5 m. + 2 h. 8 m.) = 9 h. 13 m.; and we have this as the point of time before sunrise on Wednesday when the tithi begins.

				a.	b.	c.
For sunrise (as before)				1335	943	439
	a.	b.	c.
For 9 hours (Table V.)	127	14	1
For 13 minutes (Do.)	3	0	0
Deduct	130	14	1	130	14	1
				1205	929	438
Equation for ( $b$ ) (929) (Table VI.)				79
Equation forDo. ( $c$ ) (438) (Table VII.)				37
				1667	$=t$ .

(The beginning of the tithi) 1333 − 1321 = 12 = Table X.) 51 m. after the above time (9 h 13 m.), and this gives 8 h. 22 m. before sunrise. We proceed again.

	a.	b.	c.
For 9 h. 13 m. before sunrise (found above)	1205	929	438
Plus for 51 minutes (Table V.)	12	1	0
	1217	930	438
Equation for ( $b$ ) (930) (Table VI.)	80
Equation forDo. ( $c$ ) (438) (Table VII.)	37
	1334	$=t$ .

1334 − 1333 = 1 = 4m. before the above time (viz., 8 h. 22 m.) i.e., 8h. 26 m. before sunrise. Proceed again.

	a.	b.	c.
For 8 h. 22 m. before sunrise (found above )	1335	943	439
Deduct for 4 m. (Table V.)	1	0
	1216	930	438
Equation for ( $b$ ) (930) (Table VI.)	80
Equation forDo. ( $c$ ) (438) (Table VII.)	37
	1333	$=t$ .

The result is precisely the same as the beginning point of the tithi (Table VIII.), and we know that the tithi actually began 8 hours 26 minutes before sunrise on Wednesday, or at 15 h. 34 m. after sunrise on Tuesday, 6th June.

Example ii. Required the week-day and equivalent A.D. of Jyeshṭha śuk. dasamî (10th) of the southern Vikrama year 1836 expired, 1837 current. The given year is not Chaitrâdi. Referring to Table II., Parts ii., and iii., we find, by comparing the non-Chaitradi Vikrama year with the Śaka, that the corresponding Śaka year is 1703 current, that is the same as in the first example. We know that the months are amânta.

	d.	w.	a.	b.	c.
State the figures for the initial day (Table I., cols. 19, 20, 23, 24, 25)	96	4	1	657	267
The number of intervened tithis down to end of Vaiśâkha, 60, (Table III.) + the number of the given date minus 1, is 69; reduced by a 60th part = 68, and by Table IV. we have	68	5	3027	468	186
	164	2	3028	125	453
Equation for ( $b$ ) 125 (Table VI.)			90
Equation forDo. ( $c$ ) 453 (Table VII.)			38
			1463	$=t$ .

$(d)(164)-1$ (N.B. iii., Art. 147) = 163.

The result, 3309, fixes the day as sukla 10th (Table VIII., cols. 2, 3), the same as given.

Answer.—(By Table IX.) 163 = June 12th, 2 = Monday. The year is A.D. 1780 (Table II., Part ii.). The tithi will end at (3333 − 3309 = 24, or by Table X.) 1 h. 42 m. after sunrise, since 3309 represents the state of that tithi at sunrise, and it then had 24 lunation-parts to run. Note that this ( $t$ ) (3309) is less by 24 than 3333, the ending point of the 10th tithi; that 24 is less than 40; and that the equation for ( $b$ ) is increasing. This shows that an expunction of a tithi will shortly occur (Art. 142.)

Example iii. Required the week-day and equivalent A.D. of Jyeshtha śukla ekâdaśî (11th) of the same Śaka year as in example 2, i.e., Ś. 1703 current.

	d.	w.	a.	b.	c.
See (Table I.) example 2	96	4	1	657	267
Intervened days (to end of Vaiśâkha 59, + 11 given days − 1) = 69. By Table IV.	69	6	3366	504	189
	165	3	3367	161	456
Equation for ( $b$ ) (161) (Table VI.)			258
Equation forDo. ( $c$ ) (456) (Table VII.)			43
			3668 $=t$ .

This figure ( $t=3668$ ) by Table VIII., cols. 2, 3, indicates śukla 12th.

$d-1$ (N.B. iii., Art. 147) = 164 and Table IX. gives this as June 13th. The ( $w$ ) is 3 = Tuesday. The year (Table II. Part iii.) is 1780 A.D.

The figure of ( $t$ ), 3668, shows that the 12th tithi and not the required tithi (11th) was current at sunrise on Tuesday; but we found in example 2 that the 10th tithi was current at sunrise on Monday, June 12th, and we therefore learn that the 11th tithi was expunged. It commenced 1 h. 42 min. after sunrise on Monday and ended 4 minutes before sunrise on Tuesday, 13th June.^[138] The corresponding day answering to śukla 10th is therefore Monday, June 12th, and that answering to śukla 12 is Tuesday the 13th June.

Example iv. Required the week-day and equivalent A.D. of the pûrṇimânta Âshâḍha kṛishṇa dvitîyâ (2) of the Northern Vikrama year 1837 expired, 1838 current. The northern Vikrama is a Chaitrâdi year, and so the year is the same as in the previous example, viz., A.D. 1780—1 (Table II., Part iii.). The corresponding amânta month is Jyeshṭha (Table II., Part i.). Work therefore for Jyeshṭha krishna 2nd in A.D. 1780—1 (Table I.).

	d.	w.	a.	b.	c.
See example I (Table I.)	96	4	1	657	267
60 (coll. dur. to end Vaiś.) + 15 (for kṛishṇa fortnight) + 1 (given date minus 1) = 76 tithis = 75 days (as before); Table IV. gives	75	5	5397	722	205
	171	2	5398	379	472
Equation for ( $b$ ) (379) (Table VI.)			237
Equation forDo. ( $c$ ) (472) (Table VII.)			50
			5685 $=t$ .

$(d)-1$ (N.B. iii., Art. 147) = 170 = (Table IX.) 19th June. (2) = Monday. The year is 1780 A.D.

So far we have Monday, 19th June, A.D. 1780. But the figure 5685 for ( $t$ ) shows that kṛi. 3rd and not the 2nd was current at sunrise on Monday the 19th June. It commenced (5685 − 5667 = 18 =) 1 h. 17 m. before sunrise on Monday. ( $t$ ) being greater, but within 40, than the ending point of kṛi. 2nd, and the equation for ( $b$ ) decreasing, it appears that a repetition of a tithi will shortly follow (but not precede). And thus we know that Sunday the 18th June is the equivalent of kṛi. 2nd.

Example v. Required the week-day and equivalent A.D. of the amânta Jyeshṭha kṛi. 3rd of the Śaka year 1703 current, the same as in the last 4 examples.

	d.	w.	a.	b.	c.
(See example 1)	96	4	1	657	267
60 (coll. dur. to end Vaiś.) + 15 + 2 = 77 tithis = 76 days. (Table IV.)	76	6	5736	758	208
	172	3	5737	415	475
Equation for ( $b$ ) (415) (Table VI.)			211
Equation forDo. ( $c$ ) (475) (Table VII.)			51
			5999

This indicates kṛishṇa 3rd, the same tithi as given, $(d)-1=171=$ 20th June, 1780 A.D.

From these last two examples we learn that kṛishṇa 3rd stands at sunrise on Tuesday 20th as well as Monday 19th. It is therefore a repeated or vṛiddhi tithi, and both days 19th and 20th correspond to it. It ends on Tuesday (6000 − 5999 = 1 =) 4 minutes after sunrise.

Example vi. Required the week-day and A.D. equivalent of Kârṭṭika śukla 5th of the Northern Vikrama year 1833 expired (1834 current). (See example 2, page 70.)

The given year is Chaitrâdi. It matters not whether the month is amânta or pûrṇimânta because the given tithi is in the śukla fortnight. The initial day of the given year falls on (Table I., col. 19) 20th March (80), (col. 20) 4 Wednesday; and looking in Table I. along the line of the given year, we find in col. 8 that the month Bhâdrapada was intercalated or added (adhika) in it. So the number of months which intervened between the beginning of the year and the given tithi was 8, one more than in ordinary year.

	d.	w.	a.	b.	c.
(Table I., cols. 19, 20, 23, 24, 25)	80	4	9841	54	223
(Coll. dur.) 240 + 4 = 244 = 240 days. (Table IV.)	240	2	1272	710	657
	320	6	1113	764	880
Equation for ( $b$ ) (764) (Table VI.)			0
Equation forDo. ( $c$ ) (880) (Table VII.)			102
			1215	$=t$ .
This indicates, not kṛi. 5 as given, but kṛi. 4 (Table VIII.)
Adding 1 to ( $d$ ) and ( $w$ ) (see Rule above. Art. 139)	321	0

$a-1$ (N.B. iii., Art. 147) 320 = (Table IX.) Nov. 16th, A.D. 1776. 0 = Saturday.

( $t$ ) being not within 40 of the ending point of the tithi there is no probability of a repetition or expunction shortly preceding or following, and therefore Saturday the 16th November, 1776 A.D., is the equivalent of the given tithi.

Example vii. Required the week-day and A.D. equivalent of amânta Mâgha kṛishṇa 1st of Kali 4923 expired, 4924 current. (See example 4, page 71.)

The given year is Chaitrâdi. Looking in Table I. along the line of the given year, we see that its initial day falls on 24th March (83), 1822 A.D., 1 Sunday, and that (col. 8) the month (7) Âśvina was intercalated and (10) Pausha expunged. So that, in counting, the number of intervened months is the same, viz., 10, as in an ordinary year, Mâgha coming after Pausha.

	d.	w.	a.	b.	c.
(Table I., cols. 19, 20, 23, 24, 23)	83	1	212	899	229
(Coll. dur.) 310 + 15 (śukla paksha) + (1 − 1 =) 0 = 315 tithis = 310 days. (Table IV.)	310	2	4976	250	849
	393	3	5188	149	78
Equation for ( $b$ ) (149) (Table VI.)			252
Equation forDo. ( $c$ ) (78) (Table VII.)			32
			5472	$=t$ .

The figure 5472 indicates (Table VIII.) kṛi. 2nd, i.e., not the same as given (1st), but the tithi following. We therefore subtract 1 from ( $d$ ) and ( $w$ ) (Art. 139) making them 392 and 2.

Since ( $t$ ) is not within 40 of the ending point of the tithi, there is no probability of a kshaya or vṛiddhi shortly following or preceding, ( $w$ ) 2 = Monday. 392 = (Table IX.) 27th January. And therefore 27th January, A.D. 1823, Monday, is the equivalent of the given tithi.

Example viii. Required the week-day and the A.D. equivalent of śukla 13th of the Tuḷu month Puntelu, Kali year 4853 expired, 4854 current, "Aṅgiras samvatsara" in the luni-solar or southern 60-year cycle. (See example 5, page 72.)

The initial day (Table I.) is Old Style 5th March (65), A.D. 1752, a leap-year, (5) Thursday; and Âshâḍha was intercalated. The Tuḷu month Puntelu corresponds to the Sanskrit Pausha (Table II., Part ii.), ordinarily the 10th, but now the 11th, month on account of the intercalated Âshâḍha.

	d.	w.	a.	b.	c.
(Table I., cols. 19, 20, 23, 24, 25)	65	5	39	777	213
(Coll. dur.) 300 + 12 (given tithi minus 1) = 312 tithis = 307 days. (Table IV.)	307	6	3960	142	840
	372	4	3999	919	53
Equation for ( $b$ ) (919)			71
Equation forDo. ( $c$ ) (53)			40
			4110	$=t$ .

The result, 4110, indicates śukla 13th, i.e., the same tithi as that given.

$(d)-1$ (N.B. iii., Art. 147) = 371 = (by Table IX.) January 6th, A.D. 1753.

We must add 11 days to this to make it a New Style date, because it falls after September 2nd, 1752, and before 4th April, 1753, the week-day remaining unaltered (see N.B. ii., Art. 147), and 17th January, 1753 A.D., is therefore the equivalent of the given date.

(B.) Conversion of Hindu solar dates into dates A.D.

149. To calculate the week-day and the equivalent date A.D. Turn the given year into a Meshâdi Kali, Śaka, or Vikrama year, and the name of the given month into a sign-name, if they are not already given as such, and find the corresponding year A.D. by the aid of columns 1 to 5, Table I., and Table II., Parts ii., and iii. Looking in Table I. along the line of the Meshâdi year so obtained, write down in a horizontal line the following three quantities corresponding to the commencement of that (Meshadi) year, viz., ( $d$ ) the date-indicator given in brackets after the day and month A.D. in col. 13, ( $w$ ) the week-day number (col. 14), and the time—either in ghaṭikâs and palas, or in hours and minutes as desired—of the Mesha saṅkrânti according to the Ârya-Siddhânta (cols. 15, or 17). For a Bengali date falling between A.D. 1100 and 1900, take the time by the Sûrya-Siddhânta from cols. 15a or 17a. When the result is wanted for a place not on the meridian of Ujjain, apply to the Mesha saṅkrânti time the correction given in Table XI. Under these items write from Table 111., cols. 6, 7, 8, or 9 as the case may be, the collective duration of time from the beginning of the year up to the end of the month preceding the given one—days under ( $d$ ), week-day under ( $w$ ), and hours and minutes or ghaṭikâs and palas under h. m., or gh. p. respectively. Add together the three quantities. If the sum of hours exceeds 24, or if the sum of ghaṭikâs exceeds 60, write down the remainder only, and add one each to ( $w$ ) and ( $d$ ). If the sum of ( $w$ ) exceeds 7, cast out sevens from it. The result is the time of the astronomical beginning of the current (given) month. Determine its civil beginning by the rules given in Art. 28 above.

When the month begins civilly on the same day as, on the day following, or on the third day after, the saṅkrânti day, subtract 1 from, or add 0, or1, to both ( $d$ ) and ( $w$ ), and then to each of them add the number of the given day, casting out sevens from it in the case of ( $w$ ). ( $w$ ) is then the required week-day, and ( $d$ ) will show, by Table IX., the A.D. equivalent of the given day.

N.B. i. When it is not certain whether the given year is Meshâdi or of another kind, or what rule for the civil beginning of the month applies, all possible ways must be tried.

N.B. ii. See N.B. ii., iii., iv., Art. 147, under the rules for the conversion of luni-solar dates.

Example ix. Required the week-day and the date A.D. corresponding to (Tamil) 18th Puraṭṭâśi of Rudhirodgârin, Kali year 4904 expired, (4905 current). (See example 7, p. 73.)

The given year, taken as a solar year, is Meshâdi. The month Puraṭṭâdi, or Puraṭṭâśi, corresponds to Kanyâ (Table II., Part ii. ), and the year is a Tamil (Southern) one, to which the Ârya Siddhânta is applicable (see Art. 21). Looking in Table I. along the line of the given year, we find that it commenced on 11th April (col. 13), A.D. 1803, and we write as follows:—

	d.	w.	h.	m.
(Table I., cols. 13, 14, 17)	101	2	10	7
(Table III., col. 7) collective duration up to the end of Siṁha	156	2	10	28
	257	4	20	35
This shows that the Kanyâ saṅkrânti took place on a (4) Wednesday, at 20 h. 35 m. after sunrise, or 2.35 a.m. on the European Thursday. (Always remember that the Hindu week-day begins at sunrise.) The month Kanyâ, therefore, begins civilly on Thursday.^[139] (Rule 2(a), Art. 28.) We add, therefore 0 to ( $d$ ) and ( $w$ )	0	0
Add 18, the serial number of the given day, to ( $d$ ) and, casting out sevens from the same figure, 18, add 4 to ( $w$ )	18	4
	275	1

Then $(w)=1$ , i.e., Sunday, and 275 = (Table IX.) 2nd October.

Answer.—Sunday, 2nd October, 1803 A.D.

Example x. Required the week-day and A.D. date corresponding to the 20th day of the Bengali (solar) month Phâlguna of Śaka 1776 expired, 1777 current, at Calcutta.

The year is Meshâdi and from Bengal, to which the Sûrya Siddhânta applies (see Art. 21). The Bengâli month Phâlguna corresponds to Kumbha (Table II., Part ii.). The year commenced on 11th April, 1854, A.D. (Table I.)

	d.	w.	h.	m.
(Table I., cols. 13, 14, 17a)	101	3	17	13
Difference of longitude for Calcutta (Table XI.)				+ 50
Collective duration up to the end of Makara (Table III., col. 9.)	305	422	2	2
	406	0	20	5
This result represents the moment of the astronomical beginning of Kumbha, which is after midnight on Saturday, for 20 h. 5 m. after sunrise is 2.5 a.m. on the European Sunday morning. The month, therefore, begins civilly on Monday (Art. 28, Rule 1 above). Add, therefore, 1 to ( $d$ ) and ( $w$ )	1	1
Add 20 (given day) to ( $d$ ), and, casting out sevens from 20, add 6 to ( $w$ )	20	6
0 = Saturday, 427 = 3rd March (Table IX.)	427	0

Answer.—Saturday, 3rd March, A.D. 1855.

Example xi. Required the week-day and A.D. date corresponding to the Tinnevelly Âṇḍu 1024, 20th day of Âvaṇi. (See example 8, p. 73.)

The year is South Indian. It is not Meshâdi, but Siṁhâdi. Its corresponding Śaka year is 1771 current; and the sign-name of the month corresponding to Âvaṇi is Siṁha (Table I., and Table II., Parts ii., and iii.) The Śaka year 1771 commenced on 11th April (102), A.D. 1848 (a leap-year), on (3) Tuesday. Work by the Ârya-Siddhânta (Art. 21).

	d.	w.	h.	m.
(Table I., cols. 13, 14, 17)	102	3	1	30
Collective duration up to the end of Karka	125	6	9	38
	227	2	11	8
The month begins civilly on the same day by one of the South Indian systems (Art. 28, Rule 2, a); therefore subtract 1 from both ( $d$ ) and ( $w$ )	1	1
	226	1
Add 20, the serial number of the given day, to ( $d$ ) and (less sevens) to ( $w$ )	20	6
	246	0
Deduct 1 for 29th February (N.B. ii., Art. 149 and N.B. iii., Art. 147)	1
	245

0 = Saturday. 245 = (Table IX.) Sept. 2nd.

Answer.—Saturday, September 2nd, 1848 A.D.

Example xii. Required the week-day and A.D. date corresponding to the South Malayâḷam Âṇḍu 1024, 19th Chiṅgam. (The calculations in Example xi. shew that the South-Malayâḷam month Chiṅgam began civilly one day later (Art. 28, Rule 2b). Therefore the Tamil 20th Âvaṇi was the 19th South-Malayâḷam.)

Referring to Table II., Part ii., we see that the date is the same as in the last example.

Example xiii. Required the week-day and A.D. date corresponding to the North Malayâḷam Âṇḍu 1023, 20th Chiṅgam.

Referring to Table II., Part ii., we see that the date is the same as in the last two examples.

(c.) Conversion into dates A.D. of tithis which are coupled with solar months.

150. Many inscriptions have been discovered containing dates, in expressing which a tithi has been coupled, not with a lunar, but with a solar month. We therefore find it necessary to give rules for the conversion of such dates.

Parts of two lunar months corresponding to each solar month are noted in Table II., Part ii., col. 14. Determine by Art. 119, or in doubtful cases by direct calculation made under Arts. 149 and 151, to which of these two months the given tithi of the given fortnight belongs, and then proceed according to the rules given in Art. 139.

It sometimes happens that the same solar month contains the given tithi of both the lunar months noted in Table II., Part ii., col. 14, one occurring at the beginning of it and the other at the end. Thus, suppose that in a certain year the solar month Mesha commenced on the luni-solar tithi Chaitra śukla ashṭamî (8th) and ended on Vaiśâkha śukla daśamî (10th). In this case the tithi śukla navamî (9th) of both the lunar months Chaitra and Vaiśâkha fell in the same solar month Mesha. In such a case the exact corresponding lunar month cannot be determined unless the vâra (week-day), nakshatra, or yoga is given, as well as the tithi. If it is given, examine the date for both months, and after ascertaining when the given details agree with the given tithi, determine the date accordingly.

Example xiv. Required the A.D. year, month, and day corresponding to a date given as follows;—"Śaka 1187, on the day of the nakshatra Rohiṇî, which fell on Saturday the thirteenth tithi of the second fortnight in the month of Mithuna."^[140]

It is not stated whether the Śaka year is expired or current. We will therefore try it first as expired. The current year therefore is 1188. Turning to Table I. we find that its initial day, Chaitra śukla 1st, falls on 20th March (79), Friday (6), A.D. 1265. From Table II., Part ii., col. 14, we find that parts of the lunar months Jyeshṭha and Âshâḍha correspond to the solar month Mithuna. The Mesha saṅkrânti in that year falls on (Table I., col. 13) 25th March, Wednesday, that is on or about Chaitra śukla shashṭhî (6th), and therefore the Mithuna saṅkrânti falls on (about) Jyeshṭha śukla daśamî (10th) and the Karka saṅkrânti on (about) Âshâḍha sukla dvâdaśî (12th) (see Art. 119). Thus we see that the thirteenth tithi of the second fortnight falling in the solar month of Mithuna of the given date must belong to amânta Jyeshṭha.

	d.	w.	a.	b.	c.
S. 1188, Chaitra ś. 1st (Table I., cols. 19, 20, 23, 24, 25)	79	6	288	879	265
Approximate number of days from Ch. ś. 1st to Jyesh. kṛi. 13th (87 tithis reduced by 60th part = 86) with its ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) (Table IV.)	86	2	9122	121	235
	165	1	9409	0	500
Equation for ( $b$ ) (0) (Table VI.)			140
Equation forDo. ( $c$ ) (500) TableVII.)			60
			9609	$=t$ .
The resulting number 9609 fixes the tithi as kṛishṇa 14th (Table VIII., cols. 2, 3), i.e., the tithi immediately following the given tithi. There is no probability of a kshaya or vṛiddhi shortly before or after this (Art 142). Deduct, therefore, 1 from ( $d$ ) and ( $w$ )	1	1
	164	0

164 = (Table IX.) 13th June; 0 = Saturday.

Answer.—13th June, 1265 A.D., Saturday, (as required).^[141]

(d.) Conversion of dates A.D.^[142] into Hindu luni-solar dates.

151. Given a year, month, and date A.D., write down in a horizontal line ( $w$ ) the weekday number, and ( $a$ ), ( $b$ ), ( $c$ ) (Table I., cols. 20, 23, 24. 25) of the initial day (Chaitra ś. 1) of the Hindu Chaitrâdi (Śaka) year corresponding to the given year; remembering that if the given date A.D. is earlier than such initial day, the ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) of the previous Hindu year^[143] must be taken. Subtract the date-indicator of the initial date (in brackets. Table I., col. 19) from the date number of the given date (Table IX.), remembering that, if the initial day of the previous Hindu year has been taken, the number to be taken from Table IX. is that on the right-hand side, and not that on the left (see also N.B. ii. below). The remainder is the number of days which have intervened between the beginning of the Hindu year and the required date. Write down, under their respective heads, the ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) of the number of intervening days from Table IV., and add them together as before (see rules for conversion of luni-solar dates into dates A.D.). Add to ( $a$ ) the equation for ( $b$ ) and ( $c$ ) (Tables VI., VII.) and the sum ( $t$ ) will indicate the tithi (Table VIII.) at sunrise of the given day; ( $w$ ) is its week-day. To the number of intervening days add its sixtieth^[144] part. See the number of tithis next lower than this total^[145] (Table III., col. 3) and the lunar month along the same line (col. 2). Then this month is the month preceding the required month, and the following month is the required month.

When there is an added month in the year, as shown along the line in col. 8 or 8a of Table I., if it comes prior to the resulting month, the month next preceding the resulting month is the required month; if the added month is the same as the resulting month, the date belongs to that added month itself; and if the resulting month comes earlier than the added month, the result is not affected.

When there is a suppressed month in the year, if it is the same as, or prior to, the resulting month, the month next following the resulting month is the required month. If it is subsequent to the resulting month the result is not affected. If the resulting month falls after both an added and suppressed month the result is unaffected.

From the date in a Chaitrâdi year thus found, any other Hindu year corresponding to it can be found, if required, by reference to Table II., Parts ii., and iii.

The tithi thus found is the tithi corresponding to the given date A.D.; but sometimes a tithi which is current at any moment of an A.D. date may be said to be its corresponding tithi.

N.B. i. See N.B. ii.. Art. 147; but for "+ 11" read "− 11".

N.B. ii. If the given A.D. date falls in a leap-year after 29th February, or if its date-number is more than 365 (taken from the right-hand side of Table IX.) and the year next preceding it was a leap-year, add i to the date-number before subtracting the date-indicator from it.

Example xv. Required the tithi and month in the Saka year corresponding to 7th June, 1780 A.D.

The Śaka year corresponding to the given date is 1703 current. Its initial day falls on (4) Wednesday, 5th April, the date-indicator being 96.

			w.	a.	b.	c.
(Table I., cols. 20, 23, 24, 25)			4	1	657	267
7th June =	158	(Table IX.)
Add	+1	for leap-year (N.B. ii.)
	159
Deduct	96	the ( $d$ ) of the initial date (Table I., col. 19).
Days that have intervened	63.	By Table IV. 63 =	0	1334	286	172
			4	1335	943	439
Equation for ( $b$ ) (943) (Table VI.)				90
Equation forDo. ( $c$ ) (439) (Table VII.)				38
			4	1463	$=t$ .

Śukla 5th (Table VIII.) is the required tithi, and (4) Wednesday is the week-day. Now 63 + 63/60 = 64 3/60. The next lowest number in col. 3, Table III., is 60, which shows Vaiśâkha to be the preceding month. Jyeshṭha is therefore the required month.

Answer.—Śaka 1703 current, Jyeshṭha śukla 5th, Wednesday.

If the exact beginning or ending time of the tithi is required, proceed as in example 1 above (Art. 148.)

We have seen in example 1 above (Art. 148) that this Jyeshṭha 5th ended, and śukla 6th commenced, at 13 h. 11 m. after sunrise on the given date; and after that hour śukla 6th corresponded with the given date. Śukla 6th therefore may be sometimes said to correspond to the given date as well as śukla 5th.

Example xvi.—Required the tithi and month in the southern Vikrama year corresponding to 12th September, 1776 A.D.

The Śaka year corresponding to the given date is 1699 current. Its initial date falls on 20th March (80), 4 Wednesday, A.D. 1776. Bhâdrapada was intercalated in that year.

			w.	a.	b.	c.
(Table I., cols. 20, 23, 24, 25)			4	9841	54	223
12 September =	255	(Table IX.)
Add	1	for leap-year (N.B. ii.)
	256
Deduct	80	the ( $d$ ) of the initial day.
Days that have intervened	176.	= (Table IV.)	1	9599	387	482
			5	9440	441	705
Equation for ( $b$ ) (441) (Table VI.)				191
Equation forDo. ( $c$ ) (705) (Table VII.)				118
			5	9749	$=t$ .

This indicates (Table VIII.) kṛishṇa 30th (amâvâsyâ, or new moon day), Thursday.

The intervening tithis are 176 + 176/60 = 179. The number next below this in col. 3, Table III., is 150, and shows that Śravaṇa preceded the required month. But Bhâdrapada was intercalated this year and it immediately followed Śravaṇa. Therefore the resulting tithi belongs to the intercalated or adhika Bhâdrapada.

Answer.—Adhika Bhâdrapada kṛi: 30th of Śaka 1699 current, that is adhika Bhâdrapada kṛi. 30th of the Southern Vikrama Kârttikâdi year 1833 current, 1832 expired. (Table II., Part ii.).

Example xvii. Required the Telugu and Tuḷu equivalents of December 1st, 1822 A.D. The corresponding Telugu or Tuḷu Chaitrâdi Śaka year is 1745 current. Âśvina was intercalary and Pausha was expunged (col. 8, Table I.). Its initial date falls on 24 March (83), A.D. 1822, (1) Sunday.

			w.	a.	b.	c.
(Table I., cols. 20, 23, 24, 25)			1	212	899	229
1st December =	335	(Table IX.)
Deduct	83	(The $d$ . of the initial day)
Days that have intervened	252.	= (Table IV.)	0	5335	145	690
			1	5547	44	919
Equation for ( $b$ ) (44) (Table VI.)				180
Equation forDo. ( $c$ ) (919) (Do. VII.)				90
The results give us kṛishṇa 3, Sunday (1), (Table VIII.)			1	5817	$=t$ .

252 + 252/60 = 256. The number next below 256 in col. 3, Table III., is 240. and shews that Kârttika preceded the required month, and the required month would therefore be Mârgaśîrsha. But Âśvina, which is prior to Mârgaśîrsha, was intercalated. Kârttika therefore is the required month. Pausha was expunged, but being later than Kârttika the result is not affected.

Answer.—Sunday, Kârttika (Telugu), or Jârde (Tuḷu) (Table II., Part ii.), kṛ. 3rd of the year Chitrabhânu, Śaka 1745 (1744 expired). Kali year 4923 expired.

Example xviii. Required the tithi and pûrṇimânta month in the Śaka year corresponding to 18th January, 1541 A.D.

The given date is prior to Chaitra śukla 1 in the given year. We take therefore the initial day in the previous year, A.D. 1540, which falls on Tuesday the 9th March (69). The corresponding Saka year is 1463 current.

			w.	a.	b.	c.
(Table I., cols. 20, 23, 24, 25)			3	108	756	229
18th January =	383	(Table IX.)
Add	1	for leap-year (N.B. ii., latter part.)
	384
Deduct	69	(The $d$ . of the initial day.)
No. of intervening days	315.	= (by Table IV.)	0	6669	432	862
			3	6777	188	91
Equation for ( $b$ ) (188) (Table VI.)				269
Equation forDo. ( $c$ ) (91) (Do. VII.)				28
			5	9749	$=t$ .

The result gives us kṛishṇa 7th, Tuesday (3) (Table VIII.).

315 + 315/60 = 320 tithis. The next lower number to 320 in col. 3, Table III., is 300, which shews Pausha as preceding the required month, and the required month would therefore be Mâgha. Âśvina, however, which is prior to Mâgha, was intercalary in this year; Pausha, therefore, would be the required month; but it was expunged; Mâgha, therefore, becomes again the required month. Adhika Âśvina and kshaya Pausha being both prior to Mâgha, they do not affect the result. By Table II. amânta Mâgha kṛishṇa is pûrṇimânta Phâlguna kṛishṇa. Therefore pûrṇimânta Phâlguna kṛishṇa 7th, Tuesday, Śaka 1463 current, is the required date.

(e.) Conversion of A.D. dates into Hindu solar dates.

152. Given a year, month, and date A.D., write down from Table I. in a horizontal line the ( $d$ ) ( $w$ ) and ( $h$ ) ( $m$ ) (the time) of the Mesha saṅkrânti, by the Ârya or Sûrya-Siddhânta^[146] as the case may require, of the Hindu Meshâdi year, remembering that if the given day A.D. is earlier than the Mesha saṅkranti day in that year the previous^[147] Hindu year must be taken. Subtract the date-indicator of the Mesha saṅkrânti day from the date-number of the given date (Table IX.), remembering that if the Mesha saṅkrânti time of the previous Hindu year is taken the number to be taken from Table IX. is that on the right-hand side, and not that on the left (see also Art. 151, N.B. ii.); the remainder is the number of days which intervened between the Mesha saṅkrânti and the given day. Find from Table III., cols. 6, 7, 8 or 9, as the case may be, the number next below that number of intervening days. Write its three quantities ( $d$ ), ( $w$ ), and the time of the saṅkrânti (h. m.), under their respective heads, and add together the three quantities separately (See Art. 149 above). The sum is the time of the astronomical beginning of the required month, and the month next following that given in col. 5, on the line of the next lowest number, is the month required.

Ascertain the day of the civil beginning of the current required month by the rules in Art. 28. When it falls on the same day as the saṅkrânti day, or the following, or the third day, respectively, subtract 1 from, or add 0 or 1 to, both ( $d$ ) and ( $w$ ). Subtract ( $d$ ) from the date-number of the given date. The remainder is the required Hindu day. Add that remainder, casting out sevens from it, to ( $w$ ). The sum is the week-day required.

From the Meshâdi year and the sign-name of the month thus found, any other corresponding Hindu year can be found by reference to Table III., Parts ii., and iii.

Observe the cautions contained in N.B. i. and ii. to Art. 151.

Example xix. Required the Tamil, Tinnevelly, and South and North Malayâḷam equivalents of 30th May, 1803 A.D. (See example 14, p. 76.)

The corresponding Meshâdi Śaka year current is 1726. Its Mesha saṅkrânti falls on April 11th (101), 2 Monday. The Ârya Siddhânta applies. (See Art. 21.)

			d.	w.	h.	m.
(Table I., cols. 13, 14, 17)			101	2	10	7
May 30th =	150	(Table IX.)
Deduct	101,	the ( $d$ ) of the initial day.
Intervening days	49
The number next below 49, (Table III., col. 7), for the end of Mesha and beginning of Vṛishabha, is 30, and we have			30	2	22	12
[Total of hours = 32. 1 day of 24 hours carried over to ( $d$ ) and ( $w$ ).] Astronomical beginning of Vṛishabha			132	5	8	19
By all South Indian reckonings, except that in the South Malayâḷam country, the month begins civilly on the same day as the saṅkrânti. Subtract, therefore, 1 from ( $d$ ) and ( $w$ )			1	1
			131	4
Subtract 131 ( $d$ ) from the number of the given date			150
Remainder, 19, is the required date in the month of Vrishabha.			19
Add 19, casting out sevens, to ( $w$ )				5
Required week-day				2

Answer.—Monday, 19th day of the month Vṛishabha, Tamil Vaigâśi, of Śaka 1726 current (1725 expired); Kali 4904 expired (Table I., or Table II., Part iii.); Tinnevelly Âṇḍu 978, Vaigâśi 19th; North Malayâḷam Âṇḍu 978, Eḍavam 19th.

The Vrishabha saṅkrânti took place 8 h. 19 m. after sunrise, viz., not within the first 3/5ths of the day. Therefore by the South Malayâḷam system the month Vṛishabha began civilly, not on (5) Thursday, but on the following day (6) Friday. Therefore we have to add or subtract nothing from 132 and 5. Subtracting 132 from 150, the remainder, 18th, is the required day. Adding (18 ÷ 7) to 5 ( $w$ ) we get (2) Monday as the required week-day. Therefore Monday 18th of Eḍavam, Kollam Âṇḍu 978, is the required South Malayâḷam equivalent.

Example xx. Required the week-day and Bengali date at Calcutta corresponding to March 3rd, 1855 A.D. The Sûrya-Siddhânta is the authority in Bengal. The given day is earlier than the Mesha saṅkrânti in the year given. We must take therefore as our starting-point the Mesha saṅkrânti of the previous year, which falls on 11th April (101), Tuesday, (3) Śaka 1777 current, A.D. 1854.

			d.	w.	h.	m.
(Table I., cols. 13, 14, 17a)			101	3	17	13
Difference of longitude for Calcultta						+ 50
March 3rd, 1855 =	427	(Table IX.)
Deduct ( $d$ ) of the initial day	101
Intervening days	326
The number next below 326, (Table III., col. 9), for the end of Makara and beginning of Kumbha is			305	4	2	2
The astronomical beginning of Kumbha, after midnight on Saturday =			406	0	20	5
The civil beginning falls on the third day, Monday (Art. 28). We add therefore 1 to ( $d$ ) and ( $w$ )			1	1
The last civil day of Makara =			407	1
Subtract ( $d$ ) 407 from date number of 3rd March			427
Remainder 20, and the required date is 20th Kumbha.			20
Add 20 to ( $w$ ) casting out sevens				6
The required week-day is Saturday				0

The Bengali month corresponding to Kumbha is Phâlguna (Table II., Part ii.).

Answer.— The 20th day of Phâlguna, Saturday, Śaka, 1776 expired. (See example x above.)

Example xxi. Required the South Indian solar dates equivalent to 2nd September, 1848 A.D. The corresponding Meshâdi Śaka year (current) is 1771. It commenced on 11th April (102), Tuesday (3).

			d.	w.	h.	m.
(Table I., cols. 13, 14, 17)			102	3	1	30
2nd September =	245	(Table IX.)
Add 1 for leap-year	1
Date-number of the given day	246
Deduct ( $d$ ) of the initial day	102
Intervening days	144
The number next below 144, (col. 7, Table III.), for the end of Karka and beginning of Siṁha is 125, and we have			125	6	9	38
The astronomical beginning of Siṁha is			227	2	11	8

This is the civil beginning by one of the Southern systems.

	d.	w.	h.	m.
(Brought over)	277	2	11	8
Subtract 1 from ( $d$ ) and ( $w$ )	1	1
Last civil day of Karka =	226	1
Subtract 226 from the date number 246 (Table IX.) of the given day	246
Required date in the month Siṁha	20
Add this to ( $w$ ) casting out sevens		6
The required week-day is Saturday		0

The equivalents are therefore:— (see Table II., Part ii.)

Saturday 19th Chiṅgam,	South Malayâḷam Âṇḍu	1024 (See example XII., p. 89.)
SaturdayDo. 20th ChingamDo.	North Malayalam AnduDo.	1023
SaturdayDo. 20th Âvaṇi	Tinnevelly Âṇḍu	1024
SaturdayDo. 20th ChingamDo.	Tamil Śaka year	1771 (current).

(f.) Determination of Karaṇas.

153. We now proceed to give rules for finding the karaṇas on a given day,—the exact moments of their beginning and ending, and the karaṇa current at sunrise on any given day, or at any moment of any given day.

The karaṇas^[148] of a given tithi may be found by the following rule. Multiply the number of expired tithis by two. Divide this by 7; and the remainder is the karaṇa for the current half of the tithi. Example.— Find the karaṇa for the second half of kṛishṇa 8th. The number of expired tithis from the beginning of the month is (15 + 7 1/2 =) 22 1/2. 22 1/2 × 2 = 45. Casting out sevens the 3rd, or Kaulava, is the required karaṇa.

154. To find the exact moments on which the karaṇas corresponding to a given tithi begin and end. Find the duration of the tithi from its beginning and ending moments, as calculated by the method given in Arts. 139, 144, and 145 above. The first half of the tithi is the period of duration of its first karaṇa, and the second half that of the second.

Example xxii. Find the karaṇas, and the periods of their duration, current on Jyeshṭha śukla paṅchâmî (5th) of the Śaka year 1702 expired (1703 current). From Table VIII., cols. 4 and 5 we observe that (1) Bava is the first, and (2) Bâlava is the second, karaṇa corresponding to the 5th tithi. In the first example above (Art. 148) we have found that the tithi commenced on Tuesday, 6th June, A.D. 1780, at 15 h. 34 m. after mean sunrise, and that it ended on Wednesday, 7th June, at 13 h. 11 m. after mean sunrise. It lasted therefore for 21 h. 37 m. (8 h. 26 m. on Tuesday and 13 h. 11 m. on Wednesday). Half of this duration is 10 h. 48 m. The Bava karaṇa lasted therefore from 15 h. 34 m. after mean sunrise on Tuesday, June 6th, to 2 h. 22 m. after mean sunrise on Wednesday, June 7th, and the Bâlava karaṇa lasted thence to the end of the tithi.

155. The karaṇa at sunrise or at any other time can of course easily be found by the above method. It can also be calculated independently by finding the ( $t$ ) for the time given. Its beginning or ending time also can be found, with its index, by the same method as is used for that of a tithi. The index of a karaṇa can be easily found from that of a tithi by finding the middle point of the latter. For example, the index of the middle point of śukla 14th is 4500, or 4333 + half the difference between 4333 and 4667 {Table VIII.), and therefore the indices for the beginning and ending of the 5th karaṇa on śukla 14th are 4333 and 4500, and of the 6th karaṇa on the same tithi 4500 and 4667.

Example xxii(a). Find the karaṇa at sunrise on Wednesday the 7th June, A.D. 1780, Jyeshṭha śukla 5th, Śaka 1702 expired (1703 current).

In examples i. and xv. above we have found ( $t$ ) at the given sunrise to be 1463. Turning with this to Table VIII. we see that the karaṇa was the 1st or 2nd. The index of the first is 1333 to 1500, and therefore the first karaṇa, Bava, was current at the given sunrise.

(g) Determination of Nakshatras.

156. To find the nakshatra at sunrise, or at any other moment, of an Indian or European date. If the given date be other than a tithi or a European date, turn it into one or other of these. Find the ( $a$ ) ( $b$ ) ( $c$ ) and ( $t$ ) for the given moment by the method given in Arts. 139, 148 or 151, (Examples i. or xv.) above. Multiply ( $c$ ) by ten; add 7207 to the product, and from this sum subtract the equation for ( $c$ ) (Table VII.). Call the remainder ( $s$ ). Add ( $s$ ) to ( $t$ ). Call the result ( $n$ ). Taken as an index, ( $n$ ) shows, by Table VIII., col. 6, 7, 8, the nakshatra current at the given moment as calculated by the ordinary system.

157. If the nakshatra according to the Garga or Brahma Siddhânta system is required, use cols. 9 or 10 respectively of Table VIII.

158. The beginning or ending time of the nakshatra can be calculated in the same manner as that of a tithi. Since (r) is expressed in loooths, and looooths of it are neglected, the time will not be absolutely correct.

Example xxiii. Find the nakshatra current at sunrise on Wednesday, Jyeshṭha śukla 5th, Śaka 1702 expired, (7th June, 1780 A.D.)

	t.		c.		Equation for c. (Table VII.)
As calculated in Example i. or xv. above	1463		439		38
Multiply ( $c$ ) by 10			439	× 10 =	4390
Add					7207
1597
Subtract equation for ( $c$ )					38
Add ( $s$ ) to ( $t$ )	1559				1559	$=(s)$
	3022	$=(n)$

This result ( $n$ ) gives Aśleshâ (Table VIII., cols. 6, 7, 8) as the required current nakshatra

The ( $n$ ) so found 3022 − 2963 (index to beginning point of Aśleshâ) = 59. Therefore Aslesha begins 3 h. 52 m. (Table X., col. 4) before sunrise on the Wednesday. 3333 (end of Aśleshâ) − 3022 ( $n$ ) = 311, and therefore Aśleshâ ends (19h. 40 m. + 43 m. =) 20 h. 23 m. after sunrise on the Wednesday.

For greater accuracy we may proceed as in Example 1 (Art. 148.)

(h.) Determination of Yogas.

159. The next problem is to find the yoga at sunrise or at any other moment of an Indian or European date. If the given date is other than a tithi or a European date, turn it into one or the other of these. Find ( $a$ ) ( $b$ ) ( $c$ ) ( $t$ ) ( $s$ ) ( $n$ ) for the given moment as above (Art. 156). Add ( $s$ ) to ( $n$ ). Call the sum ( $y$ ). This, as index, shews by Table VIII., cols, 11, 12, 13, the yoga current at the given moment.

Example xxiv. Find the yoga at sunrise on Jyeshṭha śukla 5th, Śaka 1702 expired, 7th June, 1780 A.D.

As calculated in example xviii.	$(s)=1559$	$(n)=3022$
Add ( $n$ ) to ( $s$ )	$(n)=3022$
Required yoga ( $y$ )	4581	= (13) Vyâghâta (Table VIII.).

We find the beginning point of Vyâghâta from this.

The ( $y$ ) so found 4581 − 4444 (beginning point of Vyâghâta) = 137 = (6 h. 6 m. + 2 h. 15 m. =) 8 h. 21 m. before sunrise on Wednesday (Table X., col. 5).

The end of Vyâghâta is found thus:

(End of Vyâghâta) 4815 − 4581 ( $y$ ) = 234 = (12 h. 12 m. + 2 h. 4 m. =) 14 h. 16 m. after sunrise on Wednesday.

(i.) Verification of Indian dates.

160. (See Art. 132.) The following is an example of the facility afforded by the Tables in this volume for verifying Indian dates.

Example xxv. Suppose an inscription to contain the following record of its date,—"Śaka 666, Kârttika kṛishṇa amâvâsyâ (30), Sunday, nakshatra Hasta." The problem is to verify this date and find its equivalent A.D. There is nothing here to shew whether the given year is current or expired, whether the given month is amânta or pûrṇimânta, and whether, if the year be the current one, the intercalary month in it was taken as true or mean.^[149]

First let us suppose that the year is an expired one (667 current) and the month amânta. There was no intercalary month in that year. The given month would therefore be the eighth, and the number of intervening months from the beginning of the year is 7.

	d.	w.	a.	b.	c.
Śaka 667 current. (Table I., cols. 19, 20, 23, 24, 25)	80	6	324	773	278
210 (7 months) + 15 (śukla) + 14 (kṛ. amâvâsyâ is 15, and 1 must be substracted by rule) = 239 tithis = 235 days	235	4	9578	59	643
	315	3	9902	302	921
Equation for ( $b$ ) (302) (Table VI.)			271
Equation forDo. ( $c$ ) (921) (Table VII.)			90
		3	263	$=t$ .

This gives us Tuesday, śukla 1st (Table VIII.). Index, $t=263$ , proves that 263 parts of the tithi had expired at sunrise on Tuesday, and thence we learn that this śukla 1st commenced on Monday, and that the preceding tithi kṛi. 30 would possibly commence on Sunday. If so, can we connect the tithi kṛi. 30 with the Sunday? Let us see.

	d.	w.	a.	b.	c.
Already obtained	315	3	9902	302	921
Subtract value for two days (Table IV.)	2	2	677	73	5
	313	1	9225	229	916
Equation for ( $b$ ) (229) (Table VI.)			279
Equation forDo. ( $c$ ) (916) (Do. VII.)			91
		1	9595	$=t$ .

This index gives us kṛishṇa 14th (Table VIII.) as current at sunrise on Sunday (1). The tithi ended and kṛi. 30 commenced (9667 − 9595 = 72 =) 5 h. 6 m. after sunrise on Sunday. This kṛi. 30 therefore can be connected with a Sunday, and if the nakshatra comes right—Hasta—then this would be the given date. We calculate the nakshatra at sunrise on Sunday.

	t.	c.
As calculated above	9595	916
( $c$ ) multiplied by 10		916	× 10 =	9160
Add constant				7207
				6367
Subtract the equation for ( $c$ ) (786) (Table VII.)				91
Add ( $s$ ) to ( $t$ )	6276			6276	$=(s)$
	5871	$=(n)$

This index ( $n$ ) gives nakshatra No. 16 Visâkhâ (Table VIII., col. 6, 7, 8). Therefore No. 13 Hasta had already passed, and this proves that the date obtained above is incorrect. Now if Kârttika in the given record be pûrṇimânta, the amânta month corresponding (Table II., Part i) would be Asvina, the 7th month, and it is possible that Âśvina kṛi. 30, falling back as it does 29 or 30 days from the date calculated, might fall on a Sunday. Let us see if it did so.

	d.	w.	a.	b.	c.
Chaitra śukla 1, Śaka 667 current (as above)	80	6	324	773	278
180 (6 expired months) + 15 (śukla) + 14 (see above) = 209 tithis = 206 days	206	3	9758	476	564
	286	2	82	249	842
Equation for ( $b$ ) (249) (Table VI.)			280
Equation forDo. ( $c$ ) (842) (Table VII.)			111
		2	473	$=(t)$

The result gives us Monday, śukla 2nd. ^[150]

	d.	w.	a.	b.	c.
State the figures for this	286	2	82	249	842
Subtract value for two days (Table IV.)	2	2	677	73	5
	284	0	9405	176	837
Equation for ( $b$ ) (176) (Table VI.)			265
Equation forDo. ( $c$ ) (842) (Do. VII.)			112
		0	9782

This gives Saturday kṛishṇa (30), amâvâsyâ. i.e., that tithi had (10,000 − 9782) 218 parts to run at sunrise on Saturday. Therefore it ended on Saturday, and cannot be connected with a Sunday. Here again we have not the correct date.

Now let us suppose that the given year 666 is a current amânta year. Then the given month, Kârttika, is amânta, and the intercalary month was Bhâdrapada. The given month would be the 9th.

	d.	w.	a.	b.	c.
Chaitra śukla 1st, Śaka 666 current (Table I.)	61	0	289	837	227
240 (for 8 months) + 15 (śukla) + 14 (as above) = 269 tithies = 265 days (Table IV.)	265	6	9737	617	726
	326	6	26	454	953
Equation for ( $b$ ) (454) (Table VI.)			180
Equation forDo. ( $c$ ) (953) (Do. VII.)			78
		6	284	$=(t)$

This gives us Friday, śukla 1st. The preceding day is kṛishṇa amâvâsyâ, and this therefore ends on Thursday and can in no way be connected with a Sunday. This date is therefore again wrong. The amâvâsyâ of the previous month (29 days back) would end on a Wednesday or perhaps Tuesday, so that cannot help us. If we go back yet a month more, it is possible that the kṛishṇa amâvâsyâ might fall on a Sunday. That month could only be called Kârttika if it were treated according to the pûrṇimânta system and if there were no intercalary month. The given month would then be the 7th in the year. We test this as usual.

	d.	w.	a.	b.	c.
Chaitra śukla 1st, Śaka 666 current	61	0	289	837	227
180 (6 expired months) + 15 śukla + 14 (as before) = 209 tithis = 206 days (Table IV.)	206	3	9758	476	564
	267	3	47	313	791
Equation for ( $b$ ) (313) (Table VI.)			269
Equation forDo. ( $c$ ) (791) (Do. VII.)			119
		3	435	$=t$ .

This gives Tuesday,^[151] śukla 2nd, two tithis in advance of the required one.

We may either subtract the value of ( $w$ ) ( $a$ ) ( $b$ ) ( $c$ ) for two days from their value as already obtained, or may add the value for (206 − 2 =) 204 days to the value at the beginning of the year. We try the latter.

	d.	w.	a.	b.	c.
Chaitra śukla 1st, Śaka 666 current. (Table I., cols. 19, 20, 23, 24, 25)	61	0	289	837	227
204 days (Table IV.)	204	1	9081	403	559
	265	1	9370	240	786
Equation for ( $b$ ) (240) (Table VI.)			280
Equation forDo. ( $c$ ) (786) (Do. VII.)			119
		1	9869	$=t$ .

This gives us kṛishṇa amâvâsyâ, (1) Sunday, as required.

$(d)=265=$ (Table IX.) 22nd September, 743 A.D. (Table I.). From Table XIII. we see that the week-day is right. If the nakshatra Hasta comes right, then this is the given date. We calculate it according to rule.

	t.	c.
As already obtained	9769	786
( $c$ ) multiplied by 10		7860
Add constant		7207
		5067
Subtract the equation for ( $c$ ) (786) (Table VII.)		119
Add ( $s$ ) to ( $t$ )	4948	4948	$=(s)$
	4717	$=(n)$

This result gives No. 13 Hasta (Table VIII.) as required.

This therefore is the given date. Its equivalent A.D. is 22nd September, 743 A.D. The data were imaginary. If they had been taken from an actual record they would have proved that mean and not true intercalary months were in use in A.D. 743, because we have found that there was no intercalary month prior to the given month Kârttika. The mean intercalary month in that year (Table I.) was the 9th month, Mârgaśîrsha, and of course Kârttika was unaffected by it.

160(A). See page of Addenda and Errata.

PART V.

The Muhammadan Calendar.

161. The Muhammadan era of the Hijra, or "flight," dates from the flight of Muhammad (Anglicé Mahomet) which took place, according to the Hissabi or astronomical reckoning, on the evening of July 15th, A.D. 622. But in the Helali, or chronological reckoning, Friday, July 16th, is made the initial date. The era was introduced by the Khalif Umar.

162. The year is purely lunar, and the month begins with the first heliacal rising of the moon after the new moon. The year is one of 354 days, and of 355 in intercalary years. The months have alternately 30 and 29 days each (but see below), with an extra day added to the last month eleven times in a cycle of thirty years. These are usually taken as the 2nd, 5th, 7th, 10th, 13th, 15th, 18th, 2ist, 24th, 26th, and 29th in the cycle, but Jervis gives the 8th, 16th, 19th, and 27th as intercalary instead of the 7th, 15th, 18th and 26th, though he mentions the usual list. Ulug Beg mentions the 16th as a leap-year. It may be taken as certain that the practice varies in different countries, and sometimes even at different periods in the same country.

30 years are equal to (354 x 30 + 11 =) 10,631 days and the mean length of the year is 354 11/30 days.^[152]

Since each Hijra year begins 10 or 11 civil days earlier than the last, in the course of 33 years the beginning of the Muhammadan year runs through the whole course of the seasons.

163. Table XVI. gives a complete list of the initial dates of the Muhammadan Hijra years from A.D. 300 to A.D. 1 900. The asterisk in col. 1 shews the leap-years, when the year consists of 355 days, an extra day being added to the last month Zî'l-ḥijjat. The numbers in brackets following the date in col. 3 refer to Table IX. (see above, Art. 95), and are for purposes of calculation as shewn below.

Muhammadan Months.
		Days.	Collective duration.			Days.	Collective duration.
1	2	3	4	1	2	3	4
1	Muḥarram	30	30	7	Rajab	30	207
2	Śafar	29	59	8	Sha'bân	29	236
3	Rabî-ul awwal	30	89	9	Ramazân	30	266
4	Rabî-ul âkhir, or Rabi-uś śânî	29	118	10	Shawwâl	29	295
5	Jumâda'l awwal	30	148	11	Zî-l-ka'da	30	325
6	Jumâda'l âkhir, or Jumâda-ś śânî	29	177	12	Zî'l-ḥijja	29	354
					In leap-years	30	355

164. Since the Muhammadan year invariably begins with the heliacal rising of the moon, or her first observed appearance on the western horizon shortly after the sunset following the new-moon (the amâvâsyâ day of the Hindu luni-solar calendar), it follows that this rising is due about the end of the first tithi (śukla pratipadâ) of every lunar month, and that she is actually seen on the evening of the civil day corresponding to the 1st or 2nd tithi of the śukla (bright) fortnight. As, however, the Muhammadan day—contrary to Hindu practice, which counts the day from sunrise to sunrise—consists of the period from sunset to sunset, the first date of a Muhammadan month is always entered in Hindu almanacks as corresponding with the next following Hindu civil day. For instance, if the heliacal rising of the moon takes place shortly after sunset on a Saturday, the 1st day of the Muhammadan month is, in Hindu pañchâṅgs, coupled with the Sunday which begins at the next sunrise. Rut the Muhammadan day and the first day of the Muhammadan month begin with the Saturday sunset. (See Art. 30, and the pañchâṅg extract attached.)

165. It will be well to note that where the first tithi of a month ends not less than 5 ghaṭikâs, about two hours, before sunset, the heliacal rising of the moon will most probably take place on the same evening; but where the first tithi ends 5 ghaṭikâs or more after sunset the heliacal rising will probably not take place till the following evening. When the first tithi ends within these two periods, i.e., 5 ghaṭikâs before or after sunset, the day of the heliacal rising can only be ascertained by elaborate calculations. In the pañchâṅg extract appended to Art. 30 it is noted that the heliacal rising of the moon takes place on the day corresponding to September 1st.

166. It must also be specially noted that variation of latitude and longitude sometimes causes a difference in the number of days in a month; for since the beginning of the Muhammadan month depends on the heliacal rising of the moon, the month may begin a day earlier at one place than at another, and therefore the following month may contain in one case a day more than in the other. Hence it is not right to lay down a law for all places in the world where Muhammadan reckoning is used, asserting that invariably months have alternately 29 and 30 days. The month Śafar, for instance, is said to have 29 days, but in the pañchâṅg extract given above (Art. 30) it has 30 days. No universal rule can be made, therefore, and each case can only be a matter of calculation.^[153] The rule may be accepted as fairly accurate.

167. The days of the week are named as in the following Table.

**Days of the Week.**
	Hindustani.	Persian.	Arabic.	Hindî.
1. Sun.	Itwâr.	Yak-shamba.	Yaumu'l-aḥad.	Rabî-bâr.
2. Mon.	Somwâr, or Pîr.	Do-shamba.	Yaumu-l„-iśnain.	Som-bâr.
3. Tues.	Mangal.	Sih-shamba.	Yaumu-l„-śalâsa'.	Mangal-bâr.
4. Wed.	Budh.	Chahâr-shamba.	Yaumu-l„-arbâ'.	Budh-bâr.
5. Thurs.	Jum'a-rât.	Panj-shamba.	Yaumu-l„-khamîs.	Brihaspati-bâr.
6. Fri.	Jum'a.	Âdîna.	Yaumu-l„-Jum'ah.	Śukra-bâr.
7. Sat.	Sanîchar.	Shamba, or Hafta.	Yaumu's-sab't.	Sanî-bâr.

Old and New style.

168. The New Style was introduced into all the Roman Catholic countries in Europe from October 5th, 1582 A.D., the year 1600 remaining a leap-year, while it was ordained that 1700, 1800, and 1900 should be common and not leap-years. This was not introduced into England till September 3rd, A.D. 1752. In the Table of Muhammadan initial dates we have given the comparative dates according to English computation, and if it is desired to assimilate the date to that of any Catholic country, 10 days must be added to the initial dates given by us from Hijra 991 to Hijra 1111 inclusive, and 11 days from H. 1112 to 1165 inclusive. Thus, for Catholic countries H. 1002 must be taken as beginning on September 27th, A.D. 1593.

The Catholic dates will be found in Professor R. Wüstenfeld's "Vergleichungs-Tabellen der Muhammadanischen und Christlichen Zeitrechnung" (Leipzic 1854).

To convert a date A.H. into a date A.D.

169. Rule I. Given a Muhammadan year, month, and date. Take down (w) the week- day number of the initial day of the given year from Table XVI., col. 2, and {d) the date-indicator in brackets given in col. 3 of the same Table (Art. 163 and 95 above.) Add to each the collective duration up to the end of the month preceding the one given, as also the moment of the given date minus i (Table in Art. 163 above). Of the two totals the first gives the day of the week by casting out sevens, and the second gives the day of the month with reference to Table IX.

Rule 2. Where the day indicated by the second total falls on or after February 29th in an English leap-year, reduce the total by one day.

Rule 3. For Old and New Style between Hijra 991 and 1165 see the preceding article.

Example 1. Required the English equivalent of 20th Muharram, A.H. 1260.

A.H. 1260 begins (Table XVI.) January 22nd, 1844.

{w) Col. 2 (d) Col. 3 2 22 Given date minus i rr 19 19 21 41 = (Table IX.) Feb. loth. Cast out sevens = 21 o =: Saturday.

Answer.—Saturday, February 10th, A.D. 1844.

Example 2. Required the English equivalent of 9th Rajab, A.H. 1311. A.H. 1311 begins July 15th, 1893. w. d. o 196 9th Rajab = (177 -f 8)= 185 185 7 I 185 381 =Jan. 1 6th, 1S94. (26) 3 — Tuesday. Answer.—Tuesday, January i6th, A.D. 1894.

This last example has been designedly introduced to prove the point we have insisted on viz., that care must be exercised in dealing with Muhammadan dates. According to Traill's Indian Diary, Comparative Table of Dates, giving the correspondence of English, Bengali, N.W. Fasali, "Samvat", Muhammadan, and Burmese dates, Rajab 1st corresponded with January 9th, and therefore Rajab 9th was Wednesday, January 17th, but Letts and Whitaker give Rajab 1st as corresponding with January 8th, and therefore Rajab 9th—Tuesday, January 16th, as by our Tables.

To convert a date A.D. into a date A.H.

170. Rule 1. Take down ( $w$ ) the week-day number of the initial day of the corresponding Muhammadan year, or the year previous if the given date falls before its initial date, from Table XVI., col. 2, and ( $d$ ) the corresponding date-indicator in brackets as given in col. 3. Subtract ( $d$ ) from the collective duration up to the given A.D. date, as given in Table IX., Parts i. or ii. as the case may be. Add the remainder to ( $w$ ). From the same remainder subtract the collective duration given in the Table in Art. 163 above which is next lowest, and add 1. Of these two totals ( $w$ ) gives, by casting out sevens, the day of the week, and ( $d$ ) the date of the Muhammadan month following that whose collective duration was taken.

Rule 2. When the given English date is in a leap-year, and falls on or after February 29th, or when its date-number is more than 365 (taken from the right-hand side of Table IX.), and the year preceding it was a leap-year, add i to the collective duration given in Table IX.

Rule 3. For Old and New Style see above. Art. 167.

Example. Required the Muhammadan equivalent of January 16th, 894 A.D.

Since by Table XVI. we see that A.H. 1312 began July 5th, 1894 A.D., it is clear that we must take the figures of the previous year. This gives us the following : o 196 Jan. 16th (Table IX.) -381 — 196 185 185 7 I 185 (26) 3:= Tuesday. Coll. dur. (Art. 163)— 177 8 + I 9

Answer.—Tuesday, Rajab 9th, A.H. 1311.

Perpetual Muhammadan Calendar.

By the kindness of Dr. J. Burgess we are able to publish the following perpetual Muham- madan Calendar, which is very simple and may be found of use. Where the week-day is known this Calendar gives a choice of four or five days in the month. But where it is not known it must be found, and in that case our own process will be the simpler, besides fixing the day exactly instead of merely giving a choice of several days.

PERPETUAL MUHAMMADAN CALENDAR.

Years A.H.

0

30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

480

510

540

570

600

630

660

690

720

750

780

810

840

870

900

930

960

990

1020

1050

1080

1110

1140

1170

1200

1230

For odd years.

1260

1290

1320

1350

1380

1410

1440

Dominical Letters.

0

5*

8

13*

21*

29*

G

B

D

F

A

C

E

1

9

17

25

C

E

G

B

D

F

A

2*

10*

18*

26*

F

A

C

E

G

B

D

3

11

16*

19

24*

27

A

C

E

G

B

D

F

4

12

20

28

B

D

F

A

C

E

G

6

14

22

D

F

A

C

E

G

B

7*

15

23

E

G

B

D

F

A

C

1 Muharram
10 Shawwal

A

G

F

E

D

C

B

2 Śafar
7 Rajab

C

B

A

G

F

E

D

3 Rabî'l-âwwal
12 Zî'l-ḥijjat

D

C

B

A

G

F

E

4 Rabî'l-âkhir
9 Ramaḍan

F

E

D

C

B

A

G

5 Jamâda-l-âwwal

G

F

E

D

C

B

A

6 Jamâda-l-âkhir
11 Zî'l-ka'dat

B

A

G

F

E

D

C

8 Sha'bân

E

D

C

B

A

G

F

1

8

15

22

29

Sun.

Mon.

Tues.

Wed.

Thur.

Fri.

Sat.

2

9

16

23

30

Mon.

Tues.

Wed.

Thur.

Fri.

Sat.

Sun.

3

10

17

24

Tues.

Wed.

Thur.

Fri.

Sat.

Sun.

Mon.

4

11

18

25

Wed.

Thur.

Fri.

Sat.

Sun.

Mon.

Tues.

5

12

19

26

Thur.

Fri.

Sat.

Sun.

Mon.

Tues.

Wed.

6

13

20

27

Fri.

Sat.

Sun.

Mon.

Tues.

Wed.

Thur.

7

14

21

28

Sat.

Sun.

Mon.

Tues.

Wed.

Thur.

Fri.

From the Hijra date subtract the next greatest at the head of the first Table, and in that column find the Dominical letter corresponding to the remainder. In the second Table, with the Dominical letter opposite the given month, run down to the week-days, and on the left will be found the dates and vice versa.

Example. For Ramaḍan, A.H. 1310. The nearest year above is 1290, difference 20; in the same column with 1290, and in line with 20, is F. In line with Ramaḍan and the column F we find Sunday 1st, 8th, 15th, 22nd, 29th, etc.

In the 11 years marked with an asterisk the month Zîl-ka'dat has 30 days; in all others 29. Thus AH. 1300 (1290 + 16) had 355 days, the 30th of Zîl-ka'dat being Sunday.

↑ See Art. 126 below.
↑ It seems almost certain that both systems had a common origin in Chaldæa. The first is the day of the sun, the second of the moon, the third of Mars, the fourth of Mercury, the fifth of Jupiter, the sixth of Venus, the seventh of Saturn. [R. S.]
↑ The word vâra is to be affixed to each of these names; Ravi = Sun, Ravivâra = Sunday.
↑ In the Table, for convenience of addition, Saturday is styled 0.
↑ The variation is of course really in the motions of the earth and the moon. It is caused by actual alterations in rate of rapidity of motion in consequence of the elliptical form of the orbits and the moon's actual perturbations; and by apparent irregularities of motion in consequence of the plane of the moon's orbit being at an angle to the plane of the ecliptic. [R. S.]
↑ An apt title. The full moon stands as it were with the waxing half on one side and the waning half on the other. The week is an arbitrary division.

↑ The "synodic revolution" of the moon is the period during which the moon completes one series of her successive phases, roughly 29½ days. The period of her exact orbital revolution is called her "sidereal revolution". The term "synodic" was given because of the sun and moon being then together in the heavens (cf: "synod"). The sidereal revolution of the moon is less by about two days than her synodic revolution in consequence of the forward movement of the earth on the ecliptic. This will be best seen by the accompanying figure, where ST is a fixed star, S the sun, E the earth, C the ecliptic, M M' the moon. (A) the position at one new moon, (B) the position at the next new moon. The circle M to Ml representing the sidereal revolution, its synodic revolution is M to Ml plus Ml to N. [R. S.]

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C. A. Young ("General Astronomy", Edit. of 1889, p 528) gives the following as the length in days of the various lunations:

	d.	h.	m.	s.
Mean synodic month (new moon to new moon)	29	12	44	2.684
Sidereal month	27	7	43	11.545
Tropical month (equinox to equinox)	27	7	43	4.68
Anomalistic month (perigee to perigee)	27	13	18	37.44
Nodical month (node to node)	27	5	5	35.81

↑ See Fleet's Corpus Inscrip. Indic., vol III., Introduction, p. 79 note; Ind. Ant., XVII., p. 141 f.
↑ Compare the note on p. 4 on the moon's motion. [R. S]
↑ This rate of annual precession is that fixed by modern European Astronomy, but since the exact occurrence of the equinoxes can never become a matter for observation, we have, in dealing with Hindu Astronomy, to be guided by Hindu calculations alone. It must therefore be borne in mind that almost all practical Hindu works (Karaṇas) fix the annual precession at one minute, or 1/60th of a degree, while the Sûrya-Siddhânta fixes it as 54″ or 3/200 degrees. (see Art. 160a. given in the Addenda sheet.)
↑ The anomaly of a planet is its angular distance from its perihelion, or an angle contained between a line drawn from the sun to the planet, called the radius vector, and a line drawn from the sun to the perihelion point of its orbit. In the case in point, the earth, after completing its sidereal revolulion, has not arrived quite at its perihelion because the apsidal point has shifted slightly eastwards. Hence the year occupied in travelling from the old perihelion to the new perihelion is called the anomalistic year. A planet's true anomaly is the actual angle as above whatever may be the variations in the planet's velocity at different periods of its orbit. Its mean anomaly is the angle which would be obtained were its motion between perihelion and aphelion uniform in time, and subject to no variation of velucity—in other words the angle described by a uniformly revolving radius vector. The angle between the true and mean anomalies is called the equation of the centre. True anom. = mean anom. + equation of the centre.

The equation of the centre is zero at perihelion and aphelion, and a maximum midway between them. In the case of the sun its greatest value is nearly 1°.55′ for the present, the sun getting alternately that amount ahead of, and behind, the position it would occupy if its motion were uniform. (C. A. Young, General Astronomy. Edit. of 1889, p. 125.)

Prof. Jacobi's, and our, $a$ , $b$ , $c$ , (Table 1., cols. 23, 24, 25) give $a$ . the distance of the noon from the sun, expressed in 10,000ths of the unit of 360°; $b$ . the moon's mean anomaly; $c$ . the sun's mean anomaly; the two last expressed in 1000ths of the unit of 360°. The respective equations of the centre are given in Tables VI. and VII. [R. S.]
↑ "The ecliptic slightly and very slowly shifts its position among the stars, thus altering the latitudes of the stars and the angle between the ecliptic and equator, i.e., the obliquity of the ecliptic. This obliquity is at present about 24′ less than it was 2000 years ago, and it is still decreasing about half a second a year. It is computed that this diminution will continue for about 15,000 years, reducing the obliquity to 22¼°, when it will begin to increase. The whole change, according to Lagrange, can never exceed about 1° 2′ on each side of the mean." (C. A. Young, General Astronomy, p. 128.)
↑ Generally speaking an astronomical Sanskrit work, called a Siddhânta, treats of the subject theoretically. A practical work on astronomy based on a Siddhânta is called in Sanskrit a Karaṇa. The Paitâmaha and following three Siddhântas are not now extant, but are alluded to and described in the Pañchasiddhântikâ, a Karaṇa by Varâhamihira, composed in or about the Saka year 427 (A. D. 505). [S. B. D.]
↑ Two other Pauliśa Siddhântas were known to Ulpala (A.D. 966), a well-known commentator of Varâhamihira. The length of the year in them was the same as that in the original Sûrya Siddhânta. [S. B. D.]
↑ The duration of the year by the First Arya-Siddhânta is noted in the interesting chronogram mukhyaḥ
muk5hyah1 kâlomayamâtulaḥ
ka1lo3ma5ya1ma5tu6lah3. These figures are to be read from right to left; thus—365, 15, 31, 15 in Hindu notation of days, ghaṭikâs, etc. (I obtained this from Dr Burgess—R. S.)
↑ The Parâśara Siddhânta is not now extant. It is described in the second Ârya Siddhânta. The date of this latter is not given, but in my opinion it is about A.D. 950. [S. B. D.]
↑ The Râjamṛigâṅka is a Karaṇa by King Bhoja. It is dated in the Śaka year 964 expired, A.D. 1012. [S. B. D.]
↑ Karaṇas and other practical works, containing tables based on one or other of the Siddhântas, are used for these calculations. [S. B. D.]
↑ The positions and motions of the sun and moon and their apogees must necessarily be fixed and known for the correct calculation of a tithi, nakshatra, yoga or karaṇa. The length of the year is also an important clement, and in the samvatsara is governed by the movement of the planet Jupiter. In the present work we are concerned chiefly with these six elements, viz., the sun, moon, their apogees, the length of the year, and Jupiter. The sketch in the text is given chiefly keeping in view these elements. When one authority differs from another in any of the first five of these six elements the tithi as calculated by one will differ from that derived from another. [S. B. D.]
↑ It is not to be understood that as soon as a standard work comes into use its predecessors go out of use from all parts of the country. There is direct evidence to show that the original Sûrya-Siddhânta was in use till A.D. 665, the date of the Khaṇḍa-khâdya of Brahmagupta, though evidently not in all parts of the country. [S. B. D.]
↑ Whenever we allude simply to the "Brahma Siddhânta" by name, we mean the Brahma-Siddhânta of Brahmagupta.
↑ Out of the six elements alluded to in note 1 on the last age, only Jupiter has this bîja. The present Sûrya-Siddhânta had undoubtedly come into use before the date of the Bhâsvatî. [S. B. D.]
↑ It is probable that the first Ârya-Siddhânta was the standard authority for South Indian solar reckoning from the earliest times. In Bengal the Sûrya-Siddhânta is the authority since about A.D. 1100, but in earlier times the first Ârya-Siddhânta was apparently the standard. [S. B. D.]
↑ When we allude simply to the Sûrya or Ârya-Siddhânta it must be borne in mind that we mean the Present Sûrya and the First Ârya-Siddhântas.
↑ See note 1, p. 2 above. [R. S.]
↑ My present opinion is that the zodiacal-sign-names, Mesha, etc., began to be used in India between 700 B. C. and 300 B. C., not earlier than the former or later than the latter. [S. B. D.]
↑ It will be seen that the Bengal names differ from the Tamil ones. The same solar month Mesha, the first of the year, is called Vaiśâkha in Bengal and Sittirai (Chaitra) in the Tamil country, Vaiśâkha being the second month in the south. To avoid confusion, therefore, we use only the sign-names (Mesha, etc.) in framing our rules.
↑ The lengths of months by the Ârya-Siddhânta here given are somewhat different from those given by Warren. But Warren seems to have taken the longitude of the sun's apogee by the Sûrya-Siddhânta in calculating the duration of months by the Ârya-Siddhânta, which is wrong. He seems also to have taken into account the chara.^[29] (See his Kâla Saṅkalita, p. 11, art. 3, p. 22, explanation of Table III., line 4; and p. 3 of the Tables). He has used the ayanâṁśa (the uniformly increasing arc between the point of the vernal equinox each year and the fixed point in Aries) which is required for finding the chara in calculating the lengths of months. The chara is not the same at the beginning of any given solar mouth for all places or for all years. Hence it is wrong to use it for general rules and tables. The inaccuracy of Warren's lengths of solar months according to the Sûrya-Siddhânta requires no elaborate proof, for they are practically the same as those given by him according to the Ârya-Siddhânta, and that this cannot be the ease is self-evident to all who have any experience of the two Siddhântas. [S. B. D.]
↑ The chara:—"The time of rising of a heavenly body is assumed to take place six hours before it comes to the meridian. Actually this is not the case for any locality not on the equator, and the chara is the correction required in consequence, i.e., the excess or defect from six hours of the time between rising and reaching the meridian. The name is also applied to the celestial arc described in this time."
↑ The Sanskrit word for "mean" is madhyama, and that for 'true' or 'apparent' is spashṭa. The words "madhyama" and "spashṭa" are applied to many varieties of time and space; as, for instance, gati (motion), bhôga (longtitude), saṅkrânti, mâna (measure or reckoning) and kâla (time). In the English Nautical Almanac the word "apparent" is used to cover almost all cases where the Sanskrit word spashṭa would be applied, the word 'true' being sometimes, but rarely, used. "Apparent," therefore, is the best word to use in my opinion; and we have adopted it prominently, in spite of the fact that previous writers on Hindu Astronomy have chiefly used the word "true." There is as a fact a little difference in the meaning of the phrases "apparent" and "true," but it is almost unknown to Indian Astronomy, and we have therefore used the two words as synonyms. [S. B. D.]
↑ Remember that the week-day is counted from sunrise to sunrise.
↑ Brown's Ephemeris follows this rule throughout in fixing the date corresponding to 1st Mesha, and consequently his solar dates are often wrong by one day for those tracts where the 2 b rule is in use.
↑ I deduced the Bengal rule from a Calcutta Pañchâṅg for Śaka 1776 (A.D. 1854—55) in my posssession. Afterwards it was corroborated by information kindly sent to me from Howrah by Mr. G. A. Grierson through Dr. Fleet. It was also amply corroborated by a set of Bengal Chronological Tables for A.D. 1882, published under the authority of the Calcutta High Court, a copy of which was sent to rac by Mr. Scwell. I owe the Orissa Rule to the Chronological Tables published by Girishchandra Tarkâlaukar, who follows the Orissa Court Tables with regard to the Amli and Vilayati years in Orissa. Dr. J. Burgess, in a note in Mr. Kṛishṇasvâmi Naidu's "South Indian Chronological Tables" edited by Mr. Sewell. gives the 2 (a) Rule as in use in the North Malayâḷam country, but I do not know what his authority is. I ascertained from Tamil and Tinnevelly panchangs that the 2 (a) rule is in use there, and the fact is corroborated by Warren's Kâla Saṅkalita; I ascertained also from some South Malayâḷam pañchâṅgs published at Cochin and Trevandruni, and from a North Malayâḷam pañchâṅg published at Calicut, that the 2 (b) rule is followed there [S. B. D.]

Notwithstanding all this I have no certain guarantee that these are the only rules, or that they are invariably followed in the tracts mentioned. Thus I find from a Tamil solar pañchâṅg for Śaka 1815 current, published at Madras, and from a Telugu luni-solar pañchâṅg for Śaka 1109 expired, also published at Madras, in which the solar months also are given, that the rule observed is that "when a saṅkrânti occurs between sunrise and midnight the month begins on the same day, otherwise on the following day", thus differing from all the four rules given above. This varying fifth rule again is followed for all solar months of the Vilayati year as given in the above-mentioned Bengal Chronological Tables for 1882, and by its use the month regularly begins one day in advance of the Bengâli month. I find a sixth rule in some Bombay and Benares lunar pañchâṅgs, viz., that at whatever time the saṅkrânti may occur, the month begins on the next day; but this is not found in any solar pañchâṅg. The rules may be further classified as (1. a) the midnight rule (Bengal), (1. b) any time rule (Orissa), (2. a) the sunset rule (Tamil), (2. b) the afternoon rule (Malabar). The fifth rule is a variety of the midnight rule, and the sixth a variety of the any time rule. I cannot say for how many years past the rules now in use in the several provinces have been in force and effect.

An inscription at Kaṇṇaṇûr, a village 5 miles north of Srîraṅgam near Trichinopoly (see Epigraph. Indic, vol. III., p. 10, date No. V., note 3, and p. 8) is dated Tuesday the thirteenth tithi of the bright fortnight of Śrâvaṇa in the year Prajâpati, which corresponded with the 24th day of the (solar) month Âḍi (Karka.) From other sources the year of this date is known to be A.D. 1271; and on carefully calculating I find that the day corresponds with the 21st July, and that the Karka saṅkrânti took place, by the Ârya-Siddhânta, on the 27th June, Saturday, shortly before midnight. From this it follows that the month Âḍi began civilly on the 28th June, and that one or the other of the two rules at present in use in Southern India was in use in Trichinopoly in A.D. 1271. [S. B. D.]
↑ We cannot enumerate the vulgar or popular names which obtain in all parts of India, and it is not necessary that we should do so.
↑ This is an ordinary pañchâṅg in daily use. It was prepared by myself from Gaṇeśa Daivjña's Grahalâghava and Laghutithichintâmaṇi. [S. B. D.]
↑ Solar days are not given in Bombay pañchâṅgs, but I have entered them here to complete the calendar. Some entries actually printed in the pañchâṅg arc not very useful and are consequently omitted in the extract. [S. B. D.]
↑ The sum total of days that have elapsed since any other standard epoch is also called the ahargaṇa. For instance, the ahargaṇa from the beginning of the present kaliyuga is in constant use. The word means "collection of days."
↑ The Nirṇayasindhu is one of these authorative works, and is in general use at the present time in most parts of India.
↑ Any assertions or definitions by previous writers on Hindu Chronology or Astronomy contrary to the above definitions and examples are certainly erroneous, and due to misapprehension. [S. B. D.]
↑ Instead of writing at full length that such and such a tithi "ends at so many ghaṭikâs after sunrise, Indian astronomers say for brevity that the tithi "is so many ghaṭikâs". The phrase is so used in the text in this sense.
↑ In the case of kshayas in the pañchâṅg extract the ghaṭikâs of expunged tithis etc., are to be counted after the end of the previons tithi etc. In some pañchâṅgs the ghaṭikâs from sunrise—59 gh. 55pa. in the present instance— are given.
↑ Since this work was in the Press, Professor Jacobi has published in the Epigraphica Indica (Vol. II, pp. 487—498) a treatise with tables for the calculation of Hindu dates in true local time, to which we refer our readers.
↑ Here Lanka is not Ceylon, but a place supposed to be on the equator, or in lat. 0° 0′ 0″ on the meridian of Ujjain, or longitude 75° 46′. It is of great importance to know the exact east longitude of Ujjain, since upon it depends the verification of apparent phenomena throughout India. Calculation by the different Siddhântas can be checked by the best European science if that point can be certainly determined. The great Trigonometical Survey map makes the centre of the city 75° 49′ 45″ E. long, and 23° 11′ 10″ N. lat. But this is subject to two corrections; first, a correction of 1′ 9″ to reduce the longitude to the origin of the Madras Observatory taken as 80° 17′ 21″, and secondly, a farther reduction of 2′ 30″ to reduce it to the latest value, 80° 14′ 51″. of that Observatory, total 3′ 39″. This reduces the E. long, of the centre of Ujjain city to 75° 46′ 06″. I take it therefore, that amidst conflicting authorities, the best of whom vary from 75° 43′ to 75° 51′, we may for the present accept 75° 46′ as the nearest approach to the truth. The accuracy of the base, the Observatory of Madras, will before long be again tested, and whatever difference is found to exist between the new fixture and 80° 14′ 51″, Ihal difference applied to 75° 46' will give the correct value of the E. long, we require. [R. S.]
↑ The mean length of the moon's revolution among the stars is 27.32166 days (27.321674 according to the Sûrya Siddhânta). Its least duration is 27 days, 4 hours, and the greatest about 7 hours longer. The number of days is thus between 27 and 28, and therefore the number of nakshatras was sometimes taken as 28 by the ancient Indian Âryas. The extra nakshatra is called Abhijit (See Table VIII., col. 7.) [S. B. D.]
↑ These systems of nakshatras arc more fully described by me in relation to the "twelve year cycle of Jupiter" in Vol. XVII. of the Ind. Ant., (p. 2 ff.) [S. B. D.]
↑ According to the Sûrya-Siddhânta the four karaṇas are Śakuni, Nâga, Chatushpada and Kiṁstughna, but we have followed the present practice of Western India, which is supported by Varâhamihira and Brahmagupta.
↑ Madhu is "honey", "sweet spring". Mâdhava, "the sweet one". Sukra and Śuchi both mean "bright". Nabhas, the rainy season. Nabhasya, "vapoury", "rainy". Ish or Isha, "draught" or "refreshment", "fertile". Ûrj, "strength", "vigour". Sahas "strength". Sahasya "strong". Tapas "penance", "mortification", "pain", "fire". Tapasya, "produced by heat", "pain". All are Vedic words.
↑ In my opinion the sidereal names "Chaitra" and the rest, came into use about 2000 B. C. They are certainly not later than 1500 B.C., and not earlier than 4000 B.C. [S. B. D.]
↑ Professor Kielhorn is satisfied that the terms adhika and nija are quite modern, the nomenclature usually adopted in documents and inscriptions earlier then the present century being prathama (first) and dvitîyâ (second). He alluded to this in Ind. Ant., XX., p. 411. [R. S.]
↑ The scheme of pûrṇimânta months and the rule for naming the intercalated months known to have been in use from the 12th century A.D., are followed in this diagram.
↑ See his Siddhânta-Siromaṇi, madhyamâdhikâra, adhimâsanirṇaya, verse 6, and his own commentary on it. [S. B. D.]
↑ It is not to be found in either of the Brâhma-Siddhântas referred to above, but there is a third Brâhma-Siddhânta which I have not seen as yet. [S. B. D.]
↑ In Prof. Chattre's list of added and suppressed mouths, in those published in Mr. Cowasjee Patells' Chronology, and in General Sir A. Cunningham's Indian Eras it is often noted that the same mouth is both added and suppressed. But it is clear from the above rules and definitions that this is impossible. A month cannot be both added and suppressed at the same time. The mistake arose probably from resort being made to the first rule for naming adhika months, and to the second for the suppressed months.
↑ Thanks are due to Mr. Mahadeo Chiṁṇâjî Apte. B.A., L.L.B., very recently deceased, the founder of the Anandâśrama at Poona, for his discovery of a part of Śrîpati's Karaṇa named the Dhîkoṭida, from which I got Śrîpati's date. I find that it was written in Śaka 961 expired (A.D. 1039–40). [S. B. D.]
↑ Up to recently the date was considered to be about the 6th century A.D. Dr. Thibaut, one of the highest living authorities on Indian Astronomy, fixes it at 400 A.D. (See his edition of the Pañcha Siddhântikâ Introd., p. LX.). My own opinion is that it came into existence not later than the 2nd century B.C. [S. B. D.]
↑ I am inclined to believe that of the two rules for naming lunar months the second was connected with the mean system of added months, and that the first came into existence with the adoption of the true system. But I am not as yet in possession of any evidence on the point. See, however, the note to Art. 51 below. [S. B. D.]
↑ It is difficult to define the exact limit, because it varies with different Siddhântas, and even for one Siddhânta it is not always the same. It is, however, generally not more than six ghaṭikâs, or about 33 of our tithi-indices ( $t$ ). But in the case of some Siddhântas as corrected with a bîja the difference may amount sometimes to as much as 20 ghaṭikâs, or 113 of our tithi-indices. It would be very rare to find any difference in true added months; but in the case of suppressed months we might expect some divergence, a month suppressed by one authority not being the same as that suppressed by another, or there being no suppression at all by the latter in some cases. Differences in mean added months would be very rare, except in the case of the Brâhma-Siddhânta. (See Art. 88.)
↑ This relation of intervals is a distinct assistance to calculation, as it should lead us to look with suspicion on any suppression of a month which docs not conform to it.
↑ See the Siddhânta-Siromaṇi, Madhyamâdhikâra. Bhâskara wrote in Śaka 1073 (A.D. 1150). He did not give the names of the suppressed months.
↑ I have ascertained that Gaṇeśa has adopted in his Grahalâghava some of the elements of the Ârya-Siddhânta as corrected by Lalla's bîja, and by putting to test one of the years noted I find that in these calculations also the Ârya-Siddhânta as corrected by Lalla's bîja was used. Gaṇeśa was a most accurate calculator, and I feel certain that his results can be depended upon. [S. B. D.]
↑ Such an anomaly with regard to the pûrṇimânta scheme could not occur if the two rules were applied, one that "that pûrṇimânta month in which the Mesha saṅkrânti occurs is always called Chaitra, and so on in succession," and the other that "that pûrṇimânta month in which no saṅkrânti occurs is called an intercalated month." The rules were, I believe, in use in the sixth century AD. (See may remarks Ind. Ant., XX., p. 50 f.) But the added month under such rules would never agree with the amânta added months. There would be from 14 to 17 months' difference in the intercalated months between the two, and much inconvenience would arise thereby. It is for this reason probably that the pûrṇimânta scheme is not recognised in naming months, and that pûrṇimânta months are named arbitrarily, as described in the first para. of Art. 51. This arbitrary rule was certainly in use in the 11th century A.D. (See Ind. Ant., vol. VI., p. 53, where the Makara-saṅkrânti is said to have taken place in Mâgha.)

After this arbitrary rule of naming the pûrṇimânta months once came into general use, it was impossible in Northern India to continue using the second, or Brâhma-Siddhânta, rule for naming the months. For in the example in Art. 45 above the intercalated month would by that rule be named Chaitra, but if its preceding fortnight be a fortnight of Vaiśâkha it is obvious that the intercalated month cannot be named Chaitra. In Southern India the practice may have continued in use a little longer. [S. B. D.]
↑ Chaitrâdi, "beginning with Chaitra"; Kârttikâdi, "beginning with Kârttika"; Meshâdi, with Mesha; and so on.
↑ See Ind. Ant., XIX., p. 45, second paragraph of my article on the Original Sûrya-Siddhânta. [S. B. D.]
↑ I have myself seen a pañchâṅg which mentions this beginning of the year, and have also found some instances of the use of it in the present day. 1 am told that at Iḍar in Gujarât the Vikrama samvat begins on Âshâḍha kṛishṇa dvitîyâ. [S. B. D.]
↑ The passage, as translated by Sachau (Vol. II., p. 8 f), is as follows. "Those who use the Saka era, the astronomers, begin the year with the month Chaitra, whilst the inhabilunts of Kanîr, which is conterminous with Kashmîr, begin it with the month Bhâdrapada…All the people who inhabit the country between Bardarî and Mârîgala the year with the mouth Kârttika…The people living in the country of Nîrahara, behind Mârîgala, as far as the utmost frontiers of Tâkeshar and Lohâvar, begin the year with the month Mârgaśîrsha…The people of Lanbaga, i.e., Lamghân, follow their example. I have been told by the people of Multân that this system is peculiar to the people of Sindh and Kanoj, and that they used to begin the year with the new moon of Mârgaśîrsha, hut that the people of Multân only a few years ago had given up this system, and had adopted the system of the people of Kashmîr, and followed their example in beginning the year with the new moon of Chaitra."
↑ Articles 53 to 61 are applicable to Northern India only (See Art. 62).
↑ The term is one not recognized in Sanskrit works. [S. B. D.]
↑ See Ind. Ant., Vol. XIX., pp. 27, 33, 187.
↑ These points have not yet heen noticed by any European writer on Indian Astronomy. [S. B. D.]
↑ As to the mean Mesha-saṅkrânti, see Art. 26 above.
↑ By all these rules the results will be correct within two ghaṭikâs where the moment of the Mesha-saṅkrânti according to the authority used is known.
↑ The rule for the present Vasishṭha, the Śâkalya Brahma, the Romaka, and the Soma Siddhântas is exactly the same. That by the original Sûrya-Siddhânta is also similar, but in that case the result will be incorrect by about 2 ghaṭikâs (48 minutes). For all these authorities take the time of the Mesha-saṅkrânti by the present Sûrya-Siddhânta or by the Ârya-Siddhânta, whichever may be available. The moment of the Mesha-saṅkrânti according to the Sûrya-Siddhânta is given in our Table I. only for the years A.D. 1100 to 1900. The same moment for all years between A.D. 300 and 1100 can be found by the Table in Art. 96. If the Ârya-Siddhânta saṅkrânti is used for years A.D. 300 to 1100 the result will never be incorrect by more than 2 ghaṭikâs 46 palas (1 hour and 6 minutes). The Table should be referred to.
↑ In these three rules the apparent Mesha-saṅkrânti is taken. If we omit the subtraction of 108, 11, and 60, and do not add 15 p., 1 gh. 45 p., and 15 p. respectively, the result will be correct with respect to the mean Mesha-saṅkrânti.
↑ I have not seen the Jyotishatattva (or "Jyotishtava" as Warren calls it, but which seems to be a mistake), but I find the rule in the Ratnamâlâ of Śrîpati (A.D. 1039). It must be as old as that by the Ârya-Siddhânta, since both are the same. [S. B. D.]
↑ If we add 4280 instead of 4291, and add 1 gh. 45 pa. to the final result, the time so arrived at will be the period elapsed since apparent Mesha-saṅkrânti. Those who interpret the Jyotishatattva rule in any different way have failed to grasp its proper meaning. [S. B. D.]
↑ It is not stated what Mesha-saṅkrânti is meant, whether mean or apparent. The rule is here given as generally interpreted by writers both Indian and European, but in this form its origin cannot be explained. I am strongly inclined to think that Varâhamihira, the author of the Bṛihatsaṁhitâ, meant the rule to run thus: Multiply the current Śaka year by 44. Add 8582 (or 8581 or 8583). Divide the sum by 3750. To the integers of the quotient add the given current Śaka year; (and the rest as above). The result is for the mean Mesha-saṅkrânti." In this form it is the same as the Ârya-Siddhânta or the Jyotishatattva rule, and can be easily explained. (S. B. D.)
↑ In this Table the Bṛihatsaṁhitâ rule is worked as I interpret it. But as interpreted by others the expunctions will differ, the differences being in Śaka (current) 231, the 56th; 998, the 52nd; 1889, the 37th.

By the Sûrya Siddhânta the years marked with an asterisk in the Śaka column of this Table differ from those given in Table I., col. 7, being in each case one earlier; the rest are the same. (S. B. D.)
↑ See Corpus Inscrip. Indic., Vol. III., p. 80, note; Ind. Antiq., XVII., p. 142.
↑ The heliacal rising of a superior planet is its first visible rising after its conjunctions with the sun, i.e., when it is at a sufficient distance from the sun to be first seen on the horizon at its rising in the morning before sunrise, or, in the case of an inferior planet (Mercury or Venus), at its setting in the evening after sunset. For Jupiter to be visible the sun must be about 11° below the horizon. [R. S.]
↑ It is fully described by me in the Indian Antiquary, vol. XVII. [S. B. D.]
↑ In practice of course the word "current" cannot be applied to the year 0, but it is applied here to distinguish it from the year 0 complete or expired, which means year 1 current. We use the word "epoch" to mean the year 0 current. The epoch of an era given in a year of another era is useful for turning years of one into years of another era. Thus, by adding 3078 (the number of the Kali year corresponding to the Graha-parivṛitti cycle epoch) to a Graha-parivṛitti year, we can get the equivalent Kali year; and by subtracting the same from a Kali year we get the corresponding Graha-parivṛitti year.
↑ Or Aṅka.
↑ On the 11th according to some, but all the evidence tends to shew that the year begins on the 12th.
↑ The real date of the Muhammadan invasion seems to be 1568 A.D. (J. A. S. B. for 1883, LII., p. 233, note). The invasion alluded to is evidently that of the "Yavanas", but as to these dates these temple chronicles must never be believed. [R. S.]
↑ Some say that the first year is also dropped, similarly; but this appears to be the result of a misunderstanding, this year being dropped only to fit in with the system described lower down in this article. Mr. J. Beames states that "the first two years and every year that has a 6 or 0 in it are omitted", so that the 37th Oṅko of the reign of Rāmachandra is really his 28th year, since the years 1, 2, 6, 10, 16, 20, 26, 30 and 86 are omitted. (J. A. S. B. 1883, LII., p. 234, note. He appears to have been misled about the first two years.
↑ Sewell's Sketch of the Dynasties of Southern India, p. 64. Archæological Survey of Southern India, vol. II., p. 204.
↑ See 'Calculations of Hindu dates', by Dr. Fleet, in the Ind. Ant., vols. XVI. to XIX.; and my notes on the date of a Jain Purâṇa in Dr. Bhândârkar's "Report on the search for Sankrit manuscripts" for 1883—1884 A. D., p.p. 429—30 §§ 36, 37. [S. B. D.]
↑ The Vikrama era is never used by Indian astronomers. Out of 160 Vikrama dates examined by Dr. Kielhorn (Ind. Ant., XIX.), there are only six which have to be taken as current years. Is it not, however, possible that all Vikrama years are really current years, but that sometimes in writings and inscriptions the authors have made them doubly current in consequence of thinking them erroneously to be expired years. There is an instance of a Śaka year made twice current in an inscription published in the Ind. Ant., (vol. XX., p 191), The year was already 1155 current, but the number given by the writer of the inscription is 1156, as if 1155 had been the expired year.

As a matter of fact I do not think that it is positively known whether the years of the Christian era are themselves really expired or current years. Warren, the author of the Kálasaṅkalita was not certain. He calls the year corresponding to the Kali year 3101 expired "A.D. 0 complete" (p 302) or "1 current" (p. 294). Thus, by his view, the Christian year corresponding to the Kali year 3102 expired would be A.D. 1 complete or A.D. 2 current. But generally European scholars fix A. D. 1 current as corresponding to Kali 3102 expired. The current and expired years undoubtedly give rise to confusion. The years of the astronomical eras, the Kali and Śaka for instance, may, unless the contrary is proved, be assumed to be expired years, and those of the non-astronomical eras, such as the Vikrama, Gupta, and many others, may be taken as current ones. (See, however, Note 3, p. 42, below.) [S. B. D.]
↑ Corpus Inscrip. Ind., Vol. III., Introduction, p. 69, note.
↑ Ind. Ant, Vol. XX., p. 149 ff.
↑ In Bengâli pañchâṅgs the Vikrama Samvat, or Sambat, is given along with the Śaka year, and, like the North-Indian Vikrama Samvat, is Chaitrâdi pûrṇimânta.
↑ See Ind. Ant., vol. XVII., p. 93; also note 3, p 31, and connected Text.
↑ See, however, note 2 on the previous page.
↑ See Ind. Ant., vol. XII., pp. 213, 293; XI., p. 242 ff.
↑ I have seen only two examples in which authors of Karaṇas have used any other era along with the Śaka. The author of the Râma-vinoda gives, as the starting-point for calculations, the Akbar year 35 together with the Śaka year 1512 (expired), and the author of the Phattesâhaprakâśa fixes as its starting-point the 48th year of "Phattesâha" coupled with the Śaka year 1626. [S. B D.]
↑ Certain Telugu (luni-solar) and Tamil (solar) pañchâṅgs for the last few years, which I have procured, and which were printed at Madras and are clearly in use in that Presidency, as well as a Canarese pañchâṅg for A.D. 1893, (Śakâ 1816 current, 1815 expired) edited by the Palace Astronomer of H. H. the Mahârâjâ of Mysore, give the current Śaka years. But I strongly doubt whether the authors of these pañchâṅgs are themselves acquainted with the distinction between so-called current and expired years. For instance, there is a pañchâṅg annually prepared by Mr. Aṇṇa Ayyaṅgâr, a resident of Kañjnûr in the Tanjore District, which appears to be in general use in the Tamil country, and in that for the solar Meshâdi year corresponding to 1887—88 he uses the expired Śaka year, calling this 1809, while in those for two other years that I have seen the current Śaka year is used. I have conversed with several Tamil gentlemen at Poona, and learn from them that in their part of India the generality of people are acquainted only with the name of the samvatsara of the 60-year cycle, and give no numerical value to the years. Where the years are numbered, however, the expired year is in general use. I am therefore inclined to believe that the so-called current Śaka years are nowhere in use; and it becomes a question whether the so-called expired Śaka year is really an expired one. [S. B. D.]
↑ Indian Antiquary for August, 1888, vol. XVII., p. 215, and the Academy, of 10th Dec., 1887, p 394 f. I had myself calculated these same inscription-dates in March, 1887, and had, in conjunction with Dr. Fleet, arrived at nearly the same conclusions as Dr. Kielhorn's, but we did not then settle the epoch, believing that the data were not sufficiently reliable (Corpus. Inscrip. Indic., Vol. III., Introd., p 9. [S. B. D.] See also Dr. Kielhorn's Paper read before the Oriental Congress in London. [R. S.]
↑ The Vilâyatî era, as given in some Bengal Government annual chronological Tables, and in a Bengali pañchâṅg printed in Calcutta that I have seen, is made identical with this Amli era in almost every respect, except that its months are made to commence civilly in accordance with the second variety of the midnight rule (Art. 28). But facts seem to be that the Vilâyatî year commences, not on lunar Bhâdrapada śukla 12th, but with the Kanyâ saṅkrânti, while the Amli year does begin on lunar Bhâdrapada śukla 12th. It may be remarked that Warren writes—in A.D. 1825—(Kálasaṅkalita, Tables p. IX.) that the "Vilaity year is reckoned from the 1st of the krishna paksha in Chaitra", and that its numerical designation is the same with the Bengali San. [S. B. D.]
↑ Alberuni's India, English translation by Sachau, Vol. II., p. 5.
↑ Corpus Inscrip. Indic, Vol. III., Introd., p. 177 ff.
↑ Giriśa Chandra's Chronological Tables for A.D. 1764 to 1900.
↑ Warren (Kálasaṅkalita, p. 298) makes it commence in "the year 3537 of the Julian period, answering to the 1926th of the Kali yug". But this is wrong if, as we believe, the Kollam years are current years, and we know no reason to think them otherwise. Warren's account was based on that of Dr. Buchanan who made the 977th year of the third cycle commence in A.D. 1800. But according to the present Malabar use it is quite clear that the year commencing in 1800 A.D., was the 976th Kollam year.
↑ General Sir A. Cunningham's Indian Eras, p. 74.
↑ Ind. Ant., Vol. XVII., p. 246 ff.
↑ This much information is from General Cunningham's "Indian Eras"
↑ Ind. Ant., XIX., p. 1 ff.
↑ General Cunningham, in his "Indian Eras", gives it as 15th February; but that day was a Saturday.
↑ Prinsep's Indian Antiquities, II., Useful Tables, p. 171.
↑ Gen. Cunningham admittedly (p. 91) follows Cowasjee Patell's "Chronology" in this respect, and on examination I find that the added and suppressed months in these two works (setting aside some few mistakes of their own) agree throughout with Prof. Chhatre's list, even so far as to include certain instances where the latter was incorrect. Patell's "Chronology" was published fifteen years after the publication of Prof. Chhatre's list, and it is not improbable that the former was a copy of the latter. It is odd that not a single word is said in Cowasjee Patell's work to shew how his calculations were made, though in those days he would hare required months or even years of intricate calculation before he could arrive at his results. [S. B. D.]
↑ A thousandth part of a tithi is equal to 1.42 minutes, which is sufficiently minute for our purposes, but a thousandth of a lunation is equivalent to 7 hours 5 minutes, and this is too large; so that we have to take the 10000th of a lunation as our unit, which is equal to 4.25 minutes, and this suffices for all practical purposes. In this work therefore a lunation is treated of as having 10,000 parts, and a tithi 1000 parts.
↑ For finding the initial date of the luni-solar years Prof, Jacobi's Tables I. to XI. were used, and in the course of the calculations it was necessary to introduce a few alterations, and to correct some misprints which had crept in in addition to those noted in the already published errata-list. Thus, the earliest date noted in Tables I. to IV., being A.D. 354, these Tables had to be extended backwards by adding two lines more of figures above those already given. In Table VI., as corrected by the errata, the bîja is taken into account only from A.D. 1601, whereas we consider that it should be introduced from A.D. 1501 (see Art. 21). In Table VI. the century correction is given for the New (Gregorian) Style from A.D 1600 according to the practice in the most part of Europe. I have preferred, however, to introduce the New Style into our Tables from Sept. A.D. 1752 to suit English readers, and this necessitated an alteration in the century data for two centuries. [R. S.]
↑ It is the same according to Warren, but in this respect he is in error. (See note to Art. 24.)
↑ 42 calculations were thus made direct by the Sûrya-Siddhânta with and without the bîja, with the satisfactory result that the error in the final figure of the tithi-index originally arrived at was generally only of 1 or 2 units, while in some cases it was nil. It was rarely 3, and only once 4. It never exceeded 4. It may therefore be fairly assumed that our results are accurate. [S. B. D.]
↑ See Art. 21, and the first footnote appended to it.
↑ See note 1 to Art. 91.
↑ We have seen before (Arts. 45 etc. above) how months and tithis are sometimes added or expunged. Now in case of Chaitra śukla pratipadâ being current at sunrise on two successive days, as sometimes happens, the first of these civil days, i.e., the day previous to that given by us, is taken as the 8rst day of the Indian luni-solar year (see Art. 52). This does not, however, create any confusion in our method C since the quantities given in cols. 23 to 25 are correct for the day and time for which they are given; while as for our methods A and B, the day noted by us is more convenient.
↑ Calculating by Prof. Jacobi's Tables, $a$ , $b$ , $c$ , are 9980, 896 and 255, each of which is wrong by 1.

The above figures were submitted by me to Dr. Downing of ihe Nautical Almanack office, with a request that he would test the results by scientific European methods. In reply he gave me the following quantities, for the sun from Leverrier's Tables, and and for the moon from Hansen's Tables (for the epoch A.D. 300, March 8th, 6 am., for the meridian of Ujjain). Mean long of sun 345° 51′ 47″.7, Do. of sun's perigee 253° 54′ 58″.5, Do. of moon 353° 0′ 36″0, Do. of moon's perigee 36° 9′ 48″.4. He also verified the statement that the sunrise on the morning of March 8th was that immediately following new moon. The difference in result is partly caused by the fact that Leverrier's and Hansen's longitudes are tropical, and those of the Sûrya-Siddhânta sidereal. Comparing the two results we find a difference of 0° 35′ 40″.9 in " $a$ ". 5° 24′ 49″.69 in " $b$ ", 0° 11′ 15″.87 in " $c$ ". The closeness of the results obtained from the use of (1) purely Hindu (2) purely European methods is remarkable. Our Tables being for Indian documents and inscriptions we of course work by the former. [R. S.]
↑ This year Śaka 1000 is chosen for convenience of addition or subtraction when calculating other years, and therefore we have not taken into account the fact that Ś 1000 was really an intercalary year, having both an Adhika Jyeshṭha and a Nija Jyeshṭha month. That peculiarity affects only that one year and not the concurrence of other months of previous or subsequent tears in other eras.
↑ Prof. Jacobi gives this as 200.5, but after most careful calculation I find it to be 200.6. [S. B. D.]
↑ Prof. Jacobi has not explained these Tables.
↑ This Table contains Prof. Jacobi's Table 11 (Ind. Ant., XVII., p. 147) and his Table 17, p. 181, in a modified form. [S. B. D.]
↑ The Table contains Prof. Jacobi's Table 11 (Ind. Ant., XVII., p. 172), as well as his Table 17 Part II. (id. p. 181) modified and enlarged. I have also added the equivalents for tithi parts, and an explanation. [S. B. D.]
↑ Of course only two in a single case, but four during the entire period of 1600 years covered by our Tables.
↑ The exact tithi can be calcalated by Arts. 149 and 151.
↑ Art. 130 has been omitted.
↑ Equation $c$ is the equation in Table VII.
↑ Reference to the diagram in Art. 108 will make all this plain, if $PSE$ be taken as the sun's mean anomaly, and $ESE'$ the equation of the centre, $PSE'$ + longitude of the sun's perigee being the sun's true or apparent longitude.
↑ This is actually wrong by one day, owing to the approximate collective duration of days (Table III, 3a) being taken as 89. 11 might equally well be taken as 88. If it is desired to convert tithis into days (p. 75. note 2) a 64th part should be subtracted. The collective duration of the last day of Jyeshṭha in tithis is 90. 90 ÷ 64 = 1.40. 90 − 1.40 = 88.60. If taken as 88 the answer would be Saturday, July 6th, which is actually correct. This serves to shew how errors may arise in days when calculation is only made approximately.
↑ The actual date was Tuesday, amânta Chaitra kṛishṇa 3rd, the difference being caused by a tithi having been expunged in the śukla fortnight of the same month (see note to examples 6 and 12 above).
↑ The initial days in cols 13 and 19, Table I., belong to the first of the double years A.D given in col 5.
↑ It will be well for a beginner to take an example at once, and work it out according to the rule. After a little practice the calculations can be made rapidly.
↑ When the intercalary month is Chaitra, count that also. See Art. 99 above.
↑ This number is taken for easy calculation. Properly speaking, to convert tithis into days the 64th part should be subtracted. The difference does not introduce any material error.
↑ Generally with regard to ( $w$ ), ( $a$ ), ( $b$ ), ( $c$ ) in working addition sums, take only the remainder respectively over 7, 10000, 1000 and 1000; and in subtracting, if the sum to be subtracted be greater, add respectively 7, 10000, 1000 and 1000 to the figure above.
↑ Thus far the process will give the correct result if there be no probability by the rule given below of the expunction (kshaya) or repetition (vṛiddhi) of a tithi shortly preceding or following; and the ( $d$ ) and ( $w$ ) arrived at at this stage will indicate by use of Table IX. the A.D. equivalent, and the week-day of the given tithi.
↑ For the definitions of expunged and repealed tithis see Art. 32 above.
↑ See Arts. 36 and 37 in which all the points noted in this article are fully treated of.
↑ This is shown by $(t)=3668$ at sunrise, the end being indicated by 3667. Difference 1 lunation-unit, or 4 minutes.
↑ It would have so begun if the saṅkrânti occurred at 7 p.m. on the Wednesday, or at any time after sunset (6 p.m.)
↑ This date is from an actual inscription in Southern India. (See Ind. Ant., XXII., p. 219).
↑ It is found by actual calculation under Art. 156 that the given nakshatra falls on the same date, and therefore we know that the above result is correct.
↑ This problem is easier than its converse, the number of intervening days here being certain.
↑ If the Rule I(a) in Art. 104 (Table II., Part iii.) be applied, this latter part of the rule necessarily follows.
↑ A 59th part, or more properly 63rd, should be added, but by adding a 60th, which is more convenient, there will be no difference in the ultimate result. Neglect the fraction half or less, and take more than half as equivalent to one.
↑ This total is the approximate number of tithis which have intervened. When it is the same as, or very near to, the number of tithis forming the collective duration up to the end of a month (as given in col. 3, Table III.), there will be some doubt about the required month; but this difficulty will be easily solved by comparing together the resulting tithi and the number of tithis which have intervened.
↑ See Art. 21, and notes 1 and 2, and Arts. 93 and 96.
↑ See note 4, p. 90.
↑ For the definition of karaṇas, and other information regarding them, see Arts. 10 and 40.
↑ This will illustrate the danger of trusting to Tables XIV. and XV. in important cases.
↑ Note that this approximate calculation, which is the same as that by method B, comes out actually wrong by two days.
↑ In this case the result by the approximate method A or B will be wrong by two days.
↑ A year of the Hijra = 0.970223 of Gregorian year, and a Gregorian year = 1.03069 years of the Hijra. Thus 32 Gregorian years are about equal to 33 years of the Hijra, or more nearly 163 Gregorian years are within less than a day of 168 Hijra years.
↑ So far as I know no European chronologist of the present century has noticed this point. Tables could be constructed for the heliacal rising of the moon in every month of every year, but it would be too great a work for the present publication. [S. B. D.]

[1] See Art. 126 below.

[2] It seems almost certain that both systems had a common origin in Chaldæa. The first is the day of the sun, the second of the moon, the third of Mars, the fourth of Mercury, the fifth of Jupiter, the sixth of Venus, the seventh of Saturn. [R. S.]

[3] The word vâra is to be affixed to each of these names; Ravi = Sun, Ravivâra = Sunday.

[4] In the Table, for convenience of addition, Saturday is styled 0.

[5] The variation is of course really in the motions of the earth and the moon. It is caused by actual alterations in rate of rapidity of motion in consequence of the elliptical form of the orbits and the moon's actual perturbations; and by apparent irregularities of motion in consequence of the plane of the moon's orbit being at an angle to the plane of the ecliptic. [R. S.]

[6] An apt title. The full moon stands as it were with the waxing half on one side and the waning half on the other. The week is an arbitrary division.

[7] The "synodic revolution" of the moon is the period during which the moon completes one series of her successive phases, roughly 29½ days. The period of her exact orbital revolution is called her "sidereal revolution". The term "synodic" was given because of the sun and moon being then together in the heavens (cf: "synod"). The sidereal revolution of the moon is less by about two days than her synodic revolution in consequence of the forward movement of the earth on the ecliptic. This will be best seen by the accompanying figure, where ST is a fixed star, S the sun, E the earth, C the ecliptic, M M' the moon. (A) the position at one new moon, (B) the position at the next new moon. The circle M to Ml representing the sidereal revolution, its synodic revolution is M to Ml plus Ml to N. [R. S.]
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C. A. Young ("General Astronomy", Edit. of 1889, p 528) gives the following as the length in days of the various lunations:

d. h. m. s.

Mean synodic month (new moon to new moon) 29 12 44 2.684

Sidereal month 27 7 43 11.545

Tropical month (equinox to equinox) 27 7 43 4.68

Anomalistic month (perigee to perigee) 27 13 18 37.44

Nodical month (node to node) 27 5 5 35.81

[8] See Fleet's Corpus Inscrip. Indic., vol III., Introduction, p. 79 note; Ind. Ant., XVII., p. 141 f.

[9] Compare the note on p. 4 on the moon's motion. [R. S]

[10] This rate of annual precession is that fixed by modern European Astronomy, but since the exact occurrence of the equinoxes can never become a matter for observation, we have, in dealing with Hindu Astronomy, to be guided by Hindu calculations alone. It must therefore be borne in mind that almost all practical Hindu works (Karaṇas) fix the annual precession at one minute, or 1/60th of a degree, while the Sûrya-Siddhânta fixes it as 54″ or 3/200 degrees. (see Art. 160a. given in the Addenda sheet.)

[11] The anomaly of a planet is its angular distance from its perihelion, or an angle contained between a line drawn from the sun to the planet, called the radius vector, and a line drawn from the sun to the perihelion point of its orbit. In the case in point, the earth, after completing its sidereal revolulion, has not arrived quite at its perihelion because the apsidal point has shifted slightly eastwards. Hence the year occupied in travelling from the old perihelion to the new perihelion is called the anomalistic year. A planet's true anomaly is the actual angle as above whatever may be the variations in the planet's velocity at different periods of its orbit. Its mean anomaly is the angle which would be obtained were its motion between perihelion and aphelion uniform in time, and subject to no variation of velucity—in other words the angle described by a uniformly revolving radius vector. The angle between the true and mean anomalies is called the equation of the centre. True anom. = mean anom. + equation of the centre.

The equation of the centre is zero at perihelion and aphelion, and a maximum midway between them. In the case of the sun its greatest value is nearly 1°.55′ for the present, the sun getting alternately that amount ahead of, and behind, the position it would occupy if its motion were uniform. (C. A. Young, General Astronomy. Edit. of 1889, p. 125.)

Prof. Jacobi's, and our, $a$ , $b$ , $c$ , (Table 1., cols. 23, 24, 25) give $a$ . the distance of the noon from the sun, expressed in 10,000ths of the unit of 360°; $b$ . the moon's mean anomaly; $c$ . the sun's mean anomaly; the two last expressed in 1000ths of the unit of 360°. The respective equations of the centre are given in Tables VI. and VII. [R. S.]

[12] "The ecliptic slightly and very slowly shifts its position among the stars, thus altering the latitudes of the stars and the angle between the ecliptic and equator, i.e., the obliquity of the ecliptic. This obliquity is at present about 24′ less than it was 2000 years ago, and it is still decreasing about half a second a year. It is computed that this diminution will continue for about 15,000 years, reducing the obliquity to 22¼°, when it will begin to increase. The whole change, according to Lagrange, can never exceed about 1° 2′ on each side of the mean." (C. A. Young, General Astronomy, p. 128.)

[13] Generally speaking an astronomical Sanskrit work, called a Siddhânta, treats of the subject theoretically. A practical work on astronomy based on a Siddhânta is called in Sanskrit a Karaṇa. The Paitâmaha and following three Siddhântas are not now extant, but are alluded to and described in the Pañchasiddhântikâ, a Karaṇa by Varâhamihira, composed in or about the Saka year 427 (A. D. 505). [S. B. D.]

[14] Two other Pauliśa Siddhântas were known to Ulpala (A.D. 966), a well-known commentator of Varâhamihira. The length of the year in them was the same as that in the original Sûrya Siddhânta. [S. B. D.]

[15] The duration of the year by the First Arya-Siddhânta is noted in the interesting chronogram mukhyaḥ
muk5hyah1 kâlomayamâtulaḥ
ka1lo3ma5ya1ma5tu6lah3. These figures are to be read from right to left; thus—365, 15, 31, 15 in Hindu notation of days, ghaṭikâs, etc. (I obtained this from Dr Burgess—R. S.)

[16] The Parâśara Siddhânta is not now extant. It is described in the second Ârya Siddhânta. The date of this latter is not given, but in my opinion it is about A.D. 950. [S. B. D.]

[17] The Râjamṛigâṅka is a Karaṇa by King Bhoja. It is dated in the Śaka year 964 expired, A.D. 1012. [S. B. D.]

[18] Karaṇas and other practical works, containing tables based on one or other of the Siddhântas, are used for these calculations. [S. B. D.]

[19] The positions and motions of the sun and moon and their apogees must necessarily be fixed and known for the correct calculation of a tithi, nakshatra, yoga or karaṇa. The length of the year is also an important clement, and in the samvatsara is governed by the movement of the planet Jupiter. In the present work we are concerned chiefly with these six elements, viz., the sun, moon, their apogees, the length of the year, and Jupiter. The sketch in the text is given chiefly keeping in view these elements. When one authority differs from another in any of the first five of these six elements the tithi as calculated by one will differ from that derived from another. [S. B. D.]

[20] It is not to be understood that as soon as a standard work comes into use its predecessors go out of use from all parts of the country. There is direct evidence to show that the original Sûrya-Siddhânta was in use till A.D. 665, the date of the Khaṇḍa-khâdya of Brahmagupta, though evidently not in all parts of the country. [S. B. D.]

[21] Whenever we allude simply to the "Brahma Siddhânta" by name, we mean the Brahma-Siddhânta of Brahmagupta.

[22] Out of the six elements alluded to in note 1 on the last age, only Jupiter has this bîja. The present Sûrya-Siddhânta had undoubtedly come into use before the date of the Bhâsvatî. [S. B. D.]

[23] It is probable that the first Ârya-Siddhânta was the standard authority for South Indian solar reckoning from the earliest times. In Bengal the Sûrya-Siddhânta is the authority since about A.D. 1100, but in earlier times the first Ârya-Siddhânta was apparently the standard. [S. B. D.]

[24] When we allude simply to the Sûrya or Ârya-Siddhânta it must be borne in mind that we mean the Present Sûrya and the First Ârya-Siddhântas.

[25] See note 1, p. 2 above. [R. S.]

[26] My present opinion is that the zodiacal-sign-names, Mesha, etc., began to be used in India between 700 B. C. and 300 B. C., not earlier than the former or later than the latter. [S. B. D.]

[p10n2-27] It will be seen that the Bengal names differ from the Tamil ones. The same solar month Mesha, the first of the year, is called Vaiśâkha in Bengal and Sittirai (Chaitra) in the Tamil country, Vaiśâkha being the second month in the south. To avoid confusion, therefore, we use only the sign-names (Mesha, etc.) in framing our rules.

[p11n1-28] The lengths of months by the Ârya-Siddhânta here given are somewhat different from those given by Warren. But Warren seems to have taken the longitude of the sun's apogee by the Sûrya-Siddhânta in calculating the duration of months by the Ârya-Siddhânta, which is wrong. He seems also to have taken into account the chara.^[29] (See his Kâla Saṅkalita, p. 11, art. 3, p. 22, explanation of Table III., line 4; and p. 3 of the Tables). He has used the ayanâṁśa (the uniformly increasing arc between the point of the vernal equinox each year and the fixed point in Aries) which is required for finding the chara in calculating the lengths of months. The chara is not the same at the beginning of any given solar mouth for all places or for all years. Hence it is wrong to use it for general rules and tables. The inaccuracy of Warren's lengths of solar months according to the Sûrya-Siddhânta requires no elaborate proof, for they are practically the same as those given by him according to the Ârya-Siddhânta, and that this cannot be the ease is self-evident to all who have any experience of the two Siddhântas. [S. B. D.]

[p11n1a-29] The chara:—"The time of rising of a heavenly body is assumed to take place six hours before it comes to the meridian. Actually this is not the case for any locality not on the equator, and the chara is the correction required in consequence, i.e., the excess or defect from six hours of the time between rising and reaching the meridian. The name is also applied to the celestial arc described in this time."

[30] The Sanskrit word for "mean" is madhyama, and that for 'true' or 'apparent' is spashṭa. The words "madhyama" and "spashṭa" are applied to many varieties of time and space; as, for instance, gati (motion), bhôga (longtitude), saṅkrânti, mâna (measure or reckoning) and kâla (time). In the English Nautical Almanac the word "apparent" is used to cover almost all cases where the Sanskrit word spashṭa would be applied, the word 'true' being sometimes, but rarely, used. "Apparent," therefore, is the best word to use in my opinion; and we have adopted it prominently, in spite of the fact that previous writers on Hindu Astronomy have chiefly used the word "true." There is as a fact a little difference in the meaning of the phrases "apparent" and "true," but it is almost unknown to Indian Astronomy, and we have therefore used the two words as synonyms. [S. B. D.]

[31] Remember that the week-day is counted from sunrise to sunrise.

[32] Brown's Ephemeris follows this rule throughout in fixing the date corresponding to 1st Mesha, and consequently his solar dates are often wrong by one day for those tracts where the 2 b rule is in use.

[p12n3-33] I deduced the Bengal rule from a Calcutta Pañchâṅg for Śaka 1776 (A.D. 1854—55) in my posssession. Afterwards it was corroborated by information kindly sent to me from Howrah by Mr. G. A. Grierson through Dr. Fleet. It was also amply corroborated by a set of Bengal Chronological Tables for A.D. 1882, published under the authority of the Calcutta High Court, a copy of which was sent to rac by Mr. Scwell. I owe the Orissa Rule to the Chronological Tables published by Girishchandra Tarkâlaukar, who follows the Orissa Court Tables with regard to the Amli and Vilayati years in Orissa. Dr. J. Burgess, in a note in Mr. Kṛishṇasvâmi Naidu's "South Indian Chronological Tables" edited by Mr. Sewell. gives the 2 (a) Rule as in use in the North Malayâḷam country, but I do not know what his authority is. I ascertained from Tamil and Tinnevelly panchangs that the 2 (a) rule is in use there, and the fact is corroborated by Warren's Kâla Saṅkalita; I ascertained also from some South Malayâḷam pañchâṅgs published at Cochin and Trevandruni, and from a North Malayâḷam pañchâṅg published at Calicut, that the 2 (b) rule is followed there [S. B. D.]

Notwithstanding all this I have no certain guarantee that these are the only rules, or that they are invariably followed in the tracts mentioned. Thus I find from a Tamil solar pañchâṅg for Śaka 1815 current, published at Madras, and from a Telugu luni-solar pañchâṅg for Śaka 1109 expired, also published at Madras, in which the solar months also are given, that the rule observed is that "when a saṅkrânti occurs between sunrise and midnight the month begins on the same day, otherwise on the following day", thus differing from all the four rules given above. This varying fifth rule again is followed for all solar months of the Vilayati year as given in the above-mentioned Bengal Chronological Tables for 1882, and by its use the month regularly begins one day in advance of the Bengâli month. I find a sixth rule in some Bombay and Benares lunar pañchâṅgs, viz., that at whatever time the saṅkrânti may occur, the month begins on the next day; but this is not found in any solar pañchâṅg. The rules may be further classified as (1. a) the midnight rule (Bengal), (1. b) any time rule (Orissa), (2. a) the sunset rule (Tamil), (2. b) the afternoon rule (Malabar). The fifth rule is a variety of the midnight rule, and the sixth a variety of the any time rule. I cannot say for how many years past the rules now in use in the several provinces have been in force and effect.

An inscription at Kaṇṇaṇûr, a village 5 miles north of Srîraṅgam near Trichinopoly (see Epigraph. Indic, vol. III., p. 10, date No. V., note 3, and p. 8) is dated Tuesday the thirteenth tithi of the bright fortnight of Śrâvaṇa in the year Prajâpati, which corresponded with the 24th day of the (solar) month Âḍi (Karka.) From other sources the year of this date is known to be A.D. 1271; and on carefully calculating I find that the day corresponds with the 21st July, and that the Karka saṅkrânti took place, by the Ârya-Siddhânta, on the 27th June, Saturday, shortly before midnight. From this it follows that the month Âḍi began civilly on the 28th June, and that one or the other of the two rules at present in use in Southern India was in use in Trichinopoly in A.D. 1271. [S. B. D.]

[34] We cannot enumerate the vulgar or popular names which obtain in all parts of India, and it is not necessary that we should do so.

[35] This is an ordinary pañchâṅg in daily use. It was prepared by myself from Gaṇeśa Daivjña's Grahalâghava and Laghutithichintâmaṇi. [S. B. D.]

[36] Solar days are not given in Bombay pañchâṅgs, but I have entered them here to complete the calendar. Some entries actually printed in the pañchâṅg arc not very useful and are consequently omitted in the extract. [S. B. D.]

[37] The sum total of days that have elapsed since any other standard epoch is also called the ahargaṇa. For instance, the ahargaṇa from the beginning of the present kaliyuga is in constant use. The word means "collection of days."

[38] The Nirṇayasindhu is one of these authorative works, and is in general use at the present time in most parts of India.

[39] Any assertions or definitions by previous writers on Hindu Chronology or Astronomy contrary to the above definitions and examples are certainly erroneous, and due to misapprehension. [S. B. D.]

[40] Instead of writing at full length that such and such a tithi "ends at so many ghaṭikâs after sunrise, Indian astronomers say for brevity that the tithi "is so many ghaṭikâs". The phrase is so used in the text in this sense.

[41] In the case of kshayas in the pañchâṅg extract the ghaṭikâs of expunged tithis etc., are to be counted after the end of the previons tithi etc. In some pañchâṅgs the ghaṭikâs from sunrise—59 gh. 55pa. in the present instance— are given.

[42] Since this work was in the Press, Professor Jacobi has published in the Epigraphica Indica (Vol. II, pp. 487—498) a treatise with tables for the calculation of Hindu dates in true local time, to which we refer our readers.

[43] Here Lanka is not Ceylon, but a place supposed to be on the equator, or in lat. 0° 0′ 0″ on the meridian of Ujjain, or longitude 75° 46′. It is of great importance to know the exact east longitude of Ujjain, since upon it depends the verification of apparent phenomena throughout India. Calculation by the different Siddhântas can be checked by the best European science if that point can be certainly determined. The great Trigonometical Survey map makes the centre of the city 75° 49′ 45″ E. long, and 23° 11′ 10″ N. lat. But this is subject to two corrections; first, a correction of 1′ 9″ to reduce the longitude to the origin of the Madras Observatory taken as 80° 17′ 21″, and secondly, a farther reduction of 2′ 30″ to reduce it to the latest value, 80° 14′ 51″. of that Observatory, total 3′ 39″. This reduces the E. long, of the centre of Ujjain city to 75° 46′ 06″. I take it therefore, that amidst conflicting authorities, the best of whom vary from 75° 43′ to 75° 51′, we may for the present accept 75° 46′ as the nearest approach to the truth. The accuracy of the base, the Observatory of Madras, will before long be again tested, and whatever difference is found to exist between the new fixture and 80° 14′ 51″, Ihal difference applied to 75° 46' will give the correct value of the E. long, we require. [R. S.]

[44] The mean length of the moon's revolution among the stars is 27.32166 days (27.321674 according to the Sûrya Siddhânta). Its least duration is 27 days, 4 hours, and the greatest about 7 hours longer. The number of days is thus between 27 and 28, and therefore the number of nakshatras was sometimes taken as 28 by the ancient Indian Âryas. The extra nakshatra is called Abhijit (See Table VIII., col. 7.) [S. B. D.]

[45] These systems of nakshatras arc more fully described by me in relation to the "twelve year cycle of Jupiter" in Vol. XVII. of the Ind. Ant., (p. 2 ff.) [S. B. D.]

[46] According to the Sûrya-Siddhânta the four karaṇas are Śakuni, Nâga, Chatushpada and Kiṁstughna, but we have followed the present practice of Western India, which is supported by Varâhamihira and Brahmagupta.

[47] Madhu is "honey", "sweet spring". Mâdhava, "the sweet one". Sukra and Śuchi both mean "bright". Nabhas, the rainy season. Nabhasya, "vapoury", "rainy". Ish or Isha, "draught" or "refreshment", "fertile". Ûrj, "strength", "vigour". Sahas "strength". Sahasya "strong". Tapas "penance", "mortification", "pain", "fire". Tapasya, "produced by heat", "pain". All are Vedic words.

[48] In my opinion the sidereal names "Chaitra" and the rest, came into use about 2000 B. C. They are certainly not later than 1500 B.C., and not earlier than 4000 B.C. [S. B. D.]

[49] Professor Kielhorn is satisfied that the terms adhika and nija are quite modern, the nomenclature usually adopted in documents and inscriptions earlier then the present century being prathama (first) and dvitîyâ (second). He alluded to this in Ind. Ant., XX., p. 411. [R. S.]

[50] The scheme of pûrṇimânta months and the rule for naming the intercalated months known to have been in use from the 12th century A.D., are followed in this diagram.

[51] See his Siddhânta-Siromaṇi, madhyamâdhikâra, adhimâsanirṇaya, verse 6, and his own commentary on it. [S. B. D.]

[52] It is not to be found in either of the Brâhma-Siddhântas referred to above, but there is a third Brâhma-Siddhânta which I have not seen as yet. [S. B. D.]

[53] In Prof. Chattre's list of added and suppressed mouths, in those published in Mr. Cowasjee Patells' Chronology, and in General Sir A. Cunningham's Indian Eras it is often noted that the same mouth is both added and suppressed. But it is clear from the above rules and definitions that this is impossible. A month cannot be both added and suppressed at the same time. The mistake arose probably from resort being made to the first rule for naming adhika months, and to the second for the suppressed months.

[54] Thanks are due to Mr. Mahadeo Chiṁṇâjî Apte. B.A., L.L.B., very recently deceased, the founder of the Anandâśrama at Poona, for his discovery of a part of Śrîpati's Karaṇa named the Dhîkoṭida, from which I got Śrîpati's date. I find that it was written in Śaka 961 expired (A.D. 1039–40). [S. B. D.]

[55] Up to recently the date was considered to be about the 6th century A.D. Dr. Thibaut, one of the highest living authorities on Indian Astronomy, fixes it at 400 A.D. (See his edition of the Pañcha Siddhântikâ Introd., p. LX.). My own opinion is that it came into existence not later than the 2nd century B.C. [S. B. D.]

[56] I am inclined to believe that of the two rules for naming lunar months the second was connected with the mean system of added months, and that the first came into existence with the adoption of the true system. But I am not as yet in possession of any evidence on the point. See, however, the note to Art. 51 below. [S. B. D.]

[57] It is difficult to define the exact limit, because it varies with different Siddhântas, and even for one Siddhânta it is not always the same. It is, however, generally not more than six ghaṭikâs, or about 33 of our tithi-indices ( $t$ ). But in the case of some Siddhântas as corrected with a bîja the difference may amount sometimes to as much as 20 ghaṭikâs, or 113 of our tithi-indices. It would be very rare to find any difference in true added months; but in the case of suppressed months we might expect some divergence, a month suppressed by one authority not being the same as that suppressed by another, or there being no suppression at all by the latter in some cases. Differences in mean added months would be very rare, except in the case of the Brâhma-Siddhânta. (See Art. 88.)

[58] This relation of intervals is a distinct assistance to calculation, as it should lead us to look with suspicion on any suppression of a month which docs not conform to it.

[59] See the Siddhânta-Siromaṇi, Madhyamâdhikâra. Bhâskara wrote in Śaka 1073 (A.D. 1150). He did not give the names of the suppressed months.

[60] I have ascertained that Gaṇeśa has adopted in his Grahalâghava some of the elements of the Ârya-Siddhânta as corrected by Lalla's bîja, and by putting to test one of the years noted I find that in these calculations also the Ârya-Siddhânta as corrected by Lalla's bîja was used. Gaṇeśa was a most accurate calculator, and I feel certain that his results can be depended upon. [S. B. D.]

[61] Such an anomaly with regard to the pûrṇimânta scheme could not occur if the two rules were applied, one that "that pûrṇimânta month in which the Mesha saṅkrânti occurs is always called Chaitra, and so on in succession," and the other that "that pûrṇimânta month in which no saṅkrânti occurs is called an intercalated month." The rules were, I believe, in use in the sixth century AD. (See may remarks Ind. Ant., XX., p. 50 f.) But the added month under such rules would never agree with the amânta added months. There would be from 14 to 17 months' difference in the intercalated months between the two, and much inconvenience would arise thereby. It is for this reason probably that the pûrṇimânta scheme is not recognised in naming months, and that pûrṇimânta months are named arbitrarily, as described in the first para. of Art. 51. This arbitrary rule was certainly in use in the 11th century A.D. (See Ind. Ant., vol. VI., p. 53, where the Makara-saṅkrânti is said to have taken place in Mâgha.)

After this arbitrary rule of naming the pûrṇimânta months once came into general use, it was impossible in Northern India to continue using the second, or Brâhma-Siddhânta, rule for naming the months. For in the example in Art. 45 above the intercalated month would by that rule be named Chaitra, but if its preceding fortnight be a fortnight of Vaiśâkha it is obvious that the intercalated month cannot be named Chaitra. In Southern India the practice may have continued in use a little longer. [S. B. D.]

[62] Chaitrâdi, "beginning with Chaitra"; Kârttikâdi, "beginning with Kârttika"; Meshâdi, with Mesha; and so on.

[63] See Ind. Ant., XIX., p. 45, second paragraph of my article on the Original Sûrya-Siddhânta. [S. B. D.]

[64] I have myself seen a pañchâṅg which mentions this beginning of the year, and have also found some instances of the use of it in the present day. 1 am told that at Iḍar in Gujarât the Vikrama samvat begins on Âshâḍha kṛishṇa dvitîyâ. [S. B. D.]

[65] The passage, as translated by Sachau (Vol. II., p. 8 f), is as follows. "Those who use the Saka era, the astronomers, begin the year with the month Chaitra, whilst the inhabilunts of Kanîr, which is conterminous with Kashmîr, begin it with the month Bhâdrapada…All the people who inhabit the country between Bardarî and Mârîgala the year with the mouth Kârttika…The people living in the country of Nîrahara, behind Mârîgala, as far as the utmost frontiers of Tâkeshar and Lohâvar, begin the year with the month Mârgaśîrsha…The people of Lanbaga, i.e., Lamghân, follow their example. I have been told by the people of Multân that this system is peculiar to the people of Sindh and Kanoj, and that they used to begin the year with the new moon of Mârgaśîrsha, hut that the people of Multân only a few years ago had given up this system, and had adopted the system of the people of Kashmîr, and followed their example in beginning the year with the new moon of Chaitra."

[66] Articles 53 to 61 are applicable to Northern India only (See Art. 62).

[67] The term is one not recognized in Sanskrit works. [S. B. D.]

[68] See Ind. Ant., Vol. XIX., pp. 27, 33, 187.

[69] These points have not yet heen noticed by any European writer on Indian Astronomy. [S. B. D.]

[70] As to the mean Mesha-saṅkrânti, see Art. 26 above.

[71] By all these rules the results will be correct within two ghaṭikâs where the moment of the Mesha-saṅkrânti according to the authority used is known.

[72] The rule for the present Vasishṭha, the Śâkalya Brahma, the Romaka, and the Soma Siddhântas is exactly the same. That by the original Sûrya-Siddhânta is also similar, but in that case the result will be incorrect by about 2 ghaṭikâs (48 minutes). For all these authorities take the time of the Mesha-saṅkrânti by the present Sûrya-Siddhânta or by the Ârya-Siddhânta, whichever may be available. The moment of the Mesha-saṅkrânti according to the Sûrya-Siddhânta is given in our Table I. only for the years A.D. 1100 to 1900. The same moment for all years between A.D. 300 and 1100 can be found by the Table in Art. 96. If the Ârya-Siddhânta saṅkrânti is used for years A.D. 300 to 1100 the result will never be incorrect by more than 2 ghaṭikâs 46 palas (1 hour and 6 minutes). The Table should be referred to.

[73] In these three rules the apparent Mesha-saṅkrânti is taken. If we omit the subtraction of 108, 11, and 60, and do not add 15 p., 1 gh. 45 p., and 15 p. respectively, the result will be correct with respect to the mean Mesha-saṅkrânti.

[74] I have not seen the Jyotishatattva (or "Jyotishtava" as Warren calls it, but which seems to be a mistake), but I find the rule in the Ratnamâlâ of Śrîpati (A.D. 1039). It must be as old as that by the Ârya-Siddhânta, since both are the same. [S. B. D.]

[75] If we add 4280 instead of 4291, and add 1 gh. 45 pa. to the final result, the time so arrived at will be the period elapsed since apparent Mesha-saṅkrânti. Those who interpret the Jyotishatattva rule in any different way have failed to grasp its proper meaning. [S. B. D.]

[76] It is not stated what Mesha-saṅkrânti is meant, whether mean or apparent. The rule is here given as generally interpreted by writers both Indian and European, but in this form its origin cannot be explained. I am strongly inclined to think that Varâhamihira, the author of the Bṛihatsaṁhitâ, meant the rule to run thus: Multiply the current Śaka year by 44. Add 8582 (or 8581 or 8583). Divide the sum by 3750. To the integers of the quotient add the given current Śaka year; (and the rest as above). The result is for the mean Mesha-saṅkrânti." In this form it is the same as the Ârya-Siddhânta or the Jyotishatattva rule, and can be easily explained. (S. B. D.)

[77] In this Table the Bṛihatsaṁhitâ rule is worked as I interpret it. But as interpreted by others the expunctions will differ, the differences being in Śaka (current) 231, the 56th; 998, the 52nd; 1889, the 37th.

By the Sûrya Siddhânta the years marked with an asterisk in the Śaka column of this Table differ from those given in Table I., col. 7, being in each case one earlier; the rest are the same. (S. B. D.)

[78] See Corpus Inscrip. Indic., Vol. III., p. 80, note; Ind. Antiq., XVII., p. 142.

[79] The heliacal rising of a superior planet is its first visible rising after its conjunctions with the sun, i.e., when it is at a sufficient distance from the sun to be first seen on the horizon at its rising in the morning before sunrise, or, in the case of an inferior planet (Mercury or Venus), at its setting in the evening after sunset. For Jupiter to be visible the sun must be about 11° below the horizon. [R. S.]

[80] It is fully described by me in the Indian Antiquary, vol. XVII. [S. B. D.]

[81] In practice of course the word "current" cannot be applied to the year 0, but it is applied here to distinguish it from the year 0 complete or expired, which means year 1 current. We use the word "epoch" to mean the year 0 current. The epoch of an era given in a year of another era is useful for turning years of one into years of another era. Thus, by adding 3078 (the number of the Kali year corresponding to the Graha-parivṛitti cycle epoch) to a Graha-parivṛitti year, we can get the equivalent Kali year; and by subtracting the same from a Kali year we get the corresponding Graha-parivṛitti year.

[82] Or Aṅka.

[83] On the 11th according to some, but all the evidence tends to shew that the year begins on the 12th.

[84] The real date of the Muhammadan invasion seems to be 1568 A.D. (J. A. S. B. for 1883, LII., p. 233, note). The invasion alluded to is evidently that of the "Yavanas", but as to these dates these temple chronicles must never be believed. [R. S.]

[85] Some say that the first year is also dropped, similarly; but this appears to be the result of a misunderstanding, this year being dropped only to fit in with the system described lower down in this article. Mr. J. Beames states that "the first two years and every year that has a 6 or 0 in it are omitted", so that the 37th Oṅko of the reign of Rāmachandra is really his 28th year, since the years 1, 2, 6, 10, 16, 20, 26, 30 and 86 are omitted. (J. A. S. B. 1883, LII., p. 234, note. He appears to have been misled about the first two years.

[86] Sewell's Sketch of the Dynasties of Southern India, p. 64. Archæological Survey of Southern India, vol. II., p. 204.

[87] See 'Calculations of Hindu dates', by Dr. Fleet, in the Ind. Ant., vols. XVI. to XIX.; and my notes on the date of a Jain Purâṇa in Dr. Bhândârkar's "Report on the search for Sankrit manuscripts" for 1883—1884 A. D., p.p. 429—30 §§ 36, 37. [S. B. D.]

[88] The Vikrama era is never used by Indian astronomers. Out of 160 Vikrama dates examined by Dr. Kielhorn (Ind. Ant., XIX.), there are only six which have to be taken as current years. Is it not, however, possible that all Vikrama years are really current years, but that sometimes in writings and inscriptions the authors have made them doubly current in consequence of thinking them erroneously to be expired years. There is an instance of a Śaka year made twice current in an inscription published in the Ind. Ant., (vol. XX., p 191), The year was already 1155 current, but the number given by the writer of the inscription is 1156, as if 1155 had been the expired year.

As a matter of fact I do not think that it is positively known whether the years of the Christian era are themselves really expired or current years. Warren, the author of the Kálasaṅkalita was not certain. He calls the year corresponding to the Kali year 3101 expired "A.D. 0 complete" (p 302) or "1 current" (p. 294). Thus, by his view, the Christian year corresponding to the Kali year 3102 expired would be A.D. 1 complete or A.D. 2 current. But generally European scholars fix A. D. 1 current as corresponding to Kali 3102 expired. The current and expired years undoubtedly give rise to confusion. The years of the astronomical eras, the Kali and Śaka for instance, may, unless the contrary is proved, be assumed to be expired years, and those of the non-astronomical eras, such as the Vikrama, Gupta, and many others, may be taken as current ones. (See, however, Note 3, p. 42, below.) [S. B. D.]

[89] Corpus Inscrip. Ind., Vol. III., Introduction, p. 69, note.

[90] Ind. Ant, Vol. XX., p. 149 ff.

[91] In Bengâli pañchâṅgs the Vikrama Samvat, or Sambat, is given along with the Śaka year, and, like the North-Indian Vikrama Samvat, is Chaitrâdi pûrṇimânta.

[92] See Ind. Ant., vol. XVII., p. 93; also note 3, p 31, and connected Text.

[93] See, however, note 2 on the previous page.

[94] See Ind. Ant., vol. XII., pp. 213, 293; XI., p. 242 ff.

[95] I have seen only two examples in which authors of Karaṇas have used any other era along with the Śaka. The author of the Râma-vinoda gives, as the starting-point for calculations, the Akbar year 35 together with the Śaka year 1512 (expired), and the author of the Phattesâhaprakâśa fixes as its starting-point the 48th year of "Phattesâha" coupled with the Śaka year 1626. [S. B D.]

[96] Certain Telugu (luni-solar) and Tamil (solar) pañchâṅgs for the last few years, which I have procured, and which were printed at Madras and are clearly in use in that Presidency, as well as a Canarese pañchâṅg for A.D. 1893, (Śakâ 1816 current, 1815 expired) edited by the Palace Astronomer of H. H. the Mahârâjâ of Mysore, give the current Śaka years. But I strongly doubt whether the authors of these pañchâṅgs are themselves acquainted with the distinction between so-called current and expired years. For instance, there is a pañchâṅg annually prepared by Mr. Aṇṇa Ayyaṅgâr, a resident of Kañjnûr in the Tanjore District, which appears to be in general use in the Tamil country, and in that for the solar Meshâdi year corresponding to 1887—88 he uses the expired Śaka year, calling this 1809, while in those for two other years that I have seen the current Śaka year is used. I have conversed with several Tamil gentlemen at Poona, and learn from them that in their part of India the generality of people are acquainted only with the name of the samvatsara of the 60-year cycle, and give no numerical value to the years. Where the years are numbered, however, the expired year is in general use. I am therefore inclined to believe that the so-called current Śaka years are nowhere in use; and it becomes a question whether the so-called expired Śaka year is really an expired one. [S. B. D.]

[97] Indian Antiquary for August, 1888, vol. XVII., p. 215, and the Academy, of 10th Dec., 1887, p 394 f. I had myself calculated these same inscription-dates in March, 1887, and had, in conjunction with Dr. Fleet, arrived at nearly the same conclusions as Dr. Kielhorn's, but we did not then settle the epoch, believing that the data were not sufficiently reliable (Corpus. Inscrip. Indic., Vol. III., Introd., p 9. [S. B. D.] See also Dr. Kielhorn's Paper read before the Oriental Congress in London. [R. S.]

[98] The Vilâyatî era, as given in some Bengal Government annual chronological Tables, and in a Bengali pañchâṅg printed in Calcutta that I have seen, is made identical with this Amli era in almost every respect, except that its months are made to commence civilly in accordance with the second variety of the midnight rule (Art. 28). But facts seem to be that the Vilâyatî year commences, not on lunar Bhâdrapada śukla 12th, but with the Kanyâ saṅkrânti, while the Amli year does begin on lunar Bhâdrapada śukla 12th. It may be remarked that Warren writes—in A.D. 1825—(Kálasaṅkalita, Tables p. IX.) that the "Vilaity year is reckoned from the 1st of the krishna paksha in Chaitra", and that its numerical designation is the same with the Bengali San. [S. B. D.]

[99] Alberuni's India, English translation by Sachau, Vol. II., p. 5.

[100] Corpus Inscrip. Indic, Vol. III., Introd., p. 177 ff.

[101] Giriśa Chandra's Chronological Tables for A.D. 1764 to 1900.

[102] Warren (Kálasaṅkalita, p. 298) makes it commence in "the year 3537 of the Julian period, answering to the 1926th of the Kali yug". But this is wrong if, as we believe, the Kollam years are current years, and we know no reason to think them otherwise. Warren's account was based on that of Dr. Buchanan who made the 977th year of the third cycle commence in A.D. 1800. But according to the present Malabar use it is quite clear that the year commencing in 1800 A.D., was the 976th Kollam year.

[103] General Sir A. Cunningham's Indian Eras, p. 74.

[104] Ind. Ant., Vol. XVII., p. 246 ff.

[105] This much information is from General Cunningham's "Indian Eras"

[106] Ind. Ant., XIX., p. 1 ff.

[107] General Cunningham, in his "Indian Eras", gives it as 15th February; but that day was a Saturday.

[108] Prinsep's Indian Antiquities, II., Useful Tables, p. 171.

[109] Gen. Cunningham admittedly (p. 91) follows Cowasjee Patell's "Chronology" in this respect, and on examination I find that the added and suppressed months in these two works (setting aside some few mistakes of their own) agree throughout with Prof. Chhatre's list, even so far as to include certain instances where the latter was incorrect. Patell's "Chronology" was published fifteen years after the publication of Prof. Chhatre's list, and it is not improbable that the former was a copy of the latter. It is odd that not a single word is said in Cowasjee Patell's work to shew how his calculations were made, though in those days he would hare required months or even years of intricate calculation before he could arrive at his results. [S. B. D.]

[110] A thousandth part of a tithi is equal to 1.42 minutes, which is sufficiently minute for our purposes, but a thousandth of a lunation is equivalent to 7 hours 5 minutes, and this is too large; so that we have to take the 10000th of a lunation as our unit, which is equal to 4.25 minutes, and this suffices for all practical purposes. In this work therefore a lunation is treated of as having 10,000 parts, and a tithi 1000 parts.

[111] For finding the initial date of the luni-solar years Prof, Jacobi's Tables I. to XI. were used, and in the course of the calculations it was necessary to introduce a few alterations, and to correct some misprints which had crept in in addition to those noted in the already published errata-list. Thus, the earliest date noted in Tables I. to IV., being A.D. 354, these Tables had to be extended backwards by adding two lines more of figures above those already given. In Table VI., as corrected by the errata, the bîja is taken into account only from A.D. 1601, whereas we consider that it should be introduced from A.D. 1501 (see Art. 21). In Table VI. the century correction is given for the New (Gregorian) Style from A.D 1600 according to the practice in the most part of Europe. I have preferred, however, to introduce the New Style into our Tables from Sept. A.D. 1752 to suit English readers, and this necessitated an alteration in the century data for two centuries. [R. S.]

[112] It is the same according to Warren, but in this respect he is in error. (See note to Art. 24.)

[113] 42 calculations were thus made direct by the Sûrya-Siddhânta with and without the bîja, with the satisfactory result that the error in the final figure of the tithi-index originally arrived at was generally only of 1 or 2 units, while in some cases it was nil. It was rarely 3, and only once 4. It never exceeded 4. It may therefore be fairly assumed that our results are accurate. [S. B. D.]

[114] See Art. 21, and the first footnote appended to it.

[115] See note 1 to Art. 91.

[116] We have seen before (Arts. 45 etc. above) how months and tithis are sometimes added or expunged. Now in case of Chaitra śukla pratipadâ being current at sunrise on two successive days, as sometimes happens, the first of these civil days, i.e., the day previous to that given by us, is taken as the 8rst day of the Indian luni-solar year (see Art. 52). This does not, however, create any confusion in our method C since the quantities given in cols. 23 to 25 are correct for the day and time for which they are given; while as for our methods A and B, the day noted by us is more convenient.

[117] Calculating by Prof. Jacobi's Tables, $a$ , $b$ , $c$ , are 9980, 896 and 255, each of which is wrong by 1.

The above figures were submitted by me to Dr. Downing of ihe Nautical Almanack office, with a request that he would test the results by scientific European methods. In reply he gave me the following quantities, for the sun from Leverrier's Tables, and and for the moon from Hansen's Tables (for the epoch A.D. 300, March 8th, 6 am., for the meridian of Ujjain). Mean long of sun 345° 51′ 47″.7, Do. of sun's perigee 253° 54′ 58″.5, Do. of moon 353° 0′ 36″0, Do. of moon's perigee 36° 9′ 48″.4. He also verified the statement that the sunrise on the morning of March 8th was that immediately following new moon. The difference in result is partly caused by the fact that Leverrier's and Hansen's longitudes are tropical, and those of the Sûrya-Siddhânta sidereal. Comparing the two results we find a difference of 0° 35′ 40″.9 in " $a$ ". 5° 24′ 49″.69 in " $b$ ", 0° 11′ 15″.87 in " $c$ ". The closeness of the results obtained from the use of (1) purely Hindu (2) purely European methods is remarkable. Our Tables being for Indian documents and inscriptions we of course work by the former. [R. S.]

[118] This year Śaka 1000 is chosen for convenience of addition or subtraction when calculating other years, and therefore we have not taken into account the fact that Ś 1000 was really an intercalary year, having both an Adhika Jyeshṭha and a Nija Jyeshṭha month. That peculiarity affects only that one year and not the concurrence of other months of previous or subsequent tears in other eras.

[119] Prof. Jacobi gives this as 200.5, but after most careful calculation I find it to be 200.6. [S. B. D.]

[120] Prof. Jacobi has not explained these Tables.

[121] This Table contains Prof. Jacobi's Table 11 (Ind. Ant., XVII., p. 147) and his Table 17, p. 181, in a modified form. [S. B. D.]

[122] The Table contains Prof. Jacobi's Table 11 (Ind. Ant., XVII., p. 172), as well as his Table 17 Part II. (id. p. 181) modified and enlarged. I have also added the equivalents for tithi parts, and an explanation. [S. B. D.]

[123] Of course only two in a single case, but four during the entire period of 1600 years covered by our Tables.

[124] The exact tithi can be calcalated by Arts. 149 and 151.

[125] Art. 130 has been omitted.

[126] Equation $c$ is the equation in Table VII.

[127] Reference to the diagram in Art. 108 will make all this plain, if $PSE$ be taken as the sun's mean anomaly, and $ESE'$ the equation of the centre, $PSE'$ + longitude of the sun's perigee being the sun's true or apparent longitude.

[128] This is actually wrong by one day, owing to the approximate collective duration of days (Table III, 3a) being taken as 89. 11 might equally well be taken as 88. If it is desired to convert tithis into days (p. 75. note 2) a 64th part should be subtracted. The collective duration of the last day of Jyeshṭha in tithis is 90. 90 ÷ 64 = 1.40. 90 − 1.40 = 88.60. If taken as 88 the answer would be Saturday, July 6th, which is actually correct. This serves to shew how errors may arise in days when calculation is only made approximately.

[129] The actual date was Tuesday, amânta Chaitra kṛishṇa 3rd, the difference being caused by a tithi having been expunged in the śukla fortnight of the same month (see note to examples 6 and 12 above).

[130] The initial days in cols 13 and 19, Table I., belong to the first of the double years A.D given in col 5.

[131] It will be well for a beginner to take an example at once, and work it out according to the rule. After a little practice the calculations can be made rapidly.

[132] When the intercalary month is Chaitra, count that also. See Art. 99 above.

[133] This number is taken for easy calculation. Properly speaking, to convert tithis into days the 64th part should be subtracted. The difference does not introduce any material error.

[134] Generally with regard to ( $w$ ), ( $a$ ), ( $b$ ), ( $c$ ) in working addition sums, take only the remainder respectively over 7, 10000, 1000 and 1000; and in subtracting, if the sum to be subtracted be greater, add respectively 7, 10000, 1000 and 1000 to the figure above.

[135] Thus far the process will give the correct result if there be no probability by the rule given below of the expunction (kshaya) or repetition (vṛiddhi) of a tithi shortly preceding or following; and the ( $d$ ) and ( $w$ ) arrived at at this stage will indicate by use of Table IX. the A.D. equivalent, and the week-day of the given tithi.

[136] For the definitions of expunged and repealed tithis see Art. 32 above.

[137] See Arts. 36 and 37 in which all the points noted in this article are fully treated of.

[138] This is shown by $(t)=3668$ at sunrise, the end being indicated by 3667. Difference 1 lunation-unit, or 4 minutes.

[139] It would have so begun if the saṅkrânti occurred at 7 p.m. on the Wednesday, or at any time after sunset (6 p.m.)

[140] This date is from an actual inscription in Southern India. (See Ind. Ant., XXII., p. 219).

[141] It is found by actual calculation under Art. 156 that the given nakshatra falls on the same date, and therefore we know that the above result is correct.

[142] This problem is easier than its converse, the number of intervening days here being certain.

[143] If the Rule I(a) in Art. 104 (Table II., Part iii.) be applied, this latter part of the rule necessarily follows.

[144] A 59th part, or more properly 63rd, should be added, but by adding a 60th, which is more convenient, there will be no difference in the ultimate result. Neglect the fraction half or less, and take more than half as equivalent to one.

[145] This total is the approximate number of tithis which have intervened. When it is the same as, or very near to, the number of tithis forming the collective duration up to the end of a month (as given in col. 3, Table III.), there will be some doubt about the required month; but this difficulty will be easily solved by comparing together the resulting tithi and the number of tithis which have intervened.

[146] See Art. 21, and notes 1 and 2, and Arts. 93 and 96.

[147] See note 4, p. 90.

[148] For the definition of karaṇas, and other information regarding them, see Arts. 10 and 40.

[149] This will illustrate the danger of trusting to Tables XIV. and XV. in important cases.

[150] Note that this approximate calculation, which is the same as that by method B, comes out actually wrong by two days.

[151] In this case the result by the approximate method A or B will be wrong by two days.

[152] A year of the Hijra = 0.970223 of Gregorian year, and a Gregorian year = 1.03069 years of the Hijra. Thus 32 Gregorian years are about equal to 33 years of the Hijra, or more nearly 163 Gregorian years are within less than a day of 168 Hijra years.

[153] So far as I know no European chronologist of the present century has noticed this point. Tables could be constructed for the heliacal rising of the moon in every month of every year, but it would be too great a work for the present publication. [S. B. D.]

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